Paradoxes - SarahHoneychurchTeaching

Paradoxes
Situations that seems to defy intuition
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Contents
Articles
Introduction
1
Paradox
1
List of paradoxes
4
Paradoxical laughter
15
Decision theory
16
Abilene paradox
16
Chainstore paradox
18
Exchange paradox
21
Kavka's toxin puzzle
33
Necktie paradox
35
Economy
37
Allais paradox
37
Arrow's impossibility theorem
39
Bertrand paradox
50
Demographic-economic paradox
51
Dollar auction
54
Downs–Thomson paradox
55
Easterlin paradox
56
Ellsberg paradox
58
Green paradox
61
Icarus paradox
63
Jevons paradox
63
Leontief paradox
68
Lucas paradox
69
Metzler paradox
70
Paradox of thrift
71
Paradox of value
75
Productivity paradox
78
St. Petersburg paradox
83
Logic
All horses are the same color
90
90
Barbershop paradox
91
Carroll's paradox
94
Crocodile Dilemma
95
Drinker paradox
96
Infinite regress
99
Lottery paradox
100
Paradoxes of material implication
102
Raven paradox
105
Unexpected hanging paradox
117
What the Tortoise Said to Achilles
121
Mathematics
125
Accuracy paradox
125
Apportionment paradox
127
Banach–Tarski paradox
129
Berkson's paradox
137
Bertrand's box paradox
139
Bertrand paradox
143
Birthday problem
147
Borel–Kolmogorov paradox
159
Boy or Girl paradox
161
Burali-Forti paradox
168
Cantor's paradox
169
Coastline paradox
170
Cramer's paradox
172
Elevator paradox
173
False positive paradox
175
Gabriel's Horn
177
Galileo's paradox
179
Gambler's fallacy
181
Gödel's incompleteness theorems
188
Interesting number paradox
205
Kleene–Rosser paradox
207
Lindley's paradox
208
Low birth weight paradox
210
Missing square puzzle
212
Paradoxes of set theory
214
Parrondo's paradox
219
Russell's paradox
224
Simpson's paradox
230
Skolem's paradox
238
Smale's paradox
241
Thomson's lamp
243
Two envelopes problem
245
Von Neumann paradox
257
Miscellaneous
260
Bracketing paradox
260
Buridan's ass
261
Buttered cat paradox
264
Lombard's Paradox
266
Mere addition paradox
266
Navigation paradox
270
Paradox of the plankton
272
Temporal paradox
273
Tritone paradox
275
Voting paradox
277
Philosophy
278
Fitch's paradox of knowability
278
Grandfather paradox
281
Liberal paradox
285
Moore's paradox
288
Moravec's paradox
292
Newcomb's paradox
294
Omnipotence paradox
298
Paradox of hedonism
307
Paradox of nihilism
309
Paradox of tolerance
311
Predestination paradox
312
Zeno's paradoxes
319
Physics
326
Algol paradox
326
Archimedes paradox
327
Aristotle's wheel paradox
328
Bell's spaceship paradox
329
Bentley's paradox
333
Black hole information paradox
333
Braess's paradox
337
Cool tropics paradox
342
D'Alembert's paradox
344
Denny's paradox
352
Ehrenfest paradox
352
Elevator paradox
358
EPR paradox
359
Faint young Sun paradox
369
Fermi paradox
372
Feynman sprinkler
393
Gibbs paradox
395
Hardy's paradox
399
Heat death paradox
400
Irresistible force paradox
401
Ladder paradox
402
Loschmidt's paradox
410
Mpemba effect
412
Olbers' paradox
416
Ontological paradox
421
Painlevé paradox
423
Physical paradox
424
Quantum pseudo-telepathy
428
Schrödinger's cat
431
Supplee's paradox
437
Tea leaf paradox
438
Twin paradox
440
Self-reference
451
Barber paradox
451
Berry paradox
454
Epimenides paradox
456
Exception paradox
458
Grelling–Nelson paradox
459
Intentionally blank page
461
Liar paradox
464
Opposite Day
469
Paradox of the Court
470
Petronius
472
Quine's paradox
475
Richard's paradox
477
Self-reference
479
Socratic paradox
483
Yablo's paradox
484
Vagueness
485
Absence paradox
485
Bonini's paradox
485
Code-talker paradox
486
Ship of Theseus
486
References
Article Sources and Contributors
492
Image Sources, Licenses and Contributors
504
Article Licenses
License
507
1
Introduction
Paradox
A paradox is an argument that produces an inconsistency, typically within logic or common sense.[1] Most logical
paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking.[2] However some
have revealed errors in logic iteslf and have caused the rules of logic to be rewritten. (e.g. Russell's paradox[3]) Still
others, such as Curry's paradox, are not yet resolved. In common usage, the word paradox often refers to irony or
contradiction. Examples outside of logic include the Grandfather paradox from physics and the Ship of Theseus
from philosophy. Paradoxes can also take the form of images or other media. For example, M.C. Escher featured
paradoxes about perspective in many of his drawings.[4]
Logical paradox
Common themes in paradoxes include self-reference, infinite regress, circular definitions, and confusion between
different levels of abstraction.
Patrick Hughes outlines three laws of the paradox:[5]
Self-reference
An example is "This statement is false", a form of the liar paradox. The statement is referring to itself. Another
example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more
example would be "Is the answer to this question no?" In this case, replying no would be stating that the
answer is not "no". If the reply is yes, it would be stating that it is "no", as the reply was yes. But because the
question was answered with a "yes", the answer is not "no". A negative response without saying the word
"no", like "it isn't", would, however, render the question answered without bringing about a paradox. Another
example is the term 'Nothing is Impossible', meaning that it is possible for something to be impossible, thus
contradicting itself.
Contradiction
"This statement is false"; the statement cannot be false and true at the same time.
Vicious circularity, or infinite regress
"This statement is false"; if the statement is true, then the statement is false, thereby making the statement true.
Another example of vicious circularity is the following group of statements:
"The following sentence is true."
"The previous sentence is false."
"What happens when Pinocchio says, 'My nose will grow now'?"
Other paradoxes involve false statements or half-truths and the resulting biased assumptions. This form is common
in howlers.
For example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree
and the father is killed. The boy is rushed to the nearest hospital where he is prepared for emergency surgery. On
entering the surgery suite, the surgeon says, "I can't operate on this boy. He's my son."
The apparent paradox is caused by a hasty generalization, for if the surgeon is the boy's father, the statement cannot
be true. The paradox is resolved if it is revealed that the surgeon is a woman, the boy's mother.
Paradox
2
Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require
extending the context or language to lose their paradoxical quality. Paradoxes that arise from apparently intelligible
uses of language are often of interest to logicians and philosophers. This sentence is false is an example of the
famous liar paradox: it is a sentence which cannot be consistently interpreted as true or false, because if it is known
to be false then it is known that it must be true, and if it is known to be true then it is known that it must be false.
Therefore, it can be concluded that it is unknowable. Russell's paradox, which shows that the notion of the set of all
those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern
logic and set theory.
Thought experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a
time traveler were to kill his own grandfather before his mother or father was conceived, thereby preventing his own
birth. W. V. Quine (1962) distinguished between three classes of paradoxes:
• A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the
paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a
twenty-one-year-old would have had only five birthdays, if he was born on a leap day. Likewise, Arrow's
impossibility theorem demonstrates difficulties in mapping voting results to the will of the people. The Monty
Hall paradox demonstrates that a decision which has an intuitive 50-50 chance in fact is heavily biased towards
making a decision which, given the intuitive conclusion, the player would be unlikely to make.
• A falsidical paradox establishes a result that not only appears false but actually is false due to a fallacy in the
demonstration. The various invalid mathematical proofs (e.g., that 1 = 2) are classic examples, generally relying
on a hidden division by zero. Another example is the inductive form of the horse paradox, falsely generalizes
from true specific statements.
• A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly
applying accepted ways of reasoning. For example, the Grelling–Nelson paradox points out genuine problems in
our understanding of the ideas of truth and description.
A fourth kind has sometimes been described since Quine's work.
• A paradox which is both true and false at the same time in the same sense is called a dialetheism. In Western
logics it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern
traditions and in paraconsistent logics. An example might be to affirm or deny the statement "John is in the room"
when John is standing precisely halfway through the doorway. It is reasonable (by human thinking) to both affirm
and deny it ("well, he is, but he isn't"), and it is also reasonable to say that he is neither ("he's halfway in the room,
which is neither in nor out"), despite the fact that the statement is to be exclusively proven or disproven.
Paradox in philosophy
A taste for paradox is central to the philosophies of Laozi, Heraclitus, Meister Eckhart, Hegel, Kierkegaard,
Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes, in the Philosophical
Fragments, that
one must not think ill of the paradox, for the paradox is the passion of thought, and the thinker without
the paradox is like the lover without passion: a mediocre fellow. But the ultimate potentiation of every
passion is always to will its own downfall, and so it is also the ultimate passion of the understanding to
will the collision, although in one way or another the collision must become its downfall. This, then, is
the ultimate paradox of thought: to want to discover something that thought itself cannot think.[6]
Paradox
Paradoxology
Paradoxology, "the use of paradoxes."[7] As a word it originates from Thomas Browne in his book Pseudodoxia
Epidemica."[8][9]
Alexander Bard and Jan Söderqvist developed a "paradoxology" in their book Det globala imperiet ("The Global
Empire").[10] The authors emphasize paradoxes between the world as static and as ever-changing, while leaning on
loose allegories from quantum mechanics. One may also include the philosopher Derrida in a list of users of
paradoxes. Derrida's deconstructions attempt to give opposing interpretations of the same text by rhetoric arguments,
similar to how lawyers in a court case may argue from the same text, the same set of laws that is, to reach opposite
conclusions.
Footnotes
[1] http:/ / www. thefreedictionary. com/ paradox
[2] http:/ / www. eric. ed. gov/ ERICWebPortal/ search/ detailmini. jsp?_nfpb=true& _& ERICExtSearch_SearchValue_0=EJ520704&
ERICExtSearch_SearchType_0=no& accno=EJ520704
[3] http:/ / scimath. unl. edu/ MIM/ files/ MATExamFiles/ LaFleurK_MATPaperFinal_LA. pdf
[4] http:/ / aminotes. tumblr. com/ post/ 653017235/ the-mathematical-art-of-m-c-escher-for-me-it
[5] Hughes, Patrick; Brecht, George (1975). Vicious Circles and Infinity - A Panoply of Paradoxes. Garden City, New York: Doubleday.
ISBN 0-385-09917-7. Library of Congress Catalog Card Number 74-17611.
. p. 1–8.
[6] Kierkegaard, Søren. Philosophical Fragments, 1844. p. 37
[7] Webster's Revised Unabridged, 2000
[8] Sturm, Sean (5 September 2009). "Paradoxology" (http:/ / www. webcitation. org/ 5minMTvXZ). Te ipu Pakore - Escribir es nacer (to write
is to be born). wordpress.com. Archived from the original (http:/ / seansturm. wordpress. com/ tag/ erratology/ #post-41) on 12 January 2010. .
Retrieved 12 January 2010.
[9] Browne, Thomas (1672) [first published 1646]. Pseudodoxia Epidemica or Enquries into very many received tenets and commonly presumed
truths. (http:/ / penelope. uchicago. edu/ pseudodoxia/ pseudodoxia. html) (6th ed.). . Retrieved 12 January 2010. "Although whoever shall
indifferently perpend the exceeding difficulty, which either the obscurity of the subject, or unavoidable paradoxology must often put upon the
Attemptor, he will easily discern, a work of this nature is not to be performed upon one legg; and should smel of oyl, if duly and deservedly
handled."
[10] Bard, Alexander; Söderqvist, Jan (2002) (in Swedish). Det globala imperiet: informationsålderns politiska filosofi [The Global Empire].
Stockholm: Bonnier Fakta. ISBN 91-85015-03-2.; reviewed in Ingdahl, Waldemar (2003). "Informationsålderns politiska filosofi [Political
Philosophy of the Information Age]" (http:/ / www. webcitation. org/ 5miuoDqyd) (in Swedish). Svensk Tidskrift (Stockholm: Nordstedts
Tryckeri) (2). ISSN 0039-677X. OCLC 1586291. Archived from the original (http:/ / www. eudoxa. se/ content/ archives/ 2003/ 05/
informationsald. html) on 12 January 2010. . Retrieved 12 January 2010.; bibliographic entries at LIBRIS No. 8814548 (http:/ / libris. kb. se/
bib/ 8814548?language=en) ( WebCite (http:/ / www. webcitation. org/ 5miwMwGj9) 12 January 2010) and lybrarything.com (http:/ / www.
librarything. com/ work/ 3399093) ( WebCite (http:/ / www. webcitation. org/ 5miwNAQBh) 12 January 2010)
External links
• Stanford Encyclopedia of Philosophy:
• " Paradoxes and Contemporary Logic (http://plato.stanford.edu/entries/paradoxes-contemporary-logic/)" –
by Andrea Cantini.
• " Insolubles (http://plato.stanford.edu/entries/insolubles)" – by Paul Vincent Spade.
• Paradoxes (http://www.dmoz.org/Society/Philosophy/Philosophy_of_Logic/Paradoxes//) at the Open
Directory Project
• "Zeno and the Paradox of Motion" (http://www.mathpages.com/rr/s3-07/3-07.htm) at MathPages.com.
3
List of paradoxes
List of paradoxes
This is a list of paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than
one category. Because of varying definitions of the term paradox, some of the following are not considered to be
paradoxes by everyone. This list collects only scenarios that have been called a paradox by at least one source and
have their own article.
Although considered paradoxes, some of these are based on fallacious reasoning, or incomplete/faulty analysis.
This list is incomplete.
Logic
• Barbershop paradox: The supposition that if one of two simultaneous assumptions leads to a contradiction, the
other assumption is also disproved leads to paradoxical consequences.
• What the Tortoise Said to Achilles "Whatever Logic is good enough to tell me is worth writing down...," also
known as Carroll's paradox, not to be confused with the physical paradox of the same name.
• Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess what the
crocodile will do, how should the crocodile respond in the case that the father correctly guesses that the child will
not be returned?
• Catch-22 (logic): A situation in which someone is in need of something that can only be had by not being in need
of it.
• Drinker paradox: In any pub there is a customer who, if they drink, everybody in the pub drinks.
• Paradox of entailment: Inconsistent premises always make an argument valid.
• Lottery paradox: There is one winning ticket in a large lottery. It is reasonable to believe of a particular lottery
ticket that it is not the winning ticket, since the probability that it is the winner is so very small, but it is not
reasonable to believe that no lottery ticket will win.
• Raven paradox (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.
• Unexpected hanging paradox: The day of the hanging will be a surprise, so it cannot happen at all, so it will be a
surprise. The surprise examination and Bottle Imp paradox use similar logic
Self-reference
These paradoxes have in common a contradiction arising from self-reference.
• Barber paradox: A barber (who is a man) shaves all and only those men who do not shave themselves. Does he
shave himself? (Russell's popularization of his set theoretic paradox.)
• Berry paradox: The phrase "the first number not nameable in under ten words" appears to name it in nine words.
• Paradox of the Court: A law student agrees to pay his teacher after winning his first case. The teacher then sues
the student (who has not yet won a case) for payment.
• Curry's paradox: "If this sentence is true, then Santa Claus exists."
• Epimenides paradox: A Cretan says: "All Cretans are liars". This paradox works in mainly the same way as the
Liar paradox.
• Exception paradox: "If there is an exception to every rule, then every rule must have at least one exception; the
exception to this one being that it has no exception." "There's always an exception to the rule, except to the
exception of the rule—which is, in of itself, an accepted exception of the rule."
• Grelling–Nelson paradox: Is the word "heterological", meaning "not applicable to itself," a heterological word?
(Another close relative of Russell's paradox.)
• Kleene–Rosser paradox: By formulating an equivalent to Richard's paradox, untyped lambda calculus is shown to
be inconsistent.
4
List of paradoxes
• Liar paradox: "This sentence is false." This is the canonical self-referential paradox. Also "Is the answer to this
question no?" And "I'm lying."
•
•
•
•
• Card paradox: "The next statement is true. The previous statement is false." A variant of the liar paradox that
does not use self-reference.
• The Pinocchio paradox: What would happen if Pinocchio said "My nose will be growing"?[1]
• Quine's paradox: "'Yields a falsehood when appended to its own quotation' yields a falsehood when appended
to its own quotation." Shows that a sentence can be paradoxical even if it is not self-referring and does not use
demonstratives or indexicals.
• Yablo's paradox: An ordered infinite sequence of sentences, each of which says that all following sentences are
false. Uses neither self-reference nor circular reference.
Opposite Day: "It is opposite day today." Therefore it is not opposite day, but if you say it is a normal day it
would be considered a normal day.
Petronius's paradox: "Moderation in all things, including moderation" (unsourced quotation sometimes attributed
to Petronius).
Richard's paradox: We appear to be able to use simple English to define a decimal expansion in a way that is
self-contradictory.
Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
• Socratic paradox: "I know that I know nothing at all."
Vagueness
• Bonini's paradox: models or simulations that explain the workings of complex systems are seemingly impossible
to construct: As a model of a complex system becomes more complete, it becomes less understandable; for it to
be more understandable it must be less complete and therefore less accurate. When the model becomes accurate,
it is just as difficult to understand as the real-world processes it represents.
• Code-talker paradox: how can a language both enable communication and block communication?
• Ship of Theseus (a.k.a. George Washington's axe or Grandfather's old axe): It seems like you can replace any
component of a ship, and it is still the same ship. So you can replace them all, one at a time, and it is still the same
ship. However, you can then take all the original pieces, and assemble them into a ship. That, too, is the same ship
you began with.
• Sorites paradox (also known as the paradox of the heap): One grain of sand is not a heap. If you don't have a heap,
then adding only one grain of sand won't give you a heap. Then no number of grains of sand will make a heap.
Similarly, one hair can't make the difference between being bald and not being bald. But then if you remove one
hair at a time, you will never become bald. Also similar, one dollar will not make you rich, so if you keep this up,
one dollar at a time, you will never become rich,
Mathematics
• Cramer's paradox: the number of points of intersection of two higher-order curves can be greater than the number
of arbitrary points needed to define one such curve.
• Elevator paradox: Elevators can seem to be mostly going in one direction, as if they were being manufactured in
the middle of the building and being disassembled on the roof and basement.
• Interesting number paradox: The first number that can be considered "dull" rather than "interesting" becomes
interesting because of that fact.
• Nontransitive dice: You can have three dice, called A, B, and C, such that A is likely to win in a roll against B, B
is likely to win in a roll against C, and C is likely to win in a roll against A.
• Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
5
List of paradoxes
6
Statistics
• Accuracy paradox: predictive models with a given level of accuracy may have greater predictive power than
models with higher accuracy.
• Berkson's paradox: a complicating factor arising in statistical tests of proportions.
• Freedman's paradox describes a problem in model selection where predictor variables with no explanatory power
can appear artificially important
• Friendship paradox: For almost everyone, their friends have more friends than they do.
• Inspection paradox: Why one will wait longer for a bus than one should.
• Lindley's paradox: Tiny errors in the null hypothesis are magnified when large data sets are analyzed, leading to
false but highly statistically significant results.
• Low birth weight paradox: Low birth weight and mothers who smoke contribute to a higher mortality rate. Babies
of smokers have lower average birth weight, but low birth weight babies born to smokers have a lower mortality
rate than other low birth weight babies. (A special case of Simpson's paradox.)
• Will Rogers phenomenon: The mathematical concept of an average, whether defined as the mean or median, leads
to apparently paradoxical results — for example, it is possible that moving an entry from an encyclopedia to a
dictionary would increase the average entry length on both books.
Probability
• Bertrand's box paradox: A paradox of conditional probability
closely related to the Boy or Girl paradox.
• Bertrand's paradox: Different common-sense definitions of
randomness give quite different results.
• Birthday paradox: What is the chance that two people in a room
have the same birthday?
• Borel's paradox: Conditional probability density functions are not
invariant under coordinate transformations.
The Monty Hall paradox: which door do you
choose?
• Boy or Girl paradox: A two-child family has at least one boy. What
is the probability that it has a girl?
• False positive paradox: A test that is accurate the vast majority of the time could show you have a disease, but the
probability that you actually have it could still be tiny.
• Monty Hall problem: An unintuitive consequence of conditional probability.
• Necktie Paradox: A wager between two people seems to favour them both. Very similar in essence to the
Two-envelope paradox.
• Proebsting's paradox: The Kelly criterion is an often optimal strategy for maximizing profit in the long run.
Proebsting's paradox apparently shows that the Kelly criterion can lead to ruin.
• Simpson's paradox: An association in sub-populations may be reversed in the population. It appears that two sets
of data separately support a certain hypothesis, but, when considered together, they support the opposite
hypothesis.
• Sleeping Beauty problem: A probability problem that can be correctly answered as one half or one third
depending on how the question is approached.
• Three cards problem: When pulling a random card, how do you determine the color of the underside?
• Three Prisoners problem: A variation of the Monty Hall problem.
• Two-envelope paradox: You are given two indistinguishable envelopes and you are told one contains twice as
much money as the other. You may open one envelope, examine its contents, and then, without opening the other,
choose which envelope to take.
List of paradoxes
7
Infinity and infinitesimals
• Burali-Forti paradox: If the ordinal numbers formed a set, it would be an ordinal number that is smaller than
itself.
• Cantor's paradox: There is no greatest cardinal number.
• Galileo's paradox: Though most numbers are not squares, there are no more numbers than squares. (See also
Cantor's diagonal argument)
• Hilbert's paradox of the Grand Hotel: If a hotel with infinitely many rooms is full, it can still take in more guests.
• Russell's paradox: Does the set of all those sets that do not contain themselves contain itself?
• Skolem's paradox: Countably infinite models of set theory contain uncountably infinite sets.
• Supertasks can result in paradoxes such as the Ross-Littlewood paradox and Benardete's paradox.
• Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of
the half, and half of that half, and so on." (This is also a physical paradox.)
Geometry and topology
• Banach–Tarski paradox: Cut a ball into a finite number of
pieces, re-assemble the pieces to get two balls, both of equal
size to the first. The von Neumann paradox is a
two-dimensional analogue.
• Paradoxical set: A set that can be partitioned into two sets,
each of which is equivalent to the original.
The Banach–Tarski paradox: A ball can be
decomposed and reassembled into two balls the same
size as the original.
• Coastline paradox: the perimeter of a landmass is in general
ill-defined.
• Gabriel's Horn or Torricelli's trumpet: A simple object with finite volume but infinite surface area. Also, the
Mandelbrot set and various other fractals are covered by a finite area, but have an infinite perimeter (in fact, there
are no two distinct points on the boundary of the Mandelbrot set that can be reached from one another by moving
a finite distance along that boundary, which also implies that in a sense you go no further if you walk "the wrong
way" around the set to reach a nearby point). This can be represented by a Klein bottle.
• Hausdorff paradox: There exists a countable subset C of the sphere S such that S\C is equidecomposable with two
copies of itself.
• Missing square puzzle: Two similar-looking figures appear to have different areas while built from the same
pieces.
• Nikodym set: A set contained in and with the same Lebesgue measure as the unit square, yet for every one of its
points there is a straight line intersecting the Nikodym set only in that point.
• Smale's paradox: A sphere can, topologically, be turned inside out.
List of paradoxes
Decision theory
• Abilene paradox: People can make decisions based not on what they actually want to do, but on what they think
that other people want to do, with the result that everybody decides to do something that nobody really wants to
do, but only what they thought that everybody else wanted to do.
• Apportionment paradox: Some systems of apportioning representation can have unintuitive results due to
rounding
•
•
•
•
•
• Alabama paradox: Increasing the total number of seats might shrink one block's seats.
• New states paradox: Adding a new state or voting block might increase the number of votes of another.
• Population paradox: A fast-growing state can lose votes to a slow-growing state.
Arrow's paradox: Given more than two choices, no system can have all the attributes of an ideal voting system at
once.
Buridan's ass: How can a rational choice be made between two outcomes of equal value?
Chainstore paradox: Even those who know better play the so-called chain store game in an irrational manner.
Fenno's paradox: The belief that people generally disapprove of the United States Congress as a whole, but
support the Congressman from their own Congressional district.
Green paradox: Policies intending to reduce future CO2 emissions may lead to increased emissions in the present.
• Inventor's paradox: It is easier to solve a more general problem that covers the specifics of the sought-after
solution.
• Kavka's toxin puzzle: Can one intend to drink the non-deadly toxin, if the intention is the only thing needed to get
the reward?
• Morton's fork: Choosing between unpalatable alternatives.
• Navigation paradox: Increased navigational precision may result in increased collision risk.
• Newcomb's paradox: How do you play a game against an omniscient opponent?
• Paradox of hedonism: When one pursues happiness itself, one is miserable; but, when one pursues something
else, one achieves happiness.
• Paradox of tolerance: Should one tolerate intolerance if intolerance would destroy the possibility of tolerance?
• Paradox of voting: Also known as the Downs paradox. For a rational, self-interested voter the costs of voting will
normally exceed the expected benefits, so why do people keep voting?
• Parrondo's paradox: It is possible to play two losing games alternately to eventually win.
• Prevention paradox: For one person to benefit, many people have to change their behavior — even though they
receive no benefit, or even suffer, from the change.
• Prisoner's dilemma: Two people might not cooperate even if it is in both their best interests to do so.
• Relevance paradox: Sometimes relevant information is not sought out because its relevance only becomes clear
after the information is available.
• Voting paradox: Also known as Condorcet's paradox and paradox of voting. A group of separately rational
individuals may have preferences that are irrational in the aggregate.
• Willpower as a paradox: Those who kept their minds open were more goal-directed and more motivated than
those who declared their objective to themselves.
8
List of paradoxes
9
Physics
• Cool tropics paradox: A contradiction between modelled estimates
of tropical temperatures during warm, ice-free periods of the
Cretaceous and Eocene, and the colder temperatures that proxies
suggest were present.
• The holographic principle: The amount of information that can be
stored in a given volume is not proportional to the volume but to the
area that bounds that volume.
• Irresistible force paradox: What would happen if an unstoppable
force hit an immovable object?
Astrophysics
• Algol paradox: In some binaries the partners seem to have different
ages, even though they're thought to have formed at the same time.
Robert Boyle's self-flowing flask fills itself in this
diagram, but perpetual motion machines cannot
exist.
• Faint young Sun paradox: The apparent contradiction between observations of liquid water early in the Earth's
history and the astrophysical expectation that the output of the young sun would have been insufficient to melt ice
on earth.
• The GZK paradox: High-energy cosmic rays have been observed that seem to violate the
Greisen-Zatsepin-Kuzmin limit, which is a consequence of special relativity.
Classical mechanics
• Archer's paradox: An archer must, in order to hit his target, not aim directly at it, but slightly to the side.
• Archimedes paradox: A massive battleship can float in a few litres of water.
• Aristotle's wheel paradox: Rolling joined concentric wheels seem to trace the same distance with their
circumferences, even though the circumferences are different.
• Carroll's paradox: The angular momentum of a stick should be zero, but is not.
• D'Alembert's paradox: Flow of an inviscid fluid produces no net force on a solid body.
• Denny's paradox: Surface-dwelling arthropods (such as the water strider) should not be able to propel themselves
horizontally.
• Elevator paradox: Even though hydrometers are used to measure fluid density, a hydrometer will not indicate
changes of fluid density caused by changing atmospheric pressure.
• Feynman sprinkler: Which way does a sprinkler rotate when submerged in a tank and made to suck in the
surrounding fluid?
• Painlevé paradox: Rigid-body dynamics with contact and friction is inconsistent.
• Tea leaf paradox: When a cup of tea is stirred, the leaves assemble in the center, even though centrifugal force
pushes them outward.
List of paradoxes
Cosmology
• Bentley's paradox: In a Newtonian universe, gravitation should pull all matter into a single point.
• Fermi paradox: If there are, as probability would suggest, many other sentient species in the Universe, then where
are they? Shouldn't their presence be obvious?
• Heat death paradox: Since the universe is not infinitely old, it cannot be infinite in extent.
• Olbers' paradox: Why is the night sky black if there is an infinity of stars?
Electromagnetism
• Faraday paradox: An apparent violation of Faraday's law of electromagnetic induction.
Quantum mechanics
•
•
•
•
Bell's theorem: Why do measured quantum particles not satisfy mathematical probability theory?
Double-slit experiment: Matter and energy can act as a wave or as a particle depending on the experiment.
Einstein-Podolsky-Rosen paradox: Can far away events influence each other in quantum mechanics?
Extinction paradox: In the small wavelength limit, the total scattering cross section of an impenetrable sphere is
twice its geometrical cross-sectional area (which is the value obtained in classical mechanics).[2]
• Hardy's paradox: How can we make inferences about past events that we haven't observed while at the same time
acknowledge that the act of observing it affects the reality we are inferring to?
• Klein paradox: When the potential of a potential barrier becomes similar to the mass of the impinging particle, it
becomes transparent.
• The Mott problem: spherically symmetric wave functions, when observed, produce linear particle tracks.
• Quantum LC circuit paradox: Energies stored on capacitance and inductance are not equal to the ground state
energy of the quantum oscillator.
• Quantum pseudo-telepathy: Two players who can not communicate accomplish tasks that seemingly require
direct contact.
• Schrödinger's cat paradox: A quantum paradox — Is the cat alive or dead before we look?
• Uncertainty principle: Attempts to determine position must disturb momentum, and vice versa.
Relativity
• Bell's spaceship paradox: concerning relativity.
• Black hole information paradox: Black holes violate a commonly assumed tenet of science — that information
cannot be destroyed.
• Ehrenfest paradox: On the kinematics of a rigid, rotating disk.
• Ladder paradox: A classic relativity problem.
• Mocanu's velocity composition paradox: a paradox in special relativity.
• Supplee's paradox: the buoyancy of a relativistic object (such as a bullet) appears to change when the reference
frame is changed from one in which the bullet is at rest to one in which the fluid is at rest.
• Trouton-Noble or Right-angle lever paradox. Does a torque arise in static systems when changing frames?
• Twin paradox: The theory of relativity predicts that a person making a round trip will return younger than his or
her identical twin who stayed at home.
10
List of paradoxes
Thermodynamics
• Gibbs paradox: In an ideal gas, is entropy an extensive variable?
• Loschmidt's paradox: Why is there an inevitable increase in entropy when the laws of physics are invariant under
time reversal? The time reversal symmetry of physical laws appears to contradict the second law of
thermodynamics.
• Maxwell's Demon: The second law of thermodynamics seems to be violated by a cleverly operated trapdoor.[3]
• Mpemba paradox: Hot water can, under certain conditions, freeze faster than cold water, even though it must pass
the lower temperature on the way to freezing.
Biology
• Paradox of enrichment: Increasing the food available to an ecosystem may lead to instability, and even to
extinction.
• French paradox: the observation that the French suffer a relatively low incidence of coronary heart disease,
despite having a diet relatively rich in saturated fats.
• Glucose paradox: The large amount of glycogen in the liver cannot be explained by its small glucose absorption.
• Gray's paradox: Despite their relatively small muscle mass, dolphins can swim at high speeds and obtain large
accelerations.
• Hispanic paradox: The finding that Hispanics in the U.S. tend to have substantially better health than the average
population in spite of what their aggregate socio-economic indicators predict.
• Lombard's paradox: When rising to stand from a sitting or squatting position, both the hamstrings and quadriceps
contract at the same time, despite their being antagonists to each other.
• Mexican paradox: Mexican children tend to have higher birth weights than can be expected from their
socio-economic status.
• Meditation paradox: The amplitude of heart rate oscillations during meditation was significantly greater than in
the pre-meditation control state and also in three non-meditation control groups[4]
• Paradox of the pesticides: Applying pesticide to a pest may increase the pest's abundance.
• Paradox of the plankton: Why are there so many different species of phytoplankton, even though competition for
the same resources tends to reduce the number of species?
• Peto's paradox: Humans get cancer with high frequency, while larger mammals, like whales, do not. If cancer is
essentially a negative outcome lottery at the cell level, and larger organisms have more cells, and thus more
potentially cancerous cell divisions, one would expect larger organisms to be more predisposed to cancer.
• Pulsus paradoxus: Sometimes it is possible to hear, with a stethoscope, heartbeats that cannot be felt at the wrist.
Also known as the Pulse Paradox.[5]
• Sherman paradox: An anomalous pattern of inheritance in the fragile X syndrome.
• Temporal paradox (paleontology): When did the ancestors of birds live?
Chemistry
• Faraday paradox (electrochemistry): Diluted nitric acid will corrode steel, while concentrated nitric acid doesn't.
• Levinthal paradox: The length of time that it takes for a protein chain to find its folded state is many orders of
magnitude shorter than it would be if it freely searched all possible configurations.
• SAR paradox: Exceptions to the principle that a small change in a molecule causes a small change in its chemical
behaviour are frequently profound.
11
List of paradoxes
Time
• Bootstrap paradox: Can a time traveler send himself information with no outside source?
• Predestination paradox:[6] A man travels back in time to discover the cause of a famous fire. While in the building
where the fire started, he accidentally knocks over a kerosene lantern and causes a fire, the same fire that would
inspire him, years later, to travel back in time. The bootstrap paradox is closely tied to this, in which, as a result of
time travel, information or objects appear to have no beginning.
• Temporal paradox: What happens when a time traveler does things in the past that prevent him from doing them
in the first place?
• Grandfather paradox: You travel back in time and kill your grandfather before he conceives one of your
parents, which precludes your own conception and, therefore, you couldn't go back in time and kill your
grandfather.
• Hitler's murder paradox: You travel back in time and kill a famous person in history before they become
famous; but if the person had never been famous then he could not have been targeted as a famous person.
Philosophy
• Paradox of analysis: It seems that no conceptual analysis can both meet the requirement of correctness and of
informativeness.
• Buridan's bridge: Will Plato throw Socrates into the water or not?
• Paradox of fiction: How people can experience strong emotions from purely fictional things?
• Fitch's paradox: If all truths are knowable, then all truths must in fact be known.
• Paradox of free will: If God knew how we will decide when he created us, how can there be free will?
• Goodman's paradox: Why can induction be used to confirm that things are "green", but not to confirm that things
are "grue"?
• Paradox of hedonism: In seeking happiness, one does not find happiness.
• Hutton's Paradox: If asking oneself "Am I dreaming?" in a dream proves that one is, what does it prove in waking
life?
• Liberal paradox: "Minimal Liberty" is incompatible with Pareto optimality.
• Meno's paradox (Learning paradox): A man cannot search either for what he knows or for what he does not know.
• Mere addition paradox: Also known as Parfit's paradox: Is a large population living a barely tolerable life better
than a small, happy population?
• Moore's paradox: "It's raining, but I don't believe that it is."
• Newcomb's paradox: A paradoxical game between two players, one of whom can predict the actions of the other.
• Paradox of nihilism: Several distinct paradoxes share this name.
• Omnipotence paradox: Can an omnipotent being create a rock too heavy for itself to lift?
• Preface paradox: The author of a book may be justified in believing that all his statements in the book are correct,
at the same time believing that at least one of them is incorrect.
• Problem of evil (Epicurean paradox): The existence of evil seems to be incompatible with the existence of an
omnipotent, omniscient, and morally perfect God.
• Zeno's paradoxes: "You will never reach point B from point A as you must always get half-way there, and half of
the half, and half of that half, and so on..." (This is also a paradox of the infinite)
12
List of paradoxes
Mysticism
• Tzimtzum: In Kabbalah, how to reconcile self-awareness of finite Creation with Infinite Divine source, as an
emanated causal chain would seemingly nullify existence. Luria's initial withdrawal of God in Hasidic
panentheism involves simultaneous illusionism of Creation (Upper Unity) and self-aware existence (Lower
Unity), God encompassing logical opposites.
Economics
• Allais paradox: A change in a possible outcome that is shared by different alternatives affects people's choices
among those alternatives, in contradiction with expected utility theory.
• Arrow information paradox: To sell information you need to give it away before the sale.
• Bertrand paradox: Two players reaching a state of Nash equilibrium both find themselves with no profits.
• Braess's paradox: Adding extra capacity to a network can reduce overall performance.
• Demographic-economic paradox: nations or subpopulations with higher GDP per capita are observed to have
fewer children, even though a richer population can support more children.
• Diamond-water paradox (or paradox of value) Water is more useful than diamonds, yet is a lot cheaper.
• Downs–Thomson paradox: Increasing road capacity at the expense of investments in public transport can make
overall congestion on the road worse.
• Easterlin paradox: For countries with income sufficient to meet basic needs, the reported level of happiness does
not correlate with national income per person.
• Edgeworth paradox: With capacity constraints, there may not be an equilibrium.
• Ellsberg paradox: People exhibit ambiguity aversion (as distinct from risk aversion), in contradiction with
expected utility theory.
• European paradox: The perceived failure of European countries to translate scientific advances into marketable
innovations.
• Gibson's paradox: Why were interest rates and prices correlated?
• Giffen paradox: Increasing the price of bread makes poor people eat more of it.
• Icarus paradox: Some businesses bring about their own downfall through their own successes.
• Jevons paradox: Increases in efficiency lead to even larger increases in demand.
• Leontief paradox: Some countries export labor-intensive commodities and import capital-intensive commodities,
in contradiction with Heckscher–Ohlin theory.
• Lucas paradox: Capital is not flowing from developed countries to developing countries despite the fact that
developing countries have lower levels of capital per worker, and therefore higher returns to capital.
• Mandeville's paradox: Actions that may be vicious to individuals may benefit society as a whole.
• Metzler paradox: The imposition of a tariff on imports may reduce the relative internal price of that good.
• Paradox of thrift: If everyone saves more money during times of recession, then aggregate demand will fall and
will in turn lower total savings in the population.
• Paradox of toil: If everyone tries to work during times of recession, lower wages will reduce prices, leading to
more deflationary expectations, leading to further thrift, reducing demand and thereby reducing employment.
• Productive failure: Providing less guidance and structure and thereby causing more failure is likely to promote
better learning.[7]
• Productivity paradox (also known as Solow computer paradox): Worker productivity may go down, despite
technological improvements.
• Scitovsky paradox: Using the Kaldor–Hicks criterion, an allocation A may be more efficient than allocation B,
while at the same time B is more efficient than A.
• Service recovery paradox: Successfully fixing a problem with a defective product may lead to higher consumer
satisfaction than in the case where no problem occurred at all.
13
List of paradoxes
• St. Petersburg paradox: People will only offer a modest fee for a reward of infinite expected value.
• Paradox of Plenty: The Paradox of Plenty (resource curse) refers to the paradox that countries and regions with an
abundance of natural resources, specifically point-source non-renewable resources like minerals and fuels, tend to
have less economic growth and worse development outcomes than countries with fewer natural resources.
• Tullock paradox
Perception
• Tritone paradox: An auditory illusion in which a sequentially played pair of Shepard tones is heard as ascending
by some people and as descending by others.
• Blub paradox: Cognitive lock of some experienced programmers that prevents them from properly evaluating the
quality of programming languages which they do not know.[8]
Politics
• Stability-instability paradox: When two countries each have nuclear weapons, the probability of a direct war
between them greatly decreases, but the probability of minor or indirect conflicts between them increases.
History
• Georg Wilhelm Friedrich Hegel: We learn from history that we do not learn from history.[9] (paraphrased)
Psychology
• Self-absorption paradox: The contradictory association whereby higher levels of self-awareness are
simultaneously associated with higher levels of psychological distress and with psychological well-being.[10]
Notes
[1] Eldridge-Smith, Peter; Eldridge-Smith, Veronique (13 January 2010). "The Pinocchio paradox" (http:/ / analysis. oxfordjournals. org/ cgi/
content/ short/ 70/ 2/ 212). Analysis 70 (2): 212–215. doi:10.1093/analys/anp173. ISSN 1467-8284. . Retrieved 23 July 2010.
As of 2010, an image of Pinocchio with a speech bubble "My nose will grow now!" has become a minor Internet phenomenon ( Google search
(http:/ / www. google. com/ search?q="pinocchio+ paradox"), Google image search (http:/ / www. google. com/ images?q="pinocchio+
paradox")). It seems likely that this paradox has been independently conceived multiple times.
[2] Newton, Roger G. (2002). Scattering Theory of Waves and Particles, second edition. Dover Publications. p. 68. ISBN 0-486-42535-5.
[3] Carnap is quoted as saying in 1977 "... the situation with respect to Maxwell's paradox", in Leff, Harvey S.; Rex, A. F., eds. (2003). Maxwell's
Demon 2: Entropy, Classical and Quantum Information, Computing (http:/ / web. archive. org/ web/ 20051109101141/ http:/ / vlatko.
madetomeasure. biz/ Papers/ maxwell2. pdf). Institute of Physics. p. 19. ISBN 0-7503-0759-5. . Retrieved 15 March 2010.
On page 36, Leff and Rex also quote Goldstein and Goldstein as saying "Smoluchowski fully resolved the paradox of the demon in 1912" in
Goldstein, Martin; Goldstein, Inge F. (1993). The Refrigerator and The Universe (http:/ / books. google. com/ books?id=R3Eek_YZdRUC).
Universities Press (India) Pvt. Ltd. p. 228. ISBN 978-81-7371-085-8. OCLC 477206415. . Retrieved 15 March 2010.
[4] Peng, C.-K; Isaac C Henry, Joseph E Mietus, Jeffrey M Hausdorff, Gurucharan Khalsa, Herbert Benson, Ary L Goldberger (May 2004).
"Heart rate dynamics during three forms of meditation" (http:/ / www. sciencedirect. com/ science/ article/ pii/ S0167527303003504).
International Journal of Cardiology 95 (1): 19–27. . Retrieved 23 May 2012.
[5] Khasnis, A.; Lokhandwala, Y. (Jan-Mar 2002). "Clinical signs in medicine: pulsus paradoxus" (http:/ / www. jpgmonline. com/ text.
asp?2002/ 48/ 1/ 46/ 153#Pulsus paradoxus: what is the paradox?). Journal of Postgraduate Medicine (Mumbai - 400 012, India: 49) 48 (1):
46. ISSN 0022-3859. PMID 12082330. . Retrieved 21 March 2010. "The "paradox" refers to the fact that heart sounds may be heard over the
precordium when the radial pulse is not felt."
[6] See also Predestination paradoxes in popular culture
[7] Kapur, Manu; Bielaczyc, K (2012). "Designing for Productive Failure" (http:/ / www. tandfonline. com/ doi/ abs/ 10. 1080/ 10508406. 2011.
591717). Journal of the Learning Sciences 21: 45–83. doi:10.1080/10508406.2011.591717. .
[8] Hidders, J. "Expressive Power of Recursion and Aggregates in XQuery" (http:/ / win. ua. ac. be/ ~adrem/ bibrem/ pubs/ TR2005-05. pdf). .
Retrieved 23 May 2012. Chapter 1, Introduction.
[9] Hegel, Georg Wilhelm Friedrich (1832). Lectures on the Philosophy of History.
14
List of paradoxes
[10] Trapnell, P. D., & Campbell, J. D. (1999). Private self-consciousness and the Five-Factor Model of Personality: Distinguishing rumination
from reflection. Journal of Personality and Social Psychology, 76, 284-304.
Paradoxical laughter
Paradoxical laughter is an exaggerated expression of humour which is unwarranted by external events. It may be
uncontrollable laughter which may be recognised as inappropriate by the person involved. It is associated with
altered mental states or mental illness, such as mania, hypomania or schizophrenia, and can have other causes.
Paradoxical laughter is indicative of an unstable mood, often caused by the pseudobulbar affect, which can quickly
change to anger and back again, on minor external cues.
This type of laughter can also occur at times when the fight-or-flight response may otherwise be evoked.
References
• Frijda, Nico H. (1986). The Emotions (http://books.google.com/books?id=QkNuuVf-pBMC&pg=PA52&
lpg=PA52&dq="paradoxical+laughter"). Cambridge University Press. p. 52. ISBN 0-521-31600-6. Retrieved 14
November 2009.
• Rutkowski, Anne-Françoise; Rijsman, John B.; Gergen, Mary (2004). "Paradoxical Laughter at a Victim as
Communication with a Non-victim" (http://dbiref.uvt.nl/iPort?request=full_record&db=wo&language=eng&
query=142993). International Review of Social Psychology 17 (4): 5–11. ISSN 0992-986X. Retrieved
2009-11-14. ( French biobliographical record (http://cat.inist.fr/?aModele=afficheN&cpsidt=16370783) with
French translation of abstract)
15
16
Decision theory
Abilene paradox
The Abilene paradox is a paradox in which a group of people collectively decide on a course of action that is
counter to the preferences of any of the individuals in the group.[1][2] It involves a common breakdown of group
communication in which each member mistakenly believes that their own preferences are counter to the group's and,
therefore, does not raise objections. A common phrase relating to the Abilene paradox is a desire to not "rock the
boat".
Origins
The Abilene paradox was introduced by management expert Jerry B. Harvey in his article The Abilene Paradox: The
Management of Agreement.[3] The name of the phenomenon comes from an anecdote in the article which Harvey
uses to elucidate the paradox:
On a hot afternoon visiting in Coleman, Texas, the family is comfortably playing dominoes on a porch, until
the father-in-law suggests that they take a trip to Abilene [53 miles north] for dinner. The wife says, "Sounds
like a great idea." The husband, despite having reservations because the drive is long and hot, thinks that his
preferences must be out-of-step with the group and says, "Sounds good to me. I just hope your mother wants to
go." The mother-in-law then says, "Of course I want to go. I haven't been to Abilene in a long time."
The drive is hot, dusty, and long. When they arrive at the cafeteria, the food is as bad as the drive. They arrive
back home four hours later, exhausted.
One of them dishonestly says, "It was a great trip, wasn't it?" The mother-in-law says that, actually, she would
rather have stayed home, but went along since the other three were so enthusiastic. The husband says, "I wasn't
delighted to be doing what we were doing. I only went to satisfy the rest of you." The wife says, "I just went
along to keep you happy. I would have had to be crazy to want to go out in the heat like that." The
father-in-law then says that he only suggested it because he thought the others might be bored.
The group sits back, perplexed that they together decided to take a trip which none of them wanted. They each
would have preferred to sit comfortably, but did not admit to it when they still had time to enjoy the afternoon.
Groupthink
The phenomenon may be a form of groupthink. It is easily explained by social psychology theories of social
conformity and social influence which suggest that human beings are often very averse to acting contrary to the trend
of the group. Likewise, it can be observed in psychology that indirect cues and hidden motives often lie behind
peoples' statements and acts, frequently because social disincentives discourage individuals from openly voicing
their feelings or pursuing their desires.
The Abilene Paradox is related to the concept of groupthink in that both theories appear to explain the observed
behavior of groups in social contexts. The crux of the theory is that groups have just as many problems managing
their agreements as they do their disagreements. This observation rings true among many researchers in the social
sciences and tends to reinforce other theories of individual and group behavior.
Abilene paradox
Applications of the theory
The theory is often used to help explain extremely poor business decisions, especially notions of the superiority of
"rule by committee." A technique mentioned in the study and/or training of management, as well as practical
guidance by consultants, is that group members, when the time comes for a group to make decisions, should ask each
other, "Are we going to Abilene?" to determine whether their decision is legitimately desired by the group's
members or merely a result of this kind of groupthink. This anecdote was also made into a short film[4] for
management education.
References
[1] McAvoy, John; Butler, Tom (2007). "The impact of the Abilene Paradox on double-loop learning in an agile team". Information and Software
Technology 49 (6): 552–563. doi:10.1016/j.infsof.2007.02.012.
[2] McAvoy, J.; Butler, T. (2006). "Resisting the change to user stories: a trip to Abilene". International Journal of Information Systems and
Change Management 1 (1): 48–61. doi:10.1504/IJISCM.2006.008286.
[3] Harvey, Jerry B. (1974). "The Abilene paradox: the management of agreement". Organizational Dynamics 3: 63–80.
doi:10.1016/0090-2616(74)90005-9.
[4] "The Abilene Paradox, 2nd Edition" (http:/ / www. crmlearning. com/ abilene-paradox). CRM Learning. 2002. . Retrieved May 20, 2012.
Further reading
• Harvey, Jerry B. (1988). The Abilene Paradox and Other Meditations on Management. Lexington, Mass:
Lexington Books. ISBN 0-669-19179-5
• Harvey, Jerry B. (1996). The Abilene Paradox and Other Meditations on Management (paperback). San
Francisco: Jossey-Bass. ISBN 0-7879-0277-2
• Harvey, Jerry B. (1999). How Come Every Time I Get Stabbed in the Back, My Fingerprints Are on the Knife?.
San Francisco: Jossey-Bass. ISBN 0-7879-4787-3
External links
• The Abilene Paradox website by Dr. Jerry B. Harvey (http://www.abileneparadox.com)
17
Chainstore paradox
Chainstore paradox
Chainstore paradox (or "Chain-Store paradox") is a concept that purports to refute standard game theory reasoning.
The chain store game
A monopolist (Player A) has branches in 20 towns. He faces 20 potential competitors, one in each town, who will be
able to choose IN or OUT. They do so in sequential order and one at a time. If a potential competitor chooses OUT,
he receives a payoff of 1, while A receives a payoff of 5. If he chooses IN, he will receive a payoff of either 2 or 0,
depending on the response of Player A to his action. Player A, in response to a choice of IN, must choose one of two
pricing strategies, COOPERATIVE or AGGRESSIVE. If he chooses COOPERATIVE, both player A and the
competitor receive a payoff of 2, and if A chooses AGGRESSIVE, each player receives a payoff of 0.
These outcomes lead to two theories for the game, the induction (game theoretically correct version) and the
deterrence theory (weakly dominated theory):
Induction theory
Consider the decision to be made by the 20th and final competitor, of whether to choose IN or OUT. He knows that
if he chooses IN, Player A receives a higher payoff from choosing cooperate than aggressive, and being the last
period of the game, there are no longer any future competitors whom Player A needs to intimidate from the market.
Knowing this, the 20th competitor enters the market, and Player A will cooperate (receiving a payoff of 2 instead of
0).
The outcome in the final period is set in stone, so to speak. Now consider period 19, and the potential competitor's
decision. He knows that A will cooperate in the next period, regardless of what happens in period 19. Thus, if player
19 enters, an aggressive strategy will be unable to deter player 20 from entering. Player 19 knows this and chooses
IN. Player A chooses cooperate.
Of course, this process of backwards induction holds all the way back to the first competitor. Each potential
competitor chooses IN, and Player A always cooperates. A receives a payoff of 40 (2×20) and each competitor
receives 2.
Deterrence theory
This theory states that Player A will be able to get payoff of higher than 40. Suppose Player A finds the induction
argument convincing. He will decide how many periods at the end to play such a strategy, say 3. In periods 1-17, he
will decide to always be aggressive against the choice of IN. If all of the potential competitors know this, it is
unlikely potential competitors 1-17 will bother the chain store, thus risking a the safe payout of 1 ("A" will not
retaliate if they choose "OUT"). If a few do test the chain store early in the game, and see that they are greeted with
the aggressive strategy, the rest of the competitors are likely not to test any further. Assuming all 17 are deterred,
Player A receives 91 (17×5 + 2×3). Even if as many as 10 competitors enter and test Player A's will, Player A will
still receive a payoff of 41 (10×0+ 7×5 + 3×2), which is better than the induction (game theoretically correct) payoff.
18
Chainstore paradox
The chain store paradox
If Player A follows the game theory payoff matrix to achieve the optimal payoff, he or she will have a lower payoff
than with the "deterrence" strategy. This creates an apparent game theory paradox: game theory states that induction
strategy should be optimal, but it looks like "deterrence strategy" is optimal instead.
Selten's response
Reinhard Selten's response to this apparent paradox is to argue that the idea of "deterrence", while irrational by the
standards of Game Theory, is in fact an acceptable idea by the rationality that individuals actually employ. Selten
argues that individuals can make decisions of three levels: Routine, Imagination, and Reasoning.
Complete information?
If we stand by game theory, then the initial description given for the game theory payoff matrix in the chain store
game is not in fact the complete payoff matrix. The "deterrence strategy" is a valid strategy for Player A, but it is
missing in the initially presented payoff matrix. Game theory is based on the idea that each matrix is modeled with
the assumption of complete information: that "every player knows the payoffs and strategies available to other
players."
The initially presented payoff matrix is written for one payoff round instead of for all rounds in their entirety. As
described in the "deterrence strategy" section (but not in the induction section), Player A's competitors look at Player
A's actions in previous game rounds to determine what course of action to take - this information is missing from the
payoff matrix. In this case, backwards induction seems like it will fail, because each individual round payoff matrix
is dependent on the previous round. In fact, by doubling the size of the payoff matrix on each round (or, quadrupling
the amount of choices -- there are two choices and four possibilities per round), we can find the optimal strategy for
all players before the first round is played.
Selten's levels of decision making
The routine level
The individuals use their past experience of the results of decisions to guide their response to choices in the present.
"The underlying criteria of similarity between decision situations are crude and sometimes inadequate". (Selten)
The imagination level
The individual tries to visualize how the selection of different alternatives may influence the probable course of
future events. This level employs the routine level within the procedural decisions. This method is similar to a
computer simulation.
19
Chainstore paradox
The reasoning level
The individual makes a conscious effort to analyze the situation in a rational way, using both past experience and
logical thinking. This mode of decision uses simplified models whose assumptions are products of imagination, and
is the only method of reasoning permitted and expected by game theory.
Decision-making process
The predecision
One chooses which method (routine, imagination or reasoning) to use for the problem, and this decision itself is
made on the routine level.
The final decision
Depending on which level is selected, the individual begins the decision procedure. The individual then arrives at a
(possibly different) decision for each level available (if we have chosen imagination, we would arrive at a routine
decision and possible and imagination decision). Selten argues that individuals can always reach a routine decision,
but perhaps not the higher levels. Once the individuals have all their levels of decision, they can decide which
answer to use...the Final Decision. The final decision is made on the routine level and governs actual behavior.
The economy of decision effort
Decision effort is a scarce commodity, being both time consuming and mentally taxing. Reasoning is more costly
than Imagination, which, in turn is more costly than Routine. The highest level activated is not always the most
accurate since the individual may be able to reach a good decision on the routine level, but makes serious
computational mistakes on higher levels, especially Reasoning.
Selten finally argues that strategic decisions, like those made by the monopolist in the chainstore paradox, are
generally made on the level of Imagination, where deterrence is a reality, due to the complexity of Reasoning, and
the great inferiority of Routine (it does not allow the individual to see herself in the other player's position). Since
Imagination cannot be used to visualize more than a few stages of an extensive form game (like the Chain-store
game) individuals break down games into "the beginning" and "towards the end". Here, deterrence is a reality, since
it is reasonable "in the beginning", yet is not convincing "towards the end".
References
• Selten, Reinhard (1978). "The chain store paradox". Theory and Decision 9 (2): 127–159.
doi:10.1007/BF00131770. ISSN 0040-5833.
Further reading
• Relation of Chain Store Paradox to Constitutional Politics in Canada [1]
References
[1] http:/ / jtp. sagepub. com/ cgi/ content/ abstract/ 11/ 1/ 5
20
Exchange paradox
Exchange paradox
The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle or paradox in logic,
philosophy, probability, and recreational mathematics, of special interest in decision theory and for the Bayesian
interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.
A statement of the problem starts with:
Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money.
One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it
contains. You pick one envelope at random but before you open it you are offered the possibility to take the
other envelope instead.
It is possible to give arguments that show that it will be to your advantage to swap envelopes by showing that your
expected return on swapping exceeds the sum in your envelope. This leads to the absurdity that it is beneficial to
continue to swap envelopes indefinitely.
A large number of different solutions have been proposed. The usual scenario is that one writer proposes a solution
that solves the problem as stated, but then some other writer discovers that by altering the problem a little the
paradox is brought back to life again. In this way a family of closely related formulations of the problem is created
which are then discussed in the literature.
There is not yet any one proposed solution that is widely accepted as being the correct one. [1] Despite this it is
common for authors to claim that the solution to the problem is easy, even elementary.[2] However, when
investigating these elementary solutions they often differ from one author to the next. During the last two decades
several new papers have been published every year. [3]
The problem
The basic setup: You are given two indistinguishable envelopes, each
of which contains a positive sum of money. One envelope contains
twice as much as the other. You may pick one envelope and keep
whatever amount it contains. You pick one envelope at random but
before you open it you are offered the possibility to take the other
envelope instead.[4]
The switching argument: Now suppose you reason as follows:
1. I denote by A the amount in my selected envelope.
2.
3.
4.
5.
6.
7.
The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
The other envelope may contain either 2A or A/2.
If A is the smaller amount, then the other envelope contains 2A.
If A is the larger amount, then the other envelope contains A/2.
Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.
So the expected value of the money in the other envelope is
8. This is greater than A, so I gain on average by swapping.
9. After the switch, I can denote that content by B and reason in exactly the same manner as above.
10. I will conclude that the most rational thing to do is to swap back again.
11. To be rational, I will thus end up swapping envelopes indefinitely.
12. As it seems more rational to open just any envelope than to swap indefinitely, we have a contradiction.
21
Exchange paradox
The puzzle: The puzzle is to find the flaw in the very compelling line of reasoning above.
A common resolution
A common way to resolve the paradox, both in popular literature and in the academic literature in philosophy, is to
observe that A stands for different things at different places in the expected value calculation, step 7 above.[5] In the
first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a
variable in the same formula like this is said to be illegitimate, so step 7 is incorrect, and this is the cause of the
paradox.
According to this analysis, a correct alternative argument would have run on the following lines. Assume that there
are only two possible sums that might be in the envelope. Denoting the lower of the two amounts by X, we can
rewrite the expected value calculation as
Here X stands for the same thing in every term of the equation. We learn that 1.5X is the average expected value in
either of the envelopes, hence no reason to swap envelopes according to this calculation.
Mathematical details
Let us rewrite the preceding calculations in a more detailed notation which explicitly distinguishes random from
not-random quantities (that is a different distinction from the usual distinction in ordinary, deterministic,
mathematics between variables and constants). This is useful in order to compare with the next, alternative,
resolution. So far we were thinking of the two amounts of money in the two envelopes as being fixed; the only
randomness lies in which one goes into which envelope. We called the smaller amount X, let us denote the larger
amount by Y. Given the values x and y of X and Y, where y = 2x and x > 0, the problem description tells us (whether
or not x and y are known)
for all possible values x of the smaller amount X; there is a corresponding definition of the probability distribution of
B given X and Y. In our resolution of the paradox, we guessed that in Step 7 the writer was trying to compute the
expected value of B given X=x. Splitting the calculation over the two possibilities for which envelope contains the
smaller amount, it is certainly correct to write
At this point the writer correctly substitutes the value 1/2 for both of the conditional probabilities on the right hand
side of this equation (Step 2). At the same time he correctly substitutes the random variable B inside the first
conditional expectation for 2A, when taking its expectation value given B > A and X = x, and he similarly correctly
substitutes the random variable B for A/2 when taking its expectation value given B < A and X = x (Steps 4 and 5).
He would then arrive at the completely correct equation
However he now proceeds, in the first of the two terms on the right hand side, to replace the expectation value of A
given that Envelope A contains the smaller amount and given that the amounts are x and 2x, by the random quantity
A itself. Similarly, in the second term on the right hand side he replaces the expectation value of A given now that
Envelope A contains the larger amount and given that the amounts are x and 2x, also by the random quantity A itself.
The correct substitutions would have been, of course, x and 2x respectively, leading to a correct conclusion
22
Exchange paradox
23
.
Naturally this coincides with the expectation value of A given X=x.
Indeed, in the two contexts in which the random variable A appears on the right hand side, it is standing for two
different things, since its distribution has been conditioned on different events. Obviously, A tends to be larger, when
we know that it is greater than B and when the two amounts are fixed, and it tends to be smaller, when we know that
it is smaller than B and the two amounts are fixed, cf. Schwitzgebel and Dever (2007, 2008). In fact, it is exactly
twice as large in the first situation as in the second situation.
The preceding resolution was first noted by Bruss in 1996.[6] A concise exposition is given by Falk in 2009.[7]
Alternative interpretation
The first solution above doesn't explain what's wrong if the player is allowed to open the first envelope before being
offered the option to switch. In this case, A stands for the value which is seen then, throughout all subsequent
calculations. The mathematical variable A stands for any particular amount he might see there (it is a mathematical
variable, a generic possible value of a random variable). The reasoning appears to show that whatever amount he
would see there, he would decide to switch. Hence, he does not need to look in the envelope at all: he knows that if
he would look, and go through the calculations, they would tell him to switch, whatever he saw in the envelope.
In this case, at Steps 6, 7 and 8 of the reasoning, A is any fixed possible value of the amount of money in the first
envelope.
Thus, the proposed "common resolution" above breaks down and another explanation is needed.
This interpretation of the two envelopes problem appears in the first publications in which the paradox was
introduced, Gardner (1989) and Nalebuff (1989). It is common in the more mathematical literature on the problem.
The "common resolution" above depends on a particular interpretation of what the writer of the argument is trying to
calculate: namely, it assumes he is after the (unconditional) expectation value of what's in Envelope B. In the
mathematical literature on Two Envelopes Problem (and in particular, in the literature where it was first introduced
to the world), another interpretation is more common, involving the conditional expectation value (conditional on
what might be in Envelope A). In order to solve this and related interpretations or versions of the problem most
authors utilize the Bayesian interpretation of probability.
Introduction to resolutions based on Bayesian probability theory
Here the ways in which the paradox can be resolved depend to a large degree on the assumptions that are made about
the things that are not made clear in the setup and the proposed argument for switching.[8] The most usual
assumption about the way the envelopes are set up is that a sum of money is put in one envelope and twice that sum
is put in another envelope and then one of the two envelopes is selected randomly, called "Envelope A", and given to
the player. It is not made clear exactly how the first mentioned sum of money (the smaller of the two sums) is
determined and what values it could possibly take and, in particular, whether there is a maximum sum which it might
contain.[9][10] It is also not specified whether the player can look in Envelope A before deciding whether or not to
switch. A further ambiguity in the paradox is that it is not made clear in the proposed argument whether the amount
A in Envelope A is intended to be a constant, a random variable, or some other quantity.
If it assumed that there is a maximum sum that can be put in the first envelope then a very simple and
mathematically sound resolution is possible within the second interpretation. Step 6 in the proposed line of reasoning
is not always true, since if the player holds more than the maximum sum that can be put into the first envelope they
must hold the envelope containing the larger sum and are thus certain to lose by switching. Although this may not
occur often, when it does the heavy loss incurred by the player means that, on average, there is no advantage in
switching. This resolves all practical cases of the problem, whether or not the player looks in their envelope.[11]
Exchange paradox
It can be envisaged, however, that the sums in the two envelopes are not limited. This requires a more careful
mathematical analysis, and also uncovers other possible interpretations of the problem. If, for example, the smaller
of the two sums of money is considered to be equally likely to be one of infinitely many positive integers, thus
without upper limit, it means that the probability that it will be any given number is always zero. This absurd
situation is an example of what is known as an improper prior and this is generally considered to resolve the paradox
in this case.
It is possible to devise a distribution for the sums possible in the first envelope such that the maximum value is
unlimited, computation of the expectation of what's in B given what's in A seems to dictate you should switch, and
the distribution constitutes a proper prior.[12] In these cases it can be shown that the expected sum in both envelopes
is infinite. There is no gain, on average, in swapping.
The first two resolutions we present correspond, technically speaking, first to A being a random variable, and
secondly to it being a possible value of a random variable (and the expectation being computed is a conditional
expectation). At the same time, in the first resolution the two original amounts of money seem to be thought of as
being fixed, while in the second they are also thought of as varying. Thus there are two main interpretations of the
problem, and two main resolutions.
Proposed resolutions to the alternative interpretation
Nalebuff (1989), Christensen and Utts (1992), Falk and Konold (1992), Blachman, Christensen and Utts (1996),[13]
Nickerson and Falk (2006), pointed out that if the amounts of money in the two envelopes have any proper
probability distribution representing the player's prior beliefs about the amounts of money in the two envelopes, then
it is impossible that whatever the amount A=a in the first envelope might be, it would be equally likely, according to
these prior beliefs, that the second contains a/2 or 2a. Thus step 6 of the argument which leads to always switching is
a non-sequitur.
Mathematical details
According to this interpretation, the writer is carrying out the following computation, where he is conditioning now
on the value of A, the amount in Envelope A, not on the pair amounts in the two envelopes X and Y:
Completely correctly, and according to Step 5, the two conditional expectation values are evaluated as
However in Step 6 the writer is invoking Steps 2 and 3 to get the two conditional probabilities, and effectively
replacing the two conditional probabilities of Envelope A containing the smaller and larger amount, respectively,
given the amount actually in that envelope, both by the unconditional probability 1/2: he makes the substitutions
But intuitively we would expect that the larger the amount in A, the more likely it is to be the larger of the two, and
vice-versa. And it is a mathematical fact, as we will see in a moment, that it is impossible that both of these
conditional probabilities are equal to 1/2 for all possible values of a. In fact, in order for step 6 to be true, whatever a
might be, the smaller amount of money in the two envelopes must be equally likely to be between 1 and 2, as
between 2 and 4, as between 4 and 8, ... ad infinitum. But there is no way to divide total probability 1 into an infinite
number of pieces which are not only all equal to one another, but also all larger than zero. Yet the smaller amount of
money in the two envelopes must have probability larger than zero to be in at least one of the just mentioned ranges.
24
Exchange paradox
25
To see this, suppose that the chance that the smaller of the two envelopes contains an amount between 2n and 2n+1 is
p(n), where n is any whole number, positive or negative, and for definiteness we include the lower limit but exclude
the upper in each interval. It follows that the conditional probability that the envelope in our hands contains the
smaller amount of money of the two, given that its contents are between 2n and 2n+1, is
If this is equal to 1/2, it follows by simple algebra that
or p(n)=p(n-1). This has to be true for all n, an impossibility.
A new variant
Though Bayesian probability theory can resolve the alternative interpretation of the paradox above, it turns out that
examples can be found of proper probability distributions, such that the expected value of the amount in the second
envelope given that in the first does exceed the amount in the first, whatever it might be. The first such example was
already given by Nalebuff (1989). See also Christensen and Utts (1992)[14]
Denote again the amount of money in the first envelope by A and that in the second by B. We think of these as
random. Let X be the smaller of the two amounts and Y=2X be the larger. Notice that once we have fixed a
probability distribution for X then the joint probability distribution of A,B is fixed, since A,B = X,Y or Y,X each with
probability 1/2, independently of X,Y.
The bad step 6 in the "always switching" argument led us to the finding
for all a, and hence to
the recommendation to switch, whether or not we know a. Now, it turns out that one can quite easily invent proper
probability distributions for X, the smaller of the two amounts of money, such that this bad conclusion is still true!
One example is analysed in more detail, in a moment.
It cannot be true that whatever a, given A=a, B is equally likely to be a/2 or 2a, but it can be true that whatever a,
given A=a, B is larger in expected value than a.
Suppose for example (Broome, 1995)[15] that the envelope with the smaller amount actually contains 2n dollars with
probability 2n/3n+1 where n = 0, 1, 2,… These probabilities sum to 1, hence the distribution is a proper prior (for
subjectivists) and a completely decent probability law also for frequentists.
Imagine what might be in the first envelope. A sensible strategy would certainly be to swap when the first envelope
contains 1, as the other must then contain 2. Suppose on the other hand the first envelope contains 2. In that case
there are two possibilities: the envelope pair in front of us is either {1, 2} or {2, 4}. All other pairs are impossible.
The conditional probability that we are dealing with the {1, 2} pair, given that the first envelope contains 2, is
and consequently the probability it's the {2, 4} pair is 2/5, since these are the only two possibilities. In this
derivation,
is the probability that the envelope pair is the pair 1 and 2, and Envelope A happens to
contain 2;
is the probability that the envelope pair is the pair 2 and 4, and (again) Envelope A
happens to contain 2. Those are the only two ways in which Envelope A can end up containing the amount 2.
It turns out that these proportions hold in general unless the first envelope contains 1. Denote by a the amount we
imagine finding in Envelope A, if we were to open that envelope, and suppose that a = 2n for some n ≥ 1. In that case
the other envelope contains a/2 with probability 3/5 and 2a with probability 2/5.
Exchange paradox
26
So either the first envelope contains 1, in which case the conditional expected amount in the other envelope is 2, or
the first envelope contains a > 1, and though the second envelope is more likely to be smaller than larger, its
conditionally expected amount is larger: the conditionally expected amount in Envelope B is
which is more than a. This means that the player who looks in Envelope A would decide to switch whatever he saw
there. Hence there is no need to look in Envelope A in order to make that decision.
This conclusion is just as clearly wrong as it was in the preceding interpretations of the Two Envelopes Problem. But
now the flaws noted above don't apply; the a in the expected value calculation is a constant and the conditional
probabilities in the formula are obtained from a specified and proper prior distribution.
Proposed resolutions
Some writers think that the new paradox can be defused.[16] Suppose
for all a. As remarked
before, this is possible for some probability distributions of X (the smaller amount of money in the two envelopes).
Averaging over a, it follows either that
, or alternatively that
. But A and
B have the same probability distribution, and hence the same expectation value, by symmetry (each envelope is
equally likely to be the smaller of the two). Thus both have infinite expectation values, and hence so must X too.
Thus if we switch for the second envelope because its conditional expected value is larger than what actually is in
the first, whatever that might be, we are exchanging an unknown amount of money whose expectation value is
infinite for another unknown amount of money with the same distribution and the same infinite expected value. The
average amount of money in both envelopes is infinite. Exchanging one for the other simply exchanges an average of
infinity with an average of infinity.
Probability theory therefore tells us why and when the paradox can occur and explains to us where the sequence of
apparently logical steps breaks down. In this situation, Steps 6 and Steps 7 of the standard Two Envelopes argument
can be replaced by correct calculations of the conditional probabilities that the other envelope contains half or twice
what's in A, and a correct calculation of the conditional expectation of what's in B given what's in A. Indeed, that
conditional expected value is larger than what's in A. But because the unconditional expected amount in A is infinite,
this does not provide a reason to switch, because it does not guarantee that on average you'll be better off after
switching. One only has this mathematical guarantee in the situation that the unconditional expectation value of
what's in A is finite. But then the reason for switching without looking in the envelope,
for all
a, simply cannot arise.
Many economists prefer to argue that in a real-life situation, the expectation of the amount of money in an envelope
cannot be infinity, for instance, because the total amount of money in the world is bounded; therefore any probability
distribution describing the real world would have to assign probability 0 to the amount being larger than the total
amount of money on the world. Therefore the expectation of the amount of money under this distribution cannot be
infinity. The resolution of the second paradox, for such writers, is that the postulated probability distributions cannot
arise in a real-life situation. These are similar arguments as used to explain the St. Petersburg Paradox.
We certainly have a counter-intuitive situation. If the two envelopes are set up exactly as the Broome recipe requires
(not with real money, but just with numbers), then it is a fact that in many, many repetitions, the average of the
number written on the piece of paper in Envelope B, taken over all those occasions where the number in envelope A
is, say, 4, is definitely larger than 4. And the same thing holds with B and A exchanged! And the same thing holds
with the number 4 replaced by any other possible number! This might be hard to imagine, but we have now shown
how it can be arranged. There is no logical contradiction because we have seen why, logically, there is not
implication that we should switch envelopes. There does remain a conflict with our intuition.
The Broome paradox can be resolved at a purely formal level by showing where the error in the sequence of
apparently logical deductions occurs. But one still is left with a strange situation which simply does not feel right.
Exchange paradox
One can try to soften the blow by giving real-world reasons why this counter-intuitive situation could not occur in
reality (with real money). As far as practical economics is concerned, we need not worry about the insult to our
intuition.
Foundations of mathematical economics
In mathematical economics and the theory of utility, which explains economic behaviour in terms of expected utility,
there remains a problem to be resolved.[17] In the real world we presumably wouldn't indefinitely exchange one
envelope for the other (and probability theory, as just discussed, explains quite well why calculations of conditional
expectations might mislead us). Yet the expected utility based theory of economic behaviour says (or assumes) that
people do (or should) make economic decisions by maximizing expected utility, conditional on present knowledge,
and hence predicts that people would (or should) switch indefinitely.
Fortunately for mathematical economics and the theory of utility, it is generally agreed that as an amount of money
increases, its utility to the owner increases less and less, and ultimately there is a finite upper bound to the utility of
all possible amounts of money. We can pretend that the amount of money in the whole world is as large as we like,
yet the owner of all that money will not have more and more use of it, the more is in his possession. For decision
theory and utility theory, the two envelope paradox illustrates that unbounded utility does not exist in the real world,
so fortunately there is no need to build a decision theory which allows unbounded utility, let alone utility of infinite
expectation.
Controversy among philosophers
As mentioned above, any distribution producing this variant of the paradox must have an infinite mean. So before
the player opens an envelope the expected gain from switching is "∞ − ∞", which is not defined. In the words of
Chalmers this is "just another example of a familiar phenomenon, the strange behaviour of infinity".[18] Chalmers
suggests that decision theory generally breaks down when confronted with games having a diverging expectation,
and compares it with the situation generated by the classical St. Petersburg paradox.
However, Clark and Shackel argue that this blaming it all on "the strange behaviour of infinity" doesn't resolve the
paradox at all; neither in the single case nor the averaged case. They provide a simple example of a pair of random
variables both having infinite mean but where it is clearly sensible to prefer one to the other, both conditionally and
on average.[19] They argue that decision theory should be extended so as to allow infinite expectation values in some
situations. Most mathematical economists are happy to exclude infinite expected utility by assumption, hence
excluding the paradox altogether. Some try to generalise some of the existing theory to allow infinite expectations.
They have to come up with clever ways to get around the paradoxical example just given.
Non-probabilistic variant
The logician Raymond Smullyan questioned if the paradox has anything to do with probabilities at all. He did this by
expressing the problem in a way which doesn't involve probabilities. The following plainly logical arguments lead to
conflicting conclusions:
1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the
potential gain is strictly greater than the potential loss.
2. Let the amounts in the envelopes be X and 2X. Now by swapping, the player may gain X or lose X. So the
potential gain is equal to the potential loss.
27
Exchange paradox
Proposed resolutions
A number of solutions have been put forward. Careful analyses have been made by some logicians. Though solutions
differ, they all pinpoint semantic issues concerned with counterfactual reasoning. We want to compare the amount
that we would gain by switching if we would gain by switching, with the amount we would lose by switching if we
would indeed lose by switching. However, we cannot both gain and lose by switching at the same time. We are
asked to compare two incompatible situations. Only one of them can factually occur, the other will be a
counterfactual situation, somehow imaginary. In order to compare them at all, we must somehow "align" the two
situations, we must give them some definite points in common.
James Chase (2002) argues that the second argument is correct because it does correspond to the way to align two
situations (one in which we gain, the other in which we lose) which is preferably indicated by the problem
description.[20] Also Bernard Katz and Doris Olin (2007) argue this point of view.[21] In the second argument, we
consider the amounts of money in the two envelopes as being fixed; what varies is which one is first given to the
player. Because that was an arbitrary and physical choice, the counterfactual world in which the player,
counterfactually, got the other envelope to the one he was actually (factually) given is a highly meaningful
counterfactual world and hence the comparison between gains and losses in the two worlds is meaningful. This
comparison is uniquely indicated by the problem description, in which two amounts of money are put in the two
envelopes first, and only after that is one chosen arbitrarily and given to the player. In the first argument, however,
we consider the amount of money in the envelope first given to the player as fixed and consider the situations where
the second envelope contains either half or twice that amount. This would only be a reasonable counterfactual world
if in reality the envelopes had been filled as follows: first, some amount of money is placed in the specific envelope
which will be given to the player; and secondly, by some arbitrary process, the other envelope is filled (arbitrarily or
randomly) either with double or with half of that amount of money.
Byeong-Uk Yi (2009), on the other hand, argues that comparing the amount you would gain if you would gain by
switching with the amount you would lose if you would lose by switching is a meaningless exercise from the
outset.[22] According to his analysis, all three implications (switch, indifferent, don't switch) are incorrect. He
analyses Smullyan's arguments in detail, showing that intermediate steps are being taken, and pinpointing exactly
where an incorrect inference is made according to his formalization of counterfactual inference. An important
difference with Chase's analysis is that he does not take account of the part of the story where we are told that which
envelope is called Envelope A is decided completely at random. Thus Chase puts probability back into the problem
description in order to conclude that arguments 1 and 3 are incorrect, argument 2 is correct, while Yi keeps "two
envelope problem without probability" completely free of probability, and comes to the conclusion that there are no
reasons to prefer any action. This corresponds to the view of Albers et al., that without probability ingredient, there is
no way to argue that one action is better than another, anyway.
In perhaps the most recent paper on the subject, Bliss argues that the source of the paradox is that when one
mistakenly believes in the possibility of a larger payoff that does not, in actuality, exist, one is mistaken by a larger
margin than when one believes in the possibility of a smaller payoff that does not actually exist.[23] If, for example,
the envelopes contained $5.00 and $10.00 respectively, a player who opened the $10.00 envelope would expect the
possibility of a $20.00 payout that simply does not exist. Were that player to open the $5.00 envelope instead, he
would believe in the possibility of a $2.50 payout, which constitutes a smaller deviation from the true value.
Albers, Kooi, and Schaafsma (2005) consider that without adding probability (or other) ingredients to the problem,
Smullyan's arguments do not give any reason to swap or not to swap, in any case. Thus there is no paradox. This
dismissive attitude is common among writers from probability and economics: Smullyan's paradox arises precisely
because he takes no account whatever of probability or utility.
28
Exchange paradox
29
Extensions to the Problem
Since the two envelopes problem became popular, many authors have studied the problem in depth in the situation in
which the player has a prior probability distribution of the values in the two envelopes, and does look in Envelope A.
One of the most recent such publications is by McDonnell and Douglas (2009), who also consider some further
generalizations.[24]
If a priori we know that the amount in the smaller envelope is a whole number of some currency units, then the
problem is determined, as far as probability theory is concerned, by the probability mass function
describing
our prior beliefs that the smaller amount is any number x = 1,2, ... ; the summation over all values of x being equal to
1. It follows that given the amount a in Envelope A, the amount in Envelope B is certainly 2a if a is an odd number.
However, if a is even, then the amount in Envelope B is 2a with probability
, and a/2
with probability
. If one would like to switch envelopes if the expectation value of
what is in the other is larger than what we have in ours, then a simple calculation shows that one should switch if
, keep to Envelope A if
.
If on the other hand the smaller amount of money can vary continuously, and we represent our prior beliefs about it
with a probability density
, thus a function which integrates to one when we integrate over x running from
zero to infinity, then given the amount a in Envelope A, the other envelope contains 2a with probability
, and a/2 with probability
. If again we decide to
switch or not according to the expectation value of what's in the other envelope, the criterion for switching now
becomes
.
The difference between the results for discrete and continuous variables may surprise many readers. Speaking
intuitively, this is explained as follows. Let h be a small quantity and imagine that the amount of money we see when
we look in Envelope A is rounded off in such a way that differences smaller than h are not noticeable, even though
actually it varies continuously. The probability that the smaller amount of money is in an interval around a of length
h, and Envelope A contains the smaller amount is approximately
. The probability that the larger
amount of money is in an interval around a of length h corresponds to the smaller amount being in an interval of
length h/2 around a/2. Hence the probability that the larger amount of money is in a small interval around a of length
h and Envelope A contains the larger amount is approximately
. Thus, given Envelope A
contains an amount about equal to a, the probability it is the smaller of the two is roughly
.
If the player only wants to end up with the larger amount of money, and does not care about expected amounts, then
in the discrete case he should switch if a is an odd number, or if a is even and
. In the continuous
case he should switch if
.
Some authors prefer to think of probability in a frequentist sense. If the player knows the probability distribution
used by the organizer to determine the smaller of the two values, then the analysis would proceed just as in the case
when p or f represents subjective prior beliefs. However, what if we take a frequentist point of view, but the player
does not know what probability distribution is used by the organiser to fix the amounts of money in any one
instance? Thinking of the arranger of the game and the player as two parties in a two person game, puts the problem
into the range of game theory. The arranger's strategy consists of a choice of a probability distribution of x, the
smaller of the two amounts. Allowing the player also to use randomness in making his decision, his strategy is
determined by his choosing a probability of switching
for each possible amount of money a he might see in
Envelope A. In this section we so far only discussed fixed strategies, that is strategies for which q only takes the
values 0 and 1, and we saw that the player is fine with a fixed strategy, if he knows the strategy of the organizer. In
the next section we will see that randomized strategies can be useful when the organizer's strategy is not known.
Exchange paradox
Randomized solutions
Suppose as in the previous section that the player is allowed to look in the first envelope before deciding whether to
switch or to stay. We'll think of the contents of the two envelopes as being two positive numbers, not necessarily two
amounts of money. The player is allowed either to keep the number in Envelope A, or to switch and take the number
in Envelope B. We'll drop the assumption that one number is exactly twice the other, we'll just suppose that they are
different and positive. On the other hand, instead of trying to maximize expectation values, we'll just try to maximize
the chance that we end up with the larger number.
In this section we ask the question, is it possible for the player to make his choice in such a way that he goes home
with the larger number with probability strictly greater than half, however the organizer has filled the two envelopes?
We are given no information at all about the two numbers in the two envelopes, except that they are different, and
strictly greater than zero. The numbers were written down on slips of paper by the organiser, put into the two
envelopes. The envelopes were then shuffled, the player picks one, calls it Envelope A, and opens it.
We are not told any joint probability distribution of the two numbers. We are not asking for a subjectivist solution.
We must think of the two numbers in the envelopes as chosen by the arranger of the game according to some
possibly random procedure, completely unknown to us, and fixed. Think of each envelope as simply containing a
positive number and such that the two numbers are not the same. The job of the player is to end up with the envelope
with the larger number. This variant of the problem, as well as its solution, is attributed by McDonnell and Abbott,
and by earlier authors, to information theorist Thomas M. Cover.[25]
Counter-intuitive though it might seem, there is a way that the player can decide whether to switch or to stay so that
he has a larger chance than 1/2 of finishing with the bigger number, however the two numbers are chosen by the
arranger of the game. However, it is only possible with a so-called randomized algorithm, that means to say, the
player needs himself to be able to generate random numbers. Suppose he is able to think up a random number, let's
call it Z, such that the probability that Z is larger than any particular quantity z is exp(-z). Note that exp(-z) starts off
equal to 1 at z=0 and decreases strictly and continuously as z increases, tending to zero as z tends to infinity. So the
chance is 0 that Z is exactly equal to any particular number, and there is a positive probability that Z lies between any
two particular different numbers. The player compares his Z with the number in Envelope A. If Z is smaller he keeps
the envelope. If Z is larger he switches to the other envelope.
Think of the two numbers in the envelopes as fixed (though of course unknown to the player). Think of the player's
random Z as a probe with which he decides whether the number in Envelope A is small or large. If it is small
compared to Z he will switch, if it is large compared to Z he will stay.
If both numbers are smaller than the player's Z then his strategy does not help him, he ends up with the Envelope B,
which is equally likely to be the larger or the smaller of the two. If both numbers are larger than Z his strategy does
not help him either, he ends up with the first Envelope A, which again is equally likely to be the larger or the smaller
of the two. However if Z happens to be in between the two numbers, then his strategy leads him correctly to keep
Envelope A if its contents are larger than those of B, but to switch to Envelope B if A has smaller contents than B.
Altogether, this means that he ends up with the envelope with the larger number with probability strictly larger than
1/2. To be precise, the probability that he ends with the "winning envelope" is 1/2 + P(Z falls between the two
numbers)/2.
In practice, the number Z we have described could be determined to the necessary degree of accuracy as follows.
Toss a fair coin many times, and convert the sequence of heads and tails into the binary representation of a number U
between 0 and 1: for instance, HTHHTH... becomes the binary representation of u=0.101101.. . In this way, we
generate a random number U, uniformly distributed between 0 and 1. Then define Z = - ln (U) where "ln" stands for
natural logarithm, i.e., logarithm to base e. Note that we just need to toss the coin long enough to be able to see for
sure whether Z is smaller or larger than the number a in the first envelope, we do not need to go on for ever. We will
only need to toss the coin a finite (though random) number of times: at some point we can be sure that the outcomes
30
Exchange paradox
of further coin tosses is not going to change the outcome of the comparison.
The particular probability law (the so-called standard exponential distribution) used to generate the random number
Z in this problem is not crucial. Any probability distribution over the positive real numbers which assigns positive
probability to any interval of positive length will do the job.
This problem can be considered from the point of view of game theory, where we make the game a two-person
zero-sum game with outcomes win or lose, depending on whether the player ends up with the higher or lower
amount of money. The organiser chooses the joint distribution of the amounts of money in both envelopes, and the
player chooses the distribution of Z. The game does not have a "solution" (or saddle point) in the sense of game
theory. This is an infinite game and von Neumann's minimax theorem does not apply.[26]
History of the paradox
The envelope paradox dates back at least to 1953, when Belgian mathematician Maurice Kraitchik proposed a puzzle
in his book Recreational Mathematics concerning two equally rich men who meet and compare their beautiful
neckties, presents from their wives, wondering which tie actually cost more money. It is also mentioned in a 1953
book on elementary mathematics and mathematical puzzles by the mathematician John Edensor Littlewood, who
credited it to the physicist Erwin Schroedinger. Martin Gardner popularized Kraitchik's puzzle in his 1982 book Aha!
Gotcha, in the form of a wallet game:
Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the
two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the
other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the
amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll
have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me."
The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the
mistake in the reasoning of each man?
In 1988 and 1989, Barry Nalebuff presented two different two-envelope problems, each with one envelope
containing twice what's in the other, and each with computation of the expectation value 5A/4. The first paper just
presents the two problems, the second paper discusses many solutions to both of them. The second of his two
problems is the one which is nowadays the most common and which is presented in this article. According to this
version, the two envelopes are filled first, then one is chosen at random and called Envelope A. Martin Gardner
independently mentioned this same version in his 1989 book Penrose Tiles to Trapdoor Ciphers and the Return of
Dr Matrix. Barry Nalebuff's asymmetric variant, often known as the Ali Baba problem, has one envelope filled first,
called Envelope A, and given to Ali. Then a coin is tossed to decide whether Envelope B should contain half or twice
that amount, and only then given to Baba.
In the Ali-Baba problem, it is a priori clear that (even if they don't look in their envelopes) Ali should want to switch,
while Baba should want to keep what he has been given. The Ali-Baba paradox comes about by imagining Baba
working through the steps of the two-envelopes problem argument (second interpretation), and wrongly coming to
the conclusion that he too wants to switch, just like Ali.
31
Exchange paradox
Notes and references
[1] Markosian, Ned (2011). "A Simple Solution to the Two Envelope Problem". Logos & Episteme II (3): 347-357.
[2] McDonnell, Mark D; Grant, Alex J.; Land, Ingmar; Vellambi, Badri N.; Abbott, Derek; Lever, Ken (2011). "Gain from the two-envelope
problem via information asymmetry: on the suboptimality of randomized switching". Proceedings of the Royal Society, A to appear.
doi:10.1098/rspa.2010.0541.
[3] A complete list of published and unpublished sources in chronological order can be found here
[4] Falk, Ruma (2008). "The Unrelenting Exchange Paradox". Teaching Statistics 30 (3): 86–88. doi:10.1111/j.1467-9639.2008.00318.x.
[5] Eckhardt, William (2013). "The Two-Envelopes Problem". Paradoxes in Probability Theory. Springer. pp. 47–48.
[6] Bruss, F.T. (1996). "The Fallacy of the Two Envelopes Problem". The Mathematical Scientist 21 (2): 112–119..
[7] Falk, Ruma (2009). "An inside look at the two envelope paradox". Teaching Statistics 31 (2): 39–41..
[8] Casper Albers, Trying to resolve the two-envelope problem, Chapter 2 of his thesis Distributional Inference: The Limits of Reason, March
2003. (Has also appeared as Albers, Casper J.; Kooi, Barteld P. and Schaafsma, Willem (2005), Trying to resolve the two-envelope problem,
Synthese, 145(1): 89–109 p91)
[9] Ruma Falk, Raymond Nickerson, An inside look at the two envelopes paradox, Teaching Statistics 31(2): 39-41.
[10] Jeff Chen, The Puzzle of the Two-Envelope Puzzle—a Logical Approach, published online p274
[11] Barry Nalebuff, Puzzles: The Other Person’s Envelope is Always Greener, Journal of Economic Perspectives 3(1): 171–181.
[12] John Broome, The Two-envelope Paradox, Analysis 55(1): 6–11.
[13] Blachman, N. M.; Christensen, R.; Utts, J. (1996). The American Statistician 50 (1): 98–99.
[14] Christensen, R.; Utts, J. (1992). The American Statistician 46 (4): 274–276., the letters to editor and responses by Christensen and Utts by
D.A. Binder (1993; vol. 47, nr. 2, p. 160) and Ross (1994; vol. 48, nr. 3, p. 267), and a letter with corrections to the original article by N.M.
Blachman, R. Christensen and J.M. Utts (1996; vol. 50, nr. 1, pp. 98-99)
[15] Here we present some details of a famous example due to John Broome of a proper probability distribution of the amounts of money in the
two envelopes, for which
for all a. Broome, John (1995). "The Two-envelope Paradox". Analysis 55 (1): 6–11.
doi:10.1093/analys/55.1.6.
[16] Binder, D. A. (1993). The American Statistician 47 (2): 160. (letters to the editor, comment on Christensen and Utts (1992)
[17] Fallis, D. (2009). "Taking the Two Envelope Paradox to the Limit". Southwest Philosophy Review 25 (2).
[18] Chalmers, David J. (2002). "The St. Petersburg Two-Envelope Paradox". Analysis 62 (2): 155–157. doi:10.1093/analys/62.2.155.
[19] Clark, M.; Shackel, N. (2000). "The Two-Envelope Paradox". Mind 109 (435): 415–442. doi:10.1093/mind/109.435.415.
[20] Chase, James (2002). "The Non-Probabilistic Two Envelope Paradox". Analysis 62 (2): 157–160. doi:10.1093/analys/62.2.157.
[21] Katz, Bernard; Olin, Doris (2007). "A tale of two envelopes". Mind 116 (464): 903–926. doi:10.1093/mind/fzm903.
[22] Byeong-Uk Yi (2009). The Two-envelope Paradox With No Probability (http:/ / philosophy. utoronto. ca/ people/ linked-documents-people/
c two envelope with no probability. pdf). .
[23] Bliss (2012). A Concise Resolution to the Two Envelope Paradox (http:/ / arxiv. org/ abs/ 1202. 4669). .
[24] McDonnell, M. D.; Abott, D. (2009). "Randomized switching in the two-envelope problem". Proceedings of the Royal Society A 465 (2111):
3309–3322. doi:10.1098/rspa.2009.0312.
[25] Cover, Thomas M.. "Pick the largest number". In Cover, T.; Gopinath, B.. Open Problems in Communication and Computation.
Springer-Verlag.
[26] http:/ / www. mit. edu/ ~emin/ writings/ envelopes. html
32
Kavka's toxin puzzle
33
Kavka's toxin puzzle
Kavka's toxin puzzle is a thought experiment about the possibility of forming an intention to perform an act which,
following from reason, is an action one would not actually perform. It was presented by moral and political
philosopher Gregory S. Kavka in "The Toxin Puzzle" (1983), and grew out of his work in deterrence theory and
mutual assured destruction.
The puzzle
Kavka's original version of the puzzle is the following:
An eccentric billionaire places before you a vial of toxin that, if you drink it, will make you painfully ill
for a day, but will not threaten your life or have any lasting effects. The billionaire will pay you one
million dollars tomorrow morning if, at midnight tonight, you intend to drink the toxin tomorrow
afternoon. He emphasizes that you need not drink the toxin to receive the money; in fact, the money will
already be in your bank account hours before the time for drinking it arrives, if you succeed. All you
have to do is. . . intend at midnight tonight to drink the stuff tomorrow afternoon. You are perfectly free
to change your mind after receiving the money and not drink the toxin.[1]
A possible interpretation: Can you intend to drink the toxin, if you know you do not have to?
The paradox
The paradoxical nature can be stated in many ways, which may be useful for understanding analysis proposed by
philosophers:
• In line with Newcomb's paradox, an omniscient pay-off mechanism makes a person's decision known to him
before he makes the decision, but it is also assumed that the person may change his decision afterwards, of free
will.
• Similarly in line with Newcomb's paradox; Kavka's claim, that one cannot intend what one will not do, makes
pay-off mechanism an example of reverse causation.
• Pay-off for decision to drink the poison is ambiguous.
• There are two decisions for one event with different pay-offs.
Since the pain caused by the poison would be more than off-set by the money received, we can sketch the pay-off
table as follows.
Pay-offs (Initial analysis)
Intend Do not intend
Drink
90
Do not drink 100
−10
0
According to Kavka: Whether you are paid or not, drinking the poison would leave you worse off. A rational person
would know he would not drink the poison and thus could not intend to drink it.
Kavka's toxin puzzle
34
Pay-offs (According to Kavka)
Intend
Drink
Do not intend
Impossible −10
Do not drink Impossible 0
David Gauthier argues once a person intends drinking the poison one cannot entertain ideas of not drinking it.[2]
The rational outcome of your deliberation tomorrow morning is the action that will be part of your life
going as well as possible, subject to the constraint that it be compatible with your commitment-in this
case, compatible with the sincere intention that you form today to drink the toxin. And so the rational
action is to drink the toxin.
Pay-offs (According to Gauthier)
Intend
Drink
90
Do not intend
−10
Do not drink Impossible 0
One of the central tenets of the puzzle is that for a reasonable person
• There is reasonable grounds for that person to drink the toxin, since some reward may be obtained.
• Having come to the above conclusion there is no reasonable grounds for that person to drink the toxin, since no
further reward may be obtained, and no reasonable person would partake in self-harm for no benefit.
Thus a reasonable person must intend to drink the toxin by the first argument, yet if that person intends to drink the
toxin, he is being irrational by the second argument.
References
[1] Kavka, Gregory (1983). "The Toxin Puzzle". Analysis 43 (1): 33–36 [pp. 33–34]. doi:10.1093/analys/43.1.33.
[2] Gauthier, David (1994). "Assure and Threaten". Ethics 104 (4): 690–721. JSTOR 2382214.
Necktie paradox
35
Necktie paradox
The necktie paradox is a puzzle or paradox within the subjectivistic interpretation of probability theory. It is a
variation (and historically, the origin) of the two-envelope paradox.
Two men are each given a necktie by their respective wives as a Christmas present. Over drinks they start arguing
over who has the cheaper necktie. They agree to have a wager over it. They will consult their wives and find out
which necktie is more expensive. The terms of the bet are that the man with the more expensive necktie has to give it
to the other as the prize.
The first man reasons as follows: winning and losing are equally likely. If I lose, then I lose the value of my necktie.
But if I win, then I win more than the value of my necktie. Therefore the wager is to my advantage. The second man
can consider the wager in exactly the same way; thus, paradoxically, it seems both men have the advantage in the
bet. This is obviously not possible.
The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my
necktie") and what is won in the other ("more than the value of my necktie"). If we assume for simplicity that the
only possible necktie prices are $20 and $30, and that a man has equal chances of having a $20 or $30 necktie, then
four outcomes (all equally likely) are possible:
Price of 1st man's tie Price of 2nd man's tie 1st man's gain/loss
$20
$20
0
$20
$30
gain $30
$30
$20
lose $30
$30
$30
0
We see that the first man has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $30, and
a 25% chance of losing a necktie worth $30. Turning to the losing and winning scenarios: if the man loses $30, then
it is true that he has lost the value of his necktie; and if he gains $30, then it is true that he has gained more than the
value of his necktie. The win and the loss are equally likely; but what we call the value of his necktie in the losing
scenario is the same amount as what we call more than the value of his necktie in the winning scenario.
Accordingly, neither man has the advantage in the wager.
In general, what goes wrong is that when the first man is imagining the scenario that his necktie is actually worth
less than the other, his beliefs as to its value have to be revised (downwards) relatively to what they are a priori,
without such additional information. Yet in the apparently logical reasoning leading him to take the wager, he is
behaving as if his necktie is worth the same when it is worth less than the other, as when it is worth more than the
other. Of course, the price his wife actually paid for it is fixed, and doesn't change if it is revealed which tie is worth
more. The point is that this price, whatever it was, is unknown to him. It is his beliefs about the price which could not
be the same if he was given further information as to which tie was worth more. And it is on the basis of his prior
beliefs about the prices that he has to make his decision whether or not to accept the wager.
On a technical note, if the prices of the ties could in principle be arbitrarily large, then it is possible to have beliefs
about their values, such that learning which was the larger would not cause any change to one's beliefs about the
value of one's own tie. However, if one is 100% certain that neither tie can be worth more than, say $100, then
knowing which is worth the most changes one's expected value of both (one goes up, the other goes down).
This paradox is a rephrasing of the simplest case of the two envelopes problem, and the explanation of "what goes
wrong" is essentially the same.
Necktie paradox
References
• Brown, Aaron C. "Neckties, wallets, and money for nothing." Journal of Recreational Mathematics 27.2 (1995):
116–122.
• Maurice Kraitchik, Mathematical Recreations, George Allen & Unwin, London 1943
36
37
Economy
Allais paradox
The Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed
choices with the predictions of expected utility theory.
Statement of the Problem
The Allais paradox arises when comparing participants' choices in two different experiments, each of which
consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:
Experiment 1
Gamble 1A
Experiment 2
Gamble 1B
Gamble 2A
Gamble 2B
Winnings Chance Winnings Chance Winnings Chance Winnings Chance
$1 million 100%
$1 million 89%
Nothing
Nothing
$1 million 11%
1%
$5 million 10%
89%
Nothing
90%
$5 million 10%
Several studies involving hypothetical and small monetary payoffs, and recently involving health outcomes[1], have
supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A.
Likewise, when presented with a choice between 2A and 2B, most people would choose 2B. Allais further asserted
that it was reasonable to choose 1A alone or 2B alone.
However, that the same person (who chose 1A alone or 2B alone) would choose both 1A and 2B together is
inconsistent with expected utility theory. According to expected utility theory, the person should choose either 1A
and 2A or 1B and 2B.
The inconsistency stems from the fact that in expected utility theory, equal outcomes added to each of the two
choices should have no effect on the relative desirability of one gamble over the other; equal outcomes should
"cancel out". Each experiment gives the same outcome 89% of the time (starting from the top row and moving down,
both 1A and 1B give an outcome of $1 million, and both 2A and 2B give an outcome of nothing). If this 89%
‘common consequence’ is disregarded, then the gambles will be left offering the same choice.
It may help to re-write the payoffs. After disregarding the 89% chance of winning — the same outcome — then 1B
is left offering a 1% chance of winning nothing and a 10% chance of winning $5 million, while 2B is also left
offering a 1% chance of winning nothing and a 10% chance of winning $5 million. Hence, choice 1B and 2B can be
seen as the same choice. In the same manner, 1A and 2A should also now be seen as the same choice.
Allais paradox
38
Experiment 1
Gamble 1A
Experiment 2
Gamble 1B
Gamble 2A
Gamble 2B
Winnings Chance Winnings Chance Winnings Chance Winnings Chance
$1 million 89%
$1 million 89%
Nothing
89%
Nothing
89%
$1 million 11%
Nothing
$1 million 11%
Nothing
1%
1%
$5 million 10%
$5 million 10%
Allais presented his paradox as a counterexample to the independence axiom.
Independence means that if an agent is indifferent between simple lotteries
between
probability
mixed with an arbitrary simple lottery
with probability
and
and
, the agent is also indifferent
mixed with
with the same
. Violating this principle is known as the "common consequence" problem (or "common consequence"
effect). The idea of the common consequence problem is that as the prize offered by
increases,
and
become consolation prizes, and the agent will modify preferences between the two lotteries so as to minimize risk
and disappointment in case they do not win the higher prize offered by
.
Difficulties such as this gave rise to a number of alternatives to, and generalizations of, the theory, notably including
prospect theory, developed by Daniel Kahneman and Amos Tversky, weighted utility (Chew), and rank-dependent
expected utility by John Quiggin. The point of these models was to allow a wider range of behavior than was
consistent with expected utility theory.
Also relevant here is the framing theory of Daniel Kahneman and Amos Tversky. Identical items will result in
different choices if presented to agents differently (i.e. a surgery with a 70% survival rate vs. a 30% chance of death).
The main point Allais wished to make is that the independence axiom of expected utility theory may not be a valid
axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to
the analysis of the gamble as a whole. However, this overlooks the notion of complementarities, the fact your choice
in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B,
there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense
of disappointment if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you
had chosen 1A. This feeling of disappointment, however, is contingent on the outcome in the other portion of the
gamble (i.e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or
choices independently of the other choices presented, as the independence axiom requires, and thus is a poor judge
of our rational action (1B cannot be valued independently of 1A as the independence or sure thing principle requires
of us). We don't act irrationally when choosing 1A and 2B; rather expected utility theory is not robust enough to
capture such "bounded rationality" choices that in this case arise because of complementarities.
Mathematical proof of inconsistency
Using the values above and a utility function U(W), where W is wealth, we can demonstrate exactly how the paradox
manifests.
Because the typical individual prefers 1A to 1B and 2B to 2A, we can conclude that the expected utilities of the
preferred is greater than the expected utilities of the second choices, or,
Experiment 1
Allais paradox
Experiment 2
We can rewrite the latter equation (Experiment 2) as
which contradicts the first bet (Experiment 1), which shows the player prefers the sure thing over the gamble.
References
[1] Oliver, Adam (2003). "A quantitative and qualitative test of the Allais paradox using health outcomes". Journal of Economic Psychology 24
(1): 35–48. doi:10.1016/S0167-4870(02)00153-8.
• Allais, M. (1953). "Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de
l’école Américaine". Econometrica 21 (4): 503–546. JSTOR 1907921.
• Chew Soo Hong; Mao, Jennifer; Nishimura, Naoko (2005). Preference for longshot: An Experimental Study of
Demand for Sweepstakes (http://cebr.ust.hk/conference/2ndconference/nishimura.htm).
• Kahneman, Daniel; Tversky, Amos (1979). "Prospect Theory: An Analysis of Decision under Risk".
Econometrica 47 (2): 263–291. JSTOR 1914185.
• Oliver, Adam (2003). "A quantitative and qualitative test of the Allais paradox using health outcomes". Journal of
Economic Psychology 24 (1): 35–48. doi:10.1016/S0167-4870(02)00153-8.
• Quiggin, J. (1993). Generalized Expected Utility Theory:The Rank-Dependent Expected Utility model.
Amsterdam: Kluwer-Nijhoff. review (http://www.uq.edu.au/economics/johnquiggin/Books/Machina.html)
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox,
states that, when voters have three or more distinct alternatives (options), no rank order voting system can convert
the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting
a specific set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and
independence of irrelevant alternatives. The theorem is often cited in discussions of election theory as it is further
interpreted by the Gibbard–Satterthwaite theorem.
The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and
popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in
the Concept of Social Welfare".[1]
In short, the theorem states that no rank-order voting system can be designed that satisfies these three "fairness"
criteria:
• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y
will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W
change).
• There is no "dictator": no single voter possesses the power to always determine the group's preference.
Voting systems that use cardinal utility (which conveys more information than rank orders; see the subsection
discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem. [2]
The theorem can also be sidestepped by weakening the notion of independence. Arrow, like many economists,
rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference
39
Arrow's impossibility theorem
rankings.
The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one
unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in
which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice
theory started from this theorem.[3]
Statement of the theorem
The need to aggregate preferences occurs in many different disciplines: in welfare economics, where one attempts to
find an economic outcome which would be acceptable and stable; in decision theory, where a person has to make a
rational choice based on several criteria; and most naturally in voting systems, which are mechanisms for extracting
a decision from a multitude of voters' preferences.
The framework for Arrow's theorem assumes that we need to extract a preference order on a given set of options
(outcomes). Each individual in the society (or equivalently, each decision criterion) gives a particular order of
preferences on the set of outcomes. We are searching for a ranked voting system, called a social welfare function
(preference aggregation rule), which transforms the set of preferences (profile of preferences) into a single global
societal preference order. The theorem considers the following properties, assumed to be reasonable requirements of
a fair voting method:
Non-dictatorship
The social welfare function should account for the wishes of multiple voters. It cannot simply mimic the
preferences of a single voter.
Unrestricted domain
(or universality) For any set of individual voter preferences, the social welfare function should yield a unique
and complete ranking of societal choices. Thus:
• It must do so in a manner that results in a complete ranking of preferences for society.
• It must deterministically provide the same ranking each time voters' preferences are presented the same way.
Independence of irrelevant alternatives (IIA)
The social preference between x and y should depend only on the individual preferences between x and y
(Pairwise Independence). More generally, changes in individuals' rankings of irrelevant alternatives (ones
outside a certain subset) should have no impact on the societal ranking of the subset. (See Remarks below.)
Positive association of social and individual values
(or monotonicity) If any individual modifies his or her preference order by promoting a certain option, then
the societal preference order should respond only by promoting that same option or not changing, never by
placing it lower than before. An individual should not be able to hurt an option by ranking it higher.
Non-imposition
(or citizen sovereignty) Every possible societal preference order should be achievable by some set of
individual preference orders. This means that the social welfare function is surjective: It has an unrestricted
target space.
Arrow's theorem says that if the decision-making body has at least two members and at least three options to decide
among, then it is impossible to design a social welfare function that satisfies all these conditions at once.
A later (1963) version of Arrow's theorem can be obtained by replacing the monotonicity and non-imposition criteria
with:
Pareto efficiency
40
Arrow's impossibility theorem
41
(or unanimity) If every individual prefers a certain option to another, then so must the resulting societal
preference order. This, again, is a demand that the social welfare function will be minimally sensitive to the
preference profile.
The later version of this theorem is stronger—has weaker conditions—since monotonicity, non-imposition, and
independence of irrelevant alternatives together imply Pareto efficiency, whereas Pareto efficiency and independence
of irrelevant alternatives together do not imply monotonicity. (Incidentally, Pareto efficiency on its own implies
non-imposition.)
Remarks on IIA
1. The IIA condition can be justified for three reasons (Mas-Colell, Whinston, and Green, 1995, page 794): (i)
normative (irrelevant alternatives should not matter), (ii) practical (use of minimal information), and (iii) strategic
(providing the right incentives for the truthful revelation of individual preferences). Though the strategic property
is conceptually different from IIA, it is closely related.
2. Arrow's death-of-a-candidate example (1963, page 26) suggests that the agenda (the set of feasible alternatives)
shrinks from, say, X = {a, b, c} to S = {a, b} because of the death of candidate c. This example is misleading
since it can give the reader an impression that IIA is a condition involving two agenda and one profile. The fact is
that IIA involves just one agendum ({x, y} in case of Pairwise Independence) but two profiles. If the condition is
applied to this confusing example, it requires this: Suppose an aggregation rule satisfying IIA chooses b from the
agenda {a, b} when the profile is given by (cab, cba), that is, individual 1 prefers c to a to b, 2 prefers c to b to a.
Then, it must still choose b from {a, b} if the profile were, say, (abc, bac) or (acb, bca) or (acb, cba) or (abc, cba).
Formal statement of the theorem
Let
be a set of outcomes,
orderings of
by
a number of voters or decision criteria. We shall denote the set of all full linear
.
A (strict) social welfare function (preference aggregation rule) is a function
aggregates voters' preferences into a single preference order on
[4]
.
The
-tuple
which
of voters'
preferences is called a preference profile. In its strongest and simplest form, Arrow's impossibility theorem states
that whenever the set
of possible alternatives has more than 2 elements, then the following three conditions
become incompatible:
unanimity, or Pareto efficiency
If alternative a is ranked above b for all orderings
, then a is ranked higher than b by
. (Note that unanimity implies non-imposition).
non-dictatorship
There is no individual i whose preferences always prevail. That is, there is no
such that
.
independence of irrelevant alternatives
For two preference profiles
and b have the same order in
as in
and
such that for all individuals i, alternatives a
as in
, alternatives a and b have the same order in
.
Arrow's impossibility theorem
42
Informal proof
Based on the proof by John Geanakoplos of Cowles Foundation, Yale University published by Economic Theory
(journal) in 2005.[5] (An attempt to improve upon it appeared on the same journal in 2012.[6])
We wish to prove that any social choice system respecting unrestricted domain, unanimity, and independence of
irrelevant alternatives (IIA) is a dictatorship.
Part one: there is a "pivotal" voter for B
Say there are three choices for society, call them A, B, and C. Suppose first that everyone prefers option B the least.
That is, everyone prefers every other option to B. By unanimity, society must prefer every option to B. Specifically,
society prefers A and C to B. Call this situation Profile 1.
On the other hand, if everyone preferred B to everything else, then society would have to prefer B to everything else
by unanimity. So it is clear that, if we take Profile 1 and, running through the members in the society in some
arbitrary but monotonic order, move B from the bottom of each person's preference list to the top, there must be
some point at which B moves off the bottom of society's preferences as well, since we know it eventually ends up at
the top. When it happens, we call that voter the pivotal voter.
We now want to show that, at the point when the pivotal voter n moves B off the bottom of his preferences to the
top, the society's B moves to the top of its preferences as well, not to an intermediate point.
To prove this, consider what would happen if it were not true. Then, after n has moved B to the top (i.e., when voters
have B at the top and voters
still have B at the bottom) society would have some
option more preferable than B, say A, and one less preferable than B, say C.
Now if each person moves his preference for C above A, then society would prefer C to A by unanimity. But
moving C above A should not change anything about how B and C compare, by independence of irrelevant
alternatives. That is, since B is either at the very top or bottom of each person's preferences, moving C or A around
does not change how either compares with B, leaving B preferred to C. Similarly, by independence of irrelevant
alternatives society still prefers A to B because the changing of C and A does not affect how A and B compare.
Since C is above A, and A is above B, C must be above B in the social preference ranking. We have reached an
absurd conclusion.
Therefore, when the voters
have moved B from the bottom of their preferences to the top, society
moves B from the bottom all the way to the top, not some intermediate point.
Note that even with a different starting profile, say Profile 1' , if the order of moving preference of B is unchanged,
the pivotal voter remains n. That is, the pivotal voter is determined only by the moving order, and not by the starting
profile.
It can be seen as following. If we concentrate on a pair of B and one of other choices, during each step on the
process, preferences in the pair are unchanged whether we start from Profile 1 and Profile 1' for every person.
Therefore by IIA, preference in the pair should be unchanged. Since it applies to every other choices, for Profile 1' ,
the position of B remains at bottom before n and remains at top after and including n, just as Profile 1.
Arrow's impossibility theorem
Part two: voter n is a dictator for A–C
We show that voter n dictates society's decision between A and C. In other words, we show that n is a (local)
dictator over the set {A, C} in the following sense: if n prefers A to C, then the society prefers A to C and if n
prefers C to A, then the society prefers C to A.
Let p1 be any profile in which voter n prefers A to C. We show that society prefers A to C. To show that, construct
two profiles from p1 by changing the position of B as follows: In Profile 2, all voters up to (not including) n have B
at the top of their preferences and the rest (including n) have B at the bottom. In Profile 3, all voters up to (and
including) n have B at the top and the rest have B at the bottom.
Now consider the profile p4 obtained from p1 as follows: everyone up to n ranks B at the top, n ranks A above B
above C, and everyone else ranks B at the bottom. As far as the A–B decision is concerned, p4 is just as in Profile 2,
which we proved puts A above B (in Profile 2, B is actually at the bottom of the social ordering). C's new position is
irrelevant to the B–A ordering for society because of IIA. Likewise, p4 has a relationship between B and C that is
just as in Profile 3, which we proved has B above C (B is actually at the top). We can conclude from these two
observations that society puts A above B above C at p4. Since the relative rankings of A and C are the same across
p1 and p4, we conclude that society puts A above C at p1.
Similarly, we can show that if q1 is any profile in which voter n prefers C to A, then society prefers C to A. It
follows that person n is a (local) dictator over {A, C}.
Remark. Since B is irrelevant (IIA) to the decision between A and C, the fact that we assumed particular profiles
that put B in particular places does not matter. This was just a way of finding out, by example, who the dictator over
A and C was. But all we need to know is that he exists.
Part three: there can be at most one dictator
Finally, we show that the (local) dictator over {A, C} is a (global) dictator: he also dictates over {A, B} and over {B,
C}. We will use the fact (which can be proved easily) that if
is a strict linear order, then it contains no cycles
such as
. We have proved in Part two that there are (local) dictators i over {A, B}, j over {B,
C}, and k over {A, C}.
• If i, j, k are all distinct, consider any profile in which i prefers A to B, j prefers B to C and k prefers C to A. Then
the society prefers A to B to C to A, a contradiction.
• If one of i, j, k is different and the other two are equal, assume i=j without loss of generality. Consider any profile
in which i=j prefers A to B to C and k prefers C to A. Then the society prefers A to B to C to A, a contradiction.
It follows that i=j=k, establishing that the local dictator over {A, C} is a global one.
Interpretations of the theorem
Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement
such as "No voting method is fair," "Every ranked voting method is flawed," or "The only voting method that isn't
flawed is a dictatorship". These statements are simplifications of Arrow's result which are not universally considered
to be true. What Arrow's theorem does state is that a deterministic preferential voting mechanism - that is, one where
a preference order is the only information in a vote, and any possible set of votes gives a unique result - cannot
comply with all of the conditions given above simultaneously.
Arrow did use the term "fair" to refer to his criteria. Indeed, Pareto efficiency, as well as the demand for
non-imposition, seems acceptable to most people.
Various theorists have suggested weakening the IIA criterion as a way out of the paradox. Proponents of ranked
voting methods contend that the IIA is an unreasonably strong criterion. It is the one breached in most useful voting
systems. Advocates of this position point out that failure of the standard IIA criterion is trivially implied by the
possibility of cyclic preferences. If voters cast ballots as follows:
43
Arrow's impossibility theorem
• 1 vote for A > B > C
• 1 vote for B > C > A
• 1 vote for C > A > B
then the pairwise majority preference of the group is that A wins over B, B wins over C, and C wins over A: these
yield rock-paper-scissors preferences for any pairwise comparison. In this circumstance, any aggregation rule that
satisfies the very basic majoritarian requirement that a candidate who receives a majority of votes must win the
election, will fail the IIA criterion, if social preference is required to be transitive (or acyclic). To see this, suppose
that such a rule satisfies IIA. Since majority preferences are respected, the society prefers A to B (two votes for A>B
and one for B>A), B to C, and C to A. Thus a cycle is generated, which contradicts the assumption that social
preference is transitive.
So, what Arrow's theorem really shows is that any majority-wins voting system is a non-trivial game, and that game
theory should be used to predict the outcome of most voting mechanisms.[7] This could be seen as a discouraging
result, because a game need not have efficient equilibria, e.g., a ballot could result in an alternative nobody really
wanted in the first place, yet everybody voted for.
Remark: Scalar rankings from a vector of attributes and the IIA property. The IIA property might not be
satisfied in human decision-making of realistic complexity because the scalar preference ranking is effectively
derived from the weighting—not usually explicit—of a vector of attributes (one book dealing with the Arrow
theorem invites the reader to consider the related problem of creating a scalar measure for the track and field
decathlon event—e.g. how does one make scoring 600 points in the discus event "commensurable" with scoring 600
points in the 1500 m race) and this scalar ranking can depend sensitively on the weighting of different attributes,
with the tacit weighting itself affected by the context and contrast created by apparently "irrelevant" choices. Edward
MacNeal discusses this sensitivity problem with respect to the ranking of "most livable city" in the chapter
"Surveys" of his book MathSemantics: making numbers talk sense (1994).
Other possibilities
In an attempt to escape from the negative conclusion of Arrow's theorem, social choice theorists have investigated
various possibilities ("ways out"). These investigations can be divided into the following two:
• those investigating functions whose domain, like that of Arrow's social welfare functions, consists of profiles of
preferences;
• those investigating other kinds of rules.
Approaches investigating functions of preference profiles
This section includes approaches that deal with
• aggregation rules (functions that map each preference profile into a social preference), and
• other functions, such as functions that map each preference profile into an alternative.
Since these two approaches often overlap, we discuss them at the same time. What is characteristic of these
approaches is that they investigate various possibilities by eliminating or weakening or replacing one or more
conditions (criteria) that Arrow imposed.
44
Arrow's impossibility theorem
Infinitely many individuals
Several theorists (e.g., Kirman and Sondermann, 1972[8]) point out that when one drops the assumption that there are
only finitely many individuals, one can find aggregation rules that satisfy all of Arrow's other conditions.
However, such aggregation rules are practically of limited interest, since they are based on ultrafilters, highly
nonconstructive mathematical objects. In particular, Kirman and Sondermann argue that there is an "invisible
dictator" behind such a rule. Mihara (1997,[9] 1999[10]) shows that such a rule violates algorithmic computability.[11]
These results can be seen to establish the robustness of Arrow's theorem.[12]
Limiting the number of alternatives
When there are only two alternatives to choose from, May's theorem shows that only simple majority rule satisfies a
certain set of criteria (e.g., equal treatment of individuals and of alternatives; increased support for a winning
alternative should not make it into a losing one). On the other hand, when there are at least three alternatives,
Arrow's theorem points out the difficulty of collective decision making. Why is there such a sharp difference
between the case of less than three alternatives and that of at least three alternatives?
Nakamura's theorem (about the core of simple games) gives an answer more generally. It establishes that if the
number of alternatives is less than a certain integer called the Nakamura number, then the rule in question will
identify "best" alternatives without any problem; if the number of alternatives is greater or equal to the Nakamura
number, then the rule will not always work, since for some profile a voting paradox (a cycle such as alternative A
socially preferred to alternative B, B to C, and C to A) will arise. Since the Nakamura number of majority rule is 3
(except the case of four individuals), one can conclude from Nakamura's theorem that majority rule can deal with up
to two alternatives rationally. Some super-majority rules (such as those requiring 2/3 of the votes) can have a
Nakamura number greater than 3, but such rules violate other conditions given by Arrow.[13]
Remark. A common way "around" Arrow's paradox is limiting the alternative set to two alternatives. Thus,
whenever more than two alternatives should be put to the test, it seems very tempting to use a mechanism that pairs
them and votes by pairs. As tempting as this mechanism seems at first glance, it is generally far from satisfying even
Pareto efficiency, not to mention IIA. The specific order by which the pairs are decided strongly influences the
outcome. This is not necessarily a bad feature of the mechanism. Many sports use the tournament
mechanism—essentially a pairing mechanism—to choose a winner. This gives considerable opportunity for weaker
teams to win, thus adding interest and tension throughout the tournament. This means that the person controlling the
order by which the choices are paired (the agenda maker) has great control over the outcome. In any case, when
viewing the entire voting process as one game, Arrow's theorem still applies.
Domain restrictions
Another approach is relaxing the universality condition, which means restricting the domain of aggregation rules.
The best-known result along this line assumes "single peaked" preferences.
Duncan Black has shown that if there is only one dimension on which every individual has a "single-peaked"
preference, then all of Arrow's conditions are met by majority rule. Suppose that there is some predetermined linear
ordering of the alternative set. An individual's preference is single-peaked with respect to this ordering if he has
some special place that he likes best along that line, and his dislike for an alternative grows larger as the alternative
goes further away from that spot (i.e., the graph of his utility function has a single peak if alternatives are placed
according to the linear ordering on the horizontal axis). For example, if voters were voting on where to set the
volume for music, it would be reasonable to assume that each voter had their own ideal volume preference and that
as the volume got progressively too loud or too quiet they would be increasingly dissatisfied. If the domain is
restricted to profiles in which every individual has a single peaked preference with respect to the linear ordering,
then simple ([]) aggregation rules, which includes majority rule, have an acyclic (defined below) social preference,
hence "best" alternatives.[14] In particular, when there are odd number of individuals, then the social preference
45
Arrow's impossibility theorem
becomes transitive, and the socially "best" alternative is equal to the median of all the peaks of the individuals
(Black's median voter theorem[15]). Under single-peaked preferences, the majority rule is in some respects the most
natural voting mechanism.
One can define the notion of "single-peaked" preferences on higher dimensional sets of alternatives. However, one
can identify the "median" of the peaks only in exceptional cases. Instead, we typically have the destructive situation
suggested by McKelvey's Chaos Theorem (1976[16]): for any x and y, one can find a sequence of alternatives such
that x is beaten by
by a majority,
by
,
,
by y.
Relaxing transitivity
By relaxing the transitivity of social preferences, we can find aggregation rules that satisfy Arrow's other conditions.
If we impose neutrality (equal treatment of alternatives) on such rules, however, there exists an individual who has a
"veto". So the possibility provided by this approach is also very limited.
First, suppose that a social preference is quasi-transitive (instead of transitive); this means that the strict preference
("better than") is transitive: if
and
, then
. Then, there do exist non-dictatorial
aggregation rules satisfying Arrow's conditions, but such rules are oligarchic (Gibbard, 1969). This means that there
exists a coalition L such that L is decisive (if every member in L prefers x to y, then the society prefers x to y), and
each member in L has a veto (if she prefers x to y, then the society cannot prefer y to x).
Second, suppose that a social preference is acyclic (instead of transitive): there does not exist alternatives
that form a cycle (
,
,
,
,
). Then, provided that there
are at least as many alternatives as individuals, an aggregation rule satisfying Arrow's other conditions is collegial
(Brown, 1975[17]). This means that there are individuals who belong to the intersection ("collegium") of all decisive
coalitions. If there is someone who has a veto, then he belongs to the collegium. If the rule is assumed to be neutral,
then it does have someone who has a veto.[]
Finally, Brown's theorem left open the case of acyclic social preferences where the number of alternatives is less
than the number of individuals. One can give a definite answer for that case using the Nakamura number. See
#Limiting the number of alternatives.
Relaxing IIA
There are numerous examples of aggregation rules satisfying Arrow's conditions except IIA. The Borda rule is one
of them. These rules, however, are susceptible to strategic manipulation by individuals (Blair and Muller, 1983[18]).
See also Interpretations of the theorem above.
Relaxing the Pareto criterion
Wilson (1972) shows that if an aggregation rule is non-imposed and non-null, then there is either a dictator or an
inverse dictator, provided that Arrow's conditions other than Pareto are also satisfied. Here, an inverse dictator is an
individual i such that whenever i prefers x to y, then the society prefers y to x.
Remark. Amartya Sen offered both relaxation of transitivity and removal of the Pareto principle.[19] He
demonstrated another interesting impossibility result, known as the "impossibility of the Paretian Liberal". (See
liberal paradox for details). Sen went on to argue that this demonstrates the futility of demanding Pareto optimality in
relation to voting mechanisms.
46
Arrow's impossibility theorem
Social choice instead of social preference
In social decision making, to rank all alternatives is not usually a goal. It often suffices to find some alternative. The
approach focusing on choosing an alternative investigates either social choice functions (functions that map each
preference profile into an alternative) or social choice rules (functions that map each preference profile into a subset
of alternatives).
As for social choice functions, the Gibbard–Satterthwaite theorem is well-known, which states that if a social choice
function whose range contains at least three alternatives is strategy-proof, then it is dictatorial.
As for social choice rules, we should assume there is a social preference behind them. That is, we should regard a
rule as choosing the maximal elements ("best" alternatives) of some social preference. The set of maximal elements
of a social preference is called the core. Conditions for existence of an alternative in the core have been investigated
in two approaches. The first approach assumes that preferences are at least acyclic (which is necessary and sufficient
for the preferences to have a maximal element on any finite subset). For this reason, it is closely related to #Relaxing
transitivity. The second approach drops the assumption of acyclic preferences. Kumabe and Mihara (2011[20]) adopt
this approach. They make a more direct assumption that individual preferences have maximal elements, and examine
conditions for the social preference to have a maximal element. See Nakamura number for details of these two
approaches.
Rated voting systems and other approaches
Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and
transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its
value is an ordinal utility in the sense that it is meaningful so far as the greater value indicates the better alternative.
For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000,
100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997. They all represent the ordering in which a is
preferred to b to c to d. The assumption of ordinal preferences, which precludes interpersonal comparisons of utility,
is an integral part of Arrow's theorem.
For various reasons, an approach based on cardinal utility, where the utility has a meaning beyond just giving a
ranking of alternatives, is not common in contemporary economics. However, once one adopts that approach, one
can take intensities of preferences into consideration, or one can compare (i) gains and losses of utility or (ii) levels
of utility, across different individuals. In particular, Harsanyi (1955) gives a justification of utilitarianism (which
evaluates alternatives in terms of the sum of individual utilities), originating from Jeremy Bentham. Hammond
(1976) gives a justification of the maximin principle (which evaluates alternatives in terms of the utility of the
worst-off individual), originating from John Rawls.
Not all voting methods use, as input, only an ordering of all candidates.[21] Methods which don't, often called "rated"
or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") voting systems, can be viewed as using
information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of
Arrow's conditions that are reformulated.[22] Range voting is such a method.[23][24] Whether such a claim is correct
depends on how each condition is reformulated. [25] Other rated voting systems which pass certain generalizations of
Arrow's criteria include Approval voting and Majority Judgment. Note that although Arrow's theorem does not apply
to such methods, the Gibbard–Satterthwaite theorem still does: no system is fully strategy-free, so the informal
dictum that "no voting system is perfect" still has a mathematical basis.
Finally, though not an approach investigating some kind of rules, there is a criticism by James M. Buchanan and
others. It argues that it is silly to think that there might be social preferences that are analogous to individual
preferences. Arrow (1963, Chapter 8) answers this sort of criticisms seen in the early period, which come at least
partly from misunderstanding.
47
Arrow's impossibility theorem
48
Notes
[1] Arrow, K.J., " A Difficulty in the Concept of Social Welfare (http:/ / gatton. uky. edu/ Faculty/ hoytw/ 751/ articles/ arrow. pdf)", Journal of
Political Economy 58(4) (August, 1950), pp. 328–346.
[2] Interview with Dr. Kenneth Arrow (http:/ / electology. org/ interview-with-dr-kenneth-arrow/ ): "CES: Now, you mention that your theorem
applies to preferential systems or ranking systems. Dr. Arrow: Yes CES: But the system that you’re just referring to, Approval Voting, falls
within a class called cardinal systems. So not within ranking systems. Dr. Arrow: And as I said, that in effect implies more information.
[3] Suzumura, 2002,Suzumura, Kōtarō; Arrow, Kenneth Joseph; Sen, Amartya Kumar (2002). Handbook of social choice and welfare, vol 1.
Amsterdam, Netherlands: Elsevier. ISBN 978-0-444-82914-6. Introduction, page 10.
[4] Note that by definition, a social welfare function as defined here satisfies the Unrestricted domain condition. Restricting the range to the
social preferences that are never indifferent between distinct outcomes is probably a very restrictive assumption, but the goal here is to give a
simple statement of the theorem. Even if the restriction is relaxed, the impossibility result will persist.
[5] Three Brief Proofs of Arrow’s Impossibility Theorem (http:/ / ideas. repec. org/ p/ cwl/ cwldpp/ 1123r3. html)
[6] A One-shot Proof of Arrow’s Impossibility Theorem (http:/ / docs. google. com/ viewer?a=v& pid=sites&
srcid=ZGVmYXVsdGRvbWFpbnxuZWlsbmluZ3l1fGd4OjExZTIwYzljYmQ1MmZiODY)
[7] This does not mean various normative criteria will be satisfied if we use equilibrium concepts in game theory. Indeed, the mapping from
profiles to equilibrium outcomes defines a social choice rule, whose performance can be investigated by social choice theory. See
Austen-Smith and Banks (1999Austen-Smith, David; Banks, Jeffrey S. (1999). Positive political theory I: Collective preference (http:/ /
books. google. com/ books?id=nxXDn3nPxIAC& q="nakamura+ number"). Ann Arbor: University of Michigan Press.
ISBN 978-0-472-08721-1.), Section 7.2.
[8] doi: 10.1016/0022-0531(72)90106-8
This citation will be automatically completed in the next few minutes. You can jump the queue or expand by hand (http:/ / en. wikipedia. org/
wiki/ Template:cite_doi/ _10. 1016. 2f0022-0531. 2872. 2990106-8?preload=Template:Cite_doi/ preload& editintro=Template:Cite_doi/
editintro& action=edit)
[9] Mihara, H. R. (1997). "Arrow's Theorem and Turing computability" (http:/ / 129. 3. 20. 41/ eps/ pe/ papers/ 9408/ 9408001. pdf). Economic
Theory 10 (2): 257–276. doi:10.1007/s001990050157. Reprinted in K. V. Velupillai , S. Zambelli, and S. Kinsella, ed., Computable
Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.
[10] Mihara, H. R. (1999). "Arrow's theorem, countably many agents, and more visible invisible dictators" (http:/ / econpapers. repec. org/ paper/
wpawuwppe/ 9705001. htm). Journal of Mathematical Economics 32: 267–277. doi:10.1016/S0304-4068(98)00061-5. .
[11] Mihara's definition of a computable aggregation rule is based on computability of a simple game (see Rice's theorem).
[12] See Taylor (2005,Taylor, Alan D. (2005). Social choice and the mathematics of manipulation. New York: Cambridge University Press.
ISBN 0-521-00883-2. Chapter 6) for a concise discussion of social choice for infinite societies.
[13] Austen-Smith and Banks (1999, Chapter 3) gives a detailed discussion of the approach trying to limit the number of alternatives.
[14] Indeed, many different social welfare functions can meet Arrow's conditions under such restrictions of the domain. It has been proved,
however, that under any such restriction, if there exists any social welfare function that adheres to Arrow's criteria, then the majority rule will
adhere to Arrow's criteria.Campbell, D.E., Kelly, J.S., "A simple characterization of majority rule", Economic Theory 15 (2000), pp. 689–700.
[15] Black, Duncan (1968). The theory of committees and elections. Cambridge, Eng.: University Press. ISBN 0-89838-189-4.
[16] McKelvey, R. (1976). "Intransitivities in multidimensional voting models and some implications for agenda control". Journal of Economic
Theory 12 (3): 472–482. doi:10.1016/0022-0531(76)90040-5.
[17] Brown, D. J. (1975). Aggregation of Preferences. Quarterly Journal of Economics 89: 456-469.
[18] Blair, D. (1983). "Essential aggregation procedures on restricted domains of preferences*1". Journal of Economic Theory 30: 34–00.
doi:10.1016/0022-0531(83)90092-3.
[19] Sen, Amartya. 1979. Personal Utilities and Public Judgements: Or What's Wrong With Welfare Economics. The Economic Journal, 89,
537-588.
[20] doi: 10.1016/j.geb.2010.06.008
This citation will be automatically completed in the next few minutes. You can jump the queue or expand by hand (http:/ / en. wikipedia. org/
wiki/ Template:cite_doi/ _10. 1016. 2fj. geb. 2010. 06. 008?preload=Template:Cite_doi/ preload& editintro=Template:Cite_doi/ editintro&
action=edit)
[21] It is sometimes asserted that such methods may trivially fail the universality criterion. However, it is more appropriate to consider that such
methods fail Arrow's definition of an aggregation rule (or that of a function whose domain consists of preference profiles), if preference
orderings cannot uniquely translate into a ballot.
[22] However, a modified version of Arrow's theorem may still apply to such methods (e.g., Brams and Fishburn, 2002, Chapter 4, Theorem 4.2).
[23] Warren D. Smith, et al.. "How can range voting accomplish the impossible?" (http:/ / rangevoting. org/ ArrowThm. html). .
[24] New Scientist 12 April 2008 pages 30-33
[25] No voting method that nontrivially uses cardinal utility satisfies Arrow's IIA (in which preference profiles are replaced by lists of ballots or
lists of utilities). For this reason, a weakened notion of IIA is proposed (e.g., Sen, 1979,Sen, Amartya Kumar (1979). Collective choice and
social welfare. Amsterdam: North-Holland. ISBN 978-0-444-85127-7. page 129). The notion requires that the social ranking of two
alternatives depend only on the levels of utility attained by individuals at the two alternatives. (More formally, a social welfare functional
is a function that maps each list
of utility functions into a social preference.
satisfies IIA (for social welfare
Arrow's impossibility theorem
functionals) if for all lists
49
and for all alternatives
, if
and
for all
, then
weakened version of IIA.
References
• Campbell, D.E. and Kelly, J.S. (2002) Impossibility theorems in the Arrovian framework, in Handbook of social
choice and welfare (ed. by Kenneth J. Arrow, Amartya K. Sen and Kotaro Suzumura), volume 1, pages 35–94,
Elsevier. Surveys many of approaches discussed in #Approaches investigating functions of preference profiles.
• The Mathematics of Behavior by Earl Hunt, Cambridge University Press, 2007. The chapter "Defining
Rationality: Personal and Group Decision Making" has a detailed discussion of the Arrow Theorem, with proof.
URL to CUP information on this book (http://www.cambridge.org/9780521850124)
• Why flip a coin? : the art and science of good decisions by Harold W. Lewis, John Wiley, 1997. Gives explicit
examples of preference rankings and apparently anomalous results under different voting systems. States but does
not prove Arrow's theorem. ISBN 0-471-29645-7
• Sen, A. K. (1979) “Personal utilities and public judgements: or what's wrong with welfare economics?” The
Economic Journal, 89, 537-558, arguing that Arrow's theorem was wrong because it did not incorporate
non-utility information and the utility information it did allow was impoverished http://www.jstor.org/stable/
2231867
External links
•
•
•
•
Three Brief Proofs of Arrow’s Impossibility Theorem (http://ideas.repec.org/p/cwl/cwldpp/1123r3.html)
A Pedagogical Proof of Arrow’s Impossibility Theorem (http://repositories.cdlib.org/ucsdecon/1999-25/)
Another Graphical Proof of Arrow’s Impossibility Theorem (http://www.jstor.org/stable/1183438)
A One-Shot Proof of Arrow’s Impossibility Theorem (http://www.springerlink.com/content/
v00202437u066604/)
• Computer-aided Proofs of Arrow's and Other Impossibility Theorems (http://www.sciencedirect.com/science/
article/pii/S0004370209000320/)
.) M
Bertrand paradox
Bertrand paradox
In economics and commerce, the Bertrand paradox—named after its creator, Joseph Bertrand[1] —describes a
situation in which two players (firms) reach a state of Nash equilibrium where both firms charge a price equal to
marginal cost. The paradox is that in models such as Cournot competition, an increase in the number of firms is
associated with a convergence of prices to marginal costs. In these alternative models of oligopoly a small number of
firms earn positive profits by charging prices above cost. Suppose two firms, A and B, sell a homogeneous
commodity, each with the same cost of production and distribution, so that customers choose the product solely on
the basis of price. It follows that demand is infinitely price-elastic. Neither A nor B will set a higher price than the
other because doing so would yield the entire market to their rival. If they set the same price, the companies will
share both the market and profits.
On the other hand, if either firm were to lower its price, even a little, it would gain the whole market and
substantially larger profits. Since both A and B know this, they will each try to undercut their competitor until the
product is selling at zero economic profit. This is the pure-strategy Nash equilibrium. Recent work has shown that
there may be an additional mixed-strategy Nash equilibrium with positive economic profits.[2][3]
The Bertrand paradox rarely appears in practice because real products are almost always differentiated in some way
other than price (brand name, if nothing else); firms have limitations on their capacity to manufacture and distribute;
and two firms rarely have identical costs.
Bertrand's result is paradoxical because if the number of firms goes from one to two, the price decreases from the
monopoly price to the competitive price and stays at the same level as the number of firms increases further. This is
not very realistic, as in reality, markets featuring a small number of firms with market power typically charge a price
in excess of marginal cost. The empirical analysis shows that in most industries with two competitors, positive
profits are made. Solutions to the Paradox attempt to derive solutions that are more in line with solutions from the
Cournot model of competition, where two firms in a market earn positive profits that lie somewhere between the
perfectly competitive and monopoly levels.
Some reasons the Bertrand paradox do not strictly apply:
• Capacity constraints. Sometimes firms do not have enough capacity to satisfy all demand. This was a point first
raised by Francis Edgeworth[4] and gave rise to the Bertrand-Edgeworth model.
• Integer pricing. Prices higher than MC are ruled out because one firm can undercut another by an arbitrarily small
amount. If prices are discrete (for example have to take integer values) then you have to undercut by at least one
cent. This implies that the price one cent above MC is now an equilibrium: if the other firm sets the price one cent
above MC, the other firm can undercut it and capture the whole market, but this will earn it zero profits. It will
prefer to share the market 50/50 with the other firm and earn strictly positive profits.[5]
• Product differentiation. If products of different firms are differentiated, then consumers may not switch
completely to the product with lower price.
• Dynamic competition. Repeated interaction or repeated price competition can lead to the price above MC in
equilibrium.
• More money for higher price. It follows from repeated interaction: If one company sets their price slightly higher,
then they will still get about the same amount of buys but more profit for each buy, so the other company will
raise their price, and so on (only in repeated games, otherwise the price dynamics are in the other direction).
• Oligopoly. If the two companies can agree on a price, it is in their long-term interest to keep the agreement: the
revenue from cutting prices is less than twice the revenue from keeping the agreement, and lasts only until the
other firm cuts its own prices.
50
Bertrand paradox
References
[1] Bertrand J. (1883), Review of Theorie mathematique de la richesse sociale and of Recherches sur les principles mathematiques de la theorie
des richesses; Journal des Savants, volume 67, pages 499–508
[2] Kaplan T. R. and Wettstein (2000), The Possibility of Mixed-Strategy Equilibria with Constant-Returns-to-Scale Technology under Bertrand
Competition, Spanish Economic Review, volume 2, pages 65–71
[3] Baye M. R., Morgan J (1999) A folk theorem for one-shot Bertrand games, Economics Letters, volume 65, pages 59–65.
[4] Edgeworth, Francis (1889) “The pure theory of monopoly”, reprinted in Collected Papers relating to Political Economy 1925, vol.1,
Macmillan
[5] Dixon, Huw David, 1993. "Integer Pricing and Bertrand-Edgeworth Oligopoly with Strictly Convex Costs: Is It Worth More Than a Penny?"
(http:/ / ideas. repec. org/ a/ bla/ buecrs/ v45y1993i3p257-68. html), Bulletin of Economic Research, Wiley Blackwell, vol. 45(3), pages
257–68, July.
Demographic-economic paradox
The demographic-economic paradox
is the inverse correlation found
between wealth and fertility within and
between nations. The higher the degree
of education and GDP per capita of a
human population, subpopulation or
social stratum, the fewer children are
born in any industrialized country. In a
1974 UN population conference in
Bucharest, Karan Singh, a former
minister of population in India,
illustrated this trend by stating
"Development
is
the
best
[1]
contraceptive."
The term "paradox" comes from the
notion that greater means would
necessitate the production of more
Graph of Total Fertility Rate vs. GDP per capita of the corresponding country, 2009. Only
countries with over 5 Million population were plotted, to reduce outliers. Sources: CIA
offspring as suggested by the
[2]
World Fact Book. For details, see List of countries and territories by fertility rate
influential
Thomas
Malthus.
Roughly
speaking,
nations
or
subpopulations with higher GDP per capita are observed to have fewer children, even though a richer population can
support more children. Malthus held that in order to prevent widespread suffering, from famine for example, what he
called "moral restraint" (which included abstinence) was required. The demographic-economic paradox suggests that
reproductive restraint arises naturally as a consequence of economic progress.
It is hypothesized that the observed trend has come about as a response to increased life expectancy, reduced
childhood mortality, improved female literacy and independence, and urbanization that all result from increased
GDP per capita,[3] consistent with the demographic transition model.
Current information suggests that the demographic-economic paradox only holds up to a point. Recent data suggests
that once a country reaches a certain level of human development and economic prosperity the fertility rate stabilizes
and then recovers slightly to replacement rates.[4]
51
Demographic-economic paradox
52
Demographic transition
Before the 19th century demographic transition of the western world, a minority of children would survive to the age
of 20, and life expectancies were short even for those who reached adulthood. For example, in the 17th century in
York, England 15% of children were still alive at age 15 and only 10% of children survived to age 20.[3]
Birth rates were correspondingly high, resulting in slow population growth. The agricultural revolution and
improvements in hygiene then brought about dramatic reductions in mortality rates in wealthy industrialized
countries, initially without affecting birth rates. In the 20th century, birth rates of industrialized countries began to
fall, as societies became accustomed to the higher probability that their children would survive them. Cultural value
changes were also contributors, as urbanization and female employment rose.
Since wealth is what drives this demographic transition, it follows that nations that lag behind in wealth also lag
behind in this demographic transition. The developing world's equivalent Green Revolution did not begin until the
mid-twentieth century. This creates the existing spread in fertility rates as a function of GDP per capita.
Religion
Another contributor to the demographic-economic paradox may be religion. Religious societies tend to have higher
birth rates than secular ones, and richer, more educated nations tend to advance secularization.[5] This may help
explain the Israeli and Saudi Arabian exceptions, the two notable outliers in the graph of fertility versus GDP per
capita at the top of this article. In American media it is widely believed that America is also an exception to global
trends. The current fertility rate in America is 2.09, higher than in most other developed countries.[6][7] This may be
due to the United States having a high percentage of religious followers compared to Europe as a whole.[8]
Church service attendance and number of offspring according to the World Value Survey
1981–2004[9]
Church service attendance Number of offspring
never
1.67
only on holidays
1.78
once per month
2.01
once per week
2.23
more frequently
2.5
The role of different religions in determining family size is complex. For example, the Catholic countries of southern
Europe traditionally had a much higher fertility rate than was the case in Protestant northern Europe. However,
economic growth in Spain, Italy, Poland etc., has been accompanied by a particularly sharp fall in the fertility rate, to
a level below that of the Protestant north. This suggests that the demographic-economic paradox applies more
strongly in Catholic countries, although Catholic fertility started to fall when the liberalizing reforms of Vatican II
were implemented. It remains to be seen if the fertility rate among (mostly Catholic) Hispanics in the U.S. will
follow a similar pattern.
Demographic-economic paradox
United States
Another possible explanation for the "American exception" is its much higher rate of teenage pregnancies,[10]
particularly in the southern US,[11] compared to other countries with effective sexual education; this does not
contradict the religious-beliefs hypothesis.
In his book America Alone: The End of the World as We Know It, Mark Steyn asserts that the United States has
higher fertility rates because of its greater economic freedom compared to other industrialized countries. However,
the countries with the highest assessed economic freedom, Hong Kong and Singapore, have significantly lower
birthrates than the United States. According to the Index of Economic Freedom, Hong Kong is the most
economically free country in the world.[12] Hong Kong also has the world's lowest birth rate.[13]
Fertility and population density
Studies have also suggested a correlation between population density and fertility rate.[14][15][16] Hong Kong and
Singapore have the third and fourth-highest population densities in the world. This may account for their very low
birth rates despite high economic freedom. By contrast, the United States ranks 180 out of 241 countries and
dependencies by population density.
Consequences
A reduction in fertility can lead to an aging population which leads to a variety of problems, see for example the
Demographics of Japan.
A related concern is that high birth rates tend to place a greater burden of child rearing and education on populations
already struggling with poverty. Consequently, inequality lowers average education and hampers economic
growth.[17] Also, in countries with a high burden of this kind, a reduction in fertility can hamper economic growth as
well as the other way around.[18]
References
[1]
[2]
[3]
[4]
[5]
David N. Weil (2004). Economic Growth. Addison-Wesley. p. 111. ISBN 0-201-68026-2.
http:/ / www. econlib. org/ library/ Malthus/ malPlong. html EconLib-1826: An Essay on the Principle of Population,
demographic transition (http:/ / www. uwmc. uwc. edu/ geography/ Demotrans/ demtran. htm)
(http:/ / www. nature. com/ nature/ journal/ v460/ n7256/ full/ nature08230. html)
Marburg Journal of Religion (June 2006) "Religiousity as a demographic factor" (http:/ / web. uni-marburg. de/ religionswissenschaft/
journal/ mjr/ art_blume_2006. html)
[6] Watch on the West: Four Surprises in Global Demography – FPRI (http:/ / www. fpri. org/ ww/ 0505. 200407. eberstadt. demography. html)
[7] CIA – The World Factbook – Rank Order – Total fertility rate (https:/ / www. cia. gov/ library/ publications/ the-world-factbook/ rankorder/
2127rank. html)
[8] Nicholas Eberstadt – America the Fertile – washingtonpost.com May 6, 2007 (http:/ / www. washingtonpost. com/ wp-dyn/ content/ article/
2007/ 05/ 04/ AR2007050401891. html?hpid=opinionsbox2)
[9] Michael Blume (2008) "Homo religiosus" (http:/ / www. wissenschaft-online. de/ artikel/ 982875), Gehirn und Geist 04/2009, pp. 32–41.
[10] Adamson, Peter; Giorgina Brown, John Micklewright and Anna Wright (July 2001). "A League Table of Teenage Births in Rich Nations"
(http:/ / www. unicef-icdc. org/ publications/ pdf/ repcard3e. pdf) (PDF). Innocenti Report Card (Unicef) (3). ISBN 88-85401-75-9. .
[11] "National and State Trends and Trends by Race and Ethnicity" (http:/ / www. guttmacher. org/ pubs/ 2006/ 09/ 12/ USTPstats. pdf) (PDF).
U.S. Teenage Pregnancy Statistics (Guttmacher Institute). September 2006. .
[12] Index of Economic Freedom (http:/ / www. heritage. org/ index/ country. cfm?id=HongKong)
[13] CIA – The World Factbook – Rank Order – Birth rate (https:/ / www. cia. gov/ library/ publications/ the-world-factbook/ rankorder/
2054rank. html)
[14] Economic geography, fertility and migration (http:/ / www. sciencedirect. com/ science?_ob=ArticleURL& _udi=B6WMG-4KW5W75-3&
_user=10& _rdoc=1& _fmt=& _orig=search& _sort=d& view=c& _acct=C000050221& _version=1& _urlVersion=0& _userid=10&
md5=149bb91c9e66f19e5400e9ac1f5c4317) Yasuhiro Sato, Journal of Urban Economics. Published July 30, 2006. Last accessed March 31,
2008.
[15] An Estimate of the Long-Term Crude Birth Rate of the Agricultural Population of China (http:/ / links. jstor. org/
sici?sici=0070-3370(1966)3:1<204:AEOTLC>2. 0. CO;2-L) Chia-lin Pan, Demography, Volume 3, No. 1. Published 1966. Last accessed
53
Demographic-economic paradox
March 31, 2008.
[16] (http:/ / houstonstrategies. blogspot. com/ 2006/ 01/ high-density-smart-growth-population. html). Tory Gattis,
houstonstrategies.blogspot.com. Published January 15, 2006. Last accessed March 31, 2008.
[17] de la Croix, David and Matthias Doepcke: Inequality and growth: why differential fertility matters. American Economic Review 4 (2003)
1091–1113. (http:/ / www. econ. ucla. edu/ workingpapers/ wp803. pdf)
[18] UNFPA: Population and poverty. Achieving equity, equality and sustainability. Population and development series no. 8, 2003. (http:/ /
www. unfpa. org/ upload/ lib_pub_file/ 191_filename_PDS08. pdf)
Dollar auction
The dollar auction is a non-zero sum sequential game designed by economist Martin Shubik to illustrate a paradox
brought about by traditional rational choice theory in which players with perfect information in the game are
compelled to make an ultimately irrational decision based completely on a sequence of rational choices made
throughout the game.[1]
Setup
The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule: the dollar goes to
the highest bidder, who pays the amount he bids. The second-highest bidder also must pay the highest amount that he
bid, but gets nothing in return. Suppose that the game begins with one of the players bidding 1 cent, hoping to make
a 99 cent profit. He will quickly be outbid by another player bidding 2 cents, as a 98 cent profit is still desirable.
Similarly, another bidder may bid 3 cents, making a 97 cent profit. Alternatively, the first bidder may attempt to
convert his loss of 1 cent into a gain of 96 cents by bidding 4 cents. Supposing that the other player had bid 98 cents,
he now has the choice of losing the 98 cents or bidding a dollar even, which would make his profit zero. After that,
the original player has a choice of either losing 99 cents or bidding $1.01, and only losing one cent. After this point
the two players continue to bid the value up well beyond the dollar, and neither stands to profit.
References
[1] Shubik, Martin (1971). "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation". Journal of Conflict Resolution
15 (1): 109–111. doi:10.1177/002200277101500111.
Further reading
• Poundstone, William (1993). "The Dollar Auction". Prisoner's Dilemma: John Von Neumann, Game Theory, and
the Puzzle of the Bomb. New York: Oxford University Press. ISBN 0-19-286162-X.
54
DownsThomson paradox
Downs–Thomson paradox
Downs-Thomson paradox (named after Anthony Downs and J. M. Thomson), also referred to as the
Pigou–Knight–Downs paradox (after Arthur Cecil Pigou and Frank Knight), states that the equilibrium speed of car
traffic on the road network is determined by the average door-to-door speed of equivalent journeys by (rail-based or
otherwise segregated) public transport.
It follows that increasing road capacity can actually make overall congestion on the road worse. This occurs when
the shift from public transport causes a disinvestment in the mode such that the operator either reduces frequency of
service or raises fares to cover costs. This shifts additional passengers into cars. Ultimately the system may be
eliminated and congestion on the original (expanded) road is worse than before.
The general conclusion, if the paradox applies, is that expanding a road system as a remedy to congestion is not only
ineffective, but often counterproductive. This is also known as Lewis–Mogridge Position and was extensively
documented by Martin Mogridge with the case-study of London on his book Travel in towns: jam yesterday, jam
today and jam tomorrow?
An article of 1968 from Dietrich Braess now at the Faculty of Mathematics in Ruhr University, already pointed out
the existence of this counter-intuitive occurrence on networks - the Braess' paradox states that adding extra capacity
to a network, when the moving entities selfishly choose their route, can in some cases reduce overall performance.
There is a recent interest in the study of this phenomenon since the same may happen in computer networks and not
only in traffic networks. Increasing the size of the network is characterized by behaviors of users similar to that of
travelers on transportation networks, who act independently and in a decentralized manner in choosing their optimal
routes of travel between origins and their destinations.
This is an extension of the induced demand theory and consistent with Downs (1992) theory of "triple convergence".
Downs (1992) formulated this theory to explain the difficulty of removing peak-hour congestion from highways. In
response to a capacity addition three immediate effects occur. Drivers using alternative routes begin to use the
expanded highway, those previously traveling at off-peak times (either immediately before or after the peak) shift to
the peak (rescheduling behavior as defined previously), and public transport users shift to driving their vehicles.
Restrictions on validity
According to Downs this link between average speeds on public transport and private transport "only applies to
regions in which the vast majority of peak-hour commuting is done on rapid transit systems with separate rights of
way. Central London is an example, since in 2001 around 85 percent of all morning peak-period commuters into that
area used public transit (including 77 percent on separate rights of way) and only 11 percent used private cars. When
peak-hour travel equilibrium has been reached between the subway system and the major commuting roads, then the
travel time required for any given trip is roughly equal on both modes."
55
DownsThomson paradox
References
• On a Paradox of Traffic Planning, translated from the (1968) original D. Braess paper from German to English by
D. Braess, A. Nagurney, and T. Wakolbinger (2005), Transportation Science 39/4, 446-450.
• Mogridge, Martin J.H. Travel in towns: jam yesterday, jam today and jam tomorrow? Macmillan Press, London,
1990. ISBN 0-333-53204-X
• Downs, Anthony, Stuck in Traffic: Coping with Peak-Hour Traffic Congestion, The Brookings Institution:
Washington, DC. 1992. ISBN 0-8157-1923-X
• Thomson, J. M. (1972), Methods of traffic limitation in urban areas. Working Paper 3, Paris, OECD.
Easterlin paradox
The Easterlin Paradox is a key concept in happiness economics. It is named for economist and USC Professor
Richard Easterlin who discussed the factors contributing to happiness in the 1974 paper "Does Economic Growth
Improve the Human Lot? Some Empirical Evidence."[1] Easterlin found that within a given country people with
higher incomes are more likely to report being happy. However, in international comparisons, the average reported
level of happiness does not vary much with national income per person, at least for countries with income sufficient
to meet basic needs. Similarly, although income per person rose steadily in the United States between 1946 and
1970, average reported happiness showed no long-term trend and declined between 1960 and 1970.
This concept was later revived by Andrew Oswald of the University of Warwick in 1997, driving media interest in
the topic. Recent research has utilised many different forms of measuring happiness, including biological measures,
showing similar patterns of results. This goes some way to answering the problems of self-rated happiness.
The implication for government policy is often said to be that once basic needs are met, policy should focus not on
economic growth or GDP, but rather on increasing life satisfaction or Gross national happiness (GNH).
Controversy
In 2003 Ruut Veenhoven and Michael Hagerty published a new analysis based on including various sources of data,
and their conclusion was that there is no paradox and countries did indeed get happier with increasing income.[2] In
his reply Easterlin maintained his position, suggesting that his critics were using inadequate data.[3]
In 2008, economists Betsey Stevenson and Justin Wolfers, both of the University of Pennsylvania, published a paper
where they reassessed the Easterlin paradox using new time-series data. They conclude like Veenhoven et al. that,
contrary to Easterlin's claim, increases in absolute income are clearly linked to increased self-reported happiness, for
both individual people and whole countries.[2][4][5][6] The statistical relationship demonstrated is between happiness
and the logarithm of absolute income, suggesting that happiness increases more slowly than income, but no
"saturation point" is ever reached. The study provides evidence that happiness is determined not only by relative
income, but also by absolute income. That is in contrast to an extreme understanding of the hedonic treadmill theory
where "keeping up with the Joneses" is the only determinant of behavior.[6]
In 2010 Easterlin published as lead author a paper in the Proceedings of the National Academy of Sciences
reaffirming the Easterlin Paradox with data from a sample of 37 countries.[7] Also, in a report prepared for the
United Nations in 2012 [8] Richard Layard, Andrew Clark and Claudia Senik point out that when other variables,
such as indicators of social trust, are added to the regression, the importance of GDP per capita falls drastically.
56
Easterlin paradox
Notes
[1] Does Economic Growth Improve the Human Lot? Some Empirical Evidence. (http:/ / graphics8. nytimes. com/ images/ 2008/ 04/ 16/
business/ Easterlin1974. pdf)
[2] WEALTH AND HAPPINESS REVISITED Growing wealth of nations does go with greater happiness (http:/ / www2. eur. nl/ fsw/ research/
veenhoven/ Pub2000s/ 2003e-full. pdf)
[3] "FEEDING THE ILLUSION OF GROWTH AND HAPPINESS: A REPLY TO HAGERTY AND VEENHOVEN" by Richard A. Easterlin
(http:/ / www-rcf. usc. edu/ ~easterl/ papers/ HVcomment. pdf)
[4] Leonhardt, David (16 April 2008). "Maybe Money Does Buy Happiness After All" (http:/ / www. nytimes. com/ 2008/ 04/ 16/ business/
16leonhardt. html). The New York Times. .
[5] Economic Growth and Subjective Well-Being: Reassessing the Easterlin Paradox (http:/ / bpp. wharton. upenn. edu/ betseys/ papers/
Happiness. pdf)
[6] Akst, Daniel (23 November 2008). "A talk with Betsey Stevenson and Justin Wolfers" (http:/ / www. boston. com/ bostonglobe/ ideas/
articles/ 2008/ 11/ 23/ a_talk_with_betsey_stevenson_and_justin_wolfers/ ?page=full). The Boston Globe. .
[7] Alok Jha: Happiness doesn't increase with growing wealth of nations, finds study (http:/ / www. guardian. co. uk/ science/ 2010/ dec/ 13/
happiness-growing-wealth-nations-study). In: The Guardian. 13 December 2010
[8] http:/ / earth. columbia. edu/ articles/ view/ 2960
References
• Easterlin, Richard A. (1974) "Does Economic Growth Improve the Human Lot?" in Paul A. David and Melvin W.
Reder, eds., Nations and Households in Economic Growth: Essays in Honor of Moses Abramovitz, New York:
Academic Press, Inc.
• Easterlin, Richard A.; Angelescu McVey, Laura Angelescu McVey; Switek, Malgorzata; Sawangfa, Onnicha;
Smith Zweig, Jacqueline (2010), "The happiness–income paradox revisited", Proceedings of the National
Academy of Sciences of the United States of America 107 (52): 22463–22468, doi:10.1073/pnas.1015962107
• Oswald, Andrew. (2006) "The Hippies Were Right all Along about Happiness" Financial Times, January 19,
2006.
External links
• It's experts that make us miserable (http://observer.guardian.co.uk/comment/story/0,,2000672,00.html) Nick Cohen - The Guardian - January 28, 2007.
• Andrew Oswald's Website (http://www.andrewoswald.com/).
• The Hippies Were Right all Along about Happiness (http://www2.warwick.ac.uk/fac/soc/economics/staff/
faculty/oswald/fthappinessjan96.pdf) - Andrew Oswald - Financial Times - January 19, 2006.
• Happiness Is Increasing in Many Countries -- But Why? (http://pewglobal.org/commentary/display.
php?AnalysisID=1020) - Bruce Stokes - July 24, 2007.
57
Ellsberg paradox
58
Ellsberg paradox
The Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices violate
the expected utility hypothesis.[1] One interpretation is that expected utility theory does not properly describe actual
human choices.
It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg,
although a version of it was noted considerably earlier by John Maynard Keynes.[2]
Ellsberg raised two problems: 1 urn problem and 2 urn problem. Here, 1 urn problem is described, which is the better
known one.
The 1 urn paradox
Suppose you have an urn containing 30 red balls and 60 other balls that are either black or yellow. You don't know
how many black or how many yellow balls there are, but that the total number of black balls plus the total number of
yellow equals 60. The balls are well mixed so that each individual ball is as likely to be drawn as any other. You are
now given a choice between two gambles:
Gamble A
Gamble B
You receive $100 if you draw a red ball You receive $100 if you draw a black ball
Also you are given the choice between these two gambles (about a different draw from the same urn):
Gamble C
Gamble D
You receive $100 if you draw a red or yellow ball You receive $100 if you draw a black or yellow ball
This situation poses both Knightian uncertainty – whether the non-red balls are all yellow or all black, which is not
quantified – and probability – whether the ball is red or non-red, which is ⅓ vs. ⅔.
Utility theory interpretation
Utility theory models the choice by assuming that in choosing between these gambles, people assume a probability
that the non-red balls are yellow versus black, and then compute the expected utility of the two gambles.
Since the prizes are exactly the same, it follows that you will prefer Gamble A to Gamble B if and only if you
believe that drawing a red ball is more likely than drawing a black ball (according to expected utility theory). Also,
there would be no clear preference between the choices if you thought that a red ball was as likely as a black ball.
Similarly it follows that you will prefer Gamble C to Gamble D if, and only if, you believe that drawing a red or
yellow ball is more likely than drawing a black or yellow ball. It might seem intuitive that, if drawing a red ball is
more likely than drawing a black ball, then drawing a red or yellow ball is also more likely than drawing a black or
yellow ball. So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C to
Gamble D. And, supposing instead that you prefer Gamble B to Gamble A, it follows that you will also prefer
Gamble D to Gamble C.
When surveyed, however, most people strictly prefer Gamble A to Gamble B and Gamble D to Gamble C.
Therefore, some assumptions of the expected utility theory are violated.
Ellsberg paradox
59
Mathematical demonstration
Mathematically, your estimated probabilities of each color ball can be represented as: R, Y, and B. If you strictly
prefer Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of
the two gambles: specifically, it must be the case that
where
is your utility function. If
(you strictly prefer $100 to nothing), this simplifies to:
If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained:
This simplifies to:
This contradiction indicates that your preferences are inconsistent with expected-utility theory.
Generality of the paradox
Note that the result holds regardless of your utility function. Indeed, the amount of the payoff is likewise irrelevant.
Whichever gamble you choose, the prize for winning it is the same, and the cost of losing it is the same (no cost), so
ultimately, there are only two outcomes: you receive a specific amount of money, or you receive nothing. Therefore
it is sufficient to assume that you prefer receiving some money to receiving nothing (and in fact, this assumption is
not necessary — in the mathematical treatment above, it was assumed U($100) > U($0), but a contradiction can still
be obtained for U($100) < U($0) and for U($100) = U($0).
In addition, the result holds regardless of your risk aversion. All the gambles involve risk. By choosing Gamble D,
you have a 1 in 3 chance of receiving nothing, and by choosing Gamble A, you have a 2 in 3 chance of receiving
nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D
(and vice versa), so, risk is not averted in this way.
However, because the exact chances of winning are known for Gambles A and D, and not known for Gambles B and
C, this can be taken as evidence for some sort of ambiguity aversion which cannot be accounted for in expected
utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits comparison of
the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in
isolation).[3]
Possible explanations
There have been various attempts to provide decision-theoretic explanations of Ellsberg's observation. Since the
probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on
quantifying the non-probabilistic ambiguity which the decision-maker faces – see Knightian uncertainty. That is,
these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily
Bayesian) probability for possible outcomes.
One such attempt is based on info-gap decision theory. The agent is told precise probabilities of some outcomes,
though the practical meaning of the probability numbers is not entirely clear. For instance, in the gambles discussed
above, the probability of a red ball is 30/90, which is a precise number. Nonetheless, the agent may not distinguish,
intuitively, between this and, say, 30/91. No probability information whatsoever is provided regarding other
outcomes, so the agent has very unclear subjective impressions of these probabilities.
Ellsberg paradox
In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected
utility. Consequently, a choice based on maximizing the expected utility is also impossible. The info-gap approach
supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. The agent
then tries to satisfice the expected utility and to maximize the robustness against uncertainty in the imprecise
probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of
decision-makers should display precisely the preference reversal which Ellsberg observed.[4]
Another possible explanation is that this type of game triggers a deceit aversion mechanism. Many humans naturally
assume in real-world situations that if they are not told the probability of a certain event, it is to deceive them. People
make the same decisions in the experiment that they would about related but not identical real-life problems where
the experimenter would be likely to be a deceiver acting against the subject's interests. When faced with the choice
between a red ball and a black ball, the probability of 30/90 is compared to the lower part of the 0/90-60/90 range
(the probability of getting a black ball). The average person expects there to be fewer black balls than yellow balls
because in most real-world situations, it would be to the advantage of the experimenter to put fewer black balls in the
urn when offering such a gamble. On the other hand, when offered a choice between red and yellow balls and black
and yellow balls, people assume that there must be fewer than 30 yellow balls as would be necessary to deceive
them. When making the decision, it is quite possible that people simply forget to consider that the experimenter does
not have a chance to modify the contents of the urn in between the draws. In real-life situations, even if the urn is not
to be modified, people would be afraid of being deceived on that front as well.
A modification of utility theory to incorporate uncertainty as distinct from risk is Choquet expected utility, which
also proposes a solution to the paradox.
Alternative explanations
Other alternative explanations include the competence hypothesis [5] and comparative ignorance hypothesis.[6] These
theories attribute the source of the ambiguity aversion to the participant's pre-existing knowledge.
References
[1] Ellsberg, Daniel (1961). "Risk, Ambiguity, and the Savage Axioms". Quarterly Journal of Economics 75 (4): 643–669. doi:10.2307/1884324.
JSTOR 1884324.
[2] (Keynes 1921, pp. 75–76, paragraph 315, footnote 2)
[3] Fox, Craig R.; Tversky, Amos (1995). "Ambiguity Aversion and Comparative Ignorance". Quarterly Journal of Economics 110 (3): 585–603.
doi:10.2307/2946693. JSTOR 2946693.
[4] Ben-Haim, Yakov (2006). Info-gap Decision Theory: Decisions Under Severe Uncertainty (2nd ed.). Academic Press. section 11.1.
ISBN 0-12-373552-1.
[5] Chip, Health (1991). "Preference and Belief: Ambiguity and Competence in Choice under Uncertainty". Journal of Risk and Uncertainty 4:
5–28.
[6] Fox, Craig (1995). "Ambiguity Aversion and Comparative Ignorance". Quarterly Journal of Economics 110: 585–603.
• Anand, Paul (1993). Foundations of Rational Choice Under Risk. Oxford University Press. ISBN 0-19-823303-5.
• Keynes, John Maynard (1921). A Treatise on Probability. London: Macmillan.
• Schmeidler, D. (1989). "Subjective Probability and Expected Utility without Additivity". Econometrica 57 (3):
571–587. doi:10.2307/1911053. JSTOR 1911053.
60
Green paradox
Green paradox
The Green Paradox is a phrase coined by German economist Hans-Werner Sinn to describe the fact that an
environmental policy that becomes greener with the passage of time acts like an announced expropriation for the
owners of fossil fuel resources, inducing them to anticipate resource extraction and hence to accelerate global
warming.
Main line of reasoning
The Green Paradox’s line of reasoning starts by recognizing a fundamental, unavoidable fact: every carbon atom in
the gas, coal or oil extracted from the ground to be used as fuel ends up in the atmosphere, in particular if high
efficiency combustion processes ensure that no part of it ends up as soot. About a quarter of the emitted carbon will
stay in the atmosphere practically forever, contributing to the greenhouse effect that causes global warming.[1]
Apart from afforestation, only two things can mitigate the accumulation of carbon in the atmosphere: either less
carbon is extracted from the ground, or it is injected back underground after harvesting its energy.
Environmental policy efforts, however, in particular European ones, go in neither of these two directions, aiming
instead at the promotion of alternative, CO2-free energy sources and a more efficient use of energy. In other words,
they only address the demand side of the carbon market, neglecting the supply side. Despite considerable investment,
the efforts to curtail demand have not reduced the aggregate amount of CO2 emitted globally, which continues to
increase unabated.[2]
The reason behind this, according to Sinn, is that green policies, by heralding a gradual tightening of policy over the
coming decades, exert a stronger downward pressure on future prices than on current ones, decreasing thus the rate
of capital appreciation of the fossil fuel deposits. The owners of these resources regard this development with
concern and react by increasing extraction volumes, converting the proceeds into investments in the capital markets,
which offer higher yields. That is the green paradox: environmental policy slated to become greener over time acts as
an announced expropriation that provokes owners to react by accelerating the rate of extraction of their fossil fuel
stocks,[3] thus accelerating climate change.
Countries that do not partake of the efforts to curb demand have a double advantage. They burn the carbon set free
by the “green” countries (leakage effect) and they also burn the additional carbon extracted as a reaction to the
announced and expected price cuts resulting from the gradual greening of environmental policies (green paradox).[4]
Sinn writes in his abstract that: "[Demand reduction strategies] simply depress the world price of carbon and induce
the environmental sinners to consume what the Kyoto countries have economized on. Even worse, if suppliers feel
threatened by a gradual greening of economic policies in the Kyoto countries that would damage their future prices,
they will extract their stocks more rapidly, thus accelerating global warming." [5]
Sinn emphasizes that a condition for the green paradox is that the resource be scarce in the sense that its price will
always be higher than the unit extraction and exploration costs combined. He points out that this condition is likely
to be satisfied as backstop technologies will at best offer a perfect substitute for electricity, but not for fossil fuels.
The prices of coal and crude oil are currently many times higher than the corresponding exploration and extraction
costs combined.
61
Green paradox
Practicable solutions
An effective climate policy must perforce focus on the hitherto neglected supply side of the carbon market in
addition to the demand side. The ways proposed as practicable by Sinn to do this include levying a withholding tax
on the capital gains on the financial investments of fossil fuel resource owners, or the establishment of a seamless
global emissions trading system that would effectively put a cap on worldwide fossil fuel consumption, thereby
achieving the desired reduction in carbon extraction rates.
Works on the subject
Hans-Werner Sinn’s ideas on the green paradox have been presented in detail in a number of scientific articles,[6][7]
his 2007 Thünen Lecture[8] at the annual meeting of the German Economic Association (Verein für Socialpolitik),
his 2007 presidential address to the International Institute of Public Finance in Warwick, two working papers,[9][10]
and a German-language book, “Das Grüne Paradoxon” (2008).[11] They build on his earlier studies on supply
reactions of the owners of natural resources to announced price changes.[12]
Notes and references
[1] D. Archer, “Fate of Fossil Fuel CO2 in Geologic Time”, Journal of Geophysical Research 110, 2005, p. 5–11; D. Archer and V. Brovkin,
“Millennial Atmospheric Lifetime of Anthropogenic CO2”, Climate Change, mimeo, 2006; G. Hoos, R. Voss, K. Hasselmann, E. MeierReimer and F. Joos, “A Nonlinear Impulse Response Model of the Coupled Carbon Cycle-Climate System (NICCS)”, Climate Dynamics 18,
2001, p. 189–202.
[2] International Energy Agency (IEA), IEA Database, CO2 Emissions from Fuel Combustion 2007. Accessible online at: www.sourceoecd.org
(http:/ / www. sourceoecd. org); Netherlands Environmental Assessment Agency, Global CO2 Emissions: Increase Continued in 2007,
Bilthoven, June 13, 2008. Accessible online at: (http:/ / www. mnp. nl/ en/ publications/ 2008/ GlobalCO2emissionsthrough2007. html)
[3] N.V. Long, “Resource Extraction under the Uncertainty about Possible Nationalization”, Journal of Economic Theory 10, 1975, p. 42– 53. K.
A. Konrad, T. E. Olson and R. Schöb, “Resource Extraction and the Threat of Possible Expropriation: The Role of Swiss Bank Accounts”,
Journal of Environmental Economics and Management 26, 1994, p. 149–162.
[4] S. Felder and T. F. Rutherford, “Unilateral CO2 Reductions and Carbon Leakage: The Consequences of International Trade in Oil and Basic
Materials”, Journal of Environmental Economics and Management 25, 1993, p. 162–176, and J.-M. Burniaux and J. Oliveira Martins, “Carbon
Emission Leakages: A General Equilibrium View”, OECD Working Paper No. 242, 2000.
[5] Sinn, H.W. (2008). ‘Public policies against global warming’, International Tax and Public Finance, 15, 4, 360-394. Accessible online at:
(http:/ / www. cesifo-group. de/ portal/ page/ portal/ ifoContent/ N/ rts/ rts-mitarbeiter/ IFOMITARBSINNCV/ CVSinnPDF/
CVSinnPDFrefjournals2007/ ITAX-hws-2008. pdf)
[6] “Public Policies against Global Warming: A Supply Side Approach”, International Tax and Public Finance 15, 2008, p. 360–394.
[7] H.-W. Sinn, “Das grüne Paradoxon: Warum man das Angebot bei der Klimapolitik nicht vergessen darf”, Perspektiven der Wirtschaftspolitik
9, 2008, p. 109–142.
[8] http:/ / www. cesifo-group. de/ link/ _ifovideo/ thuenen-vorlesung-1007. htm
[9] H.-W. Sinn, Public Policies against Global Warming, CESifo Working Paper No. 2087 (http:/ / www. cesifo-group. de/ portal/ page/ portal/
ifoHome/ b-publ/ b3publwp/ _wp_abstract?p_file_id=14563), August 2007
[10] H.-W. Sinn, Pareto Optimality in the Extraction of Fossil Fuels and the Greenhouse Effect: A Note, CESifo Working Paper No. 2083 (http:/
/ www. cesifo-group. de/ portal/ page/ portal/ ifoHome/ b-publ/ b3publwp/ _wp_abstract?p_file_id=14562), August 2007
[11] Das grüne Paradoxon - Plädoyer für eine illusionsfreie Klimapolitik (http:/ / www. cesifo-group. de/ link/ _publsinnparadoxon), Econ:
Berlin, 2008, 480 pages.
[12] H-W. Sinn, “Absatzsteuern, Ölförderung und das Allmendeproblem” (Sales Taxes, Oil Extraction and the Common Pool Problem) (http:/ /
www. cesifo-group. de/ link/ Sinn_Abs_Oel_Allmend_1982. pdf), in: H. Siebert, ed., Reaktionen auf Energiepreisänderungen, Lang:
Frankfurt and Bern 1982, pp. 83-103; N.V. Long and H.-W. Sinn, “Surprise Price Shifts, Tax Changes and the Supply Behaviour of Resource
Extracting Firms” (http:/ / www. cesifo-group. de/ link/ Sinn_Surpr_Price_Shift_AEP_1985. pdf), Australian Economic Papers 24, 1985, pp.
278-289.
62
Icarus paradox
63
Icarus paradox
The Icarus paradox is a neologism coined by Danny Miller, and popularized by his 1990 book by the same name,[1]
for the observed phenomenon of businesses that fail abruptly after a period of apparent success. [2] In a 1992 article,
Miller noted that some businesses bring about their own downfall through their own successes, be this through
over-confidence, exaggeration, complacency. It refers to Icarus of Greek mythology who flew too close to the Sun
and melted his own wings. The book is a key source of insight in Escaping the Progress trap by Daniel O'Leary.[3]
References
[1] Michael P. Griffin (December 1, 1990). "The Icarus Paradox.-(book reviews)" (http:/ / www. accessmylibrary. com/ coms2/
summary_0286-9218419_ITM). Management Review. . Retrieved 2008-01-06.
[2] Harry Barkema (January 23, 2003). "The Icarus Paradox" (http:/ / www. uvt. nl/ univers/ nieuws/ 0203/ 17/ barkema. html). Univers. .
Retrieved 2008-01-06.
[3] Danny Miller (January–February, 1992). "The Icarus paradox: how exceptional companies bring about their own downfall". Business
Horizons.
Jevons paradox
In economics, the Jevons paradox (/ˈdʒɛvənz paɹədɒks/;
sometimes Jevons effect) is the proposition that
technological progress that increases the efficiency with
which a resource is used tends to increase (rather than
decrease) the rate of consumption of that resource.[1] In
1865, the English economist William Stanley Jevons
observed that technological improvements that increased the
efficiency of coal use led to increased consumption of coal in
a wide range of industries. He argued that, contrary to
common intuition, technological improvements could not be
relied upon to reduce fuel consumption.[2]
Coal-burning factories in 19th-century Manchester, England.
Improved technology allowed coal to fuel the Industrial
Revolution, greatly increasing the consumption of coal.
The issue has been re-examined by modern economists
studying consumption rebound effects from improved energy
efficiency. In addition to reducing the amount needed for a given use, improved efficiency lowers the relative cost of
using a resource, which tends to increase the quantity of the resource demanded, potentially counteracting any
savings from increased efficiency. Additionally, increased efficiency accelerates economic growth, further increasing
the demand for resources. The Jevons paradox occurs when the effect from increased demand predominates, causing
resource use to increase.[2]
The Jevons paradox has been used to argue that energy conservation is futile, as increased efficiency may increase
fuel use. Nevertheless, increased efficiency can improve material living standards. Further, fuel use declines if
increased efficiency is coupled with a green tax or other conservation policies that keep the cost of use the same (or
higher).[3] As the Jevons paradox applies only to technological improvements that increase fuel efficiency, policies
that impose conservation standards and increase costs do not display the paradox.
Jevons paradox
64
History
The Jevons paradox was first described by the English economist William
Stanley Jevons in his 1865 book The Coal Question. Jevons observed that
England's consumption of coal soared after James Watt introduced his coal-fired
steam engine, which greatly improved the efficiency of Thomas Newcomen's
earlier design. Watt's innovations made coal a more cost-effective power source,
leading to the increased use of the steam engine in a wide range of industries.
This in turn increased total coal consumption, even as the amount of coal
required for any particular application fell. Jevons argued that improvements in
fuel efficiency tend to increase, rather than decrease, fuel use: "It is a confusion
of ideas to suppose that the economical use of fuel is equivalent to diminished
consumption. The very contrary is the truth."[4]
William Stanley Jevons
At that time many in Britain worried that coal reserves were rapidly dwindling,
but some experts opined that improving technology would reduce coal
consumption. Jevons argued that this view was incorrect, as further increases in efficiency would tend to increase the
use of coal. Hence, improving technology would tend to increase, rather than reduce, the rate at which England's coal
deposits were being depleted.[2][4]
Cause
Rebound effect
One way to understand the Jevons
paradox is to observe that an increase
in the efficiency with which a resource
(e.g., fuel) is used causes a decrease in
the price of that resource when
measured in terms of what it can
achieve (e.g., work). Generally
speaking, a decrease in the price of a
good or service will increase the
quantity demanded (see supply and
Elastic Demand for Work: A doubling of fuel efficiency more than doubles work
demand, demand curve). Thus with a
demanded, increasing the amount of fuel used. Jevons paradox occurs.
lower price for work, more work will
be "purchased" (indirectly, by buying
more fuel). The resulting increase in the demand for fuel is known as the rebound effect. This increase in demand
may or may not be large enough to offset the original drop in demand from the increased efficiency. The Jevons
paradox occurs when the rebound effect is greater than 100%, exceeding the original efficiency gains. This effect has
been called 'backfire'.[2]
Jevons paradox
Consider a simple case: a perfectly
competitive market where fuel is the
sole input used, and the only
determinant of the cost of work. If the
price of fuel remains constant but the
efficiency of its conversion into work
is doubled, the effective price of work
is halved and twice as much work can
be purchased for the same amount of
money. If the amount of work
purchased more than doubles (i.e.,
demand for work is elastic, the price
Inelastic Demand for Work:A doubling of fuel efficiency does not double work
elasticity is greater than 1), then the
demanded, the amount of fuel used decreases. Jevons paradox does not occur.
quantity of fuel used would increase,
not decrease. If however, the demand for work is inelastic (price elasticity is less than 1), the amount of work
purchased would less than double, and the quantity of fuel used would decrease.
A full analysis would also have to take into account the fact that products (work) use more than one type of input
(e.g., fuel, labour, machinery), and that other factors besides input cost (e.g., a non-competitive market structure)
may also affect the price of work. These factors would tend to decrease the effect of fuel efficiency on the price of
work, and hence reduce the rebound effect, making the Jevons paradox less likely to occur. Additionally, any change
in the demand for fuel would have an effect on the price of fuel, and also on the effective price of work.
Khazzoom–Brookes postulate
In the 1980s, economists Daniel Khazzoom and Leonard Brookes revisited the Jevons paradox in the case of a
society's energy use. Brookes, then chief economist at the UK Atomic Energy Authority, argued that attempts to
reduce energy consumption by increasing energy efficiency would simply raise demand for energy in the economy
as a whole. Khazzoom focused on the narrower point that the potential for rebound was ignored in mandatory
performance standards for domestic appliances being set by the California Energy Commission.
In 1992, the economist Harry Saunders dubbed the hypothesis that improvements in energy efficiency work to
increase, rather than decrease, energy consumption the Khazzoom–Brookes postulate. Saunders showed that the
Khazzoom–Brookes postulate was consistent with neo-classical growth theory (the mainstream economic theory of
capital accumulation, technological progress and long-run economic growth) under a wide range of assumptions.[5]
According to Saunders, increased energy efficiency tends to increase energy consumption by two means. First,
increased energy efficiency makes the use of energy relatively cheaper, thus encouraging increased use (the direct
rebound effect). Second, increased energy efficiency leads to increased economic growth, which pulls up energy use
for the whole economy. At the microeconomic level (looking at an individual market), even with the rebound effect,
improvements in energy efficiency usually result in reduced energy consumption.[6] That is, the rebound effect is
usually less than 100 percent. However, at the macroeconomic level, more efficient (and hence comparatively
cheaper) energy leads to faster economic growth, which in turn increases energy use throughout the economy.
Saunders concludes that, taking into account both microeconomic and macroeconomic effects, technological
progress that improves energy efficiency will tend to increase overall energy use.
65
Jevons paradox
Energy conservation policy
Jevons warned that fuel efficiency gains tend to increase fuel use, but this does not imply that increased fuel
efficiency is worthless. Increased fuel efficiency enables greater production and a higher quality of material life. For
example, a more efficient steam engine allowed the cheaper transport of goods and people that contributed to the
Industrial Revolution. However, if the Khazzoom–Brookes postulate is correct, increased fuel efficiency will not
reduce the rate of depletion of fossil fuels.
The Jevons paradox is sometimes used to argue that energy conservation efforts are futile, for example, that more
efficient use of oil will lead to increased demand, and will not slow the arrival or the effects of peak oil. This
argument is usually presented as a reason not to impose environmental policies, or to increase fuel efficiency (e.g. if
cars are more efficient, it will simply lead to more driving).[7][8] Several points have been raised against this
argument. First, in the context of a mature market such as for oil in developed countries, the direct rebound effect is
usually small, and so increased fuel efficiency usually reduces resource use, other conditions remaining
constant.[6][9][10] Second, even if increased efficiency does not reduce the total amount of fuel used, there remain
other benefits associated with improved efficiency. For example, increased fuel efficiency may mitigate the price
increases, shortages and disruptions in the global economy associated with peak oil.[11] Third, environmental
economists have pointed out that fuel use will unambiguously decrease if increased efficiency is coupled with an
intervention (e.g. a green tax) that keeps the cost of fuel use the same or higher.[3]
The Jevons paradox indicates that increased efficiency by itself is unlikely to reduce fuel use, and that sustainable
energy policy must rely on other types of government interventions.[12] As the Jevons paradox applies only to
technological improvements that increase fuel efficiency, the imposition of conservation standards that
simultaneously increase costs does not cause an increase in fuel use. To ensure that efficiency enhancing
technological improvements reduce fuel use, efficiency gains must be paired with government intervention that
reduces demand (e.g., green taxes, a cap and trade programme, or higher fuel taxes). The ecological economists
Mathis Wackernagel and William Rees have suggested that any cost savings from efficiency gains be "taxed away or
otherwise removed from further economic circulation. Preferably they should be captured for reinvestment in natural
capital rehabilitation."[3] By mitigating the economic effects of government interventions designed to promote
ecologically sustainable activities, efficiency-improving technological progress may make the imposition of these
interventions more palatable, and more likely to be implemented.[13]
References
[1] Alcott, Blake (July 2005). "Jevons' paradox" (http:/ / www. sciencedirect. com/ science/ article/ B6VDY-4G7GFMG-1/ 2/
5da4f921421a31032f8fcd6971b0e177). Ecological Economics 54 (1): 9–21. doi:10.1016/j.ecolecon.2005.03.020. . Retrieved 2010-08-08.
[2] Alcott, Blake (2008). "Historical Overview of the Jevons Paradox in the Literature". In JM Polimeni, K Mayumi, M Giampietro. The Jevons
Paradox and the Myth of Resource Efficiency Improvements. Earthscan. pp. 7–78. ISBN 1-84407-462-5.
[3] Wackernagel, Mathis; Rees, William (1997). "Perceptual and structural barriers to investing in natural capital: Economics from an ecological
footprint perspective". Ecological Economics 20 (3): 3–24. doi:10.1016/S0921-8009(96)00077-8.
[4] Jevons, William Stanley (1866). "VII" (http:/ / www. econlib. org/ library/ YPDBooks/ Jevons/ jvnCQ0. html). The Coal Question (2nd ed.).
London: Macmillan and Company. . Retrieved 2008-07-21.
[5] Saunders, Harry D., "The Khazzoom–Brookes postulate and neoclassical growth." The Energy Journal, October 1, 1992.
[6] A. Greening, L; David L. Greene,Carmen Difiglio (2000). "Energy efficiency and consumption—the rebound effect—a survey". Energy
Policy 28 (6–7): 389–401. doi:10.1016/S0301-4215(00)00021-5
[7] Potter, Andrew (2007-02-13). "Planet-friendly design? Bah, humbug" (http:/ / web. archive. org/ web/ 20071214235056/ http:/ / www.
macleans. ca/ article. jsp?content=20070202_154815_4816). MacLean's 120 (5): 14. Archived from the original (http:/ / www. macleans. ca/
article. jsp?content=20070202_154815_4816) on 2007-12-14. . Retrieved 2010-09-01.
[8] Strassel, Kimberley A. (2001-05-17). "Conservation Wastes Energy" (http:/ / web. archive. org/ web/ 20051113194327/ http:/ / www.
opinionjournal. com/ columnists/ kstrassel/ ?id=95000484). Wall St. Journal (Wall St. Journal—Opinion). Archived from the original (http:/ /
www. opinionjournal. com/ columnists/ kstrassel/ ?id=95000484) on 2005-11-13. . Retrieved 2009-07-31.
[9] Small, Kenneth A.; Kurt Van Dender (2005-09-21). "The Effect of Improved Fuel Economy on Vehicle Miles Traveled: Estimating the
Rebound Effect Using U.S. State Data, 1966–2001" (http:/ / escholarship. org/ uc/ item/ 1h6141nj). Policy and Economics (University of
California Energy Institute, UC Berkeley). . Retrieved 2010-09-01.
66
Jevons paradox
[10] Gottron, Frank. "Energy Efficiency and the Rebound Effect: Does Increasing Efficiency Decrease Demand?" (http:/ / www. policyarchive.
org/ handle/ 10207/ bitstreams/ 3492. pdf). . Retrieved 2012-02-24.
[11] Hirsch, R. L., Bezdek, R. and Wendling, R. (2006), Peaking of World Oil Production and Its Mitigation. AIChE Journal, 52: 2–8. doi:
10.1002/aic.10747
[12] Giampietro, Mario; Kozo Mayumi (2008). "The Jevons Paradox: The Evolution of Complex Adaptive Systems and the Challenge for
Scientific Analysis". In JM Polimeni, K Mayumi, M Giampietro. The Jevons Paradox and the Myth of Resource Efficiency Improvements.
Earthscan. pp. 79–140. ISBN 1-84407-462-5.
[13] Laitner, John A.; Stephen J. De Canio and Irene Peters (2003). "Incorporating Behavioural, Social, and Organizational Phenomena in the
Assessment of Climate Change Mitigation Options" (http:/ / www. springerlink. com/ content/ n107734r313hh4wp/ ). Society, Behaviour, and
Climate Change Mitigation. Advances in Global Change Research 8: 1–64. doi:10.1007/0-306-48160-X_1. ISBN 0-7923-6802-9. . Retrieved
2010-08-08.
Further reading
• Jevons, William Stanley (1866). The Coal Question (http://www.econlib.org/library/YPDBooks/Jevons/
jvnCQ.html) (2nd ed.). London: Macmillan and Co..
• Lords Select Committee on Science and Technology (5 July 2005). "3: The economics of energy efficiency"
(http://www.publications.parliament.uk/pa/ld200506/ldselect/ldsctech/21/2106.htm). Select Committee on
Science and Technology Second Report. Session 2005-06. House of Lords.
• Herring, Horace (19 July 1999). "Does energy efficiency save energy? The debate and its consequences". Applied
Energy 63 (3): 209–226. doi:10.1016/S0306-2619(99)00030-6. ISSN 03062619.
• Owen, David (December 20, 2010). "Annals of Environmentalism: The Efficiency Dilemma" (http://www.
newyorker.com/reporting/2010/12/20/101220fa_fact_owen). The New Yorker: pp. 78–.
• Schipper, Lee (November 26, 1994). "Energy Efficiency Works, and It Saves Money" (http://query.nytimes.
com/gst/fullpage.html?res=9904E4D61530F935A15752C1A962958260). The New York Times.
External links
• Rocky Mountain Institute (May 1, 2008). "Beating the Energy Efficiency Paradox (Part I)" (http://www.
treehugger.com/files/2008/05/beating-energy-efficiency-paradox.php). TreeHugger.
67
Leontief paradox
Leontief paradox
Leontief's paradox in economics is that the country with the world's highest capital-per worker has a lower
capital/labor ratio in exports than in imports.
This econometric find was the result of Professor Wassily W. Leontief's attempt to test the Heckscher-Ohlin theory
empirically. In 1954, Leontief found that the U.S. (the most capital-abundant country in the world) exported
labor-intensive commodities and imported capital-intensive commodities, in contradiction with Heckscher-Ohlin
theory ("H-O theory").
Measurements
• In 1971 Robert Baldwin showed that US imports were 27% more capital-intensive than US exports in the 1962
trade data,[1] using a measure similar to Leontief's.
• In 1980 Edward Leamer questioned Leontief's original methodology on Real exchange rate grounds, but
acknowledged that the US paradox still appears in the data (for years other than 1947).[2]
• A 1999 survey of the econometric literature by Elhanan Helpman concluded that the paradox persists, but some
studies in non-US trade were instead consistent with the H-O theory.
• In 2005 Kwok & Yu used an updated methodology to argue for a lower or zero paradox in US trade statistics,
though the paradox is still derived in other developed nations.[3]
Responses to the paradox
For many economists, Leontief's paradox undermined the validity of the Heckscher-Ohlin theorem (H-O) theory,
which predicted that trade patterns would be based on countries' comparative advantage in certain factors of
production (such as capital and labor). Many economists have dismissed the H-O theory in favor of a more Ricardian
model where technological differences determine comparative advantage. These economists argue that the U.S. has
an advantage in highly skilled labor more so than capital. This can be seen as viewing "capital" more broadly, to
include human capital. Using this definition, the exports of the U.S. are very (human) capital-intensive, and not
particularly intensive in (unskilled) labor.
Some explanations for the paradox dismiss the importance of comparative advantage as a determinant of trade. For
instance, the Linder hypothesis states that demand plays a more important role than comparative advantage as a
determinant of trade--with the hypothesis that countries which share similar demands will be more likely to trade.
For instance, both the U.S. and Germany are developed countries with a significant demand for cars, so both have
large automotive industries. Rather than one country dominating the industry with a comparative advantage, both
countries trade different brands of cars between them. Similarly, New Trade Theory argues that comparative
advantages can develop separately from factor endowment variation (e.g. in industrial increasing returns to scale).
References
[1] "Leontief Paradox" (http:/ / www. econ. iastate. edu/ classes/ econ355/ choi/ leo. htm). . Retrieved 2007-11-05.
[2] Duchin, Faye (2000). "International Trade: Evolution in the Thought and Analysis of Wassily Leontief" (http:/ / www. wassily. leontief. net/
PDF/ Duchin. pdf). p. 3. .
[3] "Leontief paradox and the role of factor intensity measurement" (http:/ / editorialexpress. com/ cgi-bin/ conference/ download.
cgi?db_name=ACE2005& paper_id=224). 2005. .
68
Lucas paradox
Lucas paradox
In economics, the Lucas paradox or the Lucas puzzle is the observation that capital does not flow from developed
countries to developing countries despite the fact that developing countries have lower levels of capital per worker.[1]
Classical economic theory predicts that capital should flow from rich countries to poor countries, due to the effect of
diminishing returns of capital. Poor countries have lower levels of capital per worker – which explains, in part, why
they are poor. In poor countries, the scarcity of capital relative to labor should mean that the returns related to the
infusion of capital are higher than in developed countries. In response, savers in rich countries should look at poor
countries as profitable places in which to invest. In reality, things do not seem to work that way. Surprisingly little
capital flows from rich countries to poor countries. This puzzle, famously discussed in a paper by Robert Lucas in
1990, is often referred to as the "Lucas Paradox."
The theoretical explanations for the Lucas Paradox can be grouped into two categories.[2]
1. The first group attributes the limited amount of capital received by poorer nations to differences in fundamentals
that affect the production structure of the economy, such as technological differences, missing factors of
production, government policies, and the institutional structure.
2. The second group of explanations focuses on international capital market imperfections, mainly sovereign risk
(risk of nationalization) and asymmetric information. Although the expected return on investment might be high
in many developing countries, it does not flow there because of the high level of uncertainty associated with those
expected returns.
Examples of the Lucas Paradox: 20th century development of Third World
nations
Lucas’ seminal paper was a reaction to observed trends in international development efforts during the 20th century.
Regions characterized by poverty, such as India, China and Africa, have received particular attention with regard to
the underinvestment predicted by Lucas. Africa, with its impoverished populace and rich natural resources, has been
upheld as exemplifying the type of the nation that would, under neoclassical assumptions, be able to offer extremely
high returns to capital. The meager foreign capital Africa receives outside of the charity of multinational
corporations reveals the extent to which Lucas captured the realities of today’s global capital flows.[3]
Authors more recently have focused their explanations for the paradox on Lucas’ first category of explanation, the
difference in fundamentals of the production structure. Some have pointed to the quality of institutions as the key
determinant of capital inflows to poorer nations.[4] As evidence for the central role played by institutional stability, it
has been shown that the amount of foreign direct investment a country receives is highly correlated to the strength of
infrastructure and the stability of government in that country.
Counterexample of the Lucas Paradox; American economic development
Although Lucas’ original hypothesis has widely been accepted as descriptive of the modern period in history, the
paradox does not emerge as clearly before the 20th century. The colonial era, for instance, stands out as an age of
unimpeded capital flows. The system of imperialism produced economic conditions particularly amenable to the
movement of capital according to the assumptions of classical economics. Britain, for instance, was able to design,
impose, and control the quality of institutions in their colonies to capitalize on the high returns to capital in the new
world.[5]
Jeffrey Williamson has explored in depth this reversal of the Lucas Paradox in the colonial context. Although not
emphasized by Lucas himself, Williamson maintains that unimpeded labor migration is one way that capital flows to
the citizens of developing nations. The empire structure was particularly important for facilitating low-cost
international migration, allowing wage rates to converge across the regions in the British Empire.[6] For instance, in
69
Lucas paradox
the 17th and 18th century, England incentivized its citizens to move to the labor-scarce America, endorsing a system
of indentured servitude to make overseas migration affordable.
While Britain enabled free capital flow from old to new world, the success of the American enterprise after the
American Revolution is a good example of the role of institutional and legal frameworks for facilitating a continued
flow of capital. The American Constitution’s commitment to private property rights, rights of personal liberty; and
strong contract law enabled investment from Britain to America to continue even without the incentives of the
colonial relationship.[7] In these ways, early American economic development, both pre and post-revolution,
provides a case study for the conditions under which the Lucas Paradox is reversed. Even after the average income
level in America exceeded that of Britain, the institutions exported under imperialism and the legal frameworks
established after independence enabled long term capital flows from Europe to America.
References
[1] Lucas, Robert (1990). "Why doesn't Capital Flow from Rich to Poor Countries?". American Economic Review 80 (2): 92–96
[2] Alfaro, Laura; Kalemli‐Ozcan, Sebnem; Volosovych, Vadym (2008). "Why Doesn't Capital Flow from Rich to Poor Countries? An Empirical
Investigation". Review of Economics and Statistics 90 (2): 347–368. doi:10.1162/rest.90.2.347
[3] Montiel, Peter. "Obstacles to Investment in Africa: Explaining the Lucas Paradox" (http:/ / www. imf. org/ external/ np/ seminars/ eng/ 2006/
rppia/ pdf/ montie. pdf). Article. . Retrieved 27 February 2011.
[4] Daude, Christian. "THE QUALITY OF INSTITUTIONS AND FOREIGN DIRECT INVESTMENT". Article.
doi:10.1111/j.1468-0343.2007.00318.x.
[5] Schularick, Moritz. "The Lucas Paradox and the Quality of Institutions: Then and Now" (http:/ / www. jfki. fu-berlin. de/ faculty/ economics/
team/ persons/ schularick/ Lucas_discussion_paper_FUB. pdf). . Retrieved 21 February 2011.
[6] Williamson, Jeffrey. "Winners and Losers Over Two Centuries of Globalization". National Bureau of Economic Research.
[7] Ferguson, Niall. "The British Empire and Globalization" (http:/ / www. originofnations. org/ British_Empire/
british_empire_and_globalization. htm). . Retrieved 28 February 2011.
Metzler paradox
In economics, the Metzler paradox (named after the American economist Lloyd Metzler) is the theoretical
possibility that the imposition of a tariff on imports may reduce the relative internal price of that good.[1] It was
proposed by Lloyd Metzler in 1949 upon examination of tariffs within the Heckscher–Ohlin model.[2] The paradox
has roughly the same status as immiserizing growth and a transfer that makes the recipient worse off.[3]
The strange result could occur if the exporting country's offer curve is very inelastic. In this case, the tariff lowers the
duty-free cost of the price of the import by such a great degree that the effect of the improvement of the
tariff-imposing countries' terms of trade on relative prices exceeds the amount of the tariff. Such a tariff would not
protect the industry competing with the imported goods.
It is deemed to be unlikely in practice.[4][5]
References
[1] Casas, François R.; Choi, Eun K. (1985). "The Metzler Paradox and the Non-equivalence of Tariffs and Quotas: Further Results". Journal of
Economic Studies 12 (5): 53–57. doi:10.1108/eb002612.
[2] Metzler, Lloyd A. (1949). "Tariffs, the Terms of Trade, and the Distribution of National Income". Journal of Political Economy 57 (1): 1–29.
doi:10.1086/256766.
[3] Krugman and Obstfeld (2003), p. 112
[4] de Haan, Werner A.; Visser, Patrice (December 1979). "A note on tariffs, quotas, and the Metzler Paradox: An alternative approach". Review
of World Economics 115 (4): 736–741. doi:10.1007/bf02696743.
[5] Krugman and Obstfeld (2003), p. 113
70
Metzler paradox
Further reading
• Krugman, Paul R.; Obstfeld, Maurice (2003). "Chapter 5: The Standard Trade Model". International Economics:
Theory and Policy (6th ed.). Boston: Addison-Wesley. p. 112. ISBN 0-321-11639-9.
Paradox of thrift
The paradox of thrift (or paradox of saving) is a paradox of economics, popularized by John Maynard Keynes,
though it had been stated as early as 1714 in The Fable of the Bees,[1] and similar sentiments date to antiquity.[2][3]
The paradox states that if everyone tries to save more money during times of economic recession, then aggregate
demand will fall and will in turn lower total savings in the population because of the decrease in consumption and
economic growth. The paradox is, narrowly speaking, that total savings may fall even when individual savings
attempt to rise, and, broadly speaking, that increase in savings may be harmful to an economy.[4] Both the narrow
and broad claims are paradoxical within the assumption underlying the fallacy of composition, namely that what is
true of the parts must be true of the whole. The narrow claim transparently contradicts this assumption, and the broad
one does so by implication, because while individual thrift is generally averred to be good for the economy, the
paradox of thrift holds that collective thrift may be bad for the economy.
The paradox of thrift is a central component of Keynesian economics, and has formed part of mainstream economics
since the late 1940s, though it is criticized on a number of grounds.
Overview
The argument is that, in equilibrium, total income (and thus demand) must equal total output, and that total
investment must equal total saving. Assuming that saving rises faster as a function of income than the relationship
between investment and output, then an increase in the marginal propensity to save, ceteris paribus, will move the
equilibrium point at which income equals output and investment equals savings to lower values.
In this form it represents a prisoner's dilemma as saving is beneficial to each individual but deleterious to the general
population. This is a "paradox" because it runs contrary to intuition. One who does not know about the paradox of
thrift would fall into a fallacy of composition wherein one generalizes what is perceived to be true for an individual
within the economy to the overall population. Although exercising thrift may be good for an individual by enabling
that individual to save for a "rainy day", it may not be good for the economy as a whole.
This paradox can be explained by analyzing the place, and impact, of increased savings in an economy. If a
population saves more money (that is the marginal propensity to save increases across all income levels), then total
revenues for companies will decline. This decrease in economic growth means fewer salary increases and perhaps
downsizing. Eventually the population's total savings will have remained the same or even declined because of lower
incomes and a weaker economy. This paradox is based on the proposition, put forth in Keynesian economics, that
many economic downturns are demand based. Hypothetically, if all people will save their money, savings will rise
but there is a tendency that the macroeconomic status will fall.
71
Paradox of thrift
History
While the paradox of thrift was popularized by Keynes, and is often attributed to him,[2] it was stated by a number of
others prior to Keynes, and the proposition that spending may help and saving may hurt an economy dates to
antiquity; similar sentiments occur in the Bible verse:
There is that scattereth, and yet increaseth; and there is that withholdeth more than is meet, but it tendeth to
poverty.
—Proverbs 11:24
which has found occasional use as an epigram in underconsumptionist writings.[2][5][6][7]
Keynes himself notes the appearance of the paradox in The Fable of the Bees: or, Private Vices, Publick Benefits
(1714) by Bernard Mandeville, the title itself hinting at the paradox, and Keynes citing the passage:
As this prudent economy, which some people call Saving, is in private families the most certain method to
increase an estate, so some imagine that, whether a country be barren or fruitful, the same method if generally
pursued (which they think practicable) will have the same effect upon a whole nation, and that, for example,
the English might be much richer than they are, if they would be as frugal as some of their neighbours. This, I
think, is an error.
Keynes suggests Adam Smith was referring to this passage when he wrote "What is prudence in the conduct of every
private family can scarce be folly in that of a great Kingdom."
The problem of underconsumption and oversaving, as they saw it, was developed by underconsumptionist
economists of the 19th century, and the paradox of thrift in the strict sense that "collective attempts to save yield
lower overall savings" was explicitly stated by John M. Robertson in his 1892 book The Fallacy of Saving,[2][8]
writing:
Had the whole population been alike bent on saving, the total saved would positively have been much less,
inasmuch as (other tendencies remaining the same) industrial paralysis would have been reached sooner or
oftener, profits would be less, interest much lower, and earnings smaller and more precarious. This ... is no idle
paradox, but the strictest economic truth.
—John M. Robertson, The Fallacy of Saving, p. 131–2
Similar ideas were forwarded by William Trufant Foster and Waddill Catchings in the 1920s in The Dilemma of
Thrift [9].
Keynes distinguished between business activity/investment ("Enterprise") and savings ("Thrift") in his Treatise on
Money (1930):
...mere abstinence is not enough by itself to build cities or drain fens. ... If Enterprise is afoot, wealth
accumulates whatever may be happening to Thrift; and if Enterprise is asleep, wealth decays whatever Thrift
may be doing. Thus, Thrift may be the handmaiden of Enterprise. But equally she may not. And, perhaps, even
usually she is not.
and stated the paradox of thrift in The General Theory, 1936:
For although the amount of his own saving is unlikely to have any significant influence on his own income,
the reactions of the amount of his consumption on the incomes of others makes it impossible for all individuals
simultaneously to save any given sums. Every such attempt to save more by reducing consumption will so
affect incomes that the attempt necessarily defeats itself. It is, of course, just as impossible for the community
as a whole to save less than the amount of current investment, since the attempt to do so will necessarily raise
incomes to a level at which the sums which individuals choose to save add up to a figure exactly equal to the
amount of investment.
—John Maynard Keynes, The General Theory of Employment, Interest and Money, Chapter 7, p. 84
72
Paradox of thrift
The theory is referred to as the "paradox of thrift" in Samuelson's influential Economics of 1948, which popularized
the term.
Related concepts
The paradox of thrift has been related to the debt deflation theory of economic crises, being called "the paradox of
debt"[10] – people save not to increase savings, but rather to pay down debt. As well, a paradox of toil and a paradox
of flexibility have been proposed: A willingness to work more in a liquidity trap and wage flexibility after a debt
deflation shock may lead not only to lower wages, but lower employment.[11]
During April 2009, U.S. Federal Reserve Vice Chair Janet Yellen discussed the "Paradox of deleveraging" described
by economist Hyman Minsky: "Once this massive credit crunch hit, it didn’t take long before we were in a recession.
The recession, in turn, deepened the credit crunch as demand and employment fell, and credit losses of financial
institutions surged. Indeed, we have been in the grips of precisely this adverse feedback loop for more than a year. A
process of balance sheet deleveraging has spread to nearly every corner of the economy. Consumers are pulling back
on purchases, especially on durable goods, to build their savings. Businesses are cancelling planned investments and
laying off workers to preserve cash. And, financial institutions are shrinking assets to bolster capital and improve
their chances of weathering the current storm. Once again, Minsky understood this dynamic. He spoke of the
paradox of deleveraging, in which precautions that may be smart for individuals and firms—and indeed essential to
return the economy to a normal state—nevertheless magnify the distress of the economy as a whole."[12]
Criticisms
Within mainstream economics, non-Keynesian economists, particularly neoclassical economists, criticize this theory
on three principal grounds.
The first criticism is that, following Say's law and the related circle of ideas, if demand slackens, prices will fall
(barring government intervention), and the resulting lower price will stimulate demand (though at lower profit or
cost – possibly even lower wages). This criticism in turn has been questioned by Keynesian economists, who reject
Say's law and instead point to evidence of sticky prices as a reason why prices do not fall in recession; this remains a
debated point.
The second criticism is that savings represent loanable funds, particularly at banks, assuming the savings are held at
banks, rather than currency itself being held ("stashed under one's mattress"). Thus an accumulation of savings yields
an increase in potential lending, which will lower interest rates and stimulate borrowing. So a decline in consumer
spending is offset by an increase in lending, and subsequent investment and spending.
Two caveats are added to this criticism. Firstly, if savings are held as cash, rather than being loaned out (directly by
savers, or indirectly, as via bank deposits), then loanable funds do not increase, and thus a recession may be caused –
but this is due to holding cash, not to saving per se.[13] Secondly, banks themselves may hold cash, rather than
loaning it out, which results in the growth of excess reserves – funds on deposit but not loaned out. This is argued to
occur in liquidity trap situations, when interest rates are at a zero lower bound (or near it) and savings still exceed
investment demand. Within Keynesian economics, the desire to hold currency rather than loan it out is discussed
under liquidity preference.
Third, the paradox assumes a closed economy in which savings are not invested abroad (to fund exports of local
production abroad). Thus, while the paradox may hold at the global level, it need not hold at the local or national
level: if one nation increases savings, this can be offset by trading partners consuming a greater amount relative to
their own production, i.e., if the saving nation increases exports, and its partners increase imports. This criticism is
not very controversial, and is generally accepted by Keynesian economists as well,[14] who refer to it as "exporting
one's way out of a recession". They further note that this frequently occurs in concert with currency devaluation[15]
(hence increasing exports and decreasing imports), and cannot work as a solution to a global problem, because the
73
Paradox of thrift
global economy is a closed system – not every nation can increase exports.
Austrian School criticism
Within heterodox economics, the paradox was criticized by the Austrian School economist and Nobel Prize winner
Friedrich Hayek in a 1929 article, "The 'Paradox' of Savings", attacking the paradox as proposed by Foster and
Catchings.[16] Hayek and later Austrian School economists agree that if a population saves more money, total
revenues for companies will decline, but they deny the assertion that lower revenues lead to lower economic growth.
Austrian School economists believe the productivity of the economy is determined by the consumption-investment
ratio, and the demand for money only tells us the degree to which people prefer the utility of money (protection
against uncertainty) to the utility of goods. They argue that hoarding of money (an increase in the demand for
money) does not necessarily lead to a change in the population's consumption-investment ratio;[17] instead, it may
simply be reflected in the price level. For example, if spending falls by half and prices also uniformly fall by half, the
consumption-investment ratio and productivity would be left unchanged.
Notes
[1] Keynes, The General Theory of Employment, Interest and Money, Chapter 23. Notes on Merchantilism, the Usury Laws, Stamped Money
and Theories of Under-consumption (http:/ / www. marxists. org/ reference/ subject/ economics/ keynes/ general-theory/ ch23. htm)
[2] Nash, Robert T.; Gramm, William P. (1969). "A Neglected Early Statement the Paradox of Thrift" (http:/ / hope. dukejournals. org/ cgi/
pdf_extract/ 1/ 2/ 395). History of Political Economy 1 (2): 395–400. doi:10.1215/00182702-1-2-395. .
[3] See history section for further discussion.
[4] These two formulations are given in Campbell R. McConnell (1960: 261–62), emphasis added: "By attempting to increase its rate of saving,
society may create conditions under which the amount it can actually save is reduced. This phenomenon is called the paradox of
thrift....[T]hrift, which has always been held in high esteem in our economy, now becomes something of a social vice."
[5] English, Irish and Subversives Among the Dismal Scientists, Noel Thompson, Nigel Allington, 2010, p. 122 (http:/ / books. google. com/
books?id=4fjwxnH8VPcC& pg=PA122& dq="scattereth,+ and+ yet+ increaseth"):
"A suggestion that a more equal distribution of income might be a remedy for general stagnation – and that excess saving can be harmful – is
implicit in the quotation from the Old Testament on the Reply to Mr. Say [by John Cazenove (1788–1879)].
[6] A Reply to Mr. Say’s Letters to Mr. Malthus, by John Cazenove, uses the verse as an epigram
[7] Studies in economics, William Smart, 1895, p. 249 (http:/ / books. google. com/ books?id=YwUPAAAAQAAJ& dq="scattereth,+ and+ yet+
increaseth")
[8] Robertson, John M. (1892). The Fallacy of Saving (http:/ / www. archive. org/ stream/ fallacyofsavings00robe/ fallacyofsavings00robe_djvu.
txt). .
[9] http:/ / books. google. com/ books?id=0flCPQAACAAJ
[10] Paradox of thrift (http:/ / krugman. blogs. nytimes. com/ 2009/ 02/ 03/ paradox-of-thrift/ ), Paul Krugman
[11] Eggertsson, Gauti B.; Krugman, Paul (14 February 2011), Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach
(http:/ / faculty. wcas. northwestern. edu/ ~gep575/ seminars/ spring2011/ EK. pdf), , retrieved 2011-12-15
[12] Federal Reserve-Janet Yellen-A Minsky Meltdown-April 2009 (http:/ / www. frbsf. org/ news/ speeches/ 2009/ 0416. html)
[13] See section 9.9 and 9.11 http:/ / www. auburn. edu/ ~garriro/ cbm. htm
[14] The paradox of thrift — for real (http:/ / krugman. blogs. nytimes. com/ 2009/ 07/ 07/ the-paradox-of-thrift-for-real/ ), Paul Krugman, July 7,
2009
[15] Devaluing History (http:/ / krugman. blogs. nytimes. com/ 2010/ 11/ 24/ devaluing-history/ ), Paul Krugman, November 24, 2010
[16] Hayek on the Paradox of Saving (http:/ / mises. org/ story/ 2804)
[17] Pages 37-39 of http:/ / www. mises. org/ rothbard/ agd. pdf
74
Paradox of thrift
75
References
• Samuelson, Paul& Nordhaus, William (2005). Economics (18th ed.). New York: McGraw-Hill.
ISBN 0-07-123932-4.
External links
• The paradox of thrift explained (http://ingrimayne.saintjoe.edu/econ/Keynes/Paradox.html)
Criticisms
• The Paradox of Thrift: RIP (http://www.cato.org/pubs/journal/cj16n1-7.html), by Clifford F. Thies, The Cato
Journal, Volume 16, Number 1
• Consumers don't cause recessions (http://mises.org/story/3194) by Robert P. Murphy (an Austrian School
critique of the paradox of thrift)
Paradox of value
The paradox of value (also known as the diamond–water paradox)
is the apparent contradiction that, although water is on the whole more
useful, in terms of survival, than diamonds, diamonds command a
higher price in the market. The philosopher Adam Smith is often
considered to be the classic presenter of this paradox. Nicolaus
Copernicus,[1] John Locke, John Law[2] and others had previously tried
to explain the disparity.
Labor theory of value
In a passage of Adam Smith's An Inquiry into the Nature and Causes of
the Wealth of Nations, he discusses the concepts of value in use and
value in exchange, and notices how they tend to differ:
Water diamonds.
What are the rules which men naturally observe in exchanging them [goods] for money or for one another, I
shall now proceed to examine. These rules determine what may be called the relative or exchangeable value of
goods. The word VALUE, it is to be observed, has two different meanings, and sometimes expresses the utility
of some particular object, and sometimes the power of purchasing other goods which the possession of that
object conveys. The one may be called "value in use;" the other, "value in exchange." The things which have
the greatest value in use have frequently little or no value in exchange; on the contrary, those which have the
greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it
will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has
scarce any use-value; but a very great quantity of other goods may frequently be had in exchange for it.[3]
Furthermore, he explained the value in exchange as being determined by labor:
The real price of every thing, what every thing really costs to the man who wants to acquire it, is the toil and
trouble of acquiring it.[4]
Hence, Smith denied a necessary relationship between price and utility. Price on this view was related to a factor of
production (namely, labor) and not to the point of view of the consumer.[5] Proponents of the labor theory of value
saw that as the resolution of the paradox.
Paradox of value
The labor theory of value has lost popularity in mainstream economics and has been replaced by the theory of
marginal utility.
Marginalism
The theory of marginal utility, which is
based on the subjective theory of value, says
that the price at which an object trades in the
market is determined neither by how much
labor was exerted in its production, as in the
labor theory of value, nor on how useful it is
on a whole (total utility). Rather, its price is
determined by its marginal utility. The
marginal utility of a good is derived from its
most important use to a person. So, if
someone possesses a good, he will use it to
satisfy some need or want. Which one?
Naturally,
the
one
that
takes
highest-priority. Eugen von Böhm-Bawerk
At low levels of consumption, water has a much higher marginal utility than
illustrated this with the example of a farmer
[6]
diamonds
and thus is more valuable. People usually consume water at much higher
having five sacks of grain. With the first,
levels than they do diamonds and thus the marginal utility and price of water are
he will make bread to survive. With the
lower than that of diamonds.
second, he will make more bread, in order to
be strong enough to work. With the next, he
will feed his farm animals. The next is used to make whisky, and the last one he feeds to the pigeons. If one of those
bags is stolen, he will not reduce each of those activities by one-fifth; instead he will stop feeding the pigeons. So the
value of the fifth bag of grain is equal to the satisfaction he gets from feeding the pigeons. If he sells that bag and
neglects the pigeons, his least productive use of the remaining grain is to make whisky, so the value of a fourth bag
of grain is the value of his whisky. Only if he loses four bags of grain will he start eating less; that is the most
productive use of his grain. The last bag of grain is worth his life.
In explaining the diamond-water paradox, marginalists explain that it is not the total usefulness of diamonds or water
that matters, but the usefulness of each unit of water or diamonds. It is true that the total utility of water to people is
tremendous, because they need it to survive. However, since water is in such large supply in the world, the marginal
utility of water is low. In other words, each additional unit of water that becomes available can be applied to less
urgent uses as more urgent uses for water are satisfied. Therefore, any particular unit of water becomes worth less to
people as the supply of water increases. On the other hand, diamonds are in much lower supply. They are of such
low supply that the usefulness of one diamond is greater than the usefulness of one glass of water, which is in
abundant supply. Thus, diamonds are worth more to people. Therefore, those who want diamonds are willing to pay
a higher price for one diamond than for one glass of water, and sellers of diamonds ask a price for one diamond that
is higher than for one glass of water.
76
Paradox of value
Criticisms
George Stigler has argued that Smith's statement of the paradox is flawed, since it consisted of a comparison between
heterogeneous goods, and such comparison would have required using the concept of marginal utility of income.
And since this concept was not known in Smith's time, then the value in use and value in exchange judgement may
be meaningless:
The paradox—that value in exchange may exceed or fall short of value in use—was, strictly speaking, a
meaningless statement, for Smith had no basis (i.e., no concept of marginal utility of income or marginal price
of utility) on which he could compare such heterogeneous quantities. On any reasonable interpretation,
moreover, Smith's statement that value in use could be less than value in exchange was clearly a moral
judgment, not shared by the possessors of diamonds. To avoid the incomparability of money and utility, one
may interpret Smith to mean that the ratio of values of two commodities is not equal to the ratio of their total
utilities. Or, alternatively, that the ratio of the prices of two commodities is not equal to the ratio of their total
utilities; but this also requires an illegitimate selection of units: The price of what quantity of diamonds is to be
compared with the price of one gallon of water?
—George Stigler, The development of Utility Theory. I [7]
References
[1] Gordon, Scott (1991). "Chapter 7: The Scottish Enlightenment of the eighteenth century". History and Philosophy of Social Science: An
Introduction. Routledge. p. 141. ISBN 0-415-09670-7. "This 'paradox of value', as it was called, was frequently noted before Adam Smith (for
example, by Copernicus who wrote a bit on economic questions)..."
[2] Blaug, Mark (1962). "Chapter 2: Adam Smith". Economic Theory in Retrospect. Cambridge University Press. p. 39. ISBN 0-521-57701-2.
"Moreover, such writers as Locke, Law and Harris had contrasted the value of water with that of diamonds..."
[3] Smith, Adam (1776). "Of the Origin and Use of Money" (http:/ / www. econlib. org/ LIBRARY/ Smith/ smWN. html). An Inquiry into the
Nature and Causes of the Wealth of Nations. . Retrieved April 2006.
[4] Smith, Adam (1776). "Book I, Chapter V Of the Real and Nominal Price of Commodities, or of their Price in Labour, and their Price in
Money" (http:/ / www. econlib. org/ LIBRARY/ Smith/ smWN. html). An Inquiry into the Nature and Causes of the Wealth of Nations. .
Retrieved July 2006.
[5] Dhamee, Yousuf(1996?), Adam Smith and the division of labour (http:/ / www. victorianweb. org/ economics/ division. html) accessed
09/08/06
[6] Böhm-Bawerk, Eugen von (1891). "Book III, Chapter IV: The Marginal Utility" (http:/ / www. econlib. org/ library/ BohmBawerk/ bbPTC.
html). . . Retrieved 2006-06-20. "A colonial farmer, whose log hut stands by itself in the primeval forest, far away from the busy haunts of
men, has just harvested five sacks of corn..."
[7] Stigler, George (1950). The development of Utility Theory. I. Journal of political economy 58(4), p 308.
77
Productivity paradox
Productivity paradox
The productivity paradox was analyzed and popularized in a widely-cited article[1] by Erik Brynjolfsson, which
noted the apparent contradiction between the remarkable advances in computer power and the relatively slow growth
of productivity at the level of the whole economy, individual firms and many specific applications. The concept is
sometimes referred to as the Solow computer paradox in reference to Robert Solow's 1987 quip, "You can see the
computer age everywhere but in the productivity statistics."[2] The paradox has been defined as the “discrepancy
between measures of investment in information technology and measures of output at the national level.”[3]
It was widely believed that office automation was boosting labor productivity (or total factor productivity). However,
the growth accounts didn't seem to confirm the idea. From the early 1970s to the early 1990s there was a massive
slow-down in growth as the machines were becoming ubiquitous. (Other variables in country's economies were
changing simultaneously; growth accounting separates out the improvement in production output using the same
capital and labour resources as input by calculating growth in total factor productivity, AKA the "Solow residual".)
The productivity paradox has attracted a lot of recent controversy and has grown outside the original context of
computers and communications. Some are now arguing that technology in general is subject to diminishing returns
in its ability to increase economic growth.[4]
Explanations
Different authors have explained the paradox in different ways. In his original article, Brynjolfsson (1993) identified
four possible explanations:
• Mismeasurement: the gains are real, but our current measures miss them;
• Redistribution: there are private gains, but they come at the expense of other firms and individuals, leaving little
net gain;
• Time lags: the gains take a long time to show up; and
• Mismanagement: there are no gains because of the unusual difficulties in managing IT or information itself.
He stressed the first explanation, noting weaknesses with then-existing studies and measurement methods, and
pointing out that "a shortfall of evidence is not evidence of a shortfall."
Turban, et al. (2008), mention that understanding the paradox requires an understanding of the concept of
productivity. Pinsonneault et al. (1998) state that for untangling the paradox an “understanding of how IT usage is
related to the nature of managerial work and the context in which it is deployed” is required.
One hypothesis to explain the productivity paradox is that computers are productive, yet their productive gains are
realized only after a lag period, during which complementary capital investments must be developed to allow for the
use of computers to their full potential.[5]
Diminishing marginal returns from computers, the opposite of the time lag hypothesis, is that computers, in the form
of mainframes, were used in the most productive areas, like high volume transactions of banking, accounting and
airline reservations, over two decades before personal computers. Also, computers replaced a sophisticated system of
data processing that used unit record equipment. Therefore the important productivity opportunities were exhausted
before computers were everywhere. We were looking at the wrong time period.
Another hypothesis states that computers are simply not very productivity enhancing because they require time, a
scarce complementary human input. This theory holds that although computers perform a variety of tasks, these
tasks are not done in any particularly new or efficient manner, but rather they are only done faster. Current data does
not confirm the validity of either hypothesis. It could very well be that increases in productivity due to computers is
not captured in GDP measures, but rather in quality changes and new products.
78
Productivity paradox
Economists have done research in the productivity issue and concluded that there are three possible explanations for
the paradox. The explanations can be divided in three categories:
• Data and analytical problems hide "productivity-revenues". The ratios for input and output are sometimes difficult
to measure, especially in the service sector.
• Revenues gained by a company through productivity will be hard to notice because there might be losses in other
divisions/departments of the company. So it is again hard to measure the profits made only through investments in
productivity.
• There is complexity in designing, administering and maintaining IT systems. IT projects, especially software
development, are notorious for cost overruns and schedule delays. Adding to cost are rapid obsolescence of
equipment and software, incompatible software and network platforms and issues with security such as data theft
and viruses. This causes constant spending for replacement. One time changes also occur, such as the Year 2000
problem and the changeover from Novell NetWare by many companies.
Other economists have made a more controversial charge against the utility of computers: that they pale into
insignificance as a source of productivity advantage when compared to the industrial revolution, electrification,
infrastructures (canals and waterways, railroads, highway system), Fordist mass production and the replacement of
human and animal power with machines. [6] High productivity growth occurred from last decades of the 19th century
until the 1973, with a peak from 1929-1973, then declined to levels of the early 19th century. [7][8] There was a
rebound in productivity after 2000. Much of the productivity from 1985-2000 came in the computer and related
industries.[8]
A number of explanations of this have been advanced, including:
• The tendency – at least initially – of computer technology to be used for applications that have little impact on
overall productivity, e.g. word processing.
• Inefficiencies arising from running manual paper-based and computer-based processes in parallel, requiring two
separate sets of activities and human effort to mediate between them – usually considered a technology alignment
problem
• Poor user interfaces that confuse users, prevent or slow access to time-saving facilities, are internally inconsistent
both with each other and with terms used in work processes – a concern addressed in part by enterprise taxonomy
• Extremely poor hardware and related boot image control standards that forced users into endless "fixes" as
operating systems and applications clashed – addressed in part by single board computers and simpler more
automated re-install procedures, and the rise of software specifically to solve this problem, e.g. Norton Ghost
• Technology-driven change driven by companies such as Microsoft which profit directly from more rapid
"upgrades"
• An emphasis on presentation technology and even persuasion technology such as PowerPoint, at the direct
expense of core business processes and learning – addressed in some companies including IBM and Sun
Microsystems by creating a PowerPoint-Free Zone
• The blind assumption that introducing new technology must be good
• The fact that computers handle office functions that, in most cases, are not related to the actual production of
goods and services.
• Factories were automated decades before computers. Adding computer control to existing factories resulted in
only slight productivity gains in most cases.
A paper by Triplett (1999) reviews Solow’s paradox from seven other often given explanations. They are:
• You don’t see computers “everywhere,” in a meaningful economic sense
• You only think you see computers everywhere
• You may not see computers everywhere, but in the industrial sectors where you most see them, output is poorly
measured
• Whether or not you see computer everywhere, some of what they do is not counted in economic statistics
79
Productivity paradox
80
• You don’t see computers in the productivity yet, but wait a bit and you will
• You see computers everywhere but in the productivity statistics because computers are not as productive as you
think
• There is no paradox: some economists are counting innovations and new products on an arithmetic scale when
they should count on a logarithmic scale.
Before computers: Data processing with unit record equipment
Early IBM tabulating machine. Common
applications were accounts receivable, payroll
and billing.
When computers for general business applications
appeared in the 1950s, a sophisticated industry for data
processing existed in the form of unit record
equipment. These systems processed data on punched
cards by running the cards through tabulating
machines, the holes in the cards allowing electrical
contact to activate relays and solenoids to keep a count.
The flow of punched cards could be arranged in various
program-like sequences to allow sophisticated data
processing. Some unit record equipment was
programmable by wiring a plug board, with the plug
boards being removable allowing for quick replacement
with another pre-wired program.[9]
In 1949 vacuum tube calculators were added to unit
record equipment. In 1955 the first completely
transistorized calculator with magnetic cores for
dynamic memory, the IBM 608, was introduced.[9]
Control panel for an IBM 402 Accounting Machine
The first computers were an improvement over unit
record equipment, but not by a great amount. This was
partly due to low level software used, low performance
capability and failure of vacuum tubes and other
components. Also, the data input to early computers
also used punched cards. Most of these hardware and
software shortcomings were solved by the late 1960s,
but punched cards did not become fully displaced until
the 1980s.
Analog process control
Computers did not revolutionize manufacturing because automation, in the form of control systems, had already
been in existence for decades, although computers did allow more sophisticated control, which led to improved
product quality and process optimization. Pre-computer control was known as analog control and computerized
control is called digital.
Parasitic losses of cashless transactions
Credit card transactions now represent a large percentage of low value transactions on which credit card companies
charge merchants. Most of such credit card transactions are more of a habit than an actual need for credit and to the
extent that such purchases represent convenience or lack of planning to carry cash on the part of consumers, these
transactions add a layer of unnecessary expense. However, debit or check card transactions are cheaper than
Productivity paradox
processing paper checks.
On line commerce
Despite high expectations for on line retail sales, individual item and small quantity handling and transportation costs
more than offset the savings of not having to maintain "bricks and mortar" stores. Online retail sales main success
was in specialty items, collectibles and higher priced goods. Some airline and hotel retailers and aggregators have
been very successful.
On line commerce has been extremely successful in banking, airline, hotel, and rental car reservations, to name a
few.
Restructured office
The personal computer restructured the office by reducing the secretarial and clerical staffs. Prior to computers,
secretaries transcribed Dictaphone recordings or live speech into shorthand, and typed the information, typically a
memo or letter. All filing was done with paper copies.
A new position in the office staff was the information technologist, or department. With networking came
information overload in the form of e-mail, with some office workers receiving several hundred each day, most of
which are not necessary information for the recipient.
Some hold that one of the main productivity boosts from information technology is still to come: large-scale
reductions in traditional offices as home offices become widespread, but this requires large and major changes in
work culture and remains to be proven.
Cost overruns of software projects
It is well known by software developers that projects typically run over budget and finish behind schedule.
Software development is typically for new applications that are unique. The project's analyst is responsible for
interviewing the stakeholders, individually and in group meetings, to gather the requirements and incorporate them
into a logical format for review by the stakeholders and developers. This sequence is repeated in successive
iterations, with partially completed screens available for review in the latter stages.
Unfortunately, stakeholders often have a vague idea of what the functionality should be, and tend to add a lot of
unnecessary features, resulting in schedule delays and cost overruns.
Qualifications
By the late 1990s there were some signs that productivity in the workplace been improved by the introduction of IT,
especially in the United States. In fact, Erik Brynjolfsson and his colleagues found a significant positive relationship
between IT investments and productivity, at least when these investments were made to complement organizational
changes.[10][11][12] A large share of the productivity gains outside the IT-equipment industry itself have been in
retail, wholesale and finance.[13]
Computers revolutionized accounting, billing, record keeping and many other office functions; however, early
computers used punched cards for data and programming input. Until the 1980s it was common to receive monthly
utility bills printed on a punched card that was returned with the customer’s payment.
In 1973 IBM introduced point of sale (POS) terminals in which electronic cash registers were networked to the store
mainframe computer. By the 1980s bar code readers were added. These technologies automated inventory
management. Wal-Mart Stores was an early adopter of POS.
Computers also greatly increased productivity of the communications sector, especially in areas like the elimination
of telephone operators. In engineering, computers replaced manual drafting with CAD and software was developed
81
Productivity paradox
for calculations used in electronic circuits, stress analysis, heat and material balances, etc.
Automated teller machines (ATMs) became popular in recent decades and self checkout at retailers appeared in the
1990s.
The Airline Reservations System and banking are areas where computers are practically essential. Modern military
systems also rely on computers.
References
[1] Brynjolfsson, Erik (1993). "The productivity paradox of information technology". Communications of the ACM 36 (12): 66–77.
doi:10.1145/163298.163309. ISSN 00010782.
[2] Robert Solow, "We'd better watch out", New York Times Book Review, July 12, 1987, page 36. See here (http:/ / www. standupeconomist.
com/ blog/ economics/ solows-computer-age-quote-a-definitive-citation/ ).
[3] Wetherbe, James C.; Turban, Efraim; Leidner, Dorothy E.; McLean, Ephraim R. (2007). Information Technology for Management:
Transforming Organizations in the Digital Economy (6th ed.). New York: Wiley. ISBN 0-471-78712-4.
[4] The Debate Zone: Has the US passed peak productivity growth? (http:/ / whatmatters. mckinseydigital. com/ the_debate_zone/
has-the-us-passed-peak-productivity-growth)
[5] David P.A., "The Dynamo and the Computer: A Historical Perspective on the Modern Productivity Paradox", American Economic Review
Papers and Proceedings, 1990, 355–61
[6] Gordon, Robert J. (2000). Does the "New Economy" Measure up to the Great Inventions of the Past? , NBER Working Paper No. 7833 (http:/
/ www. nber. org/ papers/ w7833).
[7] Kendrick, John (1991). U.S. productivity performance in perspective , Business Economics, October 1, 1991 (http:/ / www. allbusiness. com/
finance/ 262030-1. html).
[8] [ |Field, Alexander J (http:/ / www. scu. edu/ business/ economics/ faculty/ field. cfm)] (2007). U.S. economic growth in the gilded age 31,
Journal of Macroeconomics (2009) 173-190
[9] Fierheller, George A. (2006). Do not fold, spindle or mutilate: the "hole" story of punched cards (http:/ / www. gfierheller. ca/ books/ pdf/
do_not_fold. pdf). Stewart Pub.. ISBN 1-894183-86-X. .
[10] E.Brynjolfsson and L.Hitt, "Beyond the Productivity Paradox: Computers are the Catalyst for Bigger Changes", CACM, August 1998
[11] E. Brynjolfsson, S. Yang, “The Intangible Costs and Benefits of Computer Investments: Evidence from the Financial Markets,” MIT Sloan
School of Management, December 1999
[12] Paolo Magrassi, A.Panarella, B.Hayward, “The 'IT and Economy' Discussion: A Review”, GartnerGroup, Stamford (CT), USA, June 2002
[1]
[13] Kevin Stiroh (2002), ‘Information Technology and the US Productivity Revival: What Do the Industry Data Say?’, American Economic
Review 92(5), 1559-76.
Further reading
• Brynjolfsson, Erik, and Lorin Hitt (June 2003). "Computing Productivity: Firm Level Evidence" (http://papers.
ssrn.com/sol3/papers.cfm?abstract_id=290325). MIT Sloan Working Paper No. 4210-01.
• Brynjolfsson, Erik, and Adam Saunders (2010). Wired for Innovation: How Information Technology is Reshaping
the Economy (http://digital.mit.edu/erik/Wired4innovation.html). MIT Press.
• Greenwood, Jeremy (1997). The Third Industrial Revolution: Technology, Productivity and Income Inequality
(http://www.econ.rochester.edu/Faculty/GreenwoodPapers/third.pdf). AEI Press.
• Landauer, Thomas K. (1995). The trouble with computers: Usefulness, usability and productivity. Cambridge,
Massachusetts: MIT Press. ISBN 0-262-62108-8.
• "Information Technology and the Nature of Managerial Work: From the Productivity paradox to the Icarus
Paradox". MIS Quarterly 22 (3): 287–311. 1998.
• Triplett, Jack E. (1999). "The solow productivity paradox: what do computers do to productivity" (http://www.
csls.ca/journals/sisspp/v32n2_04.pdf). Canadian Journal of Economics 32 (2): 309–334.
• "Does successful investment in information technology solve the productivity paradox?". Information &
Management: 113. 2000.
82
St. Petersburg paradox
St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based
on a particular (theoretical) lottery game (sometimes called St. Petersburg Lottery) that leads to a random variable
with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a
very small amount of money. The St. Petersburg paradox is a classical situation where a naïve decision criterion
(which takes only the expected value into account) would recommend a course of action that no (real) rational
person would be willing to take. Several resolutions are possible.
The paradox is named from Daniel Bernoulli's presentation of the problem and his solution, published in 1738 in the
Commentaries of the Imperial Academy of Science of Saint Petersburg (Bernoulli 1738). However, the problem was
invented by Daniel's cousin Nicolas Bernoulli who first stated it in a letter to Pierre Raymond de Montmort of
September 9, 1713 (de Montmort 1713).
The paradox
A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The pot starts at 1
dollar and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins
whatever is in the pot. Thus the player wins 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on
the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8
dollars if a head appears on the first three tosses and a tail on the fourth, and so on. In short, the player wins 2k−1
dollars if the coin is tossed k times until the first tail appears.
What would be a fair price to pay the casino for entering the game? To answer this we need to consider what would
be the average payout: With probability 1/2, the player wins 1 dollar; with probability 1/4 the player wins 2 dollars;
with probability 1/8 the player wins 4 dollars, and so on. The expected value is thus
Assuming the game can continue as long as the coin toss results in heads, in particular that the casino has unlimited
resources, this sum diverges without bound, and so the expected win for the player, at least in this idealized form, is
an infinite amount of money. Considering nothing but the expectation value of the net change in one's monetary
wealth, one should therefore play the game at any price if offered the opportunity. Yet, in published descriptions of
the game, e.g., (Martin 2004), many people expressed disbelief in the result. Martin quotes Ian Hacking as saying
"few of us would pay even $25 to enter such a game" and says most commentators would agree. The paradox is the
discrepancy between what people seem willing to pay to enter the game and the infinite expected value suggested by
the above naïve analysis.
83
St. Petersburg paradox
Solutions of the paradox
There are different approaches for solving the paradox.
Expected utility theory
The classical resolution of the paradox involved the explicit introduction of a utility function, an expected utility
hypothesis, and the presumption of diminishing marginal utility of money.
In Daniel Bernoulli's own words:
The determination of the value of an item must not be based on the price, but rather on the utility it yields….
There is no doubt that a gain of one thousand ducats is more significant to the pauper than to a rich man
though both gain the same amount.
A common utility model, suggested by Bernoulli himself, is the logarithmic function U(w) = ln(w) (known as “log
utility” [1]). It is a function of the gambler’s total wealth w, and the concept of diminishing marginal utility of money
is built into it. The expected utility hypothesis posits that a utility function exists whose expected net change is a
good criterion for real people's behavior. For each possible event, the change in utility
ln(wealth after the event) - ln(wealth before the event) will be weighted by the probability of that event occurring.
Let c be the cost charged to enter the game. The expected utility of the lottery now converges to a finite value:
This formula gives an implicit relationship between the gambler's wealth and how much he should be willing to pay
to play (specifically, any c that gives a positive expected utility). For example, with log utility a millionaire should
be willing to pay up to $10.94, a person with $1000 should pay up to $5.94, a person with $2 should pay up to $2,
and a person with $0.60 should borrow $0.87 and pay up to $1.47.
It is important for the following discussion that Daniel Bernoulli did not propose the expected net change in a utility
function as a criterion, although it is often said that he did, and he is considered the originator of expected utility
theory. Daniel Bernoulli suggested that the positivity (or negativity) of the following expression should serve as a
criterion for deciding whether to buy a ticket for a lottery. Here, the first term is the expected change in utility that
would result if the ticket were obtained free of charge, and the second term is the known decrease in utility that
would result if the ticket were bought at price c and no payout was received:
Most later researchers, starting with (Laplace 1814) and (Todhunter 1865), interpreted this as an error and
"corrected" Bernoulli implicitly, stating that he computed the expected net change in logarithmic utility.
Although unlikely, the possibility cannot be excluded that Bernoulli suggested this term on purpose. He explained its
use as follows: "in a fair game the disutility to be suffered by losing must be equal to the utility to be derived by
winning" (Bernoulli 1738), p.27.
Before Daniel Bernoulli published, in 1728, another Swiss mathematician, Gabriel Cramer, had already found parts
of this idea (also motivated by the St. Petersburg Paradox) in stating that
the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the
usage that they may make of it.
He demonstrated in a letter to Nicolas Bernoulli [2] that a square root function describing the diminishing marginal
benefit of gains can resolve the problem. However, unlike Daniel Bernoulli, he did not consider the total wealth of a
person, but only the gain by the lottery.
A common misconception, debunked in (Peters 2011b), is that the lottery can easily be changed in a way such that
the paradox reappears. The argument goes that to make the logarithmic-utility solution fail, we just need to change
84
St. Petersburg paradox
85
the game so that it gives an even larger payout, such as
or
. Again, the game should be worth an infinite a
is not correct because the expected net change in logarithmic utility becomes negatively infinite as one increases the
ticket price to a level that could lead to zero wealth (bankruptcy) after the game. Thus, there exists a finite price at
which even the modified lottery should be rejected. The flawed analysis that led to the misconception is due to Karl
Menger, who came to the invalid general conclusion that St. Petersburg-like paradoxes can be constructed for any
unbounded utility function. This, he argued, constituted a solid reason to reject unbounded utility functions (Menger
1934). Menger's error can be understood as a consequence of Daniel Bernoulli's initial error: Menger only considered
Bernoulli's first term and overlooked the negative divergence at a finite ticket price of Bernoulli's second term.
Without Bernoulli's initial error, there would have only been one term and Menger's error would have been unlikely.
Menger was an expert on Bernoulli's original Latin paper and helped with its translation (see note of thanks in
(Bernoulli 1738)). He referred to the original work and was not aware of the correction by Laplace. Menger, no
doubt himself convinced, presented his invalid argument in a convincing manner, which led it to stand unchallenged
for 77 years. Writing 43 years after Menger, Samuelson stated that "Menger 1934 is a modern classic that, despite
my quibbles, stands above all criticism" (Samuelson 1977), p.49.
Recently, expected utility theory has been extended to arrive at more behavioral decision models. In some of these
new theories, as in cumulative prospect theory, the St. Petersburg paradox again appears in certain cases, even when
the utility function is concave, but not if it is bounded (Rieger & Wang 2006).
Time resolution
A resolution of the St. Petersburg paradox is possible by considering the time-average performance of the lottery
(Peters 2011a). Although expressible in mathematical terms identical to the resolution from expected logarithmic
utility, the time resolution is obtained using a conceptually different approach. This avoids the need for utility
functions, the choice of which is to a large extent arbitrary, and the expectation values thereof, in which the interest
of an investor in the lottery has little a priori justification.
Peters pointed out that computing the naive expected payout is mathematically equivalent to considering multiple
outcomes of the same lottery in parallel universes. This is irrelevant to the individual considering whether to buy a
ticket since he exists in only one universe and is unable to exchange resources with the others. It is therefore unclear
why expected wealth should be a quantity whose maximization should lead to a sound decision theory. Indeed, the
St. Petersburg paradox is only a paradox if one accepts the premise that rational actors seek to maximize their
expected wealth. The classical resolution is to apply a utility function to the wealth, which reflects the notion that the
"usefulness" of an amount of money depends on how much of it one already has, and then to maximise the
expectation of this. The choice of utility function is often framed in terms of the individual's risk preferences and
may vary between individuals: it therefore provides a somewhat arbitrary framework for the treatment of the
problem.
An alternative premise, which is less arbitrary and makes fewer assumptions, it that the performance over time of an
investment better characterises an investor's prospects and, therefore, better informs his investment decision. In this
case, the passage of time is incorporated by identifying as the quantity of interest the average rate of exponential
growth of the player's wealth in a single round of the lottery,
per round, where
is the
th (positive finite) payout and
standard St. Petersburg lottery,
and
is the (non-zero) probability of receiving it. In the
.
Although this is an expectation value of a growth rate, and may therefore be thought of in one sense as an average
over parallel universes, it is in fact equivalent to the time average growth rate that would be obtained if repeated
lotteries were played over time (Peters 2011a). While is identical to the rate of change of the expected logarithmic
utility, it has been obtained without making any assumptions about the player's risk preferences or behaviour, other
St. Petersburg paradox
86
than that he is interested in the rate of growth of his wealth.
Under this paradigm, an individual with wealth
should buy a ticket at a price
provided
It should be noted that this strategy counsels against paying any amount of money for a ticket that admits the
possibility of bankruptcy, i.e.
for any
, since this generates a negatively divergent logarithm in the sum for
all other terms in the sum and guarantee that
which can be shown to dominate
. If we assume the smallest payout is
, then the individual
will always be advised to decline the ticket at any price greater than
regardless of the payout structure of the lottery. The ticket price for which the expected growth rate falls to zero will
be less than
but may be greater than , indicating that borrowing money to purchase a ticket for more than
one's wealth can be a sound decision. This would be the case, for example, where the smallest payout exceeds the
player's current wealth, as it does in Menger's game.
It should also be noted in the above treatment that, contrary to Menger's analysis, no higher-paying lottery can
generate a paradox which the time resolution - or, equivalently, Bernoulli's or Laplace's logarithmic resolutions - fail
to resolve, since there is always a price at which the lottery should not be entered, even though for especially
favourable lotteries this may be greater than one's worth.
Probability weighting
Nicolas Bernoulli himself proposed an alternative idea for solving the paradox. He conjectured that people will
neglect unlikely events (de Montmort 1713). Since in the St. Petersburg lottery only unlikely events yield the high
prizes that lead to an infinite expected value, this could resolve the paradox. The idea of probability weighting
resurfaced much later in the work on prospect theory by Daniel Kahneman and Amos Tversky. However, their
experiments indicated that, very much to the contrary, people tend to overweight small probability events. Therefore
the proposed solution by Nicolas Bernoulli is nowadays not considered to be satisfactory.
Cumulative prospect theory is one popular generalization of expected utility theory that can predict many behavioral
regularities (Tversky & Kahneman 1992). However, the overweighting of small probabilities introduced in
cumulative prospect theory may restore the St. Petersburg paradox. Cumulative prospect theory avoids the St.
Petersburg paradox only when the power coefficient of utility function is lower than the power coefficient of
probability weighting function (Blavatskyy 2005). Intuitively, utility function must not simply be concave, but it
must be concave relative to probability weighting function to avoid the St. Petersburg paradox.
Rejection of mathematical expectation
Various authors, including Jean le Rond d'Alembert and John Maynard Keynes, have rejected maximization of
expectation (even of utility) as a proper rule of conduct. Keynes, in particular, insisted that the relative risk of an
alternative could be sufficiently high to reject it even were its expectation enormous. In other academic fields the
limited significance of expectation values is well known. Nicolas Bernoulli, the inventor of the St Petersburg
paradox, was among the pioneers of extreme value theory. A classic situation where extremes are more important
than expectation values is the construction of a dike: in deciding how high to build the dike the expectation value of
the height of a flood is irrelevant, and only the extreme value (i.e. the highest flood) is important. In statistical
mechanics it is a central problem to understand whether time averages resulting from a long observation of a single
system are equivalent to expectation values. This is the case only for a very limited class of systems that are called
"ergodic" there. For non-ergodic systems there is no general reason why expectation values should have any
relevance.
St. Petersburg paradox
87
One cannot buy what is not sold
Some economists resolve the paradox by arguing that, even if an entity had infinite resources, the game would never
be offered. If the lottery represents an infinite expected gain to the player, then it also represents an infinite expected
loss to the host. No one could be observed paying to play the game because it would never be offered. As Paul
Samuelson describes the argument:
Paul will never be willing to give as much as Peter will demand for such a contract; and hence the indicated
activity will take place at the equilibrium level of zero intensity. (Samuelson 1960)
Finite St. Petersburg lotteries
The classical St. Petersburg lottery assumes that the casino has infinite resources. This assumption is often criticized
as unrealistic, particularly in connection with the paradox, which involves the reactions of ordinary people to the
lottery. Of course, the resources of an actual casino (or any other potential backer of the lottery) are finite. More
importantly, the expected value of the lottery only grows logarithmically with the resources of the casino. As a
result, the expected value of the lottery, even when played against a casino with the largest resources realistically
conceivable, is quite modest. If the total resources (or total maximum jackpot) of the casino are W dollars, then L = 1
+ floor(log2(W)) is the maximum number of times the casino can play before it no longer covers the next bet. The
expected value E of the lottery then becomes:
The following table shows the expected value E of the game with various potential bankers and their bankroll W
(with the assumption that if you win more than the bankroll you will be paid what the bank has):
Banker
Bankroll
Expected value of lottery
Friendly game
$100
$4.28
Millionaire
$1,000,000
$10.95
Billionaire
$1,000,000,000
$15.93
Bill Gates (2008)
$58,000,000,000 $18.84
U.S. GDP (2007)
$13.8 trillion
$22.78
World GDP (2007) $54.3 trillion
$23.77
Googolaire
$166.50
$10100
Notes: The estimated net worth of Bill Gates is from Forbes. The GDP data are as estimated for 2007 by the International Monetary Fund, where
one trillion dollars equals $1012 (one million times one million dollars). A “googolaire” is a hypothetical person worth a googol dollars ($10100).
A rational person might not find the lottery worth even the modest amounts in the above table, suggesting that the
naive decision model of the expected return causes essentially the same problems as for the infinite lottery. Even so,
the possible discrepancy between theory and reality is far less dramatic.
The assumption of infinite resources can produce other apparent paradoxes in economics. See martingale (roulette
system) and gambler's ruin.
St. Petersburg paradox
Further discussions
The St. Petersburg paradox and the theory of marginal utility have been highly disputed in the past. For a discussion
from the point of view of a philosopher, see (Martin 2004).
Works cited
• Arrow, Kenneth J. (February 1974). "The use of unbounded utility functions in expected-utility maximization:
Response" [3] (PDF). Quarterly Journal of Economics (The MIT Press) 88 (1): 136–138. doi:10.2307/1881800.
JSTOR 1881800. Handle: RePEc:tpr:qjecon:v:88:y:1974:i:1:p:136-38.
• Bernoulli, Daniel; Originally published in 1738; translated by Dr. Louise Sommer. (January 1954). "Exposition of
a New Theory on the Measurement of Risk" [4]. Econometrica (The Econometric Society) 22 (1): 22–36.
doi:10.2307/1909829. JSTOR 1909829. Retrieved 2006-05-30.
• Blavatskyy, Pavlo (April 2005). "Back to the St. Petersburg Paradox?". MANAGEMENT SCIENCE 51 (4):
677–678.
• de Montmort, Pierre Remond (1713) (in (French)). Essay d'analyse sur les jeux de hazard [Essays on the analysis
of games of chance] (Reprinted in 2006) (Second ed.). Providence, Rhode Island: American Mathematical
Society. ISBN 978-0-8218-3781-8. as translated and posted at Pulskamp, Richard J. "Correspondence of Nicolas
Bernoulli concerning the St. Petersburg Game" [5] ( PDF (88 KB)). Retrieved July 22, 2010.
• Laplace, Pierre Simon (1814) (in (French)). Théorie analytique des probabilités [Analytical theory of
probabilities] (Second ed.). Paris: Ve. Courcier.
• Martin, Robert (2004). "The St. Petersburg Paradox" [6]. In Edward N. Zalta. The Stanford Encyclopedia of
Philosophy (Fall 2004 ed.). Stanford, California: Stanford University. ISSN 1095-5054. Retrieved 2006-05-30.
• Menger, Karl (August 1934). "Das Unsicherheitsmoment in der Wertlehre Betrachtungen im Anschluß an das
sogenannte Petersburger Spiel". Zeitschrift für Nationalökonomie 5 (4): 459–485. doi:10.1007/BF01311578.
ISSN 0931-8658. (Paper) (Online).
• Peters, Ole (October 2011b). "Menger 1934 revisited" [7].
• Peters, Ole (2011a). "The time resolution of the St Petersburg paradox" [8]. Philosophical Transactions of the
Royal Society 369: 4913–4931. doi:10.1098/rsta.2011.0065.
• Rieger, Marc Oliver; Wang, Mei (August 2006). "Cumulative prospect theory and the St. Petersburg paradox".
Economic Theory 28 (3): 665–679. doi:10.1007/s00199-005-0641-6. ISSN 0938-2259. (Paper) (Online).
(Publicly accessible, older version. [9])
• Samuelson, Paul (January 1960). "The St. Petersburg Paradox as a Divergent Double Limit". International
Economic Review (Blackwell Publishing) 1 (1): 31–37. doi:10.2307/2525406. JSTOR 2525406.
• Samuelson, Paul (March 1977). "St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described".
Journal of Economic Literature (American Economic Association) 15 (1): 24–55. JSTOR 2722712.
• Todhunter, Isaac (1865). A history of the mathematical theory of probabilities. Macmillan & Co.
• Tversky, Amos; Kahneman (1992). "Advances in prospect theory: Cumulative representation of uncertainty".
Journal of Risk and Uncertainty 5: 297-323.
88
St. Petersburg paradox
Bibliography
• Aumann, Robert J. (April 1977). "The St. Petersburg paradox: A discussion of some recent comments". Journal
of Economic Theory 14 (2): 443–445. doi:10.1016/0022-0531(77)90143-0.
• Durand, David (September 1957). "Growth Stocks and the Petersburg Paradox". The Journal of Finance
(American Finance Association) 12 (3): 348–363. doi:10.2307/2976852. JSTOR 2976852.
• "Bernoulli and the St. Petersburg Paradox" [10]. The History of Economic Thought. The New School for Social
Research, New York. Retrieved 2006-05-30.
• Haigh, John (1999). Taking Chances. Oxford,UK: Oxford University Press. pp. 330. ISBN 0-19-850291-9
.(Chapter 4)
External links
• Online simulation of the St. Petersburg lottery [11]
References
[1] http:/ / www. econterms. com/ glossary. cgi?query=log+ utility
[2] http:/ / www. cs. xu. edu/ math/ Sources/ Montmort/ stpetersburg. pdf#search=%22Nicolas%20Bernoulli%22
[3] http:/ / ideas. repec. org/ a/ tpr/ qjecon/ v88y1974i1p136-38. html
[4] http:/ / www. math. fau. edu/ richman/ Ideas/ daniel. htm
[5] http:/ / www. cs. xu. edu/ math/ Sources/ Montmort/ stpetersburg. pdf
[6] http:/ / plato. stanford. edu/ archives/ fall2004/ entries/ paradox-stpetersburg/
[7] http:/ / arxiv. org/ pdf/ 1110. 1578v1. pdf
[8] http:/ / rsta. royalsocietypublishing. org/ content/ 369/ 1956/ 4913. full. pdf
[9] http:/ / www. sfb504. uni-mannheim. de/ publications/ dp04-28. pdf
[10] http:/ / cepa. newschool. edu/ het/ essays/ uncert/ bernoulhyp. htm
[11] http:/ / www. mathematik. com/ Petersburg/ Petersburg. html
89
90
Logic
All horses are the same color
The horse paradox is a falsidical paradox that arises from flawed demonstrations, which purport to use
mathematical induction, of the statement All horses are the same color. There is no actual contradiction, as these
arguments have a crucial flaw that makes them incorrect. This example was used by George Pólya as an example of
the subtle errors that can occur in attempts to prove statements by induction.
The argument
The flawed argument claims to be based on mathematical induction, and proceeds as follows:
Suppose that we have a set of five horses. We wish to prove that they are all the same color. Suppose that we had a
proof that all sets of four horses were the same color. If that were true, we could prove that all five horses are the
same color by removing a horse to leave a group of four horses. Do this in two ways, and we have two different
groups of four horses. By our supposed existing proof, since these are groups of four, all horses in them must be the
same color. For example, the first, second, third and fourth horses constitute a group of four, and thus must all be the
same color; and the second, third, fourth and fifth horses also constitute a group of four and thus must also all be the
same color. For this to occur, all five horses in the group of five must be the same color.
But how are we to get a proof that all sets of four horses are the same color? We apply the same logic again. By the
same process, a group of four horses could be broken down into groups of three, and then a group of three horses
could be broken down into groups of two, and so on. Eventually we will reach a group size of one, and it is obvious
that all horses in a group of one horse must be the same color.
By the same logic we can also increase the group size. A group of five horses can be increased to a group of six, and
so on upwards, so that all finite sized groups of horses must be the same color.
Explanation
The argument above makes the implicit assumption that the two subsets of horses to which the induction assumption
is applied have a common element. This is not true when n = 1, that is, when the original set only contains 2 horses.
Let the two horses be horse A and horse B. When horse A is removed, it is true that the remaining horses in the set
are the same color (only horse B remains). If horse B is removed instead, this leaves a different set containing only
horse A, which may or may not be the same color as horse B.
The problem in the argument is the assumption that because each of these two sets contains only one color of horses,
the original set also contained only one color of horses. Because there are no common elements (horses) in the two
sets, it is unknown whether the two horses share the same color. The proof forms a falsidical paradox; it seems to
show something manifestly false by valid reasoning, but in fact the reasoning is flawed. The horse paradox exposes
the pitfalls arising from failure to consider special cases for which a general statement may be false.
All horses are the same color
References
• Enumerative Combinatorics by George E. Martin, ISBN 0-387-95225-X
Barbershop paradox
The Barbershop paradox was proposed by Lewis Carroll in a three-page essay entitled "A Logical Paradox" which
appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses to
illustrate the paradox (although it had appeared several times in more abstract terms in his writing and
correspondence before the story was published). Carroll claimed that it illustrated "a very real difficulty in the
Theory of Hypotheticals" in use at the time.[1]
The paradox
Briefly, the story runs as follows: Uncle Joe and Uncle Jim are walking to the barber shop. There are three barbers
who live and work in the shop—Allen, Brown, and Carr—but not all of them are always in the shop. Carr is a good
barber, and Uncle Jim is keen to be shaved by him. He knows that the shop is open, so at least one of them must be
in. He also knows that Allen is a very nervous man, so that he never leaves the shop without Brown going with him.
Uncle Joe insists that Carr is certain to be in, and then claims that he can prove it logically. Uncle Jim demands the
proof. Uncle Joe reasons as follows.
Suppose that Carr is out. If Carr is out, then if Allen is also out Brown would have to be in—since someone must be
in the shop for it to be open. However, we know that whenever Allen goes out he takes Brown with him, and thus we
know as a general rule that if Allen is out, Brown is out. So if Carr is out then the statements "if Allen is out then
Brown is in" and "if Allen is out then Brown is out" would both be true at the same time.
Uncle Joe notes that this seems paradoxical; the hypotheticals seem "incompatible" with each other. So, by
contradiction, Carr must logically be in.
Simplification
Carroll wrote this story to illustrate a controversy in the field of logic that was raging at the time. His vocabulary and
writing style can easily add to the confusion of the core issue for modern readers.
Notation
When reading the original it may help to keep the following in mind:
• What Carroll called "hypotheticals" modern logicians call "logical conditionals."
• Whereas Uncle Joe concludes his proof reductio ad absurdum, modern mathematicians would more commonly
claim "proof by contradiction."
• What Carroll calls the protasis of a conditional is now known as the antecedent, and similarly the apodosis is now
called the consequent.
Symbols can be used to greatly simplify logical statements such as those inherent in this story:
91
Barbershop paradox
92
Operator (Name) Colloquial
Symbolic
Negation
NOT
not X
¬
¬X
Conjunction
AND
X and Y
∧
X∧Y
Disjunction
OR
X or Y
∨
X∨Y
Conditional
IF ... THEN if X then Y ⇒
X⇒Y
Note: X ⇒ Y (also known as "Implication") can be read many ways in English, from "X is sufficient for Y" to "Y
follows from X." See also Table of mathematical symbols.
Restatement
To aid in restating Carroll's story more simply, we will take the following atomic statements:
• A = Allen is in the shop
• B = Brown is in
• C = Carr is in
So, for instance (¬A ∧ B) represents "Allen is out and Brown is in"
Uncle Jim gives us our two axioms:
1. There is at least one barber in the shop now (A ∨ B ∨ C)
2. Allen never leaves the shop without Brown (¬A ⇒ ¬B)
Uncle Joe presents a proof:
Abbreviated English with logical markers
Mainly Symbolic
Suppose Carr is NOT in.
H0: ¬C
Given NOT C, IF Allen is NOT in THEN Brown must be in, to satisfy Axiom 1.
By H0 and A1, ¬A ⇒ B
But Axiom 2 gives that it is universally true that IF Allen
is Not in THEN Brown is Not in (it's always true that if ¬A then ¬B)
By A2, ¬A ⇒ ¬B
So far we have that NOT C yields both (Not A THEN B) AND (Not A THEN Not B). Thus ¬C ⇒ ( (¬A ⇒ B) ∧ (¬A ⇒ ¬B) )
Uncle Joe claims that these are contradictory.
⊥
Therefore Carr must be in.
∴C
Uncle Joe basically makes the argument that (¬A ⇒ B) and (¬A ⇒ ¬B) are contradictory, saying that the same
antecedent cannot result in two different consequents.
This purported contradiction is the crux of Joe's "proof." Carroll presents this intuition-defying result as a paradox,
hoping that the contemporary ambiguity would be resolved.
Discussion
In modern logic theory this scenario is not a paradox. The law of implication reconciles what Uncle Joe claims are
incompatible hypotheticals. This law states that "if X then Y" is logically identical to "X is false or Y is true" (¬X ∨
Y). For example, given the statement "if you press the button then the light comes on," it must be true at any given
moment that either you have not pressed the button, or the light is on.
In short, what obtains is not that ¬C yields a contradiction, only that it necessitates A, because ¬A is what actually
yields the contradiction.
In this scenario, that means Carr doesn't have to be in, but that if he isn't in, Allen has to be in.
Barbershop paradox
93
Simplifying to Axiom 1
Applying the law of implication to the offending conditionals shows that rather than contradicting each other one
simply reiterates the fact that since the shop is open one or more of Allen, Brown or Carr is in and the other puts very
little restriction on who can or cannot be in shop.
To see this let's attack Jim's large "contradictory" result, mainly by applying the law of implication repeatedly. First
let's break down one of the two offending conditionals:
"If Allen is out, then Brown is out"
(¬A ⇒ ¬B)
"Allen is in or Brown is out"
(A ∨ ¬B)
Substituting this into
"IF Carr is out, THEN If Allen is also out Then Brown is in AND If Allen is out Then Brown is
out."
¬C ⇒ ( (¬A ⇒ B) ∧ (¬A ⇒ ¬B)
)
Which yields, with continued application of the law of implication,
"IF Carr is out, THEN if Allen is also out, Brown is in AND either Allen is in OR Brown is out."
"IF Carr is out, THEN both of these are true: Allen is in OR Brown is in AND Allen is in OR Brown
is out."
"Carr is in OR both of these are true: Allen is in OR Brown is in AND Allen is in OR Brown is out."
¬C ⇒ ( (¬A ⇒ B) ∧ (A ∨
¬B) )
¬C ⇒ ( (A ∨ B) ∧ (A ∨ ¬B) )
C ∨ ( (A ∨ B) ∧ (A ∨ ¬B) )
And finally, (on the right we are distributing over the parentheses)
"Carr is in OR Either Allen is in OR Brown is in, AND Carr is in OR Either Allen is in OR Brown is
out."
C ∨ (A ∨ B) ∧ C ∨ (A ∨
¬B)
"Inclusively, Carr is in OR Allen is in OR Brown is in, AND Inclusively, Carr is in OR Allen is in OR
Brown is out."
(C ∨ A ∨ B) ∧ (C ∨ A ∨
¬B)
So the two statements which become true at once are: "One or more of Allen, Brown or Carr is in," which is simply
Axiom 1, and "Carr is in or Allen is in or Brown is out." Clearly one way that both of these statements can become
true at once is in the case where Allen is in (because Allen's house is the barber shop, and at some point Brown left
the shop).
Another way to describe how (X ⇒ Y) ⇔ (¬X ∨ Y) resolves this into a valid set of statements is to rephrase Jim's
statement that "If Allen is also out ..." into "If Carr is out and Allen is out then Brown is in" ( (¬C ∧ ¬A) ⇒ B).
Showing conditionals compatible
The two conditionals are not logical opposites: to prove by contradiction Jim needed to show ¬C ⇒ (Z ∧ ¬Z), where
Z happens to be a conditional.
The opposite of (A ⇒ B) is ¬(A ⇒ B), which, using De Morgan's Law, resolves to (A ∧ ¬B), which is not at all the
same thing as (¬A ∨ ¬B), which is what A ⇒ ¬B reduces to.
This confusion about the "compatibility" of these two conditionals was foreseen by Carroll, who includes a mention
of it at the end of the story. He attempts to clarify the issue by arguing that the protasis and apodosis of the
implication "If Carr is in ..." are "incorrectly divided." However, application of the Law of Implication removes the
"If ..." entirely (reducing to disjunctions), so no protasis and apodosis exist and no counter-argument is needed.
Barbershop paradox
94
Notes
[1] Carroll, Lewis (July 1894). "A Logical Paradox" (http:/ / fair-use. org/ mind/ 1894/ 07/ notes/ a-logical-paradox). Mind 3 (11): 436–438. .
Further reading
• Russell, Bertrand (1903). "Chapter II. Symbolic Logic" (http://fair-use.org/bertrand-russell/
the-principles-of-mathematics/s.19#s19n1). The Principles of Mathematics. p. § 19 n. 1. ISBN 0-415-48741-2.
Russell suggests a truth-functional notion of logical conditionals, which (among other things) entails that a false
proposition will imply all propositions. In a note he mentions that his theory of implication would dissolve
Carroll's paradox, since it not only allows, but in fact requires that both "p implies q" and "p implies not-q" be
true, so long as p is not true.
Carroll's paradox
In physics, Carroll's paradox arises when considering the motion of a falling rigid rod that is specially constrained.
Considered one way, the angular momentum stays constant; considered in a different way, it changes. It is named
after Michael M. Carroll who first published it in 1984.
Explanation
Consider two concentric circles of radius
uniform rigid heavy rod of length
and
as might be drawn on the face of a wall clock. Suppose a
is somehow constrained between these two circles so that one end
of the rod remains on the inner circle and the other remains on the outer circle. Motion of the rod along these circles,
acting as guides, is frictionless. The rod is held in the three o'clock position so that it is horizontal, then released.
Now consider the angular momentum about the centre of the rod:
1. After release, the rod falls. Being constrained, it must rotate as it moves. When it gets to a vertical six o'clock
position, it has lost potential energy and, because the motion is frictionless, will have gained kinetic energy. It
therefore possesses angular momentum.
2. The reaction force on the rod from either circular guide is frictionless, so it must be directed along the rod; there
can be no component of the reaction force perpendicular to the rod. Taking moments about the center of the rod,
there can be no moment acting on the rod, so its angular momentum remains constant. Because the rod starts with
zero angular momentum, it must continue to have zero angular momentum for all time.
An apparent resolution of this paradox is that the physical situation cannot occur. To maintain the rod in a radial
position the circles have to exert an infinite force. In real life it would not be possible to construct guides that do not
exert a significant reaction force perpendicular to the rod. Victor Namias, however, disputed that infinite forces
occur, and argued that a finitely thick rod experiences torque about its center of mass even in the limit as it
approaches zero width.
References
• Victor Namias. On an apparent paradox in the motion of a smoothly constrained rod, American Journal of
Physics, 54(5), May 1986.
• M. M. Carroll. Singular constraints in rigid-body dynamics, American Journal of Physics, 52(11), Nov 1984, pp
1010–1012.
Crocodile Dilemma
Crocodile Dilemma
The Crocodile Dilemma is a paradox in logic in the same family of paradoxes as the liar paradox. [1] The premise
states that a crocodile who has stolen a child promises the father that his son will be returned if and only if he can
correctly predict whether or not the crocodile will return the child.
The transaction is logically smooth (but unpredictable) if the father guesses that the child will be returned, but a
dilemma arises for the crocodile if he guesses that the child will not be returned. In the case that the crocodile
decides to keep the child, he violates his terms: the father's prediction has been validated, and the child should be
returned. However, in the case that the crocodile decides to give back the child, he still violates his terms, even if this
decision is based on the previous result: the father's prediction has been falsified, and the child should not be
returned. The question of what the crocodile should do is therefore paradoxical, and there is no justifiable
solution.[2][3][4]
The Crocodile Dilemma serves to expose some of the logical problems presented by metaknowledge. In this regard,
it is similar in construction to the unexpected hanging paradox, which Richard Montague (1960) used to demonstrate
that the following assumptions about knowledge are inconsistent when tested in combination:[2]
(i) If ρ is known to be true, then ρ.
(ii) It is known that (i).
(iii) If ρ implies σ, and ρ is known to be true, then σ is also known to be true.
It also bears similarities to the Liar paradox. Ancient Greek sources were the first to discuss the Crocodile
Dilemma.[1]
Notes
[1] Barile, Margherita. "Crococile Dilemma – MathWorld" (http:/ / mathworld. wolfram. com/ CrocodilesDilemma. html). . Retrieved
2009-09-05.
[2] J. Siekmann, ed. (1989). Lecture Notes in Artificial Intelligence. Springer-Verlag. p. 14. ISBN 3540530827.
[3] Young, Ronald E (2005). Traveling East. iUniverse. pp. 8–9. ISBN 0595795846.
[4] Murray, Richard (1847). Murray's Compendium of logic. p. 159.
95
Drinker paradox
96
Drinker paradox
The drinker paradox (also known as drinker's principle, drinkers' principle or (the) drinking principle) is a
theorem of classical predicate logic, usually stated in natural language as: There is someone in the pub such that, if
he is drinking, everyone in the pub is drinking. The actual theorem is
where D is an arbitrary predicate. The paradox was popularised by the mathematical logician Raymond Smullyan,
who called it the "drinking principle" in his 1978 book What Is the Name of this Book?[1]
Proofs of the paradox
The proof begins by recognizing it is true that either everyone in the pub is drinking, or at least one person in the pub
isn't drinking. Consequently, there are two cases to consider:[1][2]
1. Suppose everyone is drinking. For any particular person, it can't be wrong to say that if that particular person is
drinking, then everyone in the pub is drinking — because everyone is drinking. Because everyone is drinking,
then that one person must drink because when ' that person ' drinks ' everybody ' drinks, everybody includes that
person.[1][2]
2. Suppose that at least one person is not drinking. For any particular nondrinking person, it still cannot be wrong to
say that if that particular person is drinking, then everyone in the pub is drinking — because that person is, in
fact, not drinking. In this case the condition is false, so the statement is vacuously true due to the nature of
material implication in formal logic, which states that "If P, then Q" is always true if P (the condition or
antecedent) is false.[1][2]
Either way, there is someone in the pub such that, if he is drinking, everyone in the pub is drinking. A slightly more
formal way of expressing the above is to say that if everybody drinks then anyone can be the witness for the validity
of the theorem. And if someone doesn't drink, then that particular non-drinking individual can be the witness to the
theorem's validity.[3]
The proof above is essentially model-theoretic (can be formalized as such). A purely syntactic proof is possible and
can even be mechanized (in Otter for example), but only for an equisatisfiable rather than a equivalent negation of
the theorem.[4] Namely, the negation of the theorem is
which is equivalent with the prenex normal form
By Skolemization the above is equisatisfiable with
The resolution of the two clauses
and
results in an empty set of clauses (i.e. a contradiction),
thus proving the negation of the theorem is unsatisfiable. The resolution is slightly non-straightforward because it
involves a search based on Herbrand's theorem for ground instances that are propositionally unsatisfiable. The bound
variable x is first instantiated with a constant d (making use of the assumption that the domain is non-empty),
resulting in the Herbrand universe:[5]
One can sketch the following natural deduction:[4]
Drinker paradox
97
Or spelled out:
1. Instantiating x with d yields
2. x is then instantiated with f(d) yielding
Observe that
and
which implies
which implies
.
unify syntactically in their predicate arguments. An (automated) search
[5]
thus finishes in two steps:
1.
2.
The proof by resolution given here uses the law of excluded middle, the axiom of choice, and non-emptiness of the
domain as premises.[4]
Discussion
This proof illustrates several properties of classical predicate logic that do not always agree with ordinary language.
Excluded middle
The above proof begins by saying that either everyone is drinking, or someone is not drinking. This uses the validity
of excluded middle for the statement
"everyone is drinking", which is always available in classical logic. If the
logic does not admit arbitrary excluded middle—for example if the logic is intuitionistic—then the truth of
must first be established, i.e.,
must be shown to be decidable.[6]
Material versus indicative conditional
Most important to the paradox is that the conditional in classical (and intuitionistic) logic is the material conditional.
It has the property that
is true if B is true or if A is false (in classical logic, but not intuitionistic logic, this
is also a necessary condition).
So as it was applied here, the statement "if he is drinking, everyone is drinking" was taken to be correct in one case,
if everyone was drinking, and in the other case, if he was not drinking — even though his drinking may not have had
anything to do with anyone else's drinking.
In natural language, on the other hand, typically "if...then..." is used as an indicative conditional.
Non-empty domain
It is not necessary to assume there was anyone in the pub. The assumption that the domain is non-empty is built into
the inference rules of classical predicate logic.[7] We can deduce
from
, but of course if the domain
were empty (in this case, if there were nobody in the pub), the proposition
is not well-formed for any closed
expression .
Nevertheless, if we allow empty domains we still have something like the drinker paradox in the form of the
theorem:
Or in words:
If there is anyone in the pub at all, then there is someone such that, if they are drinking, then everyone in the
pub is drinking.
Drinker paradox
Temporal aspects
Although not discussed in formal terms by Smullyan, he hints that the verb "drinks" is also ambiguous by citing a
postcard written to him by two of his students, which contains the following dialogue (emphasis in original):[1]
Logician / I know a fellow who is such that whenever he drinks, everyone does.
Student / I just don't understand. Do you mean, everyone on earth?
Logician / Yes, naturally.
Student / That sounds crazy! You mean as soon as he drinks, at just that moment, everyone does?
Logician / Of course.
Student / But that implies that at some time, everyone was drinking at once. Surely that never happened!
History and variations
Smullyan in his 1978 book attributes the naming of "The Drinking Principle" to his graduate students.[1] He also
discusses variants (obtained by substituting D with other, more dramatic predicates):
• "there is a woman on earth such that if she becomes sterile, the whole human race will die out." Smullyan writes
that this formulation emerged from a conversation he had with philosopher John Bacon.[1]
• A "dual" version of the Principle: "there is at least one person such that if anybody drinks, then he does."[1]
As "Smullyan's ‘Drinkers’ principle" or just "Drinkers' principle" it appears in H.P. Barendregt's "The quest for
correctness" (1996), accompanied by some machine proofs.[2] Since then it has made regular appearance as an
example in publications about automated reasoning; it is sometimes used to contrast the expressiveness of proof
assistants.[8][4][5]
References
[1] Raymond Smullyan (1978). What is the Name of this Book? The Riddle of Dracula and Other Logical Puzzles. Prentice Hall. chapter 14.
How to Prove Anything. (topic) 250. The Drinking Principle. pp. 209-211. ISBN 0-13-955088-7.
[2] H.P. Barendregt (1996). "The quest for correctness" (http:/ / oai. cwi. nl/ oai/ asset/ 13544/ 13544A. pdf). Images of SMC Research 1996.
Stichting Mathematisch Centrum. pp. 54-55. ISBN 978-90-6196-462-9. .
[3] Peter J. Cameron (1999). Sets, Logic and Categories (http:/ / books. google. com/ books?id=sDfdbBQ75MQC& pg=PA91). Springer. p. 91.
ISBN 978-1-85233-056-9. .
[4] Marc Bezem , Dimitri Hendriks (2008) Clausification in Coq (http:/ / igitur-archive. library. uu. nl/ lg/ 2008-0402-200713/ preprint187. pdf)
[5] J. Harrison (2008). "Automated and Interactive Theorem Proving" (http:/ / books. google. com/ books?id=QTc3WtqXXwQC& pg=PA123).
In Orna Grumberg, Tobias Nipkow, Christian Pfaller. Formal Logical Methods for System Security and Correctness. IOS Press. pp. 123–124.
ISBN 978-1-58603-843-4. .
[6] Martin Abadi; Georges Gonthier; Benjamin Werner (1998). "Choice in Dynamic Linking". In Igor Walukiewicz. Foundations of Software
Science and Computation Structures. Springer. p. 24. ISBN 3-540-21298-1.
[7] Martín Escardó; Paulo Oliva. Searchable Sets, Dubuc-Penon Compactness, Omniscience Principles, and the Drinker Paradox (http:/ / www.
cs. bham. ac. uk/ ~mhe/ papers/ dp. pdf). Computability in Europe 2010. p. 2. .
[8] Freek Wiedijk. 2001. Mizar Light for HOL Light (http:/ / www. cs. ru. nl/ ~freek/ mizar/ miz. pdf). In Proceedings of the 14th International
Conference on Theorem Proving in Higher Order Logics (TPHOLs '01), Richard J. Boulton and Paul B. Jackson (Eds.). Springer-Verlag,
London, UK, 378-394.
98
Infinite regress
99
Infinite regress
An infinite regress in a series of propositions arises if the truth of proposition P1 requires the support of proposition
P2, the truth of proposition P2 requires the support of proposition P3, ... , and the truth of proposition Pn-1 requires
the support of proposition Pn and n approaches infinity.
Distinction is made between infinite regresses that are "vicious" and those that are not. One definition given is that a
vicious regress is "an attempt to solve a problem which re-introduced the same problem in the proposed solution. If
one continues along the same lines, the initial problem will recur infinitely and will never be solved. Not all
regresses, however, are vicious."
Aristotle's answer
Aristotle argued that knowing does not necessitate an infinite regress because some knowledge does not depend on
demonstration:
“
Some hold that, owing to the necessity of knowing the primary premises, there is no scientific knowledge. Others think there is, but that all
truths are demonstrable. Neither doctrine is either true or a necessary deduction from the premises. The first school, assuming that there is no
way of knowing other than by demonstration, maintain that an infinite regress is involved, on the ground that if behind the prior stands no
primary, we could not know the posterior through the prior (wherein they are right, for one cannot traverse an infinite series): if on the other
hand – they say – the series terminates and there are primary premises, yet these are unknowable because incapable of demonstration, which
according to them is the only form of knowledge. And since thus one cannot know the primary premises, knowledge of the conclusions which
follow from them is not pure scientific knowledge nor properly knowing at all, but rests on the mere supposition that the premises are true.
The other party agree with them as regards knowing, holding that it is only possible by demonstration, but they see no difficulty in holding that
all truths are demonstrated, on the ground that demonstration may be circular and reciprocal.
”
Our own doctrine is that not all knowledge is demonstrative: on the contrary, knowledge of the immediate premises is independent of
demonstration. (The necessity of this is obvious; for since we must know the prior premises from which the demonstration is drawn, and since
the regress must end in immediate truths, those truths must be indemonstrable.) Such, then, is our doctrine, and in addition we maintain that
besides scientific knowledge there is its original source which enables us to recognize the definitions.
— Aristotle, Posterior Analytics (Book 1, Part 3, verses 1 and 2)
Consciousness
Infinite regress in consciousness is the formation of an infinite series of "inner observers" as we ask the question of
who is observing the output of the neural correlates of consciousness in the study of subjective consciousness.
Optics
Infinite regress in optics is the formation of an infinite series of receding images created in two parallel facing
mirrors.
Lottery paradox
Lottery paradox
Henry E. Kyburg, Jr.'s lottery paradox (1961, p. 197) arises from considering a fair 1000 ticket lottery that has
exactly one winning ticket. If this much is known about the execution of the lottery it is therefore rational to accept
that some ticket will win. Suppose that an event is very likely only if the probability of it occurring is greater than
0.99. On these grounds it is presumed rational to accept the proposition that ticket 1 of the lottery will not win. Since
the lottery is fair, it is rational to accept that ticket 2 won't win either--indeed, it is rational to accept for any
individual ticket i of the lottery that ticket i will not win. However, accepting that ticket 1 won't win, accepting that
ticket 2 won't win, and so on until accepting that ticket 1000 won't win: that entails that it is rational to accept that no
ticket will win, which entails that it is rational to accept the contradictory proposition that one ticket wins and no
ticket wins.
The lottery paradox was designed to demonstrate that three attractive principles governing rational acceptance lead
to contradiction, namely that
• It is rational to accept a proposition that is very likely true,
• It is irrational to accept a proposition that is known to be inconsistent, and
• If it is rational to accept a proposition A and it is rational to accept another proposition A', then it is rational to
accept A & A',
are jointly inconsistent.
The paradox remains of continuing interest because it raises several issues at the foundations of knowledge
representation and uncertain reasoning: the relationships between fallibility, corrigible belief and logical
consequence; the roles that consistency, statistical evidence and probability play in belief fixation; the precise
normative force that logical and probabilistic consistency have on rational belief.
History
Although the first published statement of the lottery paradox appears in Kyburg's 1961 Probability and the Logic of
Rational Belief, the first formulation of the paradox appears in his "Probability and Randomness," a paper delivered
at the 1959 meeting of the Association for Symbolic Logic, and the 1960 International Congress for the History and
Philosophy of Science, but published in the journal Theoria in 1963. This paper is reprinted in Kyburg (1987).
Smullyan's variation
Raymond Smullyan presents the following variation on the lottery paradox: You are either inconsistent or conceited.
Since the human brain is finite, there are a finite number of propositions p1…pn that you believe. But unless you are
conceited, you know that you sometimes make mistakes, and that not everything you believe is true. Therefore, if
you are not conceited, you know that at least some of the pi are false. Yet you believe each of the pi individually.
This is an inconsistency.(Smullyan 1978, p. 206)
A Short Guide to the Literature
The lottery paradox has become a central topic within epistemology, and the enormous literature surrounding this
puzzle threatens to obscure its original purpose. Kyburg proposed the thought experiment to get across a feature of
his innovative ideas on probability (Kyburg 1961, Kyburg and Teng 2001), which are built around taking the first
two principles above seriously and rejecting the last. For Kyburg, the lottery paradox isn't really a paradox: his
solution is to restrict aggregation.
Even so, for orthodox probabilists the second and third principles are primary, so the first principle is rejected. Here
too you'll see claims that there is really no paradox but an error: the solution is to reject the first principle, and with it
100
Lottery paradox
the idea of rational acceptance. For anyone with basic knowledge of probability, the first principle should be
rejected: for a very likely event, the rational belief about that event is just that it is very likely, not that it is true.
Most of the literature in epistemology approaches the puzzle from the orthodox point of view and grapples with the
particular consequences faced by doing so, which is why the lottery is associated with discussions of skepticism
(e.g., Klein 1981), and conditions for asserting knowledge claims (e.g., J. P. Hawthorne 2004). It is common to also
find proposed resolutions to the puzzle that turn on particular features of the lottery thought experiment (e.g., Pollock
1986), which then invites comparisons of the lottery to other epistemic paradoxes, such as David Makinson's preface
paradox, and to "lotteries" having a different structure. This strategy is addressed in (Kyburg 1997) and also in
(Wheeler 2007). An extensive bibliography is included in (Wheeler 2007).
Philosophical logicians and AI researchers have tended to be interested in reconciling weakened versions of the three
principles, and there are many ways to do this, including Jim Hawthorne and Luc Bovens's (1999) logic of belief,
Gregory Wheeler's (2006) use of 1-monotone capacities, Bryson Brown's (1999) application of preservationist
paraconsistent logics, Igor Douven and Timothy Williamson's (2006) appeal to cumulative non-monotonic logics,
Horacio Arlo-Costa's (2007) use of minimal model (classical) modal logics, and Joe Halpern's (2003) use of
first-order probability.
Finally, philosophers of science, decision scientists, and statisticians are inclined to see the lottery paradox as an
early example of the complications one faces in constructing principled methods for aggregating uncertain
information, which is now a thriving discipline of its own, with a dedicated journal, Information Fusion, in addition
to continuous contributions to general area journals.
Selected References
• Arlo-Costa, H (2005). "Non-Adjunctive Inference and Classical Modalities", The Journal of Philosophical Logic,
34, 581-605.
• Brown, B. (1999). "Adjunction and Aggregation", Nous, 33(2), 273-283.
• Douven and Williamson (2006). "Generalizing the Lottery Paradox", The British Journal for the Philosophy of
Science, 57(4), pp. 755-779.
• Halpern, J. (2003). Reasoning about Uncertainty, Cambridge, MA: MIT Press.
• Hawthorne, J. and Bovens, L. (1999). "The Preface, the Lottery, and the Logic of Belief", Mind, 108: 241-264.
• Hawthorne, J.P. (2004). Knowledge and Lotteries, New York: Oxford University Press.
• Klein, P. (1981). Certainty: a Refutation of Scepticism, Minneapolis, MN: University of Minnesota Press.
• Kyburg, H.E. (1961). Probability and the Logic of Rational Belief, Middletown, CT: Wesleyan University Press.
• Kyburg, H. E. (1983). Epistemology and Inference, Minneapolis, MN: University of Minnesota Press.
• Kyburg, H. E. (1997). "The Rule of Adjunction and Reasonable Inference", Journal of Philosophy, 94(3),
109-125.
• Kyburg, H. E., and Teng, C-M. (2001). Uncertain Inference, Cambridge: Cambridge University Press.
• Lewis, D. (1996). "Elusive Knowledge", Australasian Journal of Philosophy, 74, pp. 549-67.
• Makinson, D. (1965). "The Paradox of the Preface", Analysis, 25: 205-207.
• Pollock, J. (1986). "The Paradox of the Preface", Philosophy of Science, 53, pp. 346-258.
• Smullyan, Raymond (1978). What is the name of this book?. Prentice-Hall. p. 206. ISBN 0-13-955088-7.
• Wheeler, G. (2006). "Rational Acceptance and Conjunctive/Disjunctive Absorption", Journal of Logic, Language,
and Information, 15(1-2): 49-53.
• Wheeler, G. (2007). "A Review of the Lottery Paradox", in William Harper and Gregory Wheeler (eds.)
Probability and Inference: Essays in Honour of Henry E. Kyburg, Jr., King's College Publications, pp. 1-31.
101
Lottery paradox
102
External links
• Links to James Hawthorne's papers on the logic of nonmonotonic conditionals (and Lottery Logic) [1]
References
[1] http:/ / faculty-staff. ou. edu/ H/ James. A. Hawthorne-1/
Paradoxes of material implication
The paradoxes of material implication are a group of formulas which are truths of classical logic, but which are
intuitively problematic. One of these paradoxes is the paradox of entailment.
The root of the paradoxes lies in a mismatch between the interpretation of the validity of logical implication in
natural language, and its formal interpretation in classical logic, dating back to George Boole's algebraic logic. In
classical logic, implication describes conditional if-then statements using a truth-functional interpretation, i.e. "p
implies q" is defined to be "it is not the case that p is true and q false". Also, "p implies q" is equivalent to "p is false
or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will
bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or
material conditional.
The paradoxes are logical statements which are true but whose truth is intuitively surprising to people who are not
familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary propositions then the main paradoxes are given
formally as follows:
1.
2.
3.
4.
5.
, p and its negation imply q. This is the paradox of entailment.
, if p is true then it is implied by every q.
, if p is false then it implies every q. This is referred to as 'explosion'.
, either q or its negation is true, so their disjunction is implied by every p.
, if p, q and r are three arbitrary propositions, then either p implies q or q implies r. This is
because if q is true then p implies it, and if it is false then q implies any other statement. Since r can be p, it
follows that given two arbitrary propositions, one must imply the other, even if they are mutually contradictory.
For instance, "Nadia is in Barcelona implies Nadia is in Madrid or Nadia is in Madrid implies Nadia is in
Barcelona." This truism sounds like nonsense in ordinary discourse.
6.
, if p does not imply q then p is true and q is false. NB if p were false then it would
imply q, so p is true. If q were also true then p would imply q, hence q is false. This paradox is particularly
surprising because it tells us that if one proposition does not imply another then the first is true and the second
false.
The paradoxes of material implication arise because of the truth-functional definition of material implication, which
is said to be true merely because the antecedent is false or the consequent is true. By this criterion, "If the moon is
made of green cheese, then the world is coming to an end," is true merely because the moon isn't made of green
cheese. By extension, any contradiction implies anything whatsoever, since a contradiction is never true. (All
paraconsistent logics must, by definition, reject (1) as false.) Also, any tautology is implied by anything whatsoever,
since a tautology is always true.
To sum up, although it is deceptively similar to what we mean by "logically follows" in ordinary usage, material
implication does not capture the meaning of "if... then".
Paradoxes of material implication
Paradox of entailment
As the most well known of the paradoxes, and most formally simple, the paradox of entailment makes the best
introduction.
In natural language, an instance of the paradox of entailment arises:
It is raining
And
It is not raining
Therefore
George Washington is made of rakes.
This arises from the principle of explosion, a law of classical logic stating that inconsistent premises always make an
argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical, as it suggests that
the above is a valid argument.
Understanding the paradox of entailment
Validity is defined in classical logic as follows:
An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in
which all the premises are true and the conclusion is false.
For example a valid argument might run:
If it is raining, water exists (1st premise)
It is raining (2nd premise)
Water exists (Conclusion)
In this example there is no possible situation in which the premises are true while the conclusion is false. Since there
is no counterexample, the argument is valid.
But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid
argument since there would be no possible situation in which all the premises are true and therefore no possible
situation in which all the premises are true and the conclusion is false.
For example an argument with inconsistent premises might run:
Matter has mass (1st premise; true)
Matter does not have mass (2nd premise; false)
All numbers are equal to 12 (Conclusion)
As there is no possible situation where both premises could be true, then there is certainly no possible situation in
which the premises could be true while the conclusion was false. So the argument is valid whatever the conclusion
is; inconsistent premises imply all conclusions.
Explaining the paradox
The strangeness of the paradox of entailment comes from the fact that the definition of validity in classical logic does
not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises
are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid
argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. A
suggested improvement to the notion of logical validity to eliminate this paradox is relevant logic.
103
Paradoxes of material implication
Simplification
The classical paradox formulas are closely tied to the formula,
•
the principle of Simplification, which can be derived from the paradox formulas rather easily (e.g. from (1) by
Importation). In addition, there are serious problems with trying to use material implication as representing the
English "if ... then ...". For example, the following are valid inferences:
1.
2.
but mapping these back to English sentences using "if" gives paradoxes. The first might be read "If John is in
London then he is in England, and if he is in Paris then he is in France. Therefore, it is either true that if John is in
London then he is in France, or that if he is in Paris then he is in England." Either John is in London or John is not in
London. If John is in London, then John is in England. Thus the proposition "if John is in Paris, then John is in
England" holds because we have prior knowledge that the conclusion is true. If John is not in London, then the
proposition "if John is in London, then John is in France" is true because we have prior knowledge that the premise
is false.
The second can be read "If both switch A and switch B are closed, then the light is on. Therefore, it is either true that
if switch A is closed, the light is on, or if switch B is closed, the light is on." If the two switches are in series, then
the premise is true but the conclusion is false. Thus, using classical logic and taking material implication to mean
if-then is an unsafe method of reasoning which can give erroneous results.
References
•
•
•
•
•
Bennett, J. A Philosophical Guide to Conditionals. Oxford: Clarendon Press. 2003.
Conditionals, ed. Frank Jackson. Oxford: Oxford University Press. 1991.
Etchemendy, J. The Concept of Logical Consequence. Cambridge: Harvard University Press. 1990.
Sanford, D. If P, Then Q: Conditionals and the Foundations of Reasoning. New York: Routledge. 1989.
Priest, G. An Introduction to Non-Classical Logic, Cambridge University Press. 2001.
104
Raven paradox
105
Raven paradox
A black raven
Non-black non-ravens
The Raven paradox, also known as Hempel's paradox or Hempel's ravens is a paradox arising from the question
of what constitutes evidence for a statement. Observing objects that are neither black nor ravens may formally
increase the likelihood that all ravens are black—even though intuitively these observations are unrelated.
This problem was proposed by the logician Carl Gustav Hempel in the 1940s to illustrate a contradiction between
inductive logic and intuition. A related issue is the problem of induction and the gap between inductive and
deductive reasoning.[1]
The paradox
Hempel describes the paradox in terms of the hypothesis:[2][3]
(1) All ravens are black.
In strict logical terms, via contraposition, this statement is equivalent to:
(2) Everything that is not black is not a raven.
It should be clear that in all circumstances where (2) is true, (1) is also true; and likewise, in all circumstances where
(2) is false (i.e. if a world is imagined in which something that was not black, yet was a raven, existed), (1) is also
false. This establishes logical equivalence.
Given a general statement such as all ravens are black, a form of the same statement that refers to a specific
observable instance of the general class would typically be considered to constitute evidence for that general
statement. For example,
(3) Nevermore, my pet raven, is black.
is evidence supporting the hypothesis that all ravens are black.
The paradox arises when this same process is applied to statement (2). On sighting a green apple, one can observe:
(4) This green (and thus not black) thing is an apple (and thus not a raven).
By the same reasoning, this statement is evidence that (2) everything that is not black is not a raven. But since (as
above) this statement is logically equivalent to (1) all ravens are black, it follows that the sight of a green apple is
evidence supporting the notion that all ravens are black. This conclusion seems paradoxical, because it implies that
information has been gained about ravens by looking at an apple.
Raven paradox
Proposed resolutions
Nicod's criterion says that only observations of ravens should affect one's view as to whether all ravens are black.
Observing more instances of black ravens should support the view, observing white or coloured ravens should
contradict it, and observations of non-ravens should not have any influence.[4]
Hempel's equivalence condition states that when a proposition, X, provides evidence in favor of another proposition
Y, then X also provides evidence in favor of any proposition which is logically equivalent to Y.
The paradox shows that Nicod's criterion and Hempel's equivalence condition are not mutually consistent. A
resolution to the paradox must reject at least one out of:[5]
1. negative instances having no influence (!PC),
2. equivalence condition (EC), or,
3. validation by positive instances (NC).
A satisfactory resolution should also explain why there naively appears to be a paradox. Solutions which accept the
paradoxical conclusion can do this by presenting a proposition which we intuitively know to be false but which is
easily confused with (PC), while solutions which reject (EC) or (NC) should present a proposition which we
intuitively know to be true but which is easily confused with (EC) or (NC).
Accepting non-ravens as relevant
Although this conclusion of the paradox seems counter-intuitive, some approaches accept that observations of
(coloured) non-ravens can in fact constitute valid evidence in support for hypotheses about (the universal blackness
of) ravens.
Hempel's resolution
Hempel himself accepted the paradoxical conclusion, arguing that the reason the result appears paradoxical is
because we possess prior information without which the observation of a non-black non-raven would indeed provide
evidence that all ravens are black.
He illustrates this with the example of the generalization "All sodium salts burn yellow", and asks us to consider the
observation which occurs when somebody holds a piece of pure ice in a colorless flame which does not turn
yellow:[2]
This result would confirm the assertion, "Whatever does not burn yellow is not sodium salt", and
consequently, by virtue of the equivalence condition, it would confirm the original formulation. Why does this
impress us as paradoxical? The reason becomes clear when we compare the previous situation with the case of
an experiment where an object whose chemical constitution is as yet unknown to us is held into a flame and
fails to turn it yellow, and where subsequent analysis reveals it to contain no sodium salt. This outcome, we
should no doubt agree, is what was to be expected on the basis of the hypothesis ... thus the data here obtained
constitute confirming evidence for the hypothesis.
In the seemingly paradoxical cases of confirmation, we are often not actually judging the relation of the given
evidence, E alone to the hypothesis H ... we tacitly introduce a comparison of H with a body of evidence
which consists of E in conjunction with an additional amount of information which we happen to have at our
disposal; in our illustration, this information includes the knowledge (1) that the substance used in the
experiment is ice, and (2) that ice contains no sodium salt. If we assume this additional information as given,
then, of course, the outcome of the experiment can add no strength to the hypothesis under consideration. But
if we are careful to avoid this tacit reference to additional knowledge ... the paradoxes vanish.
106
Raven paradox
107
The standard Bayesian solution
One of the most popular proposed resolutions is to accept the conclusion that the observation of a green apple
provides evidence that all ravens are black but to argue that the amount of confirmation provided is very small, due
to the large discrepancy between the number of ravens and the number of non-black objects. According to this
resolution, the conclusion appears paradoxical because we intuitively estimate the amount of evidence provided by
the observation of a green apple to be zero, when it is in fact non-zero but extremely small.
I J Good's presentation of this argument in 1960[6] is perhaps the best known, and variations of the argument have
been popular ever since [7] although it had been presented in 1958[8] and early forms of the argument appeared as
early as 1940.[9]
Good's argument involves calculating the weight of evidence provided by the observation of a black raven or a white
shoe in favor of the hypothesis that all the ravens in a collection of objects are black. The weight of evidence is the
logarithm of the Bayes factor, which in this case is simply the factor by which the odds of the hypothesis changes
when the observation is made. The argument goes as follows:
... suppose that there are
black, and that the
are
objects that might be seen at any moment, of which
objects each have probability 1/
of being seen. Let
non-black ravens, and suppose that the hypotheses
we happen to see a black raven, the Bayes factor in favour of
are ravens and
are
be the hypothesis that there
are initially equiprobable. Then, if
is
average
i.e. about 2 if the number of ravens in existence is known to be large. But the factor if we see a white shoe is
only
average
and this exceeds unity by only about r/(2N-2b) if N-b is large compared to r. Thus the weight of evidence
provided by the sight of a white shoe is positive, but is small if the number of ravens is known to be small
compared to the number of non-black objects.[10]
Many of the proponents of this resolution and variants of it have been advocates of Bayesian probability, and it is
now commonly called the Bayesian Solution, although, as Chihara[11] observes, "there is no such thing as the
Bayesian solution. There are many different 'solutions' that Bayesians have put forward using Bayesian techniques."
Noteworthy
approaches
using
Bayesian
techniques
include
Earman,[12] Eells,[13] Gibson,[14]
Hosaisson-Lindenbaum,[15] Howson and Urbach,[16] Mackie,[17] and Hintikka,[18] who claims that his approach is
"more Bayesian than the so-called 'Bayesian solution' of the same paradox." Bayesian approaches which make use of
Carnap's theory of inductive inference include Humburg,[19] Maher,[20] and Fitelson et al.[21] Vranas[22] introduced
the term "Standard Bayesian Solution" to avoid confusion. Siebel[23] attacks the standard Bayesian solution by
claiming that the raven paradox reappears as soon as several non-black non-ravens are observed. Schiller[24] shows
that Siebel's objection to the standard Bayesian solution is flawed.
Raven paradox
108
The Carnapian approach
Maher[25] accepts the paradoxical conclusion, and refines it:
A non-raven (of whatever color) confirms that all ravens are black because
(i) the information that this object is not a raven removes the possibility that this object is a
counterexample to the generalization, and
(ii) it reduces the probability that unobserved objects are ravens, thereby reducing the probability that
they are counterexamples to the generalization.
In order to reach (ii), he appeals to Carnap's theory of inductive probability, which is (from the Bayesian point of
view) a way of assigning prior probabilities which naturally implements induction. According to Carnap's theory, the
posterior probability,
, that an object, , will have a predicate, , after the evidence
has been
observed, is:
where
is the initial probability that
has the predicate
examined (according to the available evidence
have the predicate
If
, and
is the number of objects which have been
is the number of examined objects which turned out to
is a constant which measures resistance to generalization.
is close to zero,
have the predicate
);
;
will be very close to one after a single observation of an object which turned out to
, while if
is much larger than
,
will be very close to
regardless of
the fraction of observed objects which had the predicate .
Using this Carnapian approach, Maher identifies a proposition which we intuitively (and correctly) know to be false,
but which we easily confuse with the paradoxical conclusion. The proposition in question is the proposition that
observing non-ravens tells us about the color of ravens. While this is intuitively false and is also false according to
Carnap's theory of induction, observing non-ravens (according to that same theory) causes us to reduce our estimate
of the total number of ravens, and thereby reduces the estimated number of possible counterexamples to the rule that
all ravens are black.
Hence, from the Bayesian-Carnapian point of view, the observation of a non-raven does not tell us anything about
the color of ravens, but it tells us about the prevalence of ravens, and supports "All ravens are black" by reducing our
estimate of the number of ravens which might not be black.
The role of background knowledge
Much of the discussion of the paradox in general and the Bayesian approach in particular has centred on the
relevance of background knowledge. Surprisingly, Maher[25] shows that, for a large class of possible configurations
of background knowledge, the observation of a non-black non-raven provides exactly the same amount of
confirmation as the observation of a black raven. The configurations of background knowledge which he considers
are those which are provided by a sample proposition, namely a proposition which is a conjunction of atomic
propositions, each of which ascribes a single predicate to a single individual, with no two atomic propositions
involving the same individual. Thus, a proposition of the form "A is a black raven and B is a white shoe" can be
considered a sample proposition by taking "black raven" and "white shoe" to be predicates.
Maher's proof appears to contradict the result of the Bayesian argument, which was that the observation of a
non-black non-raven provides much less evidence than the observation of a black raven. The reason is that the
background knowledge which Good and others use can not be expressed in the form of a sample proposition - in
particular, variants of the standard Bayesian approach often suppose (as Good did in the argument quoted above) that
the total numbers of ravens, non-black objects and/or the total number of objects, are known quantities. Maher
comments that, "The reason we think there are more non-black things than ravens is because that has been true of the
things we have observed to date. Evidence of this kind can be represented by a sample proposition. But ... given any
Raven paradox
109
sample proposition as background evidence, a non-black non-raven confirms A just as strongly as a black raven does
... Thus my analysis suggests that this response to the paradox [i.e. the Standard Bayesian one] cannot be correct."
Fitelson et al.[26] examined the conditions under which the observation of a non-black non-raven provides less
evidence than the observation of a black raven. They show that, if is an object selected at random,
is the
proposition that the object is black, and
is the proposition that the object is a raven, then the condition:
is sufficient for the observation of a non-black non-raven to provide less evidence than the observation of a black
raven. Here, a line over a proposition indicates the logical negation of that proposition.
This condition does not tell us how large the difference in the evidence provided is, but a later calculation in the
same paper shows that the weight of evidence provided by a black raven exceeds that provided by a non-black
non-raven by about
. This is equal to the amount of additional information (in bits, if the
base of the logarithm is 2) which is provided when a raven of unknown color is discovered to be black, given the
hypothesis that not all ravens are black.
Fitelson et al.[26] explain that:
Under normal circumstances,
may be somewhere around 0.9 or 0.95; so
is
somewhere around 1.11 or 1.05. Thus, it may appear that a single instance of a black raven does not yield
much more support than would a non-black non-raven. However, under plausible conditions it can be shown
that a sequence of instances (i.e. of n black ravens, as compared to n non-black non-ravens) yields a ratio of
likelihood ratios on the order of
, which blows up significantly for large .
The authors point out that their analysis is completely consistent with the supposition that a non-black non-raven
provides an extremely small amount of evidence although they do not attempt to prove it; they merely calculate the
difference between the amount of evidence that a black raven provides and the amount of evidence that a non-black
non-raven provides.
Disputing the induction from positive instances
Some approaches for resolving the paradox focus on the inductive step. They dispute whether observation of a
particular instance (such as one black raven) is the kind of evidence that necessarily increases confidence in the
general hypothesis (such as that ravens are always black).
The red herring
Good[27] gives an example of background knowledge with respect to which the observation of a black raven
decreases the probability that all ravens are black:
Suppose that we know we are in one or other of two worlds, and the hypothesis, H, under consideration is that
all the ravens in our world are black. We know in advance that in one world there are a hundred black ravens,
no non-black ravens, and a million other birds; and that in the other world there are a thousand black ravens,
one white raven, and a million other birds. A bird is selected equiprobably at random from all the birds in our
world. It turns out to be a black raven. This is strong evidence ... that we are in the second world, wherein not
all ravens are black.
Good concludes that the white shoe is a "red herring": Sometimes even a black raven can constitute evidence against
the hypothesis that all ravens are black, so the fact that the observation of a white shoe can support it is not surprising
and not worth attention. Nicod's criterion is false, according to Good, and so the paradoxical conclusion does not
follow.
Hempel rejected this as a solution to the paradox, insisting that the proposition 'c is a raven and is black' must be
considered "by itself and without reference to any other information", and pointing out that it "... was emphasized in
Raven paradox
section 5.2(b) of my article in Mind ... that the very appearance of paradoxicality in cases like that of the white shoe
results in part from a failure to observe this maxim."[28]
The question which then arises is whether the paradox is to be understood in the context of absolutely no background
information (as Hempel suggests), or in the context of the background information which we actually possess
regarding ravens and black objects, or with regard to all possible configurations of background information.
Good had shown that, for some configurations of background knowledge, Nicod's criterion is false (provided that we
are willing to equate "inductively support" with "increase the probability of" - see below). The possibility remained
that, with respect to our actual configuration of knowledge, which is very different from Good's example, Nicod's
criterion might still be true and so we could still reach the paradoxical conclusion. Hempel, on the other hand, insists
that it is our background knowledge itself which is the red herring, and that we should consider induction with
respect to a condition of perfect ignorance.
Good's baby
In his proposed resolution, Maher implicitly made use of the fact that the proposition "All ravens are black" is highly
probable when it is highly probable that there are no ravens. Good had used this fact before to respond to Hempel's
insistence that Nicod's criterion was to be understood to hold in the absence of background information[29]:
...imagine an infinitely intelligent newborn baby having built-in neural circuits enabling him to deal with
formal logic, English syntax, and subjective probability. He might now argue, after defining a raven in detail,
that it is extremely unlikely that there are any ravens, and therefore it is extremely likely that all ravens are
black, that is, that
is true. 'On the other hand', he goes on to argue, 'if there are ravens, then there is a
reasonable chance that they are of a variety of colours. Therefore, if I were to discover that even a black raven
exists I would consider
to be less probable than it was initially.'
This, according to Good, is as close as one can reasonably expect to get to a condition of perfect ignorance, and it
appears that Nicod's condition is still false. Maher made Good's argument more precise by using Carnap's theory of
induction to formalize the notion that if there is one raven, then it is likely that there are many.[30]
Maher's argument considers a universe of exactly two objects, each of which is very unlikely to be a raven (a one in
a thousand chance) and reasonably unlikely to be black (a one in ten chance). Using Carnap's formula for induction,
he finds that the probability that all ravens are black decreases from 0.9985 to 0.8995 when it is discovered that one
of the two objects is a black raven.
Maher concludes that not only is the paradoxical conclusion true, but that Nicod's criterion is false in the absence of
background knowledge (except for the knowledge that the number of objects in the universe is two and that ravens
are less likely than black things).
Distinguished predicates
Quine[31] argued that the solution to the paradox lies in the recognition that certain predicates, which he called
natural kinds, have a distinguished status with respect to induction. This can be illustrated with Nelson Goodman's
example of the predicate grue. An object is grue if it is blue before (say) 2015 and green afterwards. Clearly, we
expect objects which were blue before 2015 to remain blue afterwards, but we do not expect the objects which were
found to be grue before 2015 to be grue after 2015, since after 2015 they would be green. Quine's explanation is that
"blue" is a natural kind; a privileged predicate which can be used for induction, while "grue" is not a natural kind and
using induction with it leads to error.
This suggests a resolution to the paradox - Nicod's criterion is true for natural kinds, such as "blue" and "black", but
is false for artificially contrived predicates, such as "grue" or "non-raven". The paradox arises, according to this
resolution, because we implicitly interpret Nicod's criterion as applying to all predicates when in fact it only applies
to natural kinds.
110
Raven paradox
Another approach which favours specific predicates over others was taken by Hintikka.[18] Hintikka was motivated
to find a Bayesian approach to the paradox which did not make use of knowledge about the relative frequencies of
ravens and black things. Arguments concerning relative frequencies, he contends, cannot always account for the
perceived irrelevance of evidence consisting of observations of objects of type A for the purposes of learning about
objects of type not-A.
His argument can be illustrated by rephrasing the paradox using predicates other than "raven" and "black". For
example, "All men are tall" is equivalent to "All short people are women", and so observing that a randomly selected
person is a short woman should provide evidence that all men are tall. Despite the fact that we lack background
knowledge to indicate that there are dramatically fewer men than short people, we still find ourselves inclined to
reject the conclusion. Hintikka's example is: "... a generalization like 'no material bodies are infinitely divisible'
seems to be completely unaffected by questions concerning immaterial entities, independently of what one thinks of
the relative frequencies of material and immaterial entities in one's universe of discourse."
His solution is to introduce an order into the set of predicates. When the logical system is equipped with this order, it
is possible to restrict the scope of a generalization such as "All ravens are black" so that it applies to ravens only and
not to non-black things, since the order privileges ravens over non-black things. As he puts it:
If we are justified in assuming that the scope of the generalization 'All ravens are black' can be restricted to
ravens, then this means that we have some outside information which we can rely on concerning the factual
situation. The paradox arises from the fact that this information, which colors our spontaneous view of the
situation, is not incorporated in the usual treatments of the inductive situation.[32]
Rejections of Hempel's equivalence condition
Some approaches for the resolution of the paradox reject Hempel's equivalence condition. That is, they may not
consider evidence supporting the statement all non-black objects are non-ravens to necessarily support
logically-equivalent statements such as all ravens are black.
Selective confirmation
Scheffler and Goodman[33] took an approach to the paradox which incorporates Karl Popper's view that scientific
hypotheses are never really confirmed, only falsified.
The approach begins by noting that the observation of a black raven does not prove that "All ravens are black" but it
falsifies the contrary hypothesis, "No ravens are black". A non-black non-raven, on the other hand, is consistent with
both "All ravens are black" and with "No ravens are black". As the authors put it:
... the statement that all ravens are black is not merely satisfied by evidence of a black raven but is favored by
such evidence, since a black raven disconfirms the contrary statement that all ravens are not black, i.e. satisfies
its denial. A black raven, in other words, satisfies the hypothesis that all ravens are black rather than not: it
thus selectively confirms that all ravens are black.
Selective confirmation violates the equivalence condition since a black raven selectively confirms "All ravens are
black" but not "All non-black things are non-ravens".
111
Raven paradox
112
Probabilistic or non-probabilistic induction
Scheffler and Goodman's concept of selective confirmation is an example of an interpretation of "provides evidence
in favor of," which does not coincide with "increase the probability of". This must be a general feature of all
resolutions which reject the equivalence condition, since logically equivalent propositions must always have the
same probability.
It is impossible for the observation of a black raven to increase the probability of the proposition "All ravens are
black" without causing exactly the same change to the probability that "All non-black things are non-ravens". If an
observation inductively supports the former but not the latter, then "inductively support" must refer to something
other than changes in the probabilities of propositions. A possible loophole is to interpret "All" as "Nearly all" "Nearly all ravens are black" is not equivalent to "Nearly all non-black things are non-ravens", and these
propositions can have very different probabilities.[34]
This raises the broader question of the relation of probability theory to inductive reasoning. Karl Popper argued that
probability theory alone cannot account for induction. His argument involves splitting a hypothesis,
, into a part
which is deductively entailed by the evidence,
[35]
First, consider the splitting
where
,
and
, and another part. This can be done in two ways.
:
are probabilistically independent:
and so on. The condition
which is necessary for such a splitting of H and E to be possible is
probabilistically supported by
.
Popper's observation is that the part,
, while the part of
, of
, that is, that
which receives support from
which does not follow deductively from
is
actually follows deductively from
receives no support at all from
- that is,
.
[36]
Second, the splitting
separates
:
into
, which as Popper says, "is the logically strongest part of
) that follows [deductively] from
," and
, which, he says, "contains all of
He continues:
Does
, in this case, provide any support for the factor
needed to obtain
? The answer is: No. It never does. Indeed,
(or of the content of
that goes beyond
, which in the presence of
countersupports
."
is alone
unless either
or
(which are possibilities of no interest). ...
This result is completely devastating to the inductive interpretation of the calculus of probability. All
probabilistic support is purely deductive: that part of a hypothesis that is not deductively entailed by the
evidence is always strongly countersupported by the evidence ... There is such a thing as probabilistic support;
there might even be such a thing as inductive support (though we hardly think so). But the calculus of
probability reveals that probabilistic support cannot be inductive support.
The orthodox approach
The orthodox Neyman-Pearson theory of hypothesis testing considers how to decide whether to accept or reject a
hypothesis, rather than what probability to assign to the hypothesis. From this point of view, the hypothesis that "All
ravens are black" is not accepted gradually, as its probability increases towards one when more and more
observations are made, but is accepted in a single action as the result of evaluating the data which has already been
collected. As Neyman and Pearson put it:
Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern
our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall
not be too often wrong.[37]
Raven paradox
113
According to this approach, it is not necessary to assign any value to the probability of a hypothesis, although one
must certainly take into account the probability of the data given the hypothesis, or given a competing hypothesis,
when deciding whether to accept or to reject. The acceptance or rejection of a hypothesis carries with it the risk of
error.
This contrasts with the Bayesian approach, which requires that the hypothesis be assigned a prior probability, which
is revised in the light of the observed data to obtain the final probability of the hypothesis. Within the Bayesian
framework there is no risk of error since hypotheses are not accepted or rejected; instead they are assigned
probabilities.
An analysis of the paradox from the orthodox point of view has been performed, and leads to, among other insights,
a rejection of the equivalence condition:
It seems obvious that one cannot both accept the hypothesis that all P's are Q and also reject the contrapositive,
i.e. that all non-Q's are non-P. Yet it is easy to see that on the Neyman-Pearson theory of testing, a test of "All
P's are Q" is not necessarily a test of "All non-Q's are non-P" or vice versa. A test of "All P's are Q" requires
reference to some alternative statistical hypothesis of the form of all P's are Q,
, whereas a test
of "All non-Q's are non-P" requires reference to some statistical alternative of the form of all non-Q's are
non-P,
. But these two sets of possible alternatives are different ... Thus one could have a test of
without having a test of its contrapositive.[38]
Rejecting material implication
The following propositions all imply one another: "Every object is either black or not a raven", "Every Raven is
black", and "Every non-black object is a non-raven." They are therefore, by definition, logically equivalent.
However, the three propositions have different domains: the first proposition says something about "Every object",
while the second says something about "Every raven".
The first proposition is the only one whose domain is unrestricted ("all objects"), so this is the only one which can be
expressed in first order logic. It is logically equivalent to:
and also to
where
or
indicates the material conditional, according to which "If
then
" can be understood to mean "
".
It has been argued by several authors that material implication does not fully capture the meaning of "If
then
" (see the paradoxes of material implication). "For every object, , is either black or not a raven" is true when
there are no ravens. It is because of this that "All ravens are black" is regarded as true when there are no ravens.
Furthermore, the arguments which Good and Maher used to criticize Nicod's criterion (see Good's Baby, above)
relied on this fact - that "All ravens are black" is highly probable when it is highly probable that there are no ravens.
Some approaches to the paradox have sought to find other ways of interpreting "If
then
" and "All
are
" which would eliminate the perceived equivalence between "All ravens are black" and "All non-black things are
non-ravens."
One such approach involves introducing a many-valued logic according to which "If
truth-value
, meaning "Indeterminate" or "Inappropriate" when
not automatically allowed: "If
then
" is not equivalent to "If
[39]
is false.
then
then
" has the
In such a system, contraposition is
". Consequently, "All ravens are
black" is not equivalent to "All non-black things are non-ravens".
In this system, when contraposition occurs, the modality of the conditional involved changes from the indicative ("If
that piece of butter has been heated to 32 C then it has melted") to the counterfactual ("If that piece of butter had
been heated to 32 C then it would have melted"). According to this argument, this removes the alleged equivalence
Raven paradox
114
which is necessary to conclude that yellow cows can inform us about ravens:
In proper grammatical usage, a contrapositive argument ought not to be stated entirely in the indicative. Thus:
From the fact that if this match is scratched it will light, it follows that if it does not light it was not
scratched.
is awkward. We should say:
From the fact that if this match is scratched it will light, it follows that if it were not to light it would not
have been scratched. ...
One might wonder what effect this interpretation of the Law of Contraposition has on Hempel's paradox of
confirmation. "If is a raven then is black" is equivalent to "If were not black then would not be a
raven". Therefore whatever confirms the latter should also, by the Equivalence Condition, confirm the former.
True, but yellow cows still cannot figure into the confirmation of "All ravens are black" because, in science,
confirmation is accomplished by prediction, and predictions are properly stated in the indicative mood. It is
senseless to ask what confirms a counterfactual.[40]
Differing results of accepting the hypotheses
Several commentators have observed that the propositions "All ravens are black" and "All non-black things are
non-ravens" suggest different procedures for testing the hypotheses. E.g. Good writes[41]:
As propositions the two statements are logically equivalent. But they have a different psychological effect on
the experimenter. If he is asked to test whether all ravens are black he will look for a raven and then decide
whether it is black. But if he is asked to test whether all non-black things are non-ravens he may look for a
non-black object and then decide whether it is a raven.
More recently, it has been suggested that "All ravens are black" and "All non-black things are non-ravens" can have
different effects when accepted.[42] The argument considers situations in which the total numbers or prevalences of
ravens and black objects are unknown, but estimated. When the hypothesis "All ravens are black" is accepted,
according to the argument, the estimated number of black objects increases, while the estimated number of ravens
does not change.
It can be illustrated by considering the situation of two people who have identical information regarding ravens and
black objects, and who have identical estimates of the numbers of ravens and black objects. For concreteness,
suppose that there are 100 objects overall, and, according to the information available to the people involved, each
object is just as likely to be a non-raven as it is to be a raven, and just as likely to be black as it is to be non-black:
and the propositions
are independent for different objects
,
and so on. Then the estimated number of
ravens is 50; the estimated number of black things is 50; the estimated number of black ravens is 25, and the
estimated number of non-black ravens (counterexamples to the hypotheses) is 25.
One of the people performs a statistical test (e.g. a Neyman-Pearson test or the comparison of the accumulated
weight of evidence to a threshold) of the hypothesis that "All ravens are black", while the other tests the hypothesis
that "All non-black objects are non-ravens". For simplicity, suppose that the evidence used for the test has nothing to
do with the collection of 100 objects dealt with here. If the first person accepts the hypothesis that "All ravens are
black" then, according to the argument, about 50 objects whose colors were previously in doubt (the ravens) are now
thought to be black, while nothing different is thought about the remaining objects (the non-ravens). Consequently,
he should estimate the number of black ravens at 50, the number of black non-ravens at 25 and the number of
non-black non-ravens at 25. By specifying these changes, this argument explicitly restricts the domain of "All ravens
are black" to ravens.
Raven paradox
115
On the other hand, if the second person accepts the hypothesis that "All non-black objects are non-ravens", then the
approximately 50 non-black objects about which it was uncertain whether each was a raven, will be thought to be
non-ravens. At the same time, nothing different will be thought about the approximately 50 remaining objects (the
black objects). Consequently, he should estimate the number of black ravens at 25, the number of black non-ravens
at 25 and the number of non-black non-ravens at 50. According to this argument, since the two people disagree about
their estimates after they have accepted the different hypotheses, accepting "All ravens are black" is not equivalent to
accepting "All non-black things are non-ravens"; accepting the former means estimating more things to be black,
while accepting the latter involves estimating more things to be non-ravens. Correspondingly, the argument goes, the
former requires as evidence ravens which turn out to be black and the latter requires non-black things which turn out
to be non-ravens.[43]
Existential presuppositions
A number of authors have argued that propositions of the form "All
which are
...
[44]
.
are
" presuppose that there are objects
[45]
This analysis has been applied to the raven paradox:
: "All ravens are black" and
: "All nonblack things are nonravens" are not strictly equivalent ... due
to their different existential presuppositions. Moreover, although
and
describe the same regularity -
the nonexistence of nonblack ravens - they have different logical forms. The two hypotheses have different
senses and incorporate different procedures for testing the regularity they describe.
A modified logic can take account of existential presuppositions using the presuppositional operator, '*'. For
example,
can denote "All ravens are black" while indicating that it is ravens and not non-black objects which are presupposed
to exist in this example.
... the logical form of each hypothesis distinguishes it with respect to its recommended type of supporting
evidence: the possibly true substitution instances of each hypothesis relate to different types of objects. The
fact that the two hypotheses incorporate different kinds of testing procedures is expressed in the formal
language by prefixing the operator '*' to a different predicate. The presuppositional operator thus serves as a
relevance operator as well. It is prefixed to the predicate ' is a raven' in
because the objects relevant to
the testing procedure incorporated in "All raven are black" include only ravens; it is prefixed to the predicate '
is nonblack', in
, because the objects relevant to the testing procedure incorporated in "All nonblack
things are nonravens" include only nonblack things. ... Using Fregean terms: whenever their presuppositions
hold, the two hypotheses have the same referent (truth-value), but different senses; that is, they express two
different ways to determine that truth-value.[46]
Notes
[1]
[2]
[3]
[4]
http:/ / plato. stanford. edu/ entries/ hempel/
Hempel, C. G. (1945). "Studies in the Logic of Confirmation I". Mind 54 (213): 1–26. JSTOR 2250886.
Hempel, C. G. (1945). "Studies in the Logic of Confirmation II". Mind 54 (214): 97–121. JSTOR 2250948.
Nicod had proposed that, in relation to conditional hypotheses, instances of their antecedents that are also instances of their consequents
confirm them; instances of their antecedents that are not instances of their consequents disconfirm them; and non-instantiations of their
antecedents are neutral, neither confirming nor disconfirming. Stanford Encyclopedia of Philosophy (http:/ / plato. stanford. edu/ entries/
hempel/ )
[5] Maher, P. (1999). "Inductive Logic and the Ravens Paradox". Philosophy of Science 66 (1): 50–70. JSTOR 188737.
[6] Good, IJ (1960) The Paradox of Confirmation, The British Journal for the Philosophy of Science, Vol. 11, No. 42, 145-149 JSTOR (http:/ /
links. jstor. org/ sici?sici=0007-0882(196008)11:42<145:TPOC>2. 0. CO;2-5)
[7] Fitelson, B and Hawthorne, J (2006) How Bayesian Confirmation Theory Handles the Paradox of the Ravens, in Probability in Science,
Chicago: Open Court Link (http:/ / fitelson. org/ ravens. pdf)
[8] Alexander, HG (1958) The Paradoxes of Confirmation, The British Journal for the Philosophy of Science, Vol. 9, No. 35, P. 227 JSTOR
(http:/ / www. jstor. org/ stable/ 685654?origin=JSTOR-pdf)
Raven paradox
[9] Hosaisson-Lindenbaum, J (1940) On Confirmation, The Journal of Symbolic Logic, Vol. 5, No. 4, p. 133 JSTOR (http:/ / www. jstor. org/
action/ showArticle?doi=10. 2307/ 2268173)
[10] Note: Good used "crow" instead of "raven", but "raven" has been used here throughout for consistency.
[11] Chihara, (1987) Some Problems for Bayesian Confirmation Theory, British Journal for the Philosophy of Science, Vol. 38, No. 4 LINK
(http:/ / bjps. oxfordjournals. org/ cgi/ reprint/ 38/ 4/ 551)
[12] Earman, 1992 Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory, MIT Press, Cambridge, MA.
[13] Eells, 1982 Rational Decision and Causality. New York: Cambridge University Press
[14] Gibson, 1969 On Ravens and Relevance and a Likelihood Solution of the Paradox of Confirmation, LINK (http:/ / www. jstor. org/ stable/
686720)
[15] Hosaisson-Lindenbaum 1940
[16] Howson, Urbach, 1993 Scientific Reasoning: The Bayesian Approach, Open Court Publishing Company
[17] Mackie, 1963 The Paradox of Confirmation, Brit. J. Phil. Sci. Vol. 13, No. 52, p. 265 LINK (http:/ / bjps. oxfordjournals. org/ cgi/ content/
citation/ XIII/ 52/ 265)
[18] Hintikka, 1969
[19] Humburg 1986, The solution of Hempel's raven paradox in Rudolf Carnap's system of inductive logic, Erkenntnis, Vol. 24, No. 1, pp
[20] Maher 1999
[21] Fitelson 2006
[22] Vranas (2002) Hempel's Raven Paradox: A Lacuna in the Standard Bayesian Solution LINK (http:/ / philsci-archive. pitt. edu/ archive/
00000688/ 00/ hempelacuna. doc)
[23] Siebel, M. (2004). Der Rabe und der Bayesianist. Journal for General Philosophy of Science, 35, 313–329 LINK (http:/ / www. springerlink.
com/ content/ g0r8032u5k1vtju6/ )
[24] Schiller, F. (2012). "Why Bayesians Needn’t Be Afraid of Observing Many Non-black Non-ravens". Journal for General Philosophy of
Science 34 (1). doi:10.1007/s10838-012-9179-z.
[25] Maher, 1999
[26] Fitelson, 2006
[27] Good (1967). "The White Shoe is a Red Herring". British Journal for the Philosophy of Science 17 (4): 322. JSTOR 686774.
[28] Hempel 1967, The White Shoe - No Red Herring, The British Journal for the Philosophy of Science, Vol. 18, No. 3, p. 239 JSTOR (http:/ /
www. jstor. org/ stable/ 686596)
[29] Good (1968). "The White Shoe qua Red Herring is Pink". The British Journal for the Philosophy of Science 19 (2): 156. JSTOR 686795.
[30] Maher 2004, Probability Captures the Logic of Scientific Confirmation LINK (http:/ / patrick. maher1. net/ pctl. pdf)
[31] Quine, W. V. (1969). Natural Kinds, in Ontological Relativity and other Essays. New York: Columbia University Press. p. 114.
[32] Hintikka J. 1969, Inductive Independence and the Paradoxes of Confirmation LINK (http:/ / books. google. com/
books?id=pWtPcRwuacAC& pg=PA24& lpg=PA24& ots=-1PKZt0Jbz& lr=& sig=EK2qqOZ6-cZR1P1ZKIsndgxttMs)
[33] Scheffler I, Goodman NJ, Selective Confirmation and the Ravens, Journal of Philosophy, Vol. 69, No. 3, 1972 JSTOR (http:/ / www. jstor.
org/ stable/ 2024647)
[34] Gaifman, H. (1979). "Subjective Probability, Natural Predicates and Hempel's Ravens". Erkenntnis 14 (2): 105–147.
doi:10.1007/BF00196729.
[35] Popper, K. Realism and the Aim of Science, Routlege, 1992, p. 325
[36] Popper K, Miller D, (1983) A Proof of the Impossibility of Inductive Probability, Nature, Vol. 302, p. 687 Link (http:/ / www. nature. com/
nature/ journal/ v302/ n5910/ abs/ 302687a0. html)
[37] Neyman J, Pearson ES (1933) On the Problem of the Most Efficient Tests of Statistical Hypotheses, Phil. Transactions of the Royal Society
of London. Series A, Vol. 231, p289 JSTOR (http:/ / www. jstor. org/ stable/ 91247)
[38] Giere, RN (1970) An Orthodox Statistical Resolution of the Paradox of Confirmation, Philosophy of Science, Vol. 37, No. 3, p.354 JSTOR
(http:/ / www. jstor. org/ stable/ 186464)
[39] Farrell RJ (1979) Material Implication, Confirmation and Counterfactuals LINK (http:/ / projecteuclid. org/ DPubS/ Repository/ 1. 0/
Disseminate?view=body& id=pdf_1& handle=euclid. ndjfl/ 1093882546)
[40] Farrell (1979)
[41] Good (1960)
[42] O'Flanagan (2008) Judgment LINK (http:/ / philsci-archive. pitt. edu/ archive/ 00003932/ 01/ judgment6. pdf)
[43] O'Flanagan (2008)
[44] Strawson PF (1952) Introduction to Logical Theory, methuan & Co. London, John Wiley & Sons, New York
[45] Cohen Y (1987) Ravens and Relevance, Erkenntnis LINK (http:/ / www. springerlink. com/ content/ hnn2lutn1066xw47/ fulltext. pdf)
[46] Cohen (1987)
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Raven paradox
References
• Franceschi, P. The Doomsday Argument and Hempel's Problem (http://www.paulfranceschi.com/index.
php?option=com_content&view=article&id=8:the-doomsday-argument-and-hempels-problem&
catid=1:analytic-philosophy&Itemid=2), English translation of a paper initially published in French in the
Canadian Journal of Philosophy 29, 139-156, 1999, under the title Comment l'urne de Carter et Leslie se déverse
dans celle de Hempel
• Hempel, C. G. A Purely Syntactical Definition of Confirmation. J. Symb. Logic 8, 122-143, 1943.
• Hempel, C. G. Studies in the Logic of Confirmation (I) Mind 54, 1-26, 1945.
• Hempel, C. G. Studies in the Logic of Confirmation (II) Mind 54, 97-121, 1945.
• Hempel, C. G. Studies in the Logic of Confirmation. In Marguerite H. Foster and Michael L. Martin (http://
www.bu.edu/philo/faculty/martin.html), eds. Probability, Confirmation, and Simplicity. New York: Odyssey
Press, 1966. 145-183.
• Whiteley, C. H. Hempel's Paradoxes of Confirmation. Mind 55, 156-158, 1945.
External links
• "Hempel’s Ravens Paradox," PRIME (Platonic Realms Interactive Mathematics Encyclopedia). (http://www.
mathacademy.com/pr/prime/articles/paradox_raven/index.asp) Retrieved November 29, 2010.
Unexpected hanging paradox
The unexpected hanging paradox, hangman paradox, unexpected exam paradox, surprise test paradox or
prediction paradox is a paradox about a person's expectations about the timing of a future event (e.g. a prisoner's
hanging, or a school test) which he is told will occur at an unexpected time.
Despite significant academic interest, there is no consensus on its precise nature and consequently a final 'correct'
resolution has not yet been established.[1] One approach, offered by the logical school of thought, suggests that the
problem arises in a self-contradictory self-referencing statement at the heart of the judge's sentence. Another
approach, offered by the epistemological school of thought, suggests the unexpected hanging paradox is an example
of an epistemic paradox because it turns on our concept of knowledge.[2] Even though it is apparently simple, the
paradox's underlying complexities have even led to it being called a "significant problem" for philosophy.[3]
Description of the paradox
The paradox has been described as follows:[4]
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but
that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the
executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His
reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he
hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on
Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it
cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been
eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a
Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur
on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at
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all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the
above, was an utter surprise to him. Everything the judge said came true.
Other versions of the paradox replace the death sentence with a surprise fire drill, examination, pop quiz, or a lion
behind a door.[1]
The informal nature of everyday language allows for multiple interpretations of the paradox. In the extreme case, a
prisoner who is paranoid might feel certain in his knowledge that the executioner will arrive at noon on Monday,
then certain that he will come on Tuesday and so forth, thus ensuring that every day he is not hanged really is a
"surprise" to him, but that the day of his hanging he was indeed expecting to be hanged. But even without adding this
element to the story, the vagueness of the account prohibits one from being objectively clear about which
formalization truly captures its essence. There has been considerable debate between the logical school, which uses
mathematical language, and the epistemological school, which employs concepts such as knowledge, belief and
memory, over which formulation is correct.
The logical school
Formulation of the judge's announcement into formal logic is made difficult by the vague meaning of the word
"surprise". An attempt at formulation might be:
• The prisoner will be hanged next week and the date (of the hanging) will not be deducible in advance from the
assumption that the hanging will occur during the week (A).
Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week.
However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week,
the prisoner must argue that his ability to deduce, from statement (A), that the hanging will not occur on the last day,
implies that a last-day hanging would not be surprising. But since the meaning of "surprising" has been restricted to
not deducible from the assumption that the hanging will occur during the week instead of not deducible from
statement (A), the argument is blocked.
This suggests that a better formulation would in fact be:
• The prisoner will be hanged next week and its date will not be deducible in advance using this statement as an
axiom (B).
Some authors have claimed that the self-referential nature of this statement is the source of the paradox. Fitch[5] has
shown that this statement can still be expressed in formal logic. Using an equivalent form of the paradox which
reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all
circumstances, it is in this case because the statement is self-contradictory.
Objections
The first objection often raised to the logical school's approach is that it fails to explain how the judge's
announcement appears to be vindicated after the fact. If the judge's statement is self-contradictory, how does he
manage to be right all along? This objection rests on an understanding of the conclusion to be that the judge's
statement is self-contradictory and therefore the source of the paradox. However, the conclusion is more precisely
that in order for the prisoner to carry out his argument that the judge's sentence cannot be fulfilled, he must interpret
the judge's announcement as (B). A reasonable assumption would be that the judge did not intend (B) but that the
prisoner misinterprets his words to reach his paradoxical conclusion. The judge's sentence appears to be vindicated
afterwards but the statement which is actually shown to be true is that "the prisoner will be psychologically surprised
by the hanging". This statement in formal logic would not allow the prisoner's argument to be carried out.
A related objection is that the paradox only occurs because the judge tells the prisoner his sentence (rather than
keeping it secret) — which suggests that the act of declaring the sentence is important. Some have argued that since
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Unexpected hanging paradox
this action is missing from the logical school's approach, it must be an incomplete analysis. But the action is included
implicitly. The public utterance of the sentence and its context changes the judge's meaning to something like "there
will be a surprise hanging despite my having told you that there will be a surprise hanging". The logical school's
approach does implicitly take this into account.
Leaky inductive argument
The argument that first excludes Friday, and then excludes the last remaining day of the week is an inductive one.
The prisoner assumes that by Thursday he will know the hanging is due on Friday, but he does not know that before
Thursday. By trying to carry an inductive argument backward in time based on a fact known only by Thursday the
prisoner may be making an error. The conditional statement "If I reach Thursday afternoon alive then Friday will be
the latest possible day for the hanging" does little to reassure the condemned man. The prisoner's argument in any
case carries the seeds of its own destruction because if he is right, then he is wrong, and can be hanged any day
including Friday.
The counter-argument to this is that in order to claim that a statement will not be a surprise, it is not necessary to
predict the truth or falsity of the statement at the time the claim is made, but only to show that such a prediction will
become possible in the interim period. It is indeed true that the prisoner does not know on Monday that he will be
hanged on Friday, nor that he will still be alive on Thursday. However, he does know on Monday, that if the
hangman as it turns out knocks on his door on Friday, he will have already have expected that (and been alive to do
so) since Thursday night - and thus, if the hanging occurs on Friday then it will certainly have ceased to be a surprise
at some point in the interim period between Monday and Friday. The fact that it has not yet ceased to be a surprise at
the moment the claim is made is not relevant. This works for the inductive case too. When the prisoner wakes up on
any given day, on which the last possible hanging day is tomorrow, the prisoner will indeed not know for certain that
he will survive to see tomorrow. However, he does know that if he does survive today, he will then know for certain
that he must be hanged tomorrow, and thus by the time he is actually hanged tomorrow it will have ceased to be a
surprise. This removes the leak from the argument.
In other words, his reasoning is incorrect, as if the hanging was on Friday, he will have found it unexpected because
he would have expected no hanging. It would be true even if the judge said: "You will unexpectedly be hanged
today."
The epistemological school
Various epistemological formulations have been proposed that show that the prisoner's tacit assumptions about what
he will know in the future, together with several plausible assumptions about knowledge, are inconsistent.
Chow (1998) provides a detailed analysis of a version of the paradox in which a surprise examination is to take place
on one of two days. Applying Chow's analysis to the case of the unexpected hanging (again with the week shortened
to two days for simplicity), we start with the observation that the judge's announcement seems to affirm three things:
• S1: The hanging will occur on Monday or Tuesday.
• S2: If the hanging occurs on Monday, then the prisoner will not know on Sunday evening that it will occur on
Monday.
• S3: If the hanging occurs on Tuesday, then the prisoner will not know on Monday evening that it will occur on
Tuesday.
As a first step, the prisoner reasons that a scenario in which the hanging occurs on Tuesday is impossible because it
leads to a contradiction: on the one hand, by S3, the prisoner would not be able to predict the Tuesday hanging on
Monday evening; but on the other hand, by S1 and process of elimination, the prisoner would be able to predict the
Tuesday hanging on Monday evening.
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Unexpected hanging paradox
Chow's analysis points to a subtle flaw in the prisoner's reasoning. What is impossible is not a Tuesday hanging.
Rather, what is impossible is a situation in which the hanging occurs on Tuesday despite the prisoner knowing on
Monday evening that the judge's assertions S1, S2, and S3 are all true.
The prisoner's reasoning, which gives rise to the paradox, is able to get off the ground because the prisoner tacitly
assumes that on Monday evening, he will (if he is still alive) know S1, S2, and S3 to be true. This assumption seems
unwarranted on several different grounds. It may be argued that the judge's pronouncement that something is true
can never be sufficient grounds for the prisoner knowing that it is true. Further, even if the prisoner knows something
to be true in the present moment, unknown psychological factors may erase this knowledge in the future. Finally,
Chow suggests that because the statement which the prisoner is supposed to "know" to be true is a statement about
his inability to "know" certain things, there is reason to believe that the unexpected hanging paradox is simply a
more intricate version of Moore's paradox. A suitable analogy can be reached by reducing the length of the week to
just one day. Then the judge's sentence becomes: You will be hanged tomorrow, but you do not know that.
References
[1] T. Y. Chow, "The surprise examination or unexpected hanging paradox," The American Mathematical Monthly Jan 1998 (http:/ / www-math.
mit. edu/ ~tchow/ unexpected. pdf)
[2] Stanford Encyclopedia discussion of hanging paradox together with other epistemic paradoxes (http:/ / plato. stanford. edu/ entries/
epistemic-paradoxes/ )
[3] R. A. Sorensen, Blindspots, Clarendon Press, Oxford (1988)
[4] "Unexpected Hanging Paradox" (http:/ / mathworld. wolfram. com/ UnexpectedHangingParadox. html). Wolfram. .
[5] Fitch, F., A Goedelized formulation of the prediction paradox, Amer. Phil. Quart 1 (1964), 161–164
Further reading
• O'Connor, D. J. (1948). "Pragmatic Paradoxes". Mind 57: 358–359. The first appearance of the paradox in print. The author
claims that certain contingent future tense statements cannot come true.
• Scriven, M. (1951). "Paradoxical Announcements". Mind 60: 403–407. The author critiques O'Connor and discovers the
paradox as we know it today.
• Shaw, R. (1958). "The Unexpected Examination". Mind 67: 382–384. The author claims that the prisoner's premises are
self-referring.
• Wright, C. & Sudbury, A. (1977). "the Paradox of the Unexpected Examination". Australasian Journal of
Philosophy 55: 41–58. The first complete formalization of the paradox, and a proposed solution to it.
• Margalit, A. & Bar-Hillel, M. (1983). "Expecting the Unexpected". Philosophia 13: 337–344. A history and
bibliography of writings on the paradox up to 1983.
• Chihara, C. S. (1985). "Olin, Quine, and the Surprise Examination". Philosophical Studies 47: 19–26. The author
claims that the prisoner assumes, falsely, that if he knows some proposition, then he also knows that he knows it.
• Kirkham, R. (1991). "On Paradoxes and a Surprise Exam". Philosophia 21: 31–51. The author defends and extends
Wright and Sudbury's solution. He also updates the history and bibliography of Margalit and Bar-Hillel up to 1991.
• Chow, T. Y. (1998). "The surprise examination or unexpected hanging paradox" (http://www-math.mit.edu/
~tchow/unexpected.pdf). The American Mathematical Monthly.
• Franceschi, P. (2005). "Une analyse dichotomique du paradoxe de l'examen surprise". Philosophiques 32 (2):
399–421. English translation (http://www.paulfranceschi.com/index.php?option=com_content&
view=article&id=6:a-dichotomic-analysis-of-the-surprise-examination-paradox&catid=1:analytic-philosophy&
Itemid=2).
• Gardner, M. (1969). "The Paradox of the Unexpected Hanging". The Unexpected Hanging and Other *
Mathematical Diversions. Completely analyzes the paradox and introduces other situations with similar logic.
• Quine, W. V. O. (1953). "On a So-called Paradox". Mind 62: 65–66.
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Unexpected hanging paradox
• Sorensen, R. A. (1982). "Recalcitrant versions of the prediction paradox". Australasian Journal of Philosophy 69:
355–362.
• Kacser, Claude (1986). " On the unexpected hanging paradox (http://dx.doi.org/10.1119/1.14658)".
American Journal of Physics 54 (4): 296.
• Shapiro, Stuart C. (1998). " A Procedural Solution to the Unexpected Hanging and Sorites Paradoxes (http://
www.jstor.org/stable/2659782)". Mind 107: 751-761.
External links
• "The Surprise Examination Paradox and the Second Incompleteness Theorem" (http://www.ams.org/notices/
201011/rtx101101454p.pdf) by Shira Kritchman and Ran Raz, at ams.org
• "A Simple Solution of the Unexpected Hanging Paradox: A Kind of a Leaky Inductive Argument Solution" (http:/
/ssrn.com/abstract=2027851) by Kedar Joshi, at SSRN
• "The Surprise Examination Paradox: A review of two so-called solutions in dynamic epistemic logic" (http://
staff.science.uva.nl/~grossi/DyLoPro/StudentPapers/Final_Marcoci.pdf) by Alexandru Marcoci, at Faculty
of Science: University of Amsterdam
What the Tortoise Said to Achilles
"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a
brief dialogue which problematises the foundations of logic. The title alludes to one of Zeno's paradoxes of motion,
in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to
use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails,
because the clever tortoise leads him into an infinite regression.
Summary of the dialogue
The discussion begins by considering the following logical argument:
• A: "Things that are equal to the same are equal to each other" (Euclidean relation, a weakened form of the
transitive property)
• B: "The two sides of this triangle are things that are equal to the same"
• Therefore Z: "The two sides of this triangle are equal to each other"
The Tortoise asks Achilles whether the conclusion logically follows from the premises, and Achilles grants that it
obviously does. The Tortoise then asks Achilles whether there might be a reader of Euclid who grants that the
argument is logically valid, as a sequence, while denying that A and B are true. Achilles accepts that such a reader
might exist, and that he would hold that if A and B are true, then Z must be true, while not yet accepting that A and B
are true. (A reader who denies the premises.)
The Tortoise then asks Achilles whether a second kind of reader might exist, who accepts that A and B are true, but
who does not yet accept the principle that if A and B are both true, then Z must be true. Achilles grants the Tortoise
that this second kind of reader might also exist. The Tortoise, then, asks Achilles to treat the Tortoise as a reader of
this second kind. Achilles must now logically compel the Tortoise to accept that Z must be true. (The tortoise is a
reader who denies the argument itself, the syllogism's conclusion, structure or validity.)
After writing down A, B and Z in his notebook, Achilles asks the Tortoise to accept the hypothetical:
• C: "If A and B are true, Z must be true"
The Tortoise agrees to accept C, if Achilles will write down what it has to accept in his notebook, making the new
argument:
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What the Tortoise Said to Achilles
•
•
•
•
A: "Things that are equal to the same are equal to each other"
B: "The two sides of this triangle are things that are equal to the same"
C: "If A and B are true, Z must be true"
Therefore Z: "The two sides of this triangle are equal to each other"
But now that the Tortoise accepts premise C, it still refuses to accept the expanded argument. When Achilles
demands that "If you accept A and B and C, you must accept Z," the Tortoise remarks that that's another hypothetical
proposition, and suggests even if it accepts C, it could still fail to conclude Z if it did not see the truth of:
• D: "If A and B and C are true, Z must be true"
The Tortoise continues to accept each hypothetical premise once Achilles writes it down, but denies that the
conclusion necessarily follows, since each time it denies the hypothetical that if all the premises written down so far
are true, Z must be true:
"And at last we've got to the end of this ideal racecourse! Now that you accept A and B and C and D, of course
you accept Z."
"Do I?" said the Tortoise innocently. "Let's make that quite clear. I accept A and B and C and D. Suppose I still
refused to accept Z?"
"Then Logic would take you by the throat, and force you to do it!" Achilles triumphantly replied. "Logic
would tell you, 'You can't help yourself. Now that you've accepted A and B and C and D, you must accept Z!'
So you've no choice, you see."
"Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your
notebook, please. We will call it
(E) If A and B and C and D are true, Z must be true.
Until I've granted that, of course I needn't grant Z. So it's quite a necessary step, you see?"
"I see," said Achilles; and there was a touch of sadness in his tone.
Thus, the list of premises continues to grow without end, leaving the argument always in the form:
•
•
•
•
•
•
•
(1): "Things that are equal to the same are equal to each other"
(2): "The two sides of this triangle are things that are equal to the same"
(3): (1) and (2) ⇒ (Z)
(4): (1) and (2) and (3) ⇒ (Z)
...
(n): (1) and (2) and (3) and (4) and ... and (n − 1) ⇒ (Z)
Therefore (Z): "The two sides of this triangle are equal to each other"
At each step, the Tortoise argues that even though he accepts all the premises that have been written down, there is
some further premise (that if all of (1)–(n) are true, then (Z) must be true) that it still needs to accept before it is
compelled to accept that (Z) is true.
122
What the Tortoise Said to Achilles
Explanation
Lewis Carroll was showing that there's a regress problem that arises from modus ponens deductions.
(1) P ⇒ Q
(2) P
--------------Therefore, Q.
The regress problem arises, because, in order to explain the logical principle, we have to then propose a prior
principle. And, once we explain that principle, then we have to introduce another principle to explain that principle.
Thus, if the causal chain is to continue, we are to fall into infinite regress. However, if we introduce a formal system
where modus ponens is simply an axiom, then we are to abide by it simply, because it is so. For example, in a chess
game there are particular rules, and the rules simply go without question. As players of the chess game, we are to
simply follow the rules. Likewise, if we are engaging in a formal system of logic, then we are to simply follow the
rules without question. Hence, introducing the formal system of logic stops the infinite regression—that is, because
the regress would stop at the axioms or rules, per se, of the given game, system, etc. Though, it does also state that
there are problems with this as well, because, within the system, no proposition or variable carries with it any
semantic content. So, the moment you add to any proposition or variable semantic content, the problem arises again,
because the propositions and variables with semantic content run outside the system. Thus, if the solution is to be
said to work, then it is to be said to work solely within the given formal system, and not otherwise.
Some logicians (Kenneth Ross, Charles Wright) draw a firm distinction between the conditional connective (the
syntactic sign "→"), and the implication relation (the formal object denoted by the double arrow symbol "⇒"). These
logicians use the phrase not p or q for the conditional connective and the term implies for the implication relation.
Some explain the difference by saying that the conditional is the contemplated relation while the implication is the
asserted relation. In most fields of mathematics, it is treated as a variation in the usage of the single sign "⇒," not
requiring two separate signs. Not all of those who use the sign "→" for the conditional connective regard it as a sign
that denotes any kind of object, but treat it as a so-called syncategorematic sign, that is, a sign with a purely syntactic
function. For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign
notation, but allow the sign "→" to denote the boolean function that is associated with the truth table of the material
conditional.
These considerations result in the following scheme of notation.
The paradox ceases to exist the moment we replace informal logic with propositional logic. The Turtle and Achilles
don't agree on any definition of logical implication. In propositional logic the logical implication is defined as
follows:
P ⇒ Q if and only if the proposition P → Q is a tautology
hence de modus ponens [P ∧ (P → Q)] ⇒ Q, is a valid logical implication according to the definition of logical
implication just stated. There is no need to recurse since the logical implication can be translated into symbols, and
propositional operators such as →. Demonstrating the logical implication simply translates into verifying that the
compound truth table is producing a tautology.
123
What the Tortoise Said to Achilles
Discussion
Several philosophers have tried to resolve the Carroll paradox. Bertrand Russell discussed the paradox briefly in § 38
of The Principles of Mathematics [1] (1903), distinguishing between implication (associated with the form "if p, then
q"), which he held to be a relation between unasserted propositions, and inference (associated with the form "p,
therefore q"), which he held to be a relation between asserted propositions; having made this distinction, Russell
could deny that the Tortoise's attempt to treat inferring Z from A and B is equivalent to, or dependent on, agreeing to
the hypothetical "If A and B are true, then Z is true."
The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation
to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference,
which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to
infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do
something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to
the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a
set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts"
(p. 53).
Where to find the article
• Carroll, Lewis. "What the Tortoise Said to Achilles". Mind, n.s., 4 (1895), pp. 278–80.
• Hofstadter, Douglas. Gödel, Escher, Bach: an Eternal Golden Braid. See the second dialogue, entitled "Two-Part
Invention." Dr. Hofstadter appropriated the characters of Achilles and the Tortoise for other, original, dialogues in
the book which alternate contrapuntally with prose chapters. Hofstadter's Tortoise is of the male sex, though the
Tortoise's sex is never specified by Carroll. The French translation of the book rendered the Tortoise's name as
"Madame Tortue."
• A number of websites, including "What the Tortoise Said to Achilles" [2] at the Lewis Carroll Society of North
America [3], "What the Tortoise Said to Achilles" [4] at Digital Text International [5], and "What the Tortoise Said
to Achilles" [6] at Fair Use Repository [7].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
http:/ / fair-use. org/ bertrand-russell/ the-principles-of-mathematics/ s. 38
http:/ / www. lewiscarroll. org/ achilles. html
http:/ / www. lewiscarroll. org
http:/ / www. ditext. com/ carroll/ tortoise. html
http:/ / www. ditext. com/
http:/ / fair-use. org/ mind/ 1895/ 04/ what-the-tortoise-said-to-achilles
http:/ / fair-use. org
124
125
Mathematics
Accuracy paradox
The accuracy paradox for predictive analytics states that predictive models with a given level of accuracy may have
greater predictive power than models with higher accuracy. It may be better to avoid the accuracy metric in favor of
other metrics such as precision and recall.
Accuracy is often the starting point for analyzing the quality of a predictive model, as well as an obvious criterion for
prediction. Accuracy measures the ratio of correct predictions to the total number of cases evaluated. It may seem
obvious that the ratio of correct predictions to cases should be a key metric. A predictive model may have high
accuracy, but be useless.
In an example predictive model for an insurance fraud application, all cases that are predicted as high-risk by the
model will be investigated. To evaluate the performance of the model, the insurance company has created a sample
data set of 10,000 claims. All 10,000 cases in the validation sample have been carefully checked and it is known
which cases are fraudulent. To analyze the quality of the model, the insurance uses the table of confusion. The
definition of accuracy, the table of confusion for model M1Fraud, and the calculation of accuracy for model M1Fraud is
shown below.
where
TN is the number of true negative cases
FP is the number of false positive cases
FN is the number of false negative cases
TP is the number of true positive cases
Formula 1: Definition of Accuracy
Predicted Negative Predicted Positive
Negative Cases 9,700
150
Positive Cases
100
50
Table 1: Table of Confusion for Fraud Model M1Fraud.
Formula 2: Accuracy for model M1Fraud
With an accuracy of 98.0% model M1Fraud appears to perform fairly well. The paradox lies in the fact that accuracy
can be easily improved to 98.5% by always predicting "no fraud". The table of confusion and the accuracy for this
trivial “always predict negative” model M2Fraud and the accuracy of this model are shown below.
Accuracy paradox
126
Predicted Negative Predicted Positive
Negative Cases 9,850
0
Positive Cases
0
150
Table 2: Table of Confusion for Fraud Model M2Fraud.
Formula 3: Accuracy for model M2Fraud
Model M2Fraudreduces the rate of inaccurate predictions from 2% to 1.5%. This is an apparent improvement of 25%.
The new model M2Fraud shows fewer incorrect predictions and markedly improved accuracy, as compared to the
original model M1Fraud, but is obviously useless.
The alternative model M2Fraud does not offer any value to the company for preventing fraud. The less accurate model
is more useful than the more accurate model.
Model improvements should not be measured in terms of accuracy gains. It may be going too far to say that accuracy
is irrelevant, but caution is advised when using accuracy in the evaluation of predictive models.
Bibliography
• Zhu, Xingquan (2007), Knowledge Discovery and Data Mining: Challenges and Realities [1], IGI Global,
pp. 118–119, ISBN 978-1-59904-252-7
• doi:10.1117/12.785623
• pp 86-87 of this Master's thesis [2]
References
[1] http:/ / books. google. com/ ?id=zdJQAAAAMAAJ& q=data+ mining+ challenges+ and+ realities& dq=data+ mining+ challenges+ and+
realities
[2] http:/ / www. utwente. nl/ ewi/ trese/ graduation_projects/ 2009/ Abma. pdf
Apportionment paradox
Apportionment paradox
An apportionment paradox exists when the rules for apportionment in a political system produce results which are
unexpected or seem to violate common sense.
To apportion is to divide into parts according to some rule, the rule typically being one of proportion. Certain
quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole
numbers will do. In the latter case, there is an inherent tension between our desire to obey the rule of proportion as
closely as possible and the constraint restricting the size of each portion to discrete values. This results, at times, in
unintuitive observations, or paradoxes.
Several paradoxes related to apportionment, also called fair division, have been identified. In some cases, simple
adjustments to an apportionment methodology can resolve observed paradoxes. Others, such as those relating to the
United States House of Representatives, call into question notions that mathematics alone can provide a single, fair
resolution.
History
The Alabama paradox was discovered in 1880, when it was found that increasing the total number of seats would
decrease Alabama's share from 8 to 7. There was more to come: when Oklahoma became a state in 1907, a
recomputation of apportionment showed that the number of seats due to other states would be affected even though
Oklahoma would be given a fair share of seats and the total number of seats increased by that number.
The method for apportionment used during this period, originally put forth by Alexander Hamilton but not adopted
until 1852, was as follows (after meeting the requirements of the United States Constitution, wherein each state must
be allocated at least one seat in the House of Representatives, regardless of population):
• First, the fair share of each state, i.e. the proportional share of seats that each state would get if fractional values
were allowed, is computed.
• Next, the fair shares are rounded down to whole numbers, resulting in unallocated "leftover" seats. These seats are
allocated, one each, to the states whose fair share exceeds the rounded-down number by the highest amount.
Impossibility result
In 1982 two mathematicians, Michel Balinski and Peyton Young, proved that any method of apportionment will
result in paradoxes whenever there are three or more parties (or states, regions, etc.).[1][2] More precisely, their
theorem states that there is no apportionment system that has the following properties (as the example we take the
division of seats between parties in a system of proportional representation):
• It follows the quota rule: Each of the parties gets one of the two numbers closest to its fair share of seats (if the
party's fair share is 7.34 seats, it gets either 7 or 8).
• It does not have the Alabama paradox: If the total number of seats is increased, no party's number of seats
decreases.
• It does not have the population paradox: If party A gets more votes and party B gets fewer votes, no seat will be
transferred from A to B.
127
Apportionment paradox
128
Examples of paradoxes
Alabama paradox
The Alabama paradox was the first of the apportionment paradoxes to be discovered. The US House of
Representatives is constitutionally required to allocate seats based on population counts, which are required every 10
years. The size of the House is set by statute.
After the 1880 census, C. W. Seaton, chief clerk of the United States Census Bureau, computed apportionments for
all House sizes between 275 and 350, and discovered that Alabama would get 8 seats with a House size of 299 but
only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where
increasing the total number of items would decrease one of the shares. A similar exercise by the Census Bureau after
the 1900 census computed apportionments for all House sizes between 350 and 400: Colorado would have received
three seats in all cases, except with a House size of 357 in which case it would have received two.[3]
The following is a simplified example (following the largest remainder method) with three states and 10 seats and 11
seats.
With 10 seats
With 11 seats
State Population Fair share Seats Fair share Seats
A
6
4.286
4
4.714
5
B
6
4.286
4
4.714
5
C
2
1.429
2
1.571
1
Observe that state C's share decreases from 2 to 1 with the added seat.
This occurs because increasing the number of seats increases the fair share faster for the large states than for the
small states. In particular, large A and B had their fair share increase faster than small C. Therefore, the fractional
parts for A and B increased faster than those for C. In fact, they overtook C's fraction, causing C to lose its seat, since
the Hamilton method examines which states have the largest fraction.
New states paradox
Given a fixed number of total representatives (as determined by the United States House of Representatives), adding
a new state would in theory reduce the number of representatives for existing states, as under the United States
Constitution each state is entitled to at least one representative regardless of its population. However, because of how
the particular apportionment rules deal with rounding methods, it is possible for an existing state to get more
representatives than if the new state were not added.
Population paradox
The population paradox is a counterintuitive result of some procedures for apportionment. When two states have
populations increasing at different rates, a small state with rapid growth can lose a legislative seat to a big state with
slower growth.
The paradox arises because of rounding in the procedure for dividing the seats. See the apportionment rules for the
United States Congress for an example.
Apportionment paradox
External links
• The Constitution and Paradoxes [4]
• Alabama Paradox [5]
• New States Paradox [6]
• Population Paradox [7]
• Apportionment: Balinski and Young's Contribution [8]
References
[1] Balinski, Michel; H. Peyton Young (1982). Fair Representation: Meeting the Ideal of One Man, One Vote. Yale Univ Pr.
ISBN 0-300-02724-9.
[2] Balinski, Michel; H. Peyton Young (2001). Fair Representation: Meeting the Ideal of One Man, One Vote (2nd ed.). Brookings Institution
Press. ISBN 0-8157-0111-X.
[3] Cut-the-knot: The Constitution and Paradoxes (http:/ / www. cut-the-knot. org/ ctk/ Democracy. shtml)
[4] http:/ / www. cut-the-knot. org/ ctk/ Democracy. shtml
[5] http:/ / www. cut-the-knot. org/ ctk/ Democracy. shtml#alabama
[6] http:/ / www. cut-the-knot. org/ ctk/ Democracy. shtml#new-states
[7] http:/ / www. cut-the-knot. org/ ctk/ Democracy. shtml#population
[8] http:/ / www. ams. org/ featurecolumn/ archive/ apportionII3. html
Banach–Tarski paradox
The Banach–Tarski paradox is a
theorem in set-theoretic geometry
which states the following: Given a
solid ball in 3‑dimensional space, there
exists a decomposition of the ball into
Can a ball be decomposed into a finite number of point sets and reassembled into two
a finite number of non-overlapping
balls identical to the original?
pieces (i.e., subsets), which can then be
put back together in a different way to
yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and
rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but
infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects
(such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated colloquially
as "a pea can be chopped up and reassembled into the Sun".
The reason the Banach–Tarski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling
the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching,
bending, or adding new points, seems to be impossible, since all these operations preserve the volume, but the
volume is doubled in the end.
Unlike most theorems in geometry, this result depends in a critical way on the axiom of choice in set theory. This
axiom allows for the construction of nonmeasurable sets, collections of points that do not have a volume in the
ordinary sense and for their construction would require performing an uncountably infinite number of choices.
It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved
continuously into place without running into one another.[1]
129
BanachTarski paradox
Banach and Tarski publication
In a paper published in 1924,[2] Stefan Banach and Alfred Tarski gave a construction of such a "paradoxical
decomposition", based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical
decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning
decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general
statement, the strong form of the Banach–Tarski paradox:
Given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have
a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, A = A1 ∪ ... ∪ Ak,
B = B1 ∪ ... ∪ Bk, such that for each i between 1 and k, the sets Ai and Bi are congruent.
Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition
means that you can divide the original ball A into a certain number of pieces and then rotate and translate these
pieces in such a way that the result is the whole set B, which contains two copies of A.
The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed
that an analogous statement remains true if countably many subsets are allowed. The difference between the
dimensions 1 and 2 on the one hand, and three and higher, on the other hand, is due to the richer structure of the
group Gn of the Euclidean motions in the higher dimensions, which is solvable for n =1, 2 and contains a free group
with two generators for n ≥ 3. John von Neumann studied the properties of the group of equivalences that make a
paradoxical decomposition possible and introduced the notion of amenable groups. He also found a form of the
paradox in the plane which uses area-preserving affine transformations in place of the usual congruences.
Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist.
Formal treatment
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the
operations of partitioning into subsets, replacing a set with a congruent set, and reassembly. Its mathematical
structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the
notions of equidecomposable sets and paradoxical set. Suppose that G is a group acting on a set X. In the most
important special case, X is an n-dimensional Euclidean space, and G consists of all isometries of X, i.e. the
transformations of X into itself that preserve the distances. Two geometric figures that can be transformed into each
other are called congruent, and this terminology will be extended to the general G-action. Two subsets A and B of X
are called G-equidecomposable, or equidecomposable with respect to G, if A and B can be partitioned into the
same finite number of respectively G-congruent pieces. It is easy to see that this defines an equivalence relation
among all subsets of X. Formally, if
and there are elements g1,...,gk of G such that for each i between 1 and k, gi (Ai ) = Bi , then we will say that A and B
are G-equidecomposable using k pieces. If a set E has two disjoint subsets A and B such that A and E, as well as B
and E, are G-equidecomposable then E is called paradoxical.
Using this terminology, the Banach–Tarski paradox can be reformulated as follows:
A three-dimensional Euclidean ball is equidecomposable with two copies of itself.
In fact, there is a sharp result in this case, due to Robinson[3]: doubling the ball can be accomplished with five pieces,
and fewer than five pieces will not suffice.
The strong version of the paradox claims:
Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable.
130
BanachTarski paradox
While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a
generalization of the Bernstein–Schroeder theorem due to Banach that implies that if A is equidecomposable with a
subset of B and B is equidecomposable with a subset of A, then A and B are equidecomposable.
The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox,
there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the
language of Georg Cantor's set theory, these two sets have equal cardinality. Thus, if one enlarges the group to allow
arbitrary bijections of X then all sets with non-empty interior become congruent. Likewise, we can make one ball
into a larger or smaller ball by stretching, in other words, by applying similarity transformations. Hence if the group
G is large enough, we may find G-equidecomposable sets whose "size" varies. Moreover, since a countable set can
be made into two copies of itself, one might expect that somehow, using countably many pieces could do the trick.
On the other hand, in the Banach–Tarski paradox the number of pieces is finite and the allowed equivalences are
Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball!
While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets,
so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be
accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a
finitely-additive measure (or a Banach measure) defined on all subsets of a Euclidean space of three (and greater)
dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube. In his later
work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence
of a finitely-additive invariant measure.
The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a
Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere)
into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a
F2-paradoxical decomposition of F2, the free group with two generators. Banach and Tarski's proof relied on an
analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of
three sets B, C, D and a countable set E such that, on the one hand, B, C, D are pairwise congruent, and, on the other
hand, B is congruent with the union of C and D. This is often called the Hausdorff paradox.
Connection with earlier work and the role of the axiom of choice
Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name,
Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work. Vitali's and
Hausdorff's constructions depend on Zermelo's axiom of choice ("AC"), which is also crucial to the Banach–Tarski
paper, both for proving their paradox and for the proof of another result:
Two Euclidean polygons, one of which strictly contains the other, are not equidecomposable.
They remark:
Le rôle que joue cet axiome dans nos raisonnements nous semble mériter l'attention
(The role this axiom plays in our reasoning seems to us to deserve attention)
and point out that while the second result fully agrees with our geometric intuition, its proof uses AC in an even
more substantial way than the proof of the paradox. Thus Banach and Tarski imply that AC should not be rejected
simply because it produces a paradoxical decomposition, for such an argument also undermines proofs of
geometrically intuitive statements.
However, in 1949 A.P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory
and thus does not require the axiom of choice. In 1964, Paul Cohen proved that the axiom of choice cannot be
proved from ZF. A weaker version of an axiom of choice is the axiom of dependent choice, DC. It has been shown
that
The Banach–Tarski paradox is not a theorem of ZF, nor of ZF+DC (Wagon, Corollary 13.3).
131
BanachTarski paradox
132
Large amounts of mathematics use AC. As Stan Wagon points out at the end of his monograph, the Banach–Tarski
paradox has been more significant for its role in pure mathematics than for foundational questions: it motivated a
fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational
questions.
In 1991, using then-recent results by Matthew Foreman and Friedrich Wehrung,[4] Janusz Pawlikowski proved that
the Banach–Tarski paradox follows from ZF plus the Hahn–Banach theorem.[5] The Hahn–Banach theorem doesn't
rely on the full axiom of choice but can be proved using a weaker version of AC called the ultrafilter lemma. So
Pawlikowski proved that the set theory needed to prove the Banach–Tarski paradox, while stronger than ZF, is
weaker than full ZFC.
A sketch of the proof
Here we sketch a proof which is similar but not identical to that given by Banach and Tarski. Essentially, the
paradoxical decomposition of the ball is achieved in four steps:
1. Find a paradoxical decomposition of the free group in two generators.
2. Find a group of rotations in 3-d space isomorphic to the free group in two generators.
3. Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition
of the hollow unit sphere.
4. Extend this decomposition of the sphere to a decomposition of the solid unit ball.
We now discuss each of these steps in more detail.
The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a,
a−1, b and b−1 such that no a appears directly next to an a−1 and no b appears directly next to a b−1. Two such strings
can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with
the empty string. For instance: abab−1a−1 concatenated with abab−1a yields abab−1a−1abab−1a, which contains the
substring a−1a, and so gets reduced to abaab−1a. One can check that the set of those strings with this operation forms
a group with identity element the empty string e. We will call this group F2.
The group
can be "paradoxically decomposed" as
follows: let S(a) be the set of all strings that start with a
and define S(a−1), S(b) and S(b−1) similarly. Clearly,
but also
and
The notation aS(a−1) means take all the strings in
S(a−1) and concatenate them on the left with a.
Make sure that you understand this last line, because it
is at the core of the proof. For example, there may be a
string
in the set
which, because of
the rule that
must not appear next to
to the string
. In this way,
strings that start with
, reduces
The sets S(a−1) and aS(a−1) in the Cayley graph of F2
contains all the
. Similarly, it contains all the strings that start with
(for example the string
which reduces to
).
We have cut our group F2 into four pieces (plus the singleton {e}), then "shifted" two of them by multiplying with a
or b, then "reassembled" two pieces to make one copy of
and the other two to make another copy of
. That is
BanachTarski paradox
exactly what we want to do to the ball.
Step 2
In order to find a group of rotations of 3D space that behaves just like (or "isomorphic to") the group F2, we take two
orthogonal axes, e.g. the x and z axes, and let A be a rotation of arccos(1/3) about the first, x axis, and B be a rotation
of arccos(1/3) about the second, z axis (there are many other suitable pairs of irrational multiples of π, that could be
used here instead of arccos(1/3) and arccos(1/3), as well). It is somewhat messy but not too difficult to show that
these two rotations behave just like the elements a and b in our group F2. We shall skip it, leaving the exercise to the
reader. The new group of rotations generated by A and B will be called H. We now also have a paradoxical
decomposition of H. (This step cannot be performed in two dimensions since it involves rotations in three
dimensions. If we take two rotations about the same axis, the resulting group is commutative and doesn't have the
property required in step 1.)
Step 3
The unit sphere S2 is partitioned into orbits by the action of our group H: two points belong to the same orbit if and
only if there's a rotation in H which moves the first point into the second. (Note that the orbit of a point is a dense set
in S2.) We can use the axiom of choice to pick exactly one point from every orbit; collect these points into a set M.
Now (almost) every point in S2 can be reached in exactly one way by applying the proper rotation from H to the
proper element from M, and because of this, the paradoxical decomposition of H then yields a paradoxical
decomposition of S2 into four pieces A1, A2, A3, A4 as follows:
where:
(We didn't use the five "paradoxical" parts of F2 directly, as they would leave us with M as an extra piece after
doubling, due to the presence of the singleton {e}!)
The (majority of the) sphere has now been divided into four sets (each one dense on the sphere), and when two of
these are rotated, we end up with double what we had before:
Step 4
Finally, connect every point on S2 with a ray to the origin; the paradoxical decomposition of S2 then yields a
paradoxical decomposition of the solid unit ball minus the point at the ball's centre (this center point needs a bit more
care, see below).
N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen
to lie on the axis of some rotation in H. However, there are only countably many such points, and like the point at
the centre of the ball, it is possible to patch the proof to account for them all (see below).
133
BanachTarski paradox
Some details, fleshed out
In Step 3, we partitioned the sphere into orbits of our group H. To streamline the proof, we omitted the discussion of
points that are fixed by some rotation; since the paradoxical decomposition of F2 relies on shifting certain subsets,
the fact that some points are fixed might cause some trouble. Since any rotation of S2 (other than the null rotation)
has exactly two fixed points, and since H, which is isomorphic to F2, is countable, there are countably many points
of S2 that are fixed by some rotation in H, denote this set of fixed points D. Step 3 proves that S2 − D admits a
paradoxical decomposition.
What remains to be shown is the Claim: S2 − D is equidecomposable with S2.
Proof. Let λ be some line through the origin that does not intersect any point in D– this is possible since D is
countable. Let J be the set of angles, α, such that for some natural number n, and some P in D, r(nα)P is also in D,
where r(nα) is a rotation about λ of nα. Then J is countable so there exists an angle θ not in J. Let ρ be the rotation
about λ by θ, then ρ acts on S2 with no fixed points in D, i.e., ρn(D) is disjoint from D, and for natural m<n, ρn(D) is
disjoint from ρm(D). Let E be the disjoint union of ρn(D) over n = 0, 1, 2, .... Then S2 = E∪ (S2− E) ~ ρ(E) ∪ (S2 −
E) = (E − D) ∪ (S2 − E) = S2 − D, where ~ denotes "is equidecomposable to".
For step 4, it has already been shown that the ball minus a point admits a paradoxical decomposition; it remains to be
shown that the ball minus a point is equidecomposable with the ball. Consider a circle within the ball, containing the
point at the centre of the ball. Using an argument like that used to prove the Claim, one can see that the full circle is
equidecomposable with the circle minus the point at the ball's centre. (Basically, a countable set of points on the
circle can be rotated to give itself plus one more point.) Note that this involves the rotation about a point other than
the origin, so the Banach–Tarski paradox involves isometries of Euclidean 3-space rather than just SO(3).
We are using the fact that if A ~ B and B ~ C, then A ~ C. The decomposition of A into C can be done using number
of pieces equal to the product of the numbers needed for taking A into B and for taking B into C.
The proof sketched above requires 2×4×2 + 8 = 24 pieces, a factor of 2 to remove fixed points, a factor 4 from step
1, a factor 2 to recreate fixed points, and 8 for the center point of the second ball. But in step 1 when moving {e} and
all strings of the form an into S(a−1), do this to all orbits except one. Move {e} of this last orbit to the center point of
the second ball. This brings the total down to 16 + 1 pieces. With more algebra one can also decompose fixed orbits
into 4 sets as in step 1. This gives 5 pieces and is the best possible.
Obtaining infinitely many balls from one
Using the Banach–Tarski paradox, it is possible to obtain k copies of a ball in the Euclidean n-space from one, for
any integers n ≥ 3 and k ≥ 1, i.e. a ball can be cut into k pieces so that each of them is equidecomposable to a ball of
the same size as the original. Using the fact that the free group F2 of rank 2 admits a free subgroup of countably
infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces,
each of which is equidecomposable (with two pieces) to the Sn−1 using rotations. By using analytic properties of the
rotation group SO(n), which is a connected analytic Lie group, one can further prove that the sphere Sn−1 can be
partitioned into as many pieces as there are real numbers (that is,
pieces), so that each piece is
equidecomposable with two pieces to Sn−1 using rotations. These results then extend to the unit ball deprived of the
origin. A 2010 article by Vitaly Churkin gives a new proof of the continuous version of the Banach–Tarski
paradox.[6]
134
BanachTarski paradox
The von Neumann paradox in the Euclidean plane
In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are
necessarily of the same area, therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that
uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and
higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions,
the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive
measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions
of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be
constructed if one allowed a larger group of equivalences?
It is clear that if one permits similarities, any two squares in the plane become equivalent even without further
subdivision. This motivates restricting one's attention to the group SA2 of area-preserving affine transformations.
Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be
counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group SA2 contains
as a subgroup the special linear group SL(2,R), which in its turn contains the free group F2 with two generators as a
subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main
difficulty here lies in the fact that the unit square is not invariant under the action of the linear group SL(2, R), hence
one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the
above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example,
the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA2 including
the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group
(in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows:
Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect
to the area-preserving affine maps.
As von Neumann notes,[7]
"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das
Maß 1 hat), dass [sic] gegenüber allen Abbildungen von A2 invariant wäre."
"In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit
square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of
area-preserving affine transformations]."
To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain
transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is
preserved by translations and rotations, is not preserved by non-isometric transformations even when they do
preserve the area of polygons. The points of the plane (other than the origin) can be divided into two dense sets
which we may call A and B. If the A points of a given polygon are transformed by a certain area-preserving
transformation and the B points by another, both sets can become subsets of the A points in two new polygons. The
new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as
before (since they contain only part of the A points), and therefore there is no measure that "works".
The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be
very important for many areas of mathematics: these are amenable groups, or groups with an invariant mean, and
include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used
for equivalences in the definition of equidecomposability is not amenable.
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BanachTarski paradox
Recent progress
• 2000. Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit
square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that
such a decomposition exists.[8] More precisely, let A be the family of all bounded subsets of the plane with
non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property
that a union of finitely many translates under some elements of SL(2, R) contains a punctured neighbourhood of
the origin. Then all sets in the family A are SL(2, R)-equidecomposable, and likewise for the sets in B. It follows
that both families consist of paradoxical sets.
• Topoi do not assume the axiom of choices, so categorical proofs done on topoi sometimes re-create desired
results without the undesired assumption.
• 2003. It had been known for a long time that the full plane was paradoxical with respect to SA2, and that the
minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of
SA2. In 2003 Kenzi Satô constructed such a subgroup, confirming that four pieces suffice.[9]
Notes
[1] Wilson, Trevor M. (September 2005). "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem".
Journal of Symbolic Logic 70 (3): 946–952. doi:10.2178/jsl/1122038921. JSTOR 27588401.
[2] Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (http:/ /
matwbn. icm. edu. pl/ ksiazki/ fm/ fm6/ fm6127. pdf) (in French). Fundamenta Mathematicae 6: 244–277. .
[3] Robinson, R. M. (1947). "On the Decomposition of Spheres." Fund. Math. 34:246–260. This article, based on an analysis of the Hausdorff
paradox, settled a question put forth by von Neumann in 1929.
[4] Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (http:/ / matwbn.
icm. edu. pl/ ksiazki/ fm/ fm138/ fm13812. pdf). Fundamenta Mathematicae 138: 13–19. .
[5] Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox" (http:/ / matwbn. icm. edu. pl/ ksiazki/ fm/
fm138/ fm13813. pdf). Fundamenta Mathematicae 138: 21–22. .
[6] Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox". Algebra and Logic 49 (1): 81–89.
doi:10.1007/s10469-010-9080-y.
[7] On p. 85. Neumann, J. v. (1929). "Zur allgemeinen Theorie des Masses" (http:/ / matwbn. icm. edu. pl/ ksiazki/ fm/ fm13/ fm1316. pdf).
Fundamenta Mathematica 13: 73–116. .
[8] Laczkovich, Miklós (1999). "Paradoxical sets under SL2(R)". Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42: 141–145.
[9] Satô, Kenzi (2003). "A locally commutative free group acting on the plane". Fundamenta Mathematica 180 (1): 25–34.
References
• Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement
congruentes" (http://matwbn.icm.edu.pl/ksiazki/fm/fm6/fm6127.pdf) (PDF). Fundamenta Mathematicae 6:
244–277.
• Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox". Algebra and Logic 49
(1): 91–98. doi:10.1007/s10469-010-9080-y.
• Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 205–7, Simon & Schuster.
• Kuro5hin. "Layman's Guide to the Banach–Tarski Paradox" (http://www.kuro5hin.org/story/2003/5/23/
134430/275).
• Stromberg, Karl (March 1979). "The Banach–Tarski paradox". The American Mathematical Monthly
(Mathematical Association of America) 86 (3): 151–161. doi:10.2307/2321514. JSTOR 2321514.
• Su, Francis E.. "The Banach–Tarski Paradox" (http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf)
(PDF).
• von Neumann, John (1929). "Zur allgemeinen Theorie des Masses" (http://matwbn.icm.edu.pl/ksiazki/fm/
fm13/fm1316.pdf) (PDF). Fundamenta Mathematicae 13: 73–116.
• Wagon, Stan (1994). The Banach–Tarski Paradox. Cambridge: Cambridge University Press.
ISBN 0-521-45704-1.
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BanachTarski paradox
• Wapner, Leonard M. (2005). The Pea and the Sun: A Mathematical Paradox (http://gen.lib.rus.ec/
get?md5=59f223482492f1644b1023fccd4968f1). Wellesley, Mass.: A.K. Peters. ISBN 1-56881-213-2.
External links
• The Banach-Tarski Paradox (http://demonstrations.wolfram.com/TheBanachTarskiParadox/) by Stan Wagon
(Macalester College), the Wolfram Demonstrations Project.
• Irregular Webcomic! #2339 (http://www.irregularwebcomic.net/2339.html) by David Morgan-Mar provides a
non-technical explanation of the paradox. It includes a step-by-step demonstration of how to create two spheres
from one.
• Banach-Tarski Video (http://www.youtube.com/watch?v=uFvokQUHh08) Members of the University of
Copenhagen mathematics department exploit the Banach–Tarski paradox, decomposing and reassembling a
multitude of oranges to the (slightly modified) tune of Duck Sauce's "Barbra Streisand."
Berkson's paradox
Berkson's paradox or Berkson's fallacy is a result in conditional probability and statistics which is counterintuitive
for some people, and hence a veridical paradox. It is a complicating factor arising in statistical tests of proportions.
Specifically, it arises when there is an ascertainment bias inherent in a study design.
It is often described in the fields of medical statistics or biostatistics, as in the original description of the problem by
Joseph Berkson.
Statement
The result is that two independent events become conditionally dependent (negatively dependent) given that at least
one of them occurs. Symbolically:
if 0 < P(A) < 1 and 0 < P(B) < 1,
and P(A|B) = P(A), i.e. they are independent,
then P(A|B,C) < P(A|C) where C = A∪B (i.e. A or B).
In words, given two independent events, if you only consider outcomes where at least one occurs, then they become
negatively dependent.
Explanation
The cause is that the conditional probability of event A occurring, given that it or B occurs, is inflated: it is higher
than the unconditional probability, because we have excluded cases where neither occur.
P(A|A∪B) > P(A)
conditional probability inflated relative to unconditional
One can see this in tabular form as follows: the gray regions are the outcomes where at least one event occurs (and
~A means "not A").
137
Berkson's paradox
138
B
A
~A
A&B
~A & B
~B A & ~B ~A & ~B
For instance, if one has a sample of 100, and both A and B occur independently half the time (So P(A) = P(B) = 1/2),
one obtains:
A ~A
B
25 25
~B 25 25
So in 75 outcomes, either A or B occurs, of which 50 have A occurring, so
P(A|A∪B) = 50/75 = 2/3 > 1/2 = 50/100 = P(A).
Thus the probability of A is higher in the subset (of outcomes where it or B occurs), 2/3, than in the overall
population, 1/2.
Berkson's paradox arises because the conditional probability of A given B within this subset equals the conditional
probability in the overall population, but the unconditional probability within the subset is inflated relative to the
unconditional probability in the overall population, hence, within the subset, the presence of B decreases the
conditional probability of A (back to its overall unconditional probability):
P(A|B, A∪B) = P(A|B) = P(A)
P(A|A∪B) > P(A).
Examples
A classic illustration involves a retrospective study examining a risk factor for a disease in a statistical sample from a
hospital in-patient population. If a control group is also ascertained from the in-patient population, a difference in
hospital admission rates for the case sample and control sample can result in a spurious association between the
disease and the risk factor.
As another example, suppose a collector has 1000 postage stamps, of which 300 are pretty and 100 are rare, with 30
being both pretty and rare. 10% of all her stamps are rare and 10% of her pretty stamps are rare, so prettiness tells
nothing about rarity. She puts the 370 stamps which are pretty or rare on display. Just over 27% of the stamps on
display are rare, but still only 10% of the pretty stamps on display are rare (and 100% of the 70 not-pretty stamps on
display are rare). If an observer only considers stamps on display, he will observe a spurious negative relationship
between prettiness and rarity as a result of the selection bias (that is, not-prettiness strongly indicates rarity in the
display, but not in the total collection).
References
• Berkson, J. (1946) "Limitations of the application of fourfold tables to hospital data". Biometrics Bulletin, 2(3),
47-53. The paper is frequently miscited as Berkson, J. (1949) Biological Bulletin 2, 47-53. http://dx.doi.org/
10.2307/3002000
Bertrand's box paradox
Bertrand's box paradox
Bertrand's box paradox is a classic paradox of elementary probability theory. It was first posed by Joseph Bertrand
in his Calcul des probabilités, published in 1889.
There are three boxes:
1. a box containing two gold coins,
2. a box with two silver coins, and
3. a box with one of each.
After choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, it may seem
that the probability that the remaining coin is gold is 1⁄2; in fact, the probability is actually 2⁄3. Two problems that are
logically equivalent are the Monty Hall problem and the Three Prisoners problem.
These simple but slightly counterintuitive puzzles are used as a standard example in teaching probability theory.
Their solution illustrates some basic principles, including the Kolmogorov axioms.
Box version
There are three boxes, each with one drawer on each of two sides. Each drawer contains a coin. One box has a gold
coin on each side (GG), one a silver coin on each side (SS), and the other a gold coin on one side and a silver coin
on the other (GS). A box is chosen at random, a random drawer is opened, and a gold coin is found inside it. What is
the chance of the coin on the other side being gold?
The following reasoning appears to give a probability of 1⁄2:
•
•
•
•
Originally, all three boxes were equally likely to be chosen.
The chosen box cannot be box SS.
So it must be box GG or GS.
The two remaining possibilities are equally likely, so the probability that the box is GG, and the other coin is
also gold, is 1⁄2.
The flaw is in the last step. While those two cases were originally equally likely, the fact that you could not have
found a silver coin if you had chosen the GG box, but could if you had chosen the GS box, means they do not
remain equally likely. Specifically:
• The probability that GG would produce a gold coin is 1.
• The probability that SS would produce a gold coin is 0.
• The probability that GS would produce a gold coin is 1⁄2.
So the probability that the chosen box is GG becomes:
The correct answer of 2⁄3 can also be obtained as follows:
•
•
•
•
Originally, all six coins were equally likely to be chosen.
The chosen coin cannot be from drawer S of box GS, or from either drawer of box SS.
So it must come from the G drawer of box GS, or either drawer of box GG.
The three remaining possibilities are equally likely, so the probability that the drawer is from box GG is 2⁄3.
Alternatively, one can simply note that the chosen box has two coins of the same type 2⁄3 of the time. So, regardless
of what kind of coin is in the chosen drawer, the box has two coins of that type 2⁄3 of the time. In other words, the
problem is equivalent to asking the question "What is the probability that I will pick a box with two coins of the
same color?".
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Bertrand's box paradox
Bertrand's point in constructing this example was to show that merely counting cases is not always proper. Instead,
one should sum the probabilities that the cases would produce the observed result; and the two methods are
equivalent only if this probability is either 1 or 0 in every case. This condition is correctly applied in the second
solution method, but not in the first.
The paradox as stated by Bertrand
It can be easier to understand the correct answer if you consider the paradox as Bertrand originally described it. After
a box has been chosen, but before a drawer is opened to let you observe a coin, the probability is 2/3 that the box has
two of the same kind of coin. If the probability of "observing a gold coin" in combination with "the box has two of
the same kind of coin" is 1/2, then the probability of "observing a silver coin" in combination with "the box has two
of the same kind of coin" must also be 1/2. And if the probability that the box has two like coins changes to 1/2 no
matter what kind of coin is shown, the probability would have to be 1/2 even if you hadn't observed a coin this way.
Since we know his probability is 2/3, not 1/2, we have an apparent paradox. It can be resolved only by recognizing
how the combination of "observing a gold coin" with each possible box can only affect the probability that the box
was GS or SS, but not GG.
Card version
Suppose there are three cards:
• A black card that is black on both sides,
• A white card that is white on both sides, and
• A mixed card that is black on one side and white on the other.
All the cards are placed into a hat and one is pulled at random and placed on a table. The side facing up is black.
What are the odds that the other side is also black?
The answer is that the other side is black with probability 2⁄3. However, common intuition suggests a probability of
1
⁄2 either because there are two cards with black on them that this card could be, or because there are 3 white and 3
black sides and many people forget to eliminate the possibility of the "white card" in this situation (i.e. the card they
flipped CANNOT be the "white card" because a black side was turned over).
In a survey of 53 Psychology freshmen taking an introductory probability course, 35 incorrectly responded 1⁄2; only
3 students correctly responded 2⁄3.53
Another presentation of the problem is to say : pick a random card out of the three, what are the odds that it has the
same color on the other side ? Since only one card is mixed and two have the same color on their sides, it is easier to
understand that the probability is 2⁄3. Also note that saying that the color is black (or the coin is gold) instead of
white doesn't matter since it is symmetric: the answer is the same for white. So is the answer for the generic question
'same color on both sides'.
Preliminaries
To solve the problem, either formally or informally, one must assign probabilities to the events of drawing each of
the six faces of the three cards. These probabilities could conceivably be very different; perhaps the white card is
larger than the black card, or the black side of the mixed card is heavier than the white side. The statement of the
question does not explicitly address these concerns. The only constraints implied by the Kolmogorov axioms are that
the probabilities are all non-negative, and they sum to 1.
The custom in problems when one literally pulls objects from a hat is to assume that all the drawing probabilities are
equal. This forces the probability of drawing each side to be 1⁄6, and so the probability of drawing a given card is 1⁄3.
In particular, the probability of drawing the double-white card is 1⁄3, and the probability of drawing a different card is
2
⁄3.
140
Bertrand's box paradox
In question, however, one has already selected a card from the hat and it shows a black face. At first glance it
appears that there is a 50/50 chance (i.e. probability 1⁄2) that the other side of the card is black, since there are two
cards it might be: the black and the mixed. However, this reasoning fails to exploit all of the information; one knows
not only that the card on the table has at least one black face, but also that in the population it was selected from,
only 1 of the 3 black faces was on the mixed card.
An easy explanation is that to name the black sides as x, y and z where x and y are on the same card while z is on the
mixed card, then the probability is divided on the 3 black sides with 1⁄3 each. thus the probability that we chose either
x or y is the sum of their probabilities thus 2⁄3.
Solutions
Intuition
Intuition tells one that one is choosing a card at random. However, one is actually choosing a face at random. There
are 6 faces, of which 3 faces are white and 3 faces are black. Two of the 3 black faces belong to the same card. The
chance of choosing one of those 2 faces is 2⁄3. Therefore, the chance of flipping the card over and finding another
black face is also 2⁄3. Another way of thinking about it is that the problem is not about the chance that the other side
is black, it's about the chance that you drew the all black card. If you drew a black face, then it's twice as likely that
that face belongs to the black card than the mixed card.
Alternately, it can be seen as a bet not on a particular color, but a bet that the sides match. Betting on a particular
color regardless of the face shown, will always have a chance of 1⁄2. However, betting that the sides match is 2⁄3,
because 2 cards match and 1 does not.
Labels
One solution method is to label the card faces, for example numbers 1 through 6.Label16 Label the faces of the black
card 1 and 2; label the faces of the mixed card 3 (black) and 4 (white); and label the faces of the white card 5 and 6.
The observed black face could be 1, 2, or 3, all equally likely; if it is 1 or 2, the other side is black, and if it is 3, the
other side is white. The probability that the other side is black is 2⁄3.
Bayes' theorem
Given that the shown face is black, the other face is black if and only if the card is the black card. If the black card is
drawn, a black face is shown with probability 1. The total probability of seeing a black face is 1⁄2; the total
probability of drawing the black card is 1⁄3. By Bayes' theorem,Bayes the conditional probability of having drawn the
black card, given that a black face is showing, is
Eliminating the white card
Although the incorrect solution reasons that the white card is eliminated, one can also use that information in a
correct solution. Modifying the previous method, given that the white card is not drawn, the probability of seeing a
black face is 3⁄4, and the probability of drawing the black card is 1⁄2. The conditional probability of having drawn the
black card, given that a black face is showing, is
141
Bertrand's box paradox
Symmetry
The probability (without considering the individual colors) that the hidden color is the same as the displayed color is
clearly 2⁄3, as this holds if and only if the chosen card is black or white, which chooses 2 of the 3 cards. Symmetry
suggests that the probability is independent of the color chosen, so that the information about which color is shown
does not affect the odds that both sides have the same color. (This argument can be formalized, but requires more
advanced mathematics than yet discussed.)
Experiment
Using specially constructed cards, the choice can be tested a number of times. By constructing a fraction with the
denominator being the number of times "B" is on top, and the numerator being the number of times both sides are
"B", the experimenter will probably find the ratio to be near 2⁄3.
Note the logical fact that the B/B card contributes significantly more (in fact twice) to the number of times "B" is on
top. With the card B/W there is always a 50% chance W being on top, thus in 50% of the cases card B/W is drawn,
the draw affects neither numerator nor denominator and effectively does not count (this is also true for all times
W/W is drawn, so that card might as well be removed from the set altogether). Conclusively, the cards B/B and B/W
are not of equal chances, because in the 50% of the cases B/W is drawn, this card is simply "disqualified".
Related problems
•
•
•
•
Boy or Girl paradox
Three Prisoners problem
Two envelopes problem
Sleeping Beauty problem
Notes and references
1. Bar-Hillel and Falk (page 119)
2. Nickerson (page 158) advocates this solution as "less confusing" than other methods.
3. Bar-Hillel and Falk (page 120) advocate using Bayes' Rule.
• Bar-Hillel, Maya; Falk, Ruma (1982). "Some teasers concerning conditional probabilities". Cognition 11 (2):
109–22. doi:10.1016/0010-0277(82)90021-X. PMID 7198956.
• Nickerson, Raymond (2004). Cognition and Chance: The psychology of probabilistic reasoning, Lawrence
Erlbaum. Ch. 5, "Some instructive problems: Three cards", pp. 157–160. ISBN 0-8058-4898-3
• Michael Clark, Paradoxes from A to Z, p. 16;
• Howard Margolis, Wason, Monty Hall, and Adverse Defaults [1].
References
[1] http:/ / harrisschool. uchicago. edu/ About/ publications/ working-papers/ pdf/ wp_05_14. pdf
142
Bertrand paradox
143
Bertrand paradox
The Bertrand paradox is a problem within the classical interpretation of probability theory. Joseph Bertrand
introduced it in his work Calcul des probabilités (1888) as an example to show that probabilities may not be well
defined if the mechanism or method that produces the random variable is not clearly defined.
Bertrand's formulation of the problem
The Bertrand paradox goes as follows: Consider an equilateral triangle inscribed in a circle. Suppose a chord of the
circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?
Bertrand gave three arguments, all apparently valid, yet yielding different results.
The "random endpoints" method: Choose two random points on the
circumference of the circle and draw the chord joining them. To
calculate the probability in question imagine the triangle rotated so its
vertex coincides with one of the chord endpoints. Observe that if the
other chord endpoint lies on the arc between the endpoints of the
triangle side opposite the first point, the chord is longer than a side of
the triangle. The length of the arc is one third of the circumference of
the circle, therefore the probability that a random chord is longer than a
side of the inscribed triangle is 1/3.
Random chords, selection method 1; red = longer
than triangle side, blue = shorter
The "random radius" method: Choose a radius of the circle, choose a
point on the radius and construct the chord through this point and
perpendicular to the radius. To calculate the probability in question
imagine the triangle rotated so a side is perpendicular to the radius. The
chord is longer than a side of the triangle if the chosen point is nearer
the center of the circle than the point where the side of the triangle
intersects the radius. The side of the triangle bisects the radius,
therefore the probability a random chord is longer than a side of the
inscribed triangle is 1/2.
Random chords, selection method 2
Bertrand paradox
144
The "random midpoint" method: Choose a point anywhere within the
circle and construct a chord with the chosen point as its midpoint. The
chord is longer than a side of the inscribed triangle if the chosen point
falls within a concentric circle of radius 1/2 the radius of the larger
circle. The area of the smaller circle is one fourth the area of the larger
circle, therefore the probability a random chord is longer than a side of
the inscribed triangle is 1/4.
The selection methods can also be visualized as follows. A chord is
uniquely identified by its midpoint. Each of the three selection methods
presented above yields a different distribution of midpoints. Methods 1
and 2 yield two different nonuniform distributions, while method 3
yields a uniform distribution. On the other hand, if one looks at the
images of the chords below, the chords of method 2 give the circle a
homogeneously shaded look, while method 1 and 3 do not.
Midpoints of chords chosen at random,
method 1
Chords chosen at random, method 1
Midpoints of chords chosen at random,
method 2
Random chords, selection method 3
Midpoints of chords chosen at random,
method 3
Chords chosen at random, method 3
Other distributions can easily be imagined, many of which will yield a different proportion of chords which are
longer than a side of the inscribed triangle.
Bertrand paradox
Classical solution
The problem's classical solution thus hinges on the method by which a chord is chosen "at random". It turns out that
if, and only if, the method of random selection is specified, the problem has a well-defined solution. There is no
unique selection method, so there cannot be a unique solution. The three solutions presented by Bertrand correspond
to different selection methods, and in the absence of further information there is no reason to prefer one over another.
This and other paradoxes of the classical interpretation of probability justified more stringent formulations, including
frequency probability and subjectivist Bayesian probability.
Jaynes' solution using the "maximum ignorance" principle
In his 1973 paper The Well-Posed Problem,[1] Edwin Jaynes proposed a solution to Bertrand's paradox, based on the
principle of "maximum ignorance"—that we should not use any information that is not given in the statement of the
problem. Jaynes pointed out that Bertrand's problem does not specify the position or size of the circle, and argued
that therefore any definite and objective solution must be "indifferent" to size and position. In other words: the
solution must be both scale invariant and translation invariant.
To illustrate: assume that chords are laid at random onto a circle with a diameter of 2, for example by throwing
straws onto it from far away. Now another circle with a smaller diameter (e.g., 1.1) is laid into the larger circle. Then
the distribution of the chords on that smaller circle needs to be the same as on the larger circle. If the smaller circle is
moved around within the larger circle, the probability must not change either. It can be seen very easily that there
would be a change for method 3: the chord distribution on the small red circle looks qualitatively different from the
distribution on the large circle:
The same occurs for method 1, though it is harder to see in a graphical representation. Method 2 is the only one that
is both scale invariant and translation invariant; method 3 is just scale invariant, method 1 is neither.
However, Jaynes did not just use invariances to accept or reject given methods: this would leave the possibility that
there is another not yet described method that would meet his common-sense criteria. Jaynes used the integral
equations describing the invariances to directly determine the probability distribution. In this problem, the integral
145
Bertrand paradox
equations indeed have a unique solution, and it is precisely what was called "method 2" above, the random radius
method.
Physical experiments
"Method 2" is the only solution that fulfills the transformation invariants that are present in certain physical
systems—such as in statistical mechanics and gas physics— as well as in Jaynes's proposed experiment of throwing
straws from a distance onto a small circle. Nevertheless, one can design other practical experiments that give
answers according to the other methods. For example, in order to arrive at the solution of "method 1", the random
endpoints method, one can affix a spinner to the center of the circle, and let the results of two independent spins
mark the endpoints of the chord. In order to arrive at the solution of "method 3", one could cover the circle with
molasses and mark the first point that a fly lands on as the midpoint of the chord.[2] Several observers have designed
experiments in order to obtain the different solutions and verified the results empirically.[3][4]
Notes
[1] Jaynes, E. T. (1973), "The Well-Posed Problem" (http:/ / bayes. wustl. edu/ etj/ articles/ well. pdf) (PDF), Foundations of Physics 3: 477–493,
doi:10.1007/BF00709116,
[2] Gardner, Martin (1987), The Second Scientific American Book of Mathematical Puzzles and Diversions, The University of Chicago Press,
pp. 223–226, ISBN 978-0-226-28253-4
[3] Tissler, P.E. (March 1984), "Bertrand's Paradox", The Mathematical Gazette (The Mathematical Association) 68 (443): 15–19,
doi:10.2307/3615385
[4] Kac, Mark (May–June 1984), "Marginalia: more on randomness", American Scientist 72 (3): 282–283
References
• Michael Clark. Paradoxes from A to Z. London: Routledge, 2002.
146
Birthday problem
147
Birthday problem
In probability theory, the birthday problem or birthday paradox[1] concerns the probability that, in a set of n
randomly chosen people, some pair of them will have the same birthday. By the pigeonhole principle, the probability
reaches 100% when the number of people reaches 367 (since there are 366 possible birthdays, including February
29). However, 99% probability is reached with just 57 people, and 50% probability with 23 people. These
conclusions are based on the assumption that each day of the year (except February 29) is equally probable for a
birthday.
The mathematics behind this problem led to a well-known cryptographic attack called the birthday attack, which
uses this probabilistic model to reduce the complexity of cracking a hash function.
Understanding the
problem
The birthday problem asks whether
any of the people in a given group has
a birthday matching any of the
others — not one in particular. (See
"Same birthday as you" below for an
analysis of this much less surprising
alternative problem.)
In the example given earlier, a list of
23 people, comparing the birthday of
the first person on the list to the others
A graph showing the approximate probability of at least two people sharing a birthday
allows 22 chances for a matching
amongst a certain number of people.
birthday, the second person on the list
to the others allows 21 chances for a matching birthday, third person has 20 chances, and so on. Hence total chances
are: 22+21+20+....+1 = 253, so comparing every person to all of the others allows 253 distinct chances
(combinations): in a group of 23 people there are
pairs.
Presuming all birthdays are equally probable,[2][3][4] the probability of a given birthday for a person chosen from the
entire population at random is 1/365 (ignoring Leap Day, February 29). Although the pairings in a group of 23
people are not statistically equivalent to 253 pairs chosen independently, the birthday paradox becomes less
surprising if a group is thought of in terms of the number of possible pairs, rather than as the number of individuals.
Calculating the probability
The problem is to compute the approximate probability that in a room of n people, at least two have the same
birthday. For simplicity, disregard variations in the distribution, such as leap years, twins, seasonal or weekday
variations, and assume that the 365 possible birthdays are equally likely. Real-life birthday distributions are not
uniform since not all dates are equally likely.[5]
If P(A) is the probability of at least two people in the room having the same birthday, it may be simpler to calculate
P(A'), the probability of there not being any two people having the same birthday. Then, because A and A' are the
only two possibilities and are also mutually exclusive, P(A') = 1 − P(A).
In deference to widely published solutions concluding that 23 is the number of people necessary to have a P(A) that
is greater than 50%, the following calculation of P(A) will use 23 people as an example.
Birthday problem
148
When events are independent of each other, the probability of all of the events occurring is equal to a product of the
probabilities of each of the events occurring. Therefore, if P(A') can be described as 23 independent events, P(A')
could be calculated as P(1) × P(2) × P(3) × ... × P(23).
The 23 independent events correspond to the 23 people, and can be defined in order. Each event can be defined as
the corresponding person not sharing his/her birthday with any of the previously analyzed people. For Event 1, there
are no previously analyzed people. Therefore, the probability, P(1), that person number 1 does not share his/her
birthday with previously analyzed people is 1, or 100%. Ignoring leap years for this analysis, the probability of 1 can
also be written as 365/365, for reasons that will become clear below.
For Event 2, the only previously analyzed people is Person 1. Assuming that birthdays are equally likely to happen
on each of the 365 days of the year, the probability, P(2), that Person 2 has a different birthday than Person 1 is
364/365. This is because, if Person 2 was born on any of the other 364 days of the year, Persons 1 and 2 will not
share the same birthday.
Similarly, if Person 3 is born on any of the 363 days of the year other than the birthdays of Persons 1 and 2, Person 3
will not share their birthday. This makes the probability P(3) = 363/365.
This analysis continues until Person 23 is reached, whose probability of not sharing his/her birthday with people
analyzed before, P(23), is 343/365.
P(A') is equal to the product of these individual probabilities:
(1) P(A') = 365/365 × 364/365 × 363/365 × 362/365 × ... × 343/365
The terms of equation (1) can be collected to arrive at:
(2) P(A') = (1/365)23 × (365 × 364 × 363 × ... × 343)
Evaluating equation (2) gives P(A') = 0.492703
Therefore, P(A) = 1 − 0.492703 = 0.507297 (50.7297%)
This process can be generalized to a group of n people, where p(n) is the probability of at least two of the n people
sharing a birthday. It is easier to first calculate the probability p(n) that all n birthdays are different. According to the
pigeonhole principle, p(n) is zero when n > 365. When n ≤ 365:
where ' ! ' is the factorial operator,
is the binomial coefficient and
denotes permutation.
The equation expresses the fact that for no persons to share a birthday, a second person cannot have the same
birthday as the first (364/365), the third cannot have the same birthday as the first two (363/365), and in general the
nth birthday cannot be the same as any of the n − 1 preceding birthdays.
The event of at least two of the n persons having the same birthday is complementary to all n birthdays being
different. Therefore, its probability p(n) is
Birthday problem
149
This
probability
surpasses
1/2
for n = 23 (with value about 50.7%).
The following table shows the
probability for some other values of n
(this table ignores the existence of leap
years, as described above):
The approximate probability that no two people share a birthday in a group of n people.
Note that the vertical scale is logarithmic (each step down is 1020 times less likely).
n
p(n)
10
11.7%
20
41.1%
23
50.7%
30
70.6%
50
97.0%
57
99.0%
100 99.99997%
200 99.9999999999999999999999999998%
300 (100 − (6×10−80))%
350 (100 − (3×10−129))%
365 (100 − (1.45×10−155))%
366 100%
Birthday problem
150
Approximations
The Taylor series expansion of the exponential function (the constant e = 2.718281828, approximately)
provides a first-order approximation for ex for
x << 1:
To apply this approximation to the first
expression derived for p(n) set
. Then,
term
Then for each
in
the
formula
for
p(n)
.
For i = 1,
The first expression derived for p(n) can be
approximated as
A graph showing the accuracy of the approximation
Therefore,
An even coarser approximation is given by
which, as the graph illustrates, is still fairly accurate.
It is easy to see that the same approach can be applied to any number of "people" and "days". If rather than 365 days
there are n, if there are m persons, and if m<<n, then using the same approach as above we achieve the result that if
Pr[(n, m)] is the probability that at least two out of m people share the same birthday from a set of n available days,
then:
A simple exponentiation
The probability of any two people not having the same birthday is 364/365. In a room containing n people, there are
C(n, 2) = n(n − 1)/2 pairs of people, i.e. C(n, 2) events. The probability of no two people sharing the same birthday
can be approximated by assuming that these events are independent and hence by multiplying their probability
together. In short 364/365 can be multiplied by itself C(n, 2) times, which gives us
And if this is the probability of no one having the same birthday, then the probability of someone sharing a birthday
is
Birthday problem
151
Poisson approximation
Using the Poisson approximation for the binomial,
Again, this is over 50%.
Approximation of number of people
This can also be approximated using the following formula for the number of people necessary to have at least a
50% chance of matching:
This is a result of the good approximation that an event with 1 in k probability will have a 50% chance of occurring
at least once if it is repeated k ln 2 times.[6]
Probability table
length of
hex
string
#bits
8
32
4.3 × 109
2
16
64
1.8 × 1019
6.1
32
128
3.4 × 1038
2.6 ×
1010
8.2 ×
1011
2.6 ×
1013
8.2 ×
1014
2.6 ×
1016
8.3 ×
1017
2.6 ×
1018
1.4 ×
1019
2.2 ×
1019
3.1 ×
1019
64
256
1.2 × 1077
4.8 ×
1029
1.5 ×
1031
4.8 ×
1032
1.5 ×
1034
4.8 ×
1035
1.5 ×
1037
4.8 ×
1037
2.6 ×
1038
4.0 ×
1038
5.7 ×
1038
(96)
(384)
(3.9 ×
10115)
8.9 ×
1048
2.8 ×
1050
8.9 ×
1051
2.8 ×
1053
8.9 ×
1054
2.8 ×
1056
8.9 ×
1056
4.8 ×
1057
7.4 ×
1057
1.0 ×
1058
128
512
1.3 × 10154
1.6 ×
1068
5.2 ×
1069
1.6 ×
1071
5.2 ×
1072
1.6 ×
1074
5.2 ×
1075
1.6 ×
1076
8.8 ×
1076
1.4 ×
1077
1.9 ×
1077
hash space
size
(2#bits)
Number of hashed elements such that (probability of at least one hash collision) = p
p = 10−18 p = 10−15 p = 10−12 p = 10−9
2
2
2.9
p = 10−6
93
p=
0.1%
p = 1%
p = 25% p = 50% p = 75%
2.9 × 103 9.3 × 103 5.0 × 104 7.7 × 104 1.1 × 105
1.9 × 102 6.1 × 103 1.9 × 105 6.1 × 106 1.9 × 108 6.1 × 108 3.3 × 109 5.1 × 109 7.2 × 109
The white squares in this table show the number of hashes needed to achieve the given probability of collision
(column) given a hashspace of a certain size in bits (row). (Using the birthday analogy: the "hash space
size"(row) would be "365 days", the "probability of collision"(column) would be "50%", and the "required
number of people" would be "23"(row-col intersection).) One could of course also use this chart to determine
the minimum hash size required (given upper bounds on the hashes and probability of error), or the
probability of collision (for fixed number of hashes and probability of error).
For comparison, 10−18 to 10−15 is the uncorrectable bit error rate of a typical hard disk [7]. In theory, MD5,
128 bits, should stay within that range until about 820 billion documents, even if its possible outputs are many
more.
Birthday problem
152
An upper bound
The argument below is adapted from an argument of Paul Halmos.[8]
As stated above, the probability that no two birthdays coincide is
As in earlier paragraphs, interest lies in the smallest n such that p(n) > 1/2; or equivalently, the smallest n such that
p(n) < 1/2.
Using the inequality 1 − x < e−x in the above expression we replace 1 − k/365 with e−k/365. This yields
Therefore, the expression above is not only an approximation, but also an upper bound of p(n). The inequality
implies p(n) < 1/2. Solving for n gives
Now, 730 ln 2 is approximately 505.997, which is barely below 506, the value of n2 − n attained when n = 23.
Therefore, 23 people suffice. Solving n2 − n = 2 · 365 · ln 2 for n gives, by the way, the approximate formula of
Frank H. Mathis cited above.
This derivation only shows that at most 23 people are needed to ensure a birthday match with even chance; it leaves
open the possibility that, say, n = 22 could also work.
Generalizations
The generalized birthday problem
Given a year with d days, the generalized birthday problem asks for the minimal number n(d) such that, in a set of
n(d) randomly chosen people, the probability of a birthday coincidence is at least 50%. In other words, n(d) is the
minimal integer n such that
The classical birthday problem thus corresponds to determining n(365). The first 99 values of n(d) are given here:
d
1–2 3–5 6–9 10–16 17–23 24–32 33–42 43–54 55–68 69–82 83–99
n(d)
2
3
4
5
6
7
8
9
10
11
12
A number of bounds and formulas for n(d) have been published.[9] For any d≥1, the number n(d) satisfies[10]
These bounds are optimal in the sense that the sequence
, while it has
gets arbitrarily close to
as its maximum, taken for d=43. The bounds are
sufficiently tight to give the exact value of n(d) in 99% of all cases, for example n(365)=23. In general, it follows
from these bounds that n(d) always equals either
or
where
denotes the ceiling
function. The formula
Birthday problem
holds for 73% of all integers d.[11] The formula
holds for almost all d, i.e., for a set of integers d with asymptotic density 1.[11] The formula
holds for all d up to 1018, but it is conjectured that there are infinitely many counter-examples to this formula.[12]
The formula
holds too for all d up to 1018, and it is conjectured that this formula holds for all d.[12]
Cast as a collision problem
The birthday problem can be generalized as follows: given n random integers drawn from a discrete uniform
distribution with range [1,d], what is the probability p(n;d) that at least two numbers are the same? (d=365 gives the
usual birthday problem.)
The generic results can be derived using the same arguments given above.
Conversely, if n(p;d) denotes the number of random integers drawn from [1,d] to obtain a probability p that at least
two numbers are the same, then
The birthday problem in this more generic sense applies to hash functions: the expected number of N-bit hashes that
can be generated before getting a collision is not 2N, but rather only 2N/2. This is exploited by birthday attacks on
cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for all practical
purposes, inevitable.
The theory behind the birthday problem was used by Zoe Schnabel[13] under the name of capture-recapture statistics
to estimate the size of fish population in lakes.
153
Birthday problem
154
Generalization to multiple types
The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an
arbitrary number of types.[14] In the simplest extension there are just two types, say m "men" and n "women", and the
problem becomes characterizing the probability of a shared birthday between at least one man and one woman.
(Shared birthdays between, say, two women do not count.) The probability of no (i.e. zero) shared birthdays here is
where d = 365 and S2 are Stirling numbers of the second kind. Consequently, the desired probability is 1 − p0.
This variation of the birthday problem is interesting because there is not a unique solution for the total number of
people m + n. For example, the usual 0.5 probability value is realized for both a 32-member group of 16 men and 16
women and a 49-member group of 43 women and 6 men.
Other birthday problems
Reverse problem
For a fixed probability p:
• Find the greatest n for which the probability p(n) is smaller than the given p, or
• Find the smallest n for which the probability p(n) is greater than the given p.
Taking the above formula for d = 365 we have:
Sample calculations
p
n
n↓
p(n↓)
n↑
p(n↑)
0.01 0.14178√365 = 2.70864
2 0.00274
3 0.00820
0.05 0.32029√365 = 6.11916
6 0.04046
7 0.05624
0.1
0.45904√365 = 8.77002
8 0.07434
9 0.09462
0.2
0.66805√365 = 12.76302
12 0.16702
13 0.19441
0.3
0.84460√365 = 16.13607
16 0.28360
17 0.31501
0.5
1.17741√365 = 22.49439
22 0.47570
23 0.50730
0.7
1.55176√365 = 29.64625
29 0.68097
30 0.70632
0.8
1.79412√365 = 34.27666
34 0.79532
35 0.81438
0.9
2.14597√365 = 40.99862
40 0.89123
41 0.90315
0.95 2.44775√365 = 46.76414
46 0.94825
47 0.95477
0.99 3.03485√365 = 57.98081
57 0.99012
58 0.99166
Note: some values falling outside the bounds have been colored to show that the approximation is not always exact.
Birthday problem
155
First match
A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same
birthday as someone already in the room? That is, for what n is p(n) − p(n − 1) maximum? The answer is 20—if
there's a prize for first match, the best position in line is 20th.
Same birthday as you
Note that in the birthday problem, neither of
the two people is chosen in advance. By way
of contrast, the probability q(n) that someone
in a room of n other people has the same
birthday as a particular person (for example,
you), is given by
and for general d by
Comparing p(n) = probability of a birthday match with q(n) = probability of
matching your birthday
In the standard case of d = 365 substituting n = 23 gives about 6.1%, which is less than 1 chance in 16. For a greater
than 50% chance that one person in a roomful of n people has the same birthday as you, n would need to be at least
253. Note that this number is significantly higher than 365/2 = 182.5: the reason is that it is likely that there are some
birthday matches among the other people in the room.
It is not a coincidence that
; a similar approximate pattern can be found using a number of
possibilities different from 365, or a target probability different from 50%.
Near matches
Another generalization is to ask what is the probability of finding at least one pair in a group of n people with
birthdays within k calendar days of each other's, if there are m equally likely birthdays.
[15]
The number of people required so that the probability that some pair will have a birthday separated by k days or
fewer will be higher than 50% is:
k # people required(i.e. n) when m=365
0
23
1
14
2
11
3
9
4
8
5
8
6
7
7
7
Birthday problem
Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a
week of each other.[15]
Collision counting
The probability that the kth integer randomly chosen from [1, d] will repeat at least one previous choice equals
q(k − 1; d) above. The expected total number of times a selection will repeat a previous selection as n such integers
are chosen equals
Average number of people
In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair
with the same birthday. The problem is relevant to several hashing algorithms analyzed by Donald Knuth in his book
The Art of Computer Programming. It may be shown[16][17] that if one samples uniformly, with replacement, from a
population of size M, the number of trials required for the first repeated sampling of some individual has expected
value
, where
The function
has been studied by Srinivasa Ramanujan and has asymptotic expansion:
With M = 365 days in a year, the average number of people required to find a pair with the same birthday is
, slightly more than the number required for a 50% chance. In the best case, two people will
suffice; at worst, the maximum possible number of M + 1 = 366 people is needed; but on average, only 25 people are
required.
An informal demonstration of the problem can be made from the list of Prime Ministers of Australia, of which there
have been 27, in which Paul Keating, the 24th Prime Minister, and Edmund Barton, the first Prime Minister, share
the same birthday, 18 January.
Partition problem
A related problem is the partition problem, a variant of the knapsack problem from operations research. Some
weights are put on a balance scale; each weight is an integer number of grams randomly chosen between one gram
and one million grams (one metric ton). The question is whether one can usually (that is, with probability close to 1)
transfer the weights between the left and right arms to balance the scale. (In case the sum of all the weights is an odd
number of grams, a discrepancy of one gram is allowed.) If there are only two or three weights, the answer is very
clearly no; although there are some combinations which work, the majority of randomly selected combinations of
three weights do not. If there are very many weights, the answer is clearly yes. The question is, how many are just
sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as
it is to be impossible?
Some people's intuition is that the answer is above 100,000. Most people's intuition is that it is in the thousands or
tens of thousands, while others feel it should at least be in the hundreds. The correct answer is approximately 23.
156
Birthday problem
The reason is that the correct comparison is to the number of partitions of the weights into left and right. There are
2N−1 different partitions for N weights, and the left sum minus the right sum can be thought of as a new random
quantity for each partition. The distribution of the sum of weights is approximately Gaussian, with a peak at
1,000,000 N and width
, so that when 2N−1 is approximately equal to
the transition occurs.
223−1 is about 4 million, while the width of the distribution is only 5 million.[18]
Notes
[1] This is not a paradox in the sense of leading to a logical contradiction, but is called a paradox because the mathematical truth contradicts
naïve intuition: most people estimate that the chance of two individuals sharing the same birthday in a group of 23 is much lower than 50%.
[2] In reality, birthdays are not evenly distributed throughout the year; there are more births per day in some seasons than in others, but for the
purposes of this problem the distribution is treated as uniform.
[3] Murphy, Ron. "An Analysis of the Distribution of Birthdays in a Calendar Year" (http:/ / www. panix. com/ ~murphy/ bday. html). .
Retrieved 2011-12-27.
[4] Mathers, C D; R S Harris (1983). "Seasonal Distribution of Births in Australia" (http:/ / ije. oxfordjournals. org/ content/ 12/ 3/ 326. abstract).
International Journal of Epidemiology 12 (3): 326–331. doi:10.1093/ije/12.3.326. . Retrieved 2011-12-27.
[5] In particular, many children are born in the summer, especially the months of August and September (for the northern hemisphere) (http:/ /
scienceworld. wolfram. com/ astronomy/ LeapDay. html), and in the U.S. it has been noted that many children are conceived around the
holidays of Christmas and New Year's Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more
Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g. a class in a school), this
creates a tendency toward particular dates. In Sweden 9.3% of the population is born in March and 7.3% in November when a uniform
distribution would give 8.3% Swedish statistics board (http:/ / www. scb. se/ statistik/ BE/ BE0101/ 2006A01a/
BE0101_2006A01a_SM_BE12SM0701. pdf) Both of these factors tend to increase the chance of identical birth dates, since a denser subset
has more possible pairs (in the extreme case when everyone was born on three days, there would obviously be many identical birthdays). The
birthday problem for such non-constant birthday probabilities was first understood by Murray Klamkin in 1967. A formal proof that the
probability of two matching birthdays is least for a uniform distribution of birthdays was given by D. Bloom (1973)
[6] Mathis, Frank H. (June 1991). "A Generalized Birthday Problem". SIAM Review (Society for Industrial and Applied Mathematics) 33 (2):
265–270. doi:10.1137/1033051. ISSN 0036-1445. JSTOR 2031144. OCLC 37699182.
[7] http:/ / arxiv. org/ abs/ cs/ 0701166
[8] In his autobiography, Halmos criticized the form in which the birthday paradox is often presented, in terms of numerical computation. He
believed that it should be used as an example in the use of more abstract mathematical concepts. He wrote:
The reasoning is based on important tools that all students of mathematics should have ready access to.
The birthday problem used to be a splendid illustration of the advantages of pure thought over
mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the
multiplications would take much longer, and be much more subject to error, whether the instrument is a
pencil or an old-fashioned desk computer. What calculators do not yield is understanding, or
mathematical facility, or a solid basis for more advanced, generalized theories.
[9] D. Brink, A (probably) exact solution to the Birthday Problem, Ramanujan Journal, 2012, doi: 10.1007/s11139-011-9343-9 (http:/ / www.
springerlink. com/ content/ 1194r3627822841q/ ).
[10] Brink 2012, Theorem 2
[11] Brink 2012, Theorem 3
[12] Brink 2012, Table 3, Conjecture 1
[13] Z. E. Schnabel (1938) The Estimation of the Total Fish Population of a Lake, American Mathematical Monthly 45, 348–352.
[14] M. C. Wendl (2003) Collision Probability Between Sets of Random Variables (http:/ / dx. doi. org/ 10. 1016/ S0167-7152(03)00168-8),
Statistics and Probability Letters 64(3), 249–254.
[15] M. Abramson and W. O. J. Moser (1970) More Birthday Surprises, American Mathematical Monthly 77, 856–858
[16] D. E. Knuth; The Art of Computer Programming. Vol. 3, Sorting and Searching (Addison-Wesley, Reading, Massachusetts, 1973)
[17] P. Flajolet, P. J. Grabner, P. Kirschenhofer, H. Prodinger (1995), On Ramanujan's Q-Function, Journal of Computational and Applied
Mathematics 58, 103–116
[18] C. Borgs, J. Chayes, and B. Pittel (2001) Phase Transition and Finite Size Scaling in the Integer Partition Problem, Random Structures and
Algorithms 19(3–4), 247–288.
157
Birthday problem
References
• John G. Kemeny, J. Laurie Snell, and Gerald Thompson Introduction to Finite Mathematics . The first edition,
1957
• E. H. McKinney (1966) Generalized Birthday Problem, American Mathematical Monthly 73, 385–387.
• M. Klamkin and D. Newman (1967) Extensions of the Birthday Surprise, Journal of Combinatorial Theory 3,
279–282.
• M. Abramson and W. O. J. Moser (1970) More Birthday Surprises, American Mathematical Monthly 77,
856–858
• D. Bloom (1973) A Birthday Problem, American Mathematical Monthly 80, 1141–1142.
• Shirky, Clay Here Comes Everybody: The Power of Organizing Without Organizations, (2008.) New York.
25–27.
External links
• Coincidences: the truth is out there (http://www.rsscse-edu.org.uk/tsj/wp-content/uploads/2011/03/
matthews.pdf) Experimental test of the Birthday Paradox and other coincidences
• http://www.efgh.com/math/birthday.htm
• http://planetmath.org/encyclopedia/BirthdayProblem.html
• Weisstein, Eric W., " Birthday Problem (http://mathworld.wolfram.com/BirthdayProblem.html)" from
MathWorld.
• A humorous article explaining the paradox (http://www.damninteresting.com/?p=402)
• SOCR EduMaterials activities birthday experiment (http://wiki.stat.ucla.edu/socr/index.php/
SOCR_EduMaterials_Activities_BirthdayExperiment)
• Understanding the Birthday Problem (Better Explained) (http://betterexplained.com/articles/
understanding-the-birthday-paradox/)
• Eurobirthdays 2012. A birthday problem. (http://www.matifutbol.com/docs/units/eurobirthdays.html) A
practical football example of the birthday paradox.
158
BorelKolmogorov paradox
159
Borel–Kolmogorov paradox
In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating
to conditional probability with respect to an event of probability zero (also known as a null set). It is named after
Émile Borel and Andrey Kolmogorov.
A great circle puzzle
Suppose that a random variable has a uniform distribution on a unit sphere. What is its conditional distribution on a
great circle? Because of the symmetry of the sphere, one might expect that the distribution is uniform and
independent of the choice of coordinates. However, two analyses give contradictory results. First, note that choosing
a point uniformly on the sphere is equivalent to choosing the longitude λ uniformly from [-π,π] and choosing the
latitude φ from [-π/2,π/2] with density
.[1] Then we can look at two different great circles:
1. If the coordinates are chosen so that the great circle is an equator (latitude φ = 0), the conditional density for
a longitude λ defined on the interval [–π,π] is
2. If the great circle is a line of longitude with λ = 0, the conditional density for φ on the interval [–π/2,π/2] is
One distribution is uniform on the circle, the other is not. Yet both seem to be referring to the same great circle in
different coordinate systems.
Many quite futile arguments have raged - between otherwise competent probabilists - over which of these
results is 'correct'.
—E.T. Jaynes[1]
Explanation and implications
In case (1) above, the conditional probability that the longitude λ lies in a set E given that φ = 0 can be written P(λ ∈
E | φ = 0). Elementary probability theory suggests this can be computed as P(λ ∈ E and φ=0)/P(φ=0), but that
expression is not well-defined since P(φ=0) = 0. Measure theory provides a way to define a conditional probability,
using the family of events Rab = {φ : a < φ < b} which are horizontal rings consisting of all points with latitude
between a and b.
The resolution of the paradox is to notice that in case (2), P(φ ∈ F | λ=0) is defined using the events Lab = {λ : a < λ
< b}, which are vertical wedges (more precisely lunes), consisting of all points whose longitude varies between a and
b. So although P(λ ∈ E | φ=0) and P(φ ∈ F | λ=0) each provide a probability distribution on a great circle, one of
them is defined using rings, and the other using lunes. Thus it is not surprising after all that P(λ ∈ E | φ=0) and P(φ ∈
F | λ=0) have different distributions.
The concept of a conditional probability with regard to an isolated hypothesis whose probability equals 0 is
inadmissible. For we can obtain a probability distribution for [the latitude] on the meridian circle only if we
regard this circle as an element of the decomposition of the entire spherical surface onto meridian circles with
the given poles
—Andrey Kolmogorov[2]
… the term 'great circle' is ambiguous until we specify what limiting operation is to produce it. The intuitive
symmetry argument presupposes the equatorial limit; yet one eating slices of an orange might presuppose the
BorelKolmogorov paradox
160
other.
—E.T. Jaynes[1]
Mathematical explication
To understand the problem we need to recognize that a distribution on a continuous random variable is described by
a density f only with respect to some measure μ. Both are important for the full description of the probability
distribution. Or, equivalently, we need to fully define the space on which we want to define f.
Let Φ and Λ denote two random variables taking values in Ω1 = [-π/2,π/2] respectively Ω2 = [-π,π]. An event
{Φ=φ,Λ=λ} gives a point on the sphere S(r) with radius r. We define the coordinate transform
for which we obtain the volume element
Furthermore, if either φ or λ is fixed, we get the volume elements
Let
denote the joint measure on
If we assume that the density
Hence,
other hand,
, which has a density
with respect to
and let
is uniform, then
has a uniform density with respect to
has a uniform density with respect to
but not with respect to the Lebesgue measure. On the
and the Lebesgue measure.
BorelKolmogorov paradox
Notes
[1] Jaynes 2003, pp. 1514–1517
[2] Originally Kolmogorov (1933), translated in Kolmogorov (1956). Sourced from Pollard (2002)
References and further reading
• Jaynes, E.T. (2003). "15.7 The Borel-Kolmogorov paradox". Probability Theory: The Logic of Science.
Cambridge University Press. pp. 467–470. ISBN 0-521-59271-2. MR1992316.
• Fragmentary Edition (1994) (pp. 1514–1517) (http://omega.math.albany.edu:8008/ETJ-PS/cc15w.ps)
(PostScript format)
• Kolmogorov, Andrey (1933) (in German). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Julius
Springer.
• Translation: Kolmogorov, Andrey (1956). "Chapter V, §2. Explanation of a Borel Paradox" (http://www.
mathematik.com/Kolmogorov/0029.html). Foundations of the Theory of Probability (http://www.
mathematik.com/Kolmogorov/index.html) (2nd ed.). New York: Chelsea. pp. 50–51. ISBN 0-8284-0023-7.
• Pollard, David (2002). "Chapter 5. Conditioning, Example 17.". A User's Guide to Measure Theoretic
Probability. Cambridge University Press. pp. 122–123. ISBN 0-521-00289-3. MR1873379.
Boy or Girl paradox
The Boy or Girl paradox surrounds a well-known set of questions in probability theory which are also known as
The Two Child Problem,[1] Mr. Smith's Children[2] and the Mrs. Smith Problem. The initial formulation of the
question dates back to at least 1959, when Martin Gardner published one of the earliest variants of the paradox in
Scientific American. Titled The Two Children Problem, he phrased the paradox as follows:
• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
Gardner initially gave the answers 1/2 and 1/3, respectively; but later acknowledged that the second question was
ambiguous.[1] Its answer could be 1/2, depending on how you found out that one child was a boy. The ambiguity,
depending on the exact wording and possible assumptions, was confirmed by Bar-Hillel and Falk,[3] and
Nickerson.[4]
Other variants of this question, with varying degrees of ambiguity, have been recently popularized by Ask Marilyn in
Parade Magazine,[5] John Tierney of The New York Times,[6] and Leonard Mlodinow in Drunkard's Walk.[7] One
scientific study[2] showed that when identical information was conveyed, but with different partially ambiguous
wordings that emphasized different points, that the percentage of MBA students who answered 1/2 changed from
85% to 39%.
The paradox has frequently stimulated a great deal of controversy.[4] Many people argued strongly for both sides
with a great deal of confidence, sometimes showing disdain for those who took the opposing view. The paradox
stems from whether the problem setup is similar for the two questions.[2][7] The intuitive answer is 1/2.[2] This
answer is intuitive if the question leads the reader to believe that there are two equally likely possibilities for the sex
of the second child (i.e., boy and girl),[2][8] and that the probability of these outcomes is absolute, not conditional.[9]
161
Boy or Girl paradox
162
Common assumptions
The two possible answers share a number of assumptions. First, it is assumed that the space of all possible events can
be easily enumerated, providing an extensional definition of outcomes: {BB, BG, GB, GG}.[10] This notation
indicates that there are four possible combinations of children, labeling boys B and girls G, and using the first letter
to represent the older child. Second, it is assumed that these outcomes are equally probable.[10] This implies the
following model, a Bernoulli process with
:
1. Each child is either male or female.
2. Each child has the same chance of being male as of being female.
3. The sex of each child is independent of the sex of the other.
In reality, this is a rather inaccurate model,[10] since it ignores (amongst other factors) the fact that the ratio of boys
to girls is not exactly 50:50, the possibility of identical twins (who are always the same sex), and the possibility of an
intersex child. However, this problem is about probability and not biology. The mathematical outcome would be the
same if it were phrased in terms of a coin toss.
First question
• Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?
Under the forementioned assumptions, in this problem, a random family is selected. In this sample space, there are
four equally probable events:
Older child Younger child
Girl
Girl
Girl
Boy
Boy
Girl
Boy
Boy
Only two of these possible events meet the criteria specified in the question (e.g., GG, GB). Since both of the two
possibilities in the new sample space {GG, GB} are equally likely, and only one of the two, GG, includes two girls,
the probability that the younger child is also a girl is 1/2.
Second question
• Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?
This question is identical to question one, except that instead of specifying that the older child is a boy, it is specified
that at least one of them is a boy. In response to reader criticism of the question posed in 1959, Gardner agreed that a
precise formulation of the question is critical to getting different answers for question 1 and 2. Specifically, Gardner
argued that a "failure to specify the randomizing procedure" could lead readers to interpret the question in two
distinct ways:
• From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield
the answer of 1/3.
• From all families with two children, one child is selected at random, and the sex of that child is specified. This
would yield an answer of 1/2.[3][4]
Grinstead and Snell argue that the question is ambiguous in much the same way Gardner did.[11]
For example, if you see the children in the garden, you may see a boy. The other child may be hidden behind a tree.
In this case, the statement is equivalent to the second (the child that you can see is a boy). The first statement does
not match as one case is one boy, one girl. Then the girl may be visible. (The first statement says that it can be
Boy or Girl paradox
163
either.)
While it is certainly true that every possible Mr. Smith has at least one boy - i.e., the condition is necessary - it is not
clear that every Mr. Smith with at least one boy is intended. That is, the problem statement does not say that having a
boy is a sufficient condition for Mr. Smith to be identified as having a boy this way.
Commenting on Gardner's version of the problem, Bar-Hillel and Falk [3] note that "Mr. Smith, unlike the reader, is
presumably aware of the sex of both of his children when making this statement", i.e. that 'I have two children and at
least one of them is a boy.' If it is further assumed that Mr Smith would report this fact if it were true then the correct
answer is 1/3 as Gardner intended.
Analysis of the ambiguity
If it is assumed that this information was obtained by looking at both children to see if there is at least one boy, the
condition is both necessary and sufficient. Three of the four equally probable events for a two-child family in the
sample space above meet the condition:
Older child Younger child
Girl
Girl
Girl
Boy
Boy
Girl
Boy
Boy
Thus, if it is assumed that both children were considered while looking for a boy, the answer to question 2 is 1/3.
However, if the family was first selected and then a random, true statement was made about the gender of one child
(whether or not both were considered), the correct way to calculate the conditional probability is not to count the
cases that match. Instead, one must add the probabilities that the condition will be satisfied in each case[11]:
Older child Younger child P(this case) P("at least one boy" given this case) P(both this case, and "at least one boy")
Girl
Girl
1/4
0
0
Girl
Boy
1/4
1/2
1/8
Boy
Girl
1/4
1/2
1/8
Boy
Boy
1/4
1
1/4
The answer is found by adding the numbers in the last column wherever you would have counted that case:
(1/4)/(0+1/8+1/8+1/4)=1/2. Note that this is not necessarily the same as reporting the gender of a specific child,
although doing so will produce the same result by a different calculation. For instance, if the younger child is picked,
the calculation is (1/4)/(0+1/4+0+1/4)=1/2. In general, 1/2 is a better answer any time a Mr. Smith with a boy and a
girl could have been identified as having at least one girl.
Boy or Girl paradox
Bayesian analysis
Following classical probability arguments, we consider a large Urn containing two children. We assume equal
probability that either is a boy or a girl. The three discernible cases are thus: 1. both are girls (GG) - with probability
P(GG) = 0.25, 2. both are boys (BB) - with probability of P(BB) = 0.25, and 3. one of each (G.B) - with probability
of P(G.B) = 0.50. These are the prior probabilities.
Now we add the additional assumption that "at least one is a girl" = G. Using Bayes Theorem, we find
P(GG|G) = P(G|GG) * P(GG) / P(G) = 1 * 1/4 / 3/4 = 1/3.
where P(A|B) means "probability of A given B". P(G|GG) = probability of at least one girl given both are girls = 1.
P(GG) = probability of both girls = 1/4 from the prior distribution. P(G) = probability of at least one being a girl,
which includes cases GG and G.B = 1/4 + 1/2 = 3/4.
Note that, although the natural assumption seems to be a probability of 1/2, so the derived value of 1/3 seems low,
the actual "normal" value for P(GG) is 1/4, so the 1/3 is actually a bit higher.
The paradox arises because the second assumption is somewhat artificial, and when describing the problem in an
actual setting things get a bit sticky. Just how do we know that "at least" one is a girl? One description of the
problem states that we look into a window, see only one child and it is a girl. Sounds like the same
assumption...but...this one is equivalent to "sampling" the distribution (i.e. removing one child from the urn,
ascertaining that it is a girl, then replacing). Let's call the statement "the sample is a girl" proposition "g". Now we
have:
P(GG|g) = P(g|GG) * P(GG) / P(g) = 1 * 1/4 / 1/2 = 1/2.
The difference here is the P(g), which is just the probability of drawing a girl from all possible cases (i.e. without the
"at least"), which is clearly 0.5.
The Bayesian analysis generalizes easily to the case in which we relax the 50/50 population assumption. If we have
no information about the populations then we assume a "flat prior", i.e. P(BB) = P(GG) = P(G.B) = 1/3. In this case
the "at least" assumption produces the result P(GG|G) = 1/2, and the sampling assumption produces P(GG|g) = 2/3, a
result also derivable from the Rule of Succession.
Variants of the question
Following the popularization of the paradox by Gardner it has been presented and discussed in various forms. The
first variant presented by Bar-Hillel & Falk [3] is worded as follows:
• Mr. Smith is the father of two. We meet him walking along the street with a young boy whom he proudly
introduces as his son. What is the probability that Mr. Smith’s other child is also a boy?
Bar-Hillel & Falk use this variant to highlight the importance of considering the underlying assumptions. The
intuitive answer is 1/2 and, when making the most natural assumptions, this is correct. However, someone may argue
that “...before Mr. Smith identifies the boy as his son, we know only that he is either the father of two boys, BB, or of
two girls, GG, or of one of each in either birth order, i.e., BG or GB. Assuming again independence and
equiprobability, we begin with a probability of 1/4 that Smith is the father of two boys. Discovering that he has at
least one boy rules out the event GG. Since the remaining three events were equiprobable, we obtain a probability of
1/3 for BB.”[3]
Bar-Hillel & Falk say that the natural assumption is that Mr Smith selected the child companion at random but, if so,
the three combinations of BB, BG and GB are no longer equiprobable. For this to be the case each combination
would need to be equally likely to produce a boy companion but it can be seen that in the BB combination a boy
companion is guaranteed whereas in the other two combinations this is not the case. When the correct calculations
are made, if the walking companion was chosen at random then the probability that the other child is also a boy is
1/2. Bar-Hillel & Falk suggest an alternative scenario. They imagine a culture in which boys are invariably chosen
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Boy or Girl paradox
over girls as walking companions. With this assumption the combinations of BB, BG and GB are equally likely to be
represented by a boy walking companion and then the probability that the other child is also a boy is 1/3.
In 1991, Marilyn vos Savant responded to a reader who asked her to answer a variant of the Boy or Girl paradox that
included beagles.[5] In 1996, she published the question again in a different form. The 1991 and 1996 questions,
respectively were phrased:
• A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male,
female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath.
"Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other
one is a male?
• Say that a woman and a man (who are unrelated) each has two children. We know that at least one of the woman's
children is a boy and that the man's oldest child is a boy. Can you explain why the chances that the woman has
two boys do not equal the chances that the man has two boys?
With regard to the second formulation Vos Savant gave the classic answer that the chances that the woman has two
boys are about 1/3 whereas the chances that the man has two boys are about 1/2. In response to reader response that
questioned her analysis vos Savant conducted a survey of readers with exactly two children, at least one of which is a
boy. Of 17,946 responses, 35.9% reported two boys.[10]
Vos Savant's articles were discussed by Carlton and Stansfield[10] in a 2005 article in The American Statistician. The
authors do not discuss the possible ambiguity in the question and conclude that her answer is correct from a
mathematical perspective, given the assumptions that the likelihood of a child being a boy or girl is equal, and that
the sex of the second child is independent of the first. With regard to her survey they say it "at least validates vos
Savant’s correct assertion that the “chances” posed in the original question, though similar-sounding, are different,
and that the first probability is certainly nearer to 1 in 3 than to 1 in 2."
Carlton and Stansfield go on to discuss the common assumptions in the Boy or Girl paradox. They demonstrate that
in reality male children are actually more likely than female children, and that the sex of the second child is not
independent of the sex of the first. The authors conclude that, although the assumptions of the question run counter
to observations, the paradox still has pedagogical value, since it "illustrates one of the more intriguing applications of
conditional probability."[10] Of course, the actual probability values do not matter; the purpose of the paradox is to
demonstrate seemingly contradictory logic, not actual birth rates.
Information about the child
Suppose we were told not only that Mr. Smith has two children, and one of them is a boy, but also that the boy was
born on a Tuesday: does this change our previous analyses? Again, the answer depends on how this information
comes to us - what kind of selection process brought us this knowledge.
Following the tradition of the problem, let us suppose that out there in the population of two-child families, the sex
of the two children is independent of one another, equally likely boy or girl, and that each child is independently of
the other children born on any of the seven days of the week, each with equal probability 1/7. In that case, the chance
that a two child family consists of two boys, one (at least) born on a Tuesday, is equal to 1/4 (the probability of two
boys) times one minus 6/7 squared = 1 - 36/49 = 13/49 (one minus the probability that neither child is born on a
Tuesday). 1/4 times 13/49 equals 13/196.
The probability that a two child family consists of a boy and a girl, the boy born on a Tuesday, equals 2 (boy-girl or
girl-boy) times 1/4 (the two specified sexes) times 1/7 (the boy born on Tuesday) = 1/14. Therefore, among all two
child families with at least one boy born on a Tuesday, the fraction of families in which the other child is a girl is
1/14 divided by the sum of 1/14 plus 13/196 = 0.5185185.
It seems that we introduced quite irrelevant information, yet the probability of the sex of the other child has changed
dramatically from what it was before (the chance the other child was a girl was 2/3, when we didn't know that the
165
Boy or Girl paradox
boy was born on Tuesday).
This is still a bit bigger than a half, but close! It is not difficult to check that as we specify more and more details
about the boy child (for instance: born on January 1), the chance that the other child is a girl approaches one half.
However, is it really plausible that our child family with at least one boy born on a Tuesday was delivered to us by
choosing just one of such families at random? It is much more easy to imagine the following scenario. We know Mr.
Smith has two children. We knock at his door and a boy comes and answers the door. We ask the boy on what day of
the week he was born. Let's assume that which of the two children answers the door is determined by chance! Then
the procedure was (1) pick a two-child family at random from all two-child families (2) pick one of the two children
at random, (3) see it's a boy and ask on what day he was born. The chance the other child is a girl is 1/2. This is a
very different procedure from (1) picking a two-child family at random from all families with two children, at least
one a boy, born on a Tuesday. The chance the family consists of a boy and a girl is 0.5185815...
This variant of the boy and girl problem is discussed on many recent internet blogs and is the subject of a paper by
Ruma Falk, [12]. The moral of the story is that these probabilities don't just depend on the information we have in
front of us, but on how we came by that information.
Psychological investigation
From the position of statistical analysis the relevant question is often ambiguous and as such there is no “correct”
answer. However, this does not exhaust the boy or girl paradox for it is not necessarily the ambiguity that explains
how the intuitive probability is derived. A survey such as vos Savant’s suggests that the majority of people adopt an
understanding of Gardner’s problem that if they were consistent would lead them to the 1/3 probability answer but
overwhelmingly people intuitively arrive at the 1/2 probability answer. Ambiguity notwithstanding, this makes the
problem of interest to psychological researchers who seek to understand how humans estimate probability.
Fox & Levav (2004) used the problem (called the Mr. Smith problem, credited to Gardner, but not worded exactly
the same as Gardner's version) to test theories of how people estimate conditional probabilities.[2] In this study, the
paradox was posed to participants in two ways:
• "Mr. Smith says: 'I have two children and at least one of them is a boy.' Given this information, what is the
probability that the other child is a boy?"
• "Mr. Smith says: 'I have two children and it is not the case that they are both girls.' Given this information, what is
the probability that both children are boys?"
The authors argue that the first formulation gives the reader the mistaken impression that there are two possible
outcomes for the "other child",[2] whereas the second formulation gives the reader the impression that there are four
possible outcomes, of which one has been rejected (resulting in 1/3 being the probability of both children being boys,
as there are 3 remaining possible outcomes, only one of which is that both of the children are boys). The study found
that 85% of participants answered 1/2 for the first formulation, while only 39% responded that way to the second
formulation. The authors argued that the reason people respond differently to this question (along with other similar
problems, such as the Monty Hall Problem and the Bertrand's box paradox) is because of the use of naive heuristics
that fail to properly define the number of possible outcomes.[2]
166
Boy or Girl paradox
References
[1] Martin Gardner (1954). The Second Scientific American Book of Mathematical Puzzles and Diversions. Simon & Schuster.
ISBN 978-0-226-28253-4..
[2] Craig R. Fox & Jonathan Levav (2004). "Partition–Edit–Count: Naive Extensional Reasoning in Judgment of Conditional Probability".
Journal of Experimental Psychology 133 (4): 626–642. doi:10.1037/0096-3445.133.4.626. PMID 15584810.
[3] Maya Bar-Hillel and Ruma Falk (1982). "Some teasers concerning conditional probabilities". Cognition 11 (2): 109–122.
doi:10.1016/0010-0277(82)90021-X. PMID 7198956.
[4] Raymond S. Nickerson (May 2004). Cognition and Chance: The Psychology of Probabilistic Reasoning. Psychology Press.
ISBN 0-8058-4899-1.
[5] Ask Marilyn. Parade Magazine. October 13, 1991; January 5, 1992; May 26, 1996; December 1, 1996; March 30, 1997; July 27, 1997;
October 19, 1997.
[6] Tierney, John (2008-04-10). "The psychology of getting suckered" (http:/ / tierneylab. blogs. nytimes. com/ 2008/ 04/ 10/
the-psychology-of-getting-suckered/ ). The New York Times. . Retrieved 24 February 2009.
[7] Leonard Mlodinow (2008). Pantheon. ISBN 0-375-42404-0.
[8] Nikunj C. Oza (1993). "On The Confusion in Some Popular Probability Problems". CiteSeerX: 10.1.1.44.2448 (http:/ / citeseerx. ist. psu. edu/
viewdoc/ summary?doi=10. 1. 1. 44. 2448).
[9] P.J. Laird et al. (1999). "Naive Probability: A Mental Model Theory of Extensional Reasoning". Psychological Review.
[10] Matthew A. CARLTON and William D. STANSFIELD (2005). "Making Babies by the Flip of a Coin?". The American Statistician.
[11] Charles M. Grinstead and J. Laurie Snell. "Grinstead and Snell's Introduction to Probability" (http:/ / math. dartmouth. edu/ ~prob/ prob/
prob. pdf). The CHANCE Project. .
[12] http:/ / www. tandfonline. com/ doi/ abs/ 10. 1080/ 13546783. 2011. 613690
External links
•
•
•
•
•
Boy or Girl: Two Interpretations (http://mathforum.org/library/drmath/view/52186.html)
At Least One Girl (http://www.mathpages.com/home/kmath036.htm) at MathPages
A Problem With Two Bear Cubs (http://www.cut-the-knot.org/bears.shtml)
Lewis Carroll's Pillow Problem (http://www.cut-the-knot.org/carroll.shtml)
When intuition and math probably look wrong (http://www.sciencenews.org/view/generic/id/60598/title/
Math_Trek__When_intuition_and_math_probably_look_wrong)
167
Burali-Forti paradox
168
Burali-Forti paradox
In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that naïvely constructing "the set of all
ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It
is named after Cesare Burali-Forti, who discovered it in 1897.
Stated in terms of von Neumann ordinals
The reason is that the set of all ordinal numbers
carries all properties of an ordinal number and would have to be
considered an ordinal number itself. Then, we can construct its successor
However, this ordinal number must be an element of
since
, which is strictly greater than
.
contains all ordinal numbers, and we arrive at:
and
Stated more generally
The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John
von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the
paradox was framed by Burali-Forti. Here is an account with fewer presuppositions: suppose that we associate with
each well-ordering an object called its "order type" in an unspecified way (the order types are the ordinal numbers).
The "order types" (ordinal numbers) themselves are well-ordered in a natural way, and this well-ordering must have
an order type . It is easily shown in naïve set theory (and remains true in ZFC but not in New Foundations) that
the order type of all ordinal numbers less than a fixed is itself. So the order type of all ordinal numbers less
than is itself. But this means that , being the order type of a proper initial segment of the ordinals, is strictly
less than the order type of all the ordinals, but the latter is
itself by definition. This is a contradiction.
If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the
paradox is unavoidable: the offending proposition that the order type of all ordinal numbers less than a fixed is
itself must be true. The collection of von Neumann ordinals, like the collection in the Russell paradox, cannot be
a set in any set theory with classical logic. But the collection of order types in New Foundations (defined as
equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order
type of the ordinals less than turns out not to be .
Resolution of the paradox
Modern axiomatic set theory such as ZF and ZFC circumvents this antinomy by simply not allowing construction of
sets with unrestricted comprehension terms like "all sets with the property ", as it was for example possible in
Gottlob Frege's axiom system. New Foundations uses a different solution.
References
• Burali-Forti, Cesare (1897), "Una questione sui numeri transfiniti", Rendiconti del Circolo Matematico di
Palermo 11: 154–164, doi:10.1007/BF03015911
Burali-Forti paradox
External links
• Stanford Encyclopedia of Philosophy: "Paradoxes and Contemporary Logic [1]" -- by Andrea Cantini.
References
[1] http:/ / plato. stanford. edu/ entries/ paradoxes-contemporary-logic/
Cantor's paradox
In set theory, Cantor's paradox is derivable from the theorem that there is no greatest cardinal number, so that the
collection of "infinite sizes" is itself infinite. The difficulty is handled in axiomatic set theory by declaring that this
collection is not a set but a proper class; in von Neumann–Bernays–Gödel set theory it follows from this and the
axiom of limitation of size that this proper class must be in bijection with the class of all sets. Thus, not only are
there infinitely many infinities, but this infinity is larger than any of the infinities it enumerates.
This paradox is named for Georg Cantor, who is often credited with first identifying it in 1899 (or between 1895 and
1897). Like a number of "paradoxes" it is not actually contradictory but merely indicative of a mistaken intuition, in
this case about the nature of infinity and the notion of a set. Put another way, it is paradoxical within the confines of
naïve set theory and therefore demonstrates that a careless axiomatization of this theory is inconsistent.
Statements and proofs
In order to state the paradox it is necessary to understand that the cardinal numbers admit an ordering, so that one
can speak about one being greater or less than another. Then Cantor's paradox is:
Theorem: There is no greatest cardinal number.
This fact is a direct consequence of Cantor's theorem on the cardinality of the power set of a set.
Proof: Assume the contrary, and let C be the largest cardinal number. Then (in the von Neumann formulation
of cardinality) C is a set and therefore has a power set 2C which, by Cantor's theorem, has cardinality strictly
larger than that of C. Demonstrating a cardinality (namely that of 2C) larger than C, which was assumed to be
the greatest cardinal number, falsifies the definition of C. This contradiction establishes that such a cardinal
cannot exist.
Another consequence of Cantor's theorem is that the cardinal numbers constitute a proper class. That is, they cannot
all be collected together as elements of a single set. Here is a somewhat more general result.
Theorem: If S is any set then S cannot contain elements of all cardinalities. In fact, there is a strict upper
bound on the cardinalities of the elements of S.
Proof: Let S be a set, and let T be the union of the elements of S. Then every element of S is a subset of T, and
hence is of cardinality less than or equal to the cardinality of T. Cantor's theorem then implies that every
element of S is of cardinality strictly less than the cardinality of 2lTl.
Discussion and consequences
Since the cardinal numbers are well-ordered by indexing with the ordinal numbers (see Cardinal number, formal
definition), this also establishes that there is no greatest ordinal number; conversely, the latter statement implies
Cantor's paradox. By applying this indexing to the Burali-Forti paradox we obtain another proof that the cardinal
numbers are a proper class rather than a set, and (at least in ZFC or in von Neumann–Bernays–Gödel set theory) it
follows from this that there is a bijection between the class of cardinals and the class of all sets. Since every set is a
subset of this latter class, and every cardinality is the cardinality of a set (by definition!) this intuitively means that
169
Cantor's paradox
the "cardinality" of the collection of cardinals is greater than the cardinality of any set: it is more infinite than any
true infinity. This is the paradoxical nature of Cantor's "paradox".
Historical note
While Cantor is usually credited with first identifying this property of cardinal sets, some mathematicians award this
distinction to Bertrand Russell, who defined a similar theorem in 1899 or 1901.
References
• Anellis, I.H. (1991). Drucker, Thomas. ed. "The first Russell paradox," Perspectives on the History of
Mathematical Logic. Cambridge, Mass.: Birkäuser Boston. pp. 33–46.
• Moore, G.H. and Garciadiego, A. (1981). "Burali-Forti's paradox: a reappraisal of its origins". Historia Math 8
(3): 319–350. doi:10.1016/0315-0860(81)90070-7.
External links
• An Historical Account of Set-Theoretic Antinomies Caused by the Axiom of Abstraction [1]: report by Justin T.
Miller, Department of Mathematics, University of Arizona.
• PlanetMath.org [2]: article.
References
[1] http:/ / citeseer. ist. psu. edu/ 496807. html
[2] http:/ / planetmath. org/ encyclopedia/ CantorsParadox. html
Coastline paradox
The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a
well-defined length. This results from the fractal-like properties of coastlines.[1][2] The first recorded observation of
this phenomenon was by Lewis Fry Richardson.
More concretely, the length of the coastline depends on the method used to measure it. Since a landmass has features
at all scales, from hundreds of kilometers in size to tiny fractions of a millimeter and below, there is no obvious limit
170
Coastline paradox
to the size of the smallest feature that should not be measured around, and hence no single well-defined perimeter to
the landmass. Various approximations exist when specific assumptions are made about minimum feature size.
For practical considerations, an appropriate choice of minimum feature size is on the order of the units being used to
measure. If a coastline is measured in kilometers, then small variations much smaller than one kilometer are easily
ignored. To measure the coastline in centimeters, tiny variations the size of centimeters must be considered.
However, at scales on the order of centimeters various arbitrary and non-fractal assumptions must be made, such as
where an estuary joins the sea, or where in a broad tidal flat the coastline measurements ought to be taken. Using
different measurement methodologies for different units also destroys the usual certainty that units can be converted
by a simple multiplication.
Extreme cases of the coastline paradox include the fjord-heavy coastlines of Norway, Chile and the Pacific
Northwest of North America. From the southern tip of Vancouver Island northwards to the southern tip of the Alaska
Panhandle, the convolutions of the coastline of the Canadian province of British Columbia make it over 10% of the
entire Canadian coastline—25,725 km (15,985 mi) vs 243,042 km (151,019 mi) over a linear distance of only
965 km (600 mi), including the maze of islands of the Arctic archipelago.[3]
Notes
[1] Weisstein, Eric W., " Coastline Paradox (http:/ / mathworld. wolfram. com/ CoastlineParadox. html)" from MathWorld.
[2] Mandelbrot, Benoit (1983). The Fractal Geometry of Nature. W.H. Freeman and Co.. 25–33. ISBN 978-0-7167-1186-5.
[3] Sebert, L.M., and M. R. Munro. 1972. Dimensions and Areas of Maps of the National Topographic System of Canada. Technical Report
72-1. Ottawa: Department of Energy, Mines and Resources, Surveys and Mapping Branch.
External links
• The Atlas of Canada – Coastline and Shoreline (http://atlas.nrcan.gc.ca/site/english/learningresources/facts/
coastline.html)
• La costa infinita (animation of a coastline with fractal details) (http://cibermitanios.com.ar/2008/01/
la-costa-infinita.html)
171
Cramer's paradox
Cramer's paradox
In mathematics, Cramer's paradox is the statement that the number of points of intersection of two higher-order
curves can be greater than the number of arbitrary points needed to define one such curve.
Cramer's paradox is the result of two theorems: Bézout's theorem (the number of points of intersection of two
algebraic curves is equal to the product of their degrees) and a theorem of Cramer (a curve of degree n is determined
by n(n + 3)/2 points). Observe that for n ≥ 3, it is the case that n2 is greater than or equal to n(n + 3)/2.
History
The paradox was first published by Colin Maclaurin.[1][2] Cramer and Leonard Euler corresponded on the paradox in
letters of 1744 and 1745 and Euler explained the problem to Cramer.[3]
It has become known as Cramer's paradox after featuring in Gabriel Cramer's 1750 book Introduction à l'analyse des
lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement.[4]
At around the same time Euler published examples showing a cubic curve which was not uniquely defined by 9
points[3][5] and discussed the problem in his book Introductio in analysin infinitorum.
The result was publicized by James Stirling and explained by Julius Plücker.[6]
No paradox for lines and conics
For first order curves (that is lines) the paradox does not occur. In general two lines L1 and L2 intersect at a single
point P unless the lines are of equal gradient. A single point is not sufficient to define a line (two are needed);
through the point P there pass not only the two given lines but an infinite number of other lines as well.
Similarly two conics intersect at 4 points, and 5 points are needed to define a conic.
The paradox illustrated: cubics and higher curves
By Bézout's theorem two cubics (curves of degree 3) intersect in 9 points. By Cramer's theorem, 9 arbitrary points
define a unique cubic. At first thought it may seem that the number of intersection points is too high, defining a
unique cubic rather than the two separate cubics that meet there.
Similarly for two curves of degree 4, there will be 16 points of intersection. Through 16 points (assuming they are
arbitrarily given) we will usually not be able to draw any quartic curve (14 points suffice), let alone two intersecting
quartics.
References
[1] Maclaurin, Colin (1720). Geometria Organica. London.
[2] Tweedie, Charles (January 1891). "V.—The “Geometria Organica” of Colin Maclaurin: A Historical and Critical Survey" (http:/ / journals.
cambridge. org/ action/ displayAbstract?fromPage=online& aid=8340277). Transactions of the Royal Society of Edinburgh 36 (1-2): 87-150. .
Retrieved 28 September 2012.
[3] Struik, D. J. (1969). A Source Book in Mathematics, 1200-1800. Harvard University Press. p. 182. ISBN 0674823559.
[4] Tweedie, Charles (1915). "A Study of the Life and Writings of Colin Maclaurin". The Mathematical Gazette 8 (119): 133-151.
JSTOR 3604693.
[5] Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Academie des Sciences de Berlin 4, 219-233,
1750
[6] Weisstein, Eric W., " Cramér-Euler Paradox (http:/ / mathworld. wolfram. com/ Cramer-EulerParadox. html)" from MathWorld.
172
Cramer's paradox
External links
• Ed Sandifer "Cramer’s Paradox" (http://www.maa.org/editorial/euler/How Euler Did It 10 Cramers Paradox.
pdf)
Elevator paradox
This article refers to the elevator paradox for the transport device. For the elevator paradox for the
hydrometer, see elevator paradox (physics).
The elevator paradox is a paradox first noted by Marvin Stern and George Gamow, physicists who had offices on
different floors of a multi-story building. Gamow, who had an office near the bottom of the building noticed that the
first elevator to stop at his floor was most often going down, while Stern, who had an office near the top, noticed that
the first elevator to stop at his floor was most often going up.
At first sight, this created the impression that perhaps elevator cars were being manufactured in the middle of the
building and sent upwards to the roof and downwards to the basement to be dismantled. Clearly this was not the
case. But how could the observation be explained?
Modeling the elevator problem
Several attempts (beginning with Gamow
and Stern) were made to analyze the reason
for this phenomenon: the basic analysis is
simple, while detailed analysis is more
difficult than it would at first appear.
Simply, if one is on the top floor of a
building, all elevators will come from below
(none can come from above), and then
depart going down, while if one is on the
second from top floor, an elevator going to
the top floor will pass first on the way up,
and then shortly afterward on the way down
– thus, while an equal number will pass
Near the top floor, elevators to the top come down shortly after they go up.
going up as going down, downwards
elevators will generally shortly follow upwards elevators (unless the elevator idles on the top floor), and thus the first
elevator observed will usually be going up. The first elevator observed will be going down only if one begins
observing in the short interval after an elevator has passed going up, while the rest of the time the first elevator
observed will be going up.
In more detail, the explanation is as follows: a single elevator spends most of its time in the larger section of the
building, and thus is more likely to approach from that direction when the prospective elevator user arrives. An
observer who remains by the elevator doors for hours or days, observing every elevator arrival, rather than only
observing the first elevator to arrive, would note an equal number of elevators traveling in each direction. This then
becomes a sampling problem — the observer is sampling stochastically a non uniform interval.
To help visualize this, consider a thirty-story building, plus lobby, with only one slow elevator. The elevator is so
slow because it stops at every floor on the way up, and then on every floor on the way down. It takes a minute to
travel between floors and wait for passengers. Here is the arrival schedule for people unlucky enough to work in this
building; as depicted above, it forms a triangle wave:
173
Elevator paradox
174
Floor
Time on way up Time on way down
Lobby
8:00, 9:00, ...
n/a
1st floor
8:01, 9:01, ...
8:59, 9:59, ...
2nd floor
8:02, 9:02, ...
8:58, 9:58, ...
...
...
...
29th floor 8:29, 9:29, ...
8:31, 9:31, ...
30th floor n/a
8:30, 9:30, ...
If you were on the first floor and walked up randomly to the elevator, chances are the next elevator would be heading
down. The next elevator would be heading up only during the first two minutes at each hour, e.g., at 9:00 and 9:01.
The number of elevator stops going upwards and downwards are the same, but the odds that the next elevator is
going up is only 2 in 60.
A similar effect can be observed in railway stations where a station near the end of the line will likely have the next
train headed for the end of the line. Another visualization is to imagine sitting in bleachers near one end of an oval
racetrack: if you are waiting for a single car to pass in front of you, it will be more likely to pass on the straight-away
before entering the turn.
More than one elevator
Interestingly, if there is more than one elevator in a building, the bias decreases — since there is a greater chance
that the intending passenger will arrive at the elevator lobby during the time that at least one elevator is below them;
with an infinite number of elevators, the probabilities would be equal.[1]
In the example above, if there are 30 floors and 58 elevators, so at every minute there are 2 elevators on each floor,
one going up and one going down (save at the top and bottom), the bias is eliminated – every minute, one elevator
arrives going up and another going down. This also occurs with 30 elevators spaced 2 minutes apart – on odd floors
they alternate up/down arrivals, while on even floors they arrive simultaneously every two minutes.
Watching cars pass on an oval racetrack, one perceives little bias if the time between cars is small compared to the
time required for a car to return past the observer.
The real-world case
In a real building, there are complicated factors such as the tendency of elevators to be frequently required on the
ground or first floor, and to return there when idle. These factors tend to shift the frequency of observed arrivals, but
do not eliminate the paradox entirely. In particular, a user very near the top floor will perceive the paradox even
more strongly, as elevators are infrequently present or required above their floor.
There are other complications of a real building: such as lopsided demand where everyone wants to go down at the
end of the day; the way full elevators skip extra stops; or the effect of short trips where the elevator stays idle. These
complications make the paradox harder to visualize than the race track examples.
Elevator paradox
Popular culture
The elevator paradox was mentioned by Charlie Eppes on the television show Numb3rs in the episode entitled
"Chinese Box".[2]
References
[1] Knuth, Donald E. (1969-7). "The Gamow-Stern Elevator Problem" (http:/ / www. baywood. com/ journals/ previewjournals.
asp?Id=0022-412x). Journal of Recreational Mathematics (Baywood Publishing Company, Inc.) 2: 131–137. ISSN: 0022-412x. .
[2] Numb3rs Episode 410: Chinese Box: Wolfram Research Math Notes (http:/ / numb3rs. wolfram. com/ 410/ )
• Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, chapter 10. W H Freeman & Co.;
(October 1986). ISBN 0-7167-1799-9.
• Martin Gardner, Aha! Gotcha, page 96. W H Freeman & Co.; 1982. ISBN 0-7167-1414-0
External links
• A detailed treatment, part 1 (http://www.kwansei.ac.jp/hs/z90010/english/sugakuc/toukei/elevator/
elevator.htm) by Tokihiko Niwa
• Part 2: the multi-elevator case (http://www.kwansei.ac.jp/hs/z90010/english/sugakuc/toukei/elevator2/
elevator2.htm)
• MathWorld article (http://mathworld.wolfram.com/ElevatorParadox.html) on the elevator paradox
False positive paradox
The false positive paradox is a statistical result where false positive tests are more probable than true positive tests,
occurring when the overall population has a low incidence of a condition and the incidence rate is lower than the
false positive rate. The probability of a positive test result is determined not only by the accuracy of the test but by
the characteristics of the sampled population.[1] When the incidence, the proportion of those who have a given
condition, is lower than the test's false positive rate, even tests that have a very low chance of giving a false positive
in an individual case will give more false than true positives overall.[2] So, in a society with very few infected
people—fewer proportionately than the test gives false positives—there will actually be more who test positive for a
disease incorrectly and don't have it than those who test positive accurately and do. The paradox has surprised
many.[3]
It is especially counter-intuitive when interpreting a positive result in a test on a low-incidence population after
having dealt with positive results drawn from a high-incidence population.[2] If the false positive rate of the test is
higher than the proportion of the new population with the condition, then a test administrator whose experience has
been drawn from testing in a high-incidence population may conclude from experience that a positive test result
usually indicates a positive subject, when in fact a false positive is far more likely to have occurred.
Not adjusting to the scarcity of the condition in the new population, and concluding that a positive test result
probably indicates a positive subject, even though population incidence is below the false positive rate is a "base rate
fallacy".
175
False positive paradox
Example
High-incidence population
Imagine running an HIV test on population A, in which 200 out of 10,000 (2%) are infected. The test has a false
positive rate of .0004 (.04%) and no false negative rate. The expected outcome of a million tests on population A
would be:
Unhealthy and test indicates disease (true positive)
1,000,000 × (200/10000) = 20,000 people would receive a true positive
Healthy and test indicates disease (false positive)
1,000,000 × (9800/10000) × .0004 = 392 people would receive a false positive
(The remaining 979,608 tests are correctly negative.)
So, in population A, a person receiving a positive test could be over 98% confident (20,000/20,392) that it correctly
indicates infection.
Low-incidence population
Now consider the same test applied to population B, in which only 1 person in 10,000 (.01%) is infected . The
expected outcome of a million tests on population B would be:
Unhealthy and test indicates disease (true positive)
1,000,000 × (1/10,000) = 100 people would receive a true positive
Healthy and test indicates disease (false positive)
1,000,000 × (9999/10,000) × .0004 ≈ 400 people would receive a false positive
(The remaining 999,500 tests are correctly negative.)
In population B, only 100 of the 500 total people with a positive test result are actually infected. So, the probability
of actually being infected after you are told you are infected is only 20% (100/500) for a test that otherwise appears
to be "over 99.95% accurate".
A tester with experience of group A might find it a paradox that in group B, a result that had almost always indicated
infection is now usually a false positive. The confusion of the posterior probability of infection with the prior
probability of receiving a false negative is a natural error after receiving a life-threatening test result.
References
[1] Rheinfurth, M. H.; Howell, L. W. (March 1998). Probability and statistics in aerospace engineering (http:/ / ntrs. nasa. gov/ archive/ nasa/
casi. ntrs. nasa. gov/ 19980045313_1998119122. pdf) (pdf). NASA. p. 16. . "MESSAGE: False positive tests are more probable than true
positive tests when the overall population has a low incidence of the disease. This is called the false-positive paradox."
[2] Vacher, H. L. (May 2003). "Quantitative literacy - drug testing, cancer screening, and the identification of igneous rocks" (http:/ / findarticles.
com/ p/ articles/ mi_qa4089/ is_200305/ ai_n9252796/ pg_2/ ). Journal of Geoscience Education: 2. . "At first glance, this seems perverse: the
less the students as a whole use steroids, the more likely a student identified as a user will be a non-user. This has been called the False
Positive Paradox" - Citing: Smith, W. (1993). The cartoon guide to statistics. New York: Harper Collins. p. 49.
[3] Madison, B. L. (August 2007). "Mathematical Proficiency for Citizenship" (http:/ / books. google. com/ books?id=5gQz0akjYcwC&
pg=113#v=onepage& q& f=false). In Schoenfeld, A. H.. Assessing Mathematical Proficiency. Mathematical Sciences Research Institute
Publications (New ed.). Cambridge University Press. p. 122. ISBN 978-0-521-69766-8. . "The correct [probability estimate...] is surprising to
many; hence, the term paradox."
176
False positive paradox
177
External links
• The false positive paradox explained visually (http://www.youtube.com/watch?v=D8VZqxcu0I0) (video)
Gabriel's Horn
Gabriel's Horn (also called Torricelli's trumpet) is a geometric
figure which has infinite surface area, but finite volume. The name
refers to the tradition identifying the Archangel Gabriel as the angel
who blows the horn to announce Judgment Day, associating the divine,
or infinite, with the finite. The properties of this figure were first
studied by Italian physicist and mathematician Evangelista Torricelli.
3D illustration of Gabriel's Horn.
Mathematical definition
Gabriel's horn is formed by taking the graph of
domain
, with the
(thus avoiding the asymptote at x = 0) and rotating it in
three dimensions about the x-axis. The discovery was made using
Cavalieri's principle before the invention of calculus, but today
calculus can be used to calculate the volume and surface area of the
horn between x = 1 and x = a, where a > 1. Using integration (see Solid
of revolution and Surface of revolution for details), it is possible to find
the volume
and the surface area
:
Graph of y = 1/x
can be as large as required, but it can be seen from the equation that the volume of the part of the horn between
and
will never exceed ; however, it will get closer and closer to as becomes larger.
Mathematically, the volume approaches
volume may be expressed as:
This is so because as
approaches infinity,
as
approaches infinity. Using the limit notation of calculus, the
approaches zero. This means the volume approaches
which equals .
As for the area, the above shows that the area is greater than
bound for the natural logarithm of
surface area. That is to say;
as
or
times the natural logarithm of
(1 - 0)
. There is no upper
as it approaches infinity. That means, in this case, that the horn has an infinite
Gabriel's Horn
Apparent paradox
When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinite curve about the x-axis
generates an object of finite volume was considered paradoxical. However, the explanation is that the bounding
curve,
, is simply a special case–just like the simple harmonic series (Σx−1)–for which the successive area
2
'segments' do not decrease rapidly enough to allow for convergence to a limit. For volume segments (Σ1/x )
however, and in fact for any generally constructed higher degree curve (e.g. y = 1/x1.001), the same is not true and the
rate of decrease in the associated series is sufficiently rapid for convergence to a (finite) limiting sum.
The apparent paradox formed part of a great dispute over the nature of infinity involving many of the key thinkers of
the time including Thomas Hobbes, John Wallis and Galileo Galilei.[1]
Painter's Paradox
Since the Horn has finite volume but infinite surface area, it seems that it could be filled with a finite quantity of
paint, and yet that paint would not be sufficient to coat its inner surface – an apparent paradox. In fact, in a
theoretical mathematical sense, a finite amount of paint can coat an infinite area, provided the thickness of the coat
becomes vanishingly small "quickly enough" to compensate for the ever-expanding area, which in this case is forced
to happen to an inner-surface coat as the horn narrows. However, to coat the outer surface of the horn with a constant
thickness of paint, no matter how thin, would require an infinite amount of paint.[2]
Of course, in reality, paint is not infinitely divisible, and at some point the horn would become too narrow for even
one molecule to pass.
References
[1] Havil, Julian (2007). Nonplussed!: mathematical proof of implausible ideas. Princeton University Press. pp. 82–91. ISBN 0-691-12056-0.
[2] Clegg, Brian (2003). Infinity: The Quest to Think the Unthinkable. Robinson (Constable & Robinson Ltd). pp. 239–242.
ISBN 978-1-84119-650-3.
External links
•
•
•
•
Information and diagrams about Gabriel's Horn (http://curvebank.calstatela.edu/torricelli/torricelli.htm)
Torricelli's Trumpet at PlanetMath (http://planetmath.org/encyclopedia/TorricellisTrumpet.html)
Weisstein, Eric W., " Gabriel's Horn (http://mathworld.wolfram.com/GabrielsHorn.html)" from MathWorld.
"Gabriel's Horn" (http://demonstrations.wolfram.com/GabrielsHorn/) by John Snyder, the Wolfram
Demonstrations Project, 2007.
• Gabriel's Horn: An Understanding of a Solid with Finite Volume and Infinite Surface Area (http://www.
palmbeachstate.edu/honors/Documents/jeansergejoseph.pdf) by Jean S. Joseph.
178
Galileo's paradox
179
Galileo's paradox
Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. In his final scientific work,
Two New Sciences, Galileo Galilei made apparently contradictory statements about the positive integers. First, some
numbers are squares, while others are not; therefore, all the numbers, including both squares and non-squares, must
be more numerous than just the squares. And yet, for every square there is exactly one positive number that is its
square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other.
This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets.
Galileo concluded that the ideas of less, equal, and greater apply to finite sets, but not to infinite sets. In the
nineteenth century, using the same methods, Cantor showed that this restriction is not necessary. It is possible to
define comparisons amongst infinite sets in a meaningful way (by which definition the two sets he considers,
integers and squares, have "the same size"), and that by this definition some infinite sets are strictly larger than
others.
Galileo on infinite sets
The relevant section of Two New Sciences is excerpted below:[1]
Simplicio: Here a difficulty presents itself which appears to me insoluble. Since it is clear that we may
have one line greater than another, each containing an infinite number of points, we are forced to admit
that, within one and the same class, we may have something greater than infinity, because the infinity of
points in the long line is greater than the infinity of points in the short line. This assigning to an infinite
quantity a value greater than infinity is quite beyond my comprehension.
Salviati: This is one of the difficulties which arise when we attempt, with our finite minds, to discuss the
infinite, assigning to it those properties which we give to the finite and limited; but this I think is wrong,
for we cannot speak of infinite quantities as being the one greater or less than or equal to another. To
prove this I have in mind an argument which, for the sake of clearness, I shall put in the form of
questions to Simplicio who raised this difficulty.
I take it for granted that you know which of the numbers are squares and which are not.
Simplicio: I am quite aware that a squared number is one which results from the multiplication of
another number by itself; thus 4, 9, etc., are squared numbers which come from multiplying 2, 3, etc., by
themselves.
Salviati: Very well; and you also know that just as the products are called squares so the factors are
called sides or roots; while on the other hand those numbers which do not consist of two equal factors
are not squares. Therefore if I assert that all numbers, including both squares and non-squares, are more
than the squares alone, I shall speak the truth, shall I not?
Simplicio: Most certainly.
Salviati: If I should ask further how many squares there are one might reply truly that there are as many
as the corresponding number of roots, since every square has its own root and every root its own square,
while no square has more than one root and no root more than one square.
Simplicio: Precisely so.
Salviati: But if I inquire how many roots there are, it cannot be denied that there are as many as the
numbers because every number is the root of some square. This being granted, we must say that there
are as many squares as there are numbers because they are just as numerous as their roots, and all the
numbers are roots. Yet at the outset we said that there are many more numbers than squares, since the
larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes
Galileo's paradox
180
as we pass to larger numbers, Thus up to 100 we have 10 squares, that is, the squares constitute 1/10 part
of all the numbers; up to 10000, we find only 1/100 part to be squares; and up to a million only 1/1000
part; on the other hand in an infinite number, if one could conceive of such a thing, he would be forced
to admit that there are as many squares as there are numbers taken all together.
Sagredo: What then must one conclude under these circumstances?
Salviati: So far as I see we can only infer that the totality of all numbers is infinite, that the number of
squares is infinite, and that the number of their roots is infinite; neither is the number of squares less
than the totality of all the numbers, nor the latter greater than the former; and finally the attributes
"equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities. When therefore
Simplicio introduces several lines of different lengths and asks me how it is possible that the longer ones
do not contain more points than the shorter, I answer him that one line does not contain more or less or
just as many points as another, but that each line contains an infinite number.
— Galileo, Two New Sciences
References
[1] Galilei, Galileo (1954) [1638]. Dialogues concerning two new sciences. Transl. Crew and de Salvio. New York: Dover. pp. 31–33.
ISBN 4-86600-998-.
External links
• Philosophical Method and Galileo's Paradox of Infinity (http://philsci-archive.pitt.edu/archive/00004276/),
Matthew W. Parker, in the PhilSci Archive
Gambler's fallacy
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Gambler's fallacy
The Gambler's fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a
Monte Carlo Casino in 1913),[1][2] and also referred to as the fallacy of the maturity of chances, is the belief that if
deviations from expected behaviour are observed in repeated independent trials of some random process, future
deviations in the opposite direction are then more likely.
An example: coin-tossing
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin.
With a fair coin, the outcomes in different tosses are statistically independent and the
probability of getting heads on a single toss is exactly 1⁄2 (one in two). It follows that the
probability of getting two heads in two tosses is 1⁄4 (one in four) and the probability of
getting three heads in three tosses is 1⁄8 (one in eight). In general, if we let Ai be the event
that toss i of a fair coin comes up heads, then we have,
.
Now suppose that we have just tossed four heads in a row, so that if the next coin toss
were also to come up heads, it would complete a run of five successive heads. Since the
probability of a run of five successive heads is only 1⁄32 (one in thirty-two), a believer in
the gambler's fallacy might believe that this next flip is less likely to be heads than to be
tails. However, this is not correct, and is a manifestation of the gambler's fallacy; the
event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely,
each having probability 1⁄32. Given the first four rolls turn up heads, the probability that
the next toss is a head is in fact,
.
While a run of five heads is only 1⁄32 = 0.03125, it is only that before the coin is first
tossed. After the first four tosses the results are no longer unknown, so their probabilities
are 1. Reasoning that it is more likely that the next toss will be a tail than a head due to
the past tosses, that a run of luck in the past somehow influences the odds in the future, is
the fallacy.
Simulation of coin tosses:
Each frame, a coin is
flipped which is red on one
side and blue on the other.
The result of each flip is
added as a colored dot in
the corresponding column.
As the pie chart shows, the
proportion of red versus
blue approaches 50-50 (the
Law of Large Numbers).
But the difference between
red and blue does not
systematically decrease to
zero.
Explaining why the probability is 1/2 for a fair coin
We can see from the above that, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2,097,152.
However, the probability of flipping a head after having already flipped 20 heads in a row is simply 1⁄2. This is an
application of Bayes' theorem.
This can also be seen without knowing that 20 heads have occurred for certain (without applying of Bayes' theorem).
Consider the following two probabilities, assuming a fair coin:
• probability of 20 heads, then 1 tail = 0.520 × 0.5 = 0.521
• probability of 20 heads, then 1 head = 0.520 × 0.5 = 0.521
The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in
2,097,152. Therefore, it is equally likely to flip 21 heads as it is to flip 20 heads and then 1 tail when flipping a fair
coin 21 times. Furthermore, these two probabilities are equally as likely as any other 21-flip combinations that can
Gambler's fallacy
be obtained (there are 2,097,152 total); all 21-flip combinations will have probabilities equal to 0.521, or 1 in
2,097,152. From these observations, there is no reason to assume at any point that a change of luck is warranted
based on prior trials (flips), because every outcome observed will always have been as likely as the other outcomes
that were not observed for that particular trial, given a fair coin. Therefore, just as Bayes' theorem shows, the result
of each trial comes down to the base probability of the fair coin: 1⁄2.
Other examples
There is another way to emphasize the fallacy. As already mentioned, the fallacy is built on the notion that previous
failures indicate an increased probability of success on subsequent attempts. This is, in fact, the inverse of what
actually happens, even on a fair chance of a successful event, given a set number of iterations. Assume a fair
16-sided die, where a win is defined as rolling a 1. Assume a player is given 16 rolls to obtain at least one win
(1−p(rolling no ones)). The low winning odds are just to make the change in probability more noticeable. The
probability of having at least one win in the 16 rolls is:
However, assume now that the first roll was a loss (93.75% chance of that, 15⁄16). The player now only has 15 rolls
left and, according to the fallacy, should have a higher chance of winning since one loss has occurred. His chances of
having at least one win are now:
Simply by losing one toss the player's probability of winning dropped by 2 percentage points. By the time this
reaches 5 losses (11 rolls left), his probability of winning on one of the remaining rolls will have dropped to ~50%.
The player's odds for at least one win in those 16 rolls has not increased given a series of losses; his odds have
decreased because he has fewer iterations left to win. In other words, the previous losses in no way contribute to the
odds of the remaining attempts, but there are fewer remaining attempts to gain a win, which results in a lower
probability of obtaining it.
The player becomes more likely to lose in a set number of iterations as he fails to win, and eventually his probability
of winning will again equal the probability of winning a single toss, when only one toss is left: 6.25% in this
instance.
Some lottery players will choose the same numbers every time, or intentionally change their numbers, but both are
equally likely to win any individual lottery draw. Copying the numbers that won the previous lottery draw gives an
equal probability, although a rational gambler might attempt to predict other players' choices and then deliberately
avoid these numbers. Low numbers (below 31 and especially below 12) are popular because people play birthdays as
their so-called lucky numbers; hence a win in which these numbers are over-represented is more likely to result in a
shared payout.
A joke told among mathematicians demonstrates the nature of the fallacy. When flying on an aircraft, a man decides
to always bring a bomb with him. "The chances of an aircraft having a bomb on it are very small," he reasons, "and
certainly the chances of having two are almost none!" A similar example is in the book The World According to
Garp when the hero Garp decides to buy a house a moment after a small plane crashes into it, reasoning that the
chances of another plane hitting the house have just dropped to zero.
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Gambler's fallacy
Reverse fallacy
The reversal is also a fallacy (not to be confused with the inverse gambler's fallacy) in which a gambler may instead
decide that tails are more likely out of some mystical preconception that fate has thus far allowed for consistent
results of tails. Believing the odds to favor tails, the gambler sees no reason to change to heads. Again, the fallacy is
the belief that the "universe" somehow carries a memory of past results which tend to favor or disfavor future
outcomes.
Caveats
In most illustrations of the gambler's fallacy and the reversed gambler's fallacy, the trial (e.g. flipping a coin) is
assumed to be fair. In practice, this assumption may not hold.
For example, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2,097,152 (above). If the coin is
fair, then the probability of the next flip being heads is 1/2. However, because the odds of flipping 21 heads in a row
is so slim, it may well be that the coin is somehow biased towards landing on heads, or that it is being controlled by
hidden magnets, or similar.[3] In this case, the smart bet is "heads" because the empirical evidence—21 "heads" in a
row—suggests that the coin is likely to be biased toward "heads", contradicting the general assumption that the coin
is fair.
Childbirth
Instances of the gambler’s fallacy when applied to childbirth can be traced all the way back to 1796, in Pierre-Simon
Laplace’s A Philosophical Essay on Probabilities. Laplace wrote of the ways men calculated their probability of
having sons: "I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of
boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls
ought to be the same at the end of each month, they judged that the boys already born would render more probable
the births next of girls." In short, the expectant fathers feared that if more sons were born in the surrounding
community, then they themselves would be more likely to have a daughter.[4]
Some expectant parents believe that, after having multiple children of the same sex, they are "due" to have a child of
the opposite sex. While the Trivers–Willard hypothesis predicts that birth sex is dependent on living conditions (i.e.
more male children are born in "good" living conditions, while more female children are born in poorer living
conditions), the probability of having a child of either gender is still regarded as 50/50.
Monte Carlo Casino
The most famous example happened in a game of roulette at the Monte Carlo Casino in the summer of 1913, when
the ball fell in black 26 times in a row, an extremely uncommon occurrence (but not more nor less common than any
of the other 67,108,863 sequences of 26 red or black, neglecting the 0 slot on the wheel), and gamblers lost millions
of francs betting against black after the black streak happened. Gamblers reasoned incorrectly that the streak was
causing an "imbalance" in the randomness of the wheel, and that it had to be followed by a long streak of red.[1]
Non-examples of the fallacy
There are many scenarios where the gambler's fallacy might superficially seem to apply, but actually does not. When
the probability of different events is not independent, the probability of future events can change based on the
outcome of past events (see statistical permutation). Formally, the system is said to have memory. An example of this
is cards drawn without replacement. For example, if an ace is drawn from a deck and not reinserted, the next draw is
less likely to be an ace and more likely to be of another rank. The odds for drawing another ace, assuming that it was
the first card drawn and that there are no jokers, have decreased from 4⁄52 (7.69%) to 3⁄51 (5.88%), while the odds for
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Gambler's fallacy
each other rank have increased from 4⁄52 (7.69%) to 4⁄51 (7.84%). This type of effect is what allows card counting
schemes to work (for example in the game of blackjack).
Meanwhile, the reversed gambler's fallacy may appear to apply in the story of Joseph Jagger, who hired clerks to
record the results of roulette wheels in Monte Carlo. He discovered that one wheel favored nine numbers and won
large sums of money until the casino started rebalancing the roulette wheels daily. In this situation, the observation
of the wheel's behavior provided information about the physical properties of the wheel rather than its "probability"
in some abstract sense, a concept which is the basis of both the gambler's fallacy and its reversal. Even a biased
wheel's past results will not affect future results, but the results can provide information about what sort of results the
wheel tends to produce. However, if it is known for certain that the wheel is completely fair, then past results provide
no information about future ones.
The outcome of future events can be affected if external factors are allowed to change the probability of the events
(e.g., changes in the rules of a game affecting a sports team's performance levels). Additionally, an inexperienced
player's success may decrease after opposing teams discover his weaknesses and exploit them. The player must then
attempt to compensate and randomize his strategy. (See Game theory).
Many riddles trick the reader into believing that they are an example of the gambler's fallacy, such as the Monty Hall
problem.
Non-example: unknown probability of event
When the probability of repeated events are not known, outcomes may not be equally probable. In the case of coin
tossing, as a run of heads gets longer and longer, the likelihood that the coin is biased towards heads increases. If one
flips a coin 21 times in a row and obtains 21 heads, one might rationally conclude a high probability of bias towards
heads, and hence conclude that future flips of this coin are also highly likely to be heads. In fact, Bayesian inference
can be used to show that when the long-run proportion of different outcomes are unknown but exchangeable
(meaning that the random process from which they are generated may be biased but is equally likely to be biased in
any direction) previous observations demonstrate the likely direction of the bias, such that the outcome which has
occurred the most in the observed data is the most likely to occur again.[5]
Psychology behind the fallacy
Origins
Gambler's fallacy arises out of a belief in the law of small numbers, or the erroneous belief that small samples must
be representative of the larger population. According to the fallacy, "streaks" must eventually even out in order to be
representative.[6] Amos Tversky and Daniel Kahneman first proposed that the gambler's fallacy is a cognitive bias
produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the
probability of a certain event by assessing how similar it is to events they have experienced before, and how similar
the events surrounding those two processes are.[7][8] According to this view, "after observing a long run of red on the
roulette wheel, for example, most people erroneously believe that black will result in a more representative sequence
than the occurrence of an additional red",[9] so people expect that a short run of random outcomes should share
properties of a longer run, specifically in that deviations from average should balance out. When people are asked to
make up a random-looking sequence of coin tosses, they tend to make sequences where the proportion of heads to
tails stays closer to 0.5 in any short segment than would be predicted by chance (insensitivity to sample size);[10]
Kahneman and Tversky interpret this to mean that people believe short sequences of random events should be
representative of longer ones.[11] The representativeness heuristic is also cited behind the related phenomenon of the
clustering illusion, according to which people see streaks of random events as being non-random when such streaks
are actually much more likely to occur in small samples than people expect.[12]
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Gambler's fallacy
The gambler's fallacy can also be attributed to the mistaken belief that gambling (or even chance itself) is a fair
process that can correct itself in the event of streaks, otherwise known as the just-world hypothesis.[13] Other
researchers believe that individuals with an internal locus of control - that is, people who believe that the gambling
outcomes are the result of their own skill - are more susceptible to the gambler's fallacy because they reject the idea
that chance could overcome skill or talent.[14]
Variations of the gambler's fallacy
Some researchers believe that there are actually two types of gambler's fallacy: Type I and Type II. Type I is the
"classic" gambler's fallacy, when individuals believe that a certain outcome is "due" after a long streak of another
outcome. Type II gambler's fallacy, as defined by Gideon Keren and Charles Lewis, occurs when a gambler
underestimates how many observations are needed to detect a favorable outcome (such as watching a roulette wheel
for a length of time and then betting on the numbers that appear most often). Detecting a bias that will lead to a
favorable outcome takes an impractically large amount of time and is very difficult, if not impossible, to do,
therefore people fall prey to the Type II gambler's fallacy.[15] The two types are different in that Type I wrongly
assumes that gambling conditions are fair and perfect, while Type II assumes that the conditions are biased, and that
this bias can be detected after a certain amount of time.
Another variety, known as the retrospective gambler's fallacy, occurs when individuals judge that a seemingly rare
event must come from a longer sequence than a more common event does. For example, people believe that an
imaginary sequence of die rolls is more than three times as long when a set of three 6's is observed as opposed to
when there are only two 6's. This effect can be observed in isolated instances, or even sequentially. A real world
example is when a teenager becomes pregnant after having unprotected sex, people assume that she has been
engaging in unprotected sex for longer than someone who has been engaging in unprotected sex and is not
pregnant.[16]
Relationship to hot-hand fallacy
Another psychological perspective states that gambler's fallacy can be seen as the counterpart to basketball's
Hot-hand fallacy. In the hot-hand fallacy, people tend to predict the same outcome of the last event (positive
recency) - that a high scorer will continue to score. In gambler's fallacy, however, people predict the opposite
outcome of the last event (negative recency) - that, for example, since the roulette wheel has landed on black the last
six times, it is due to land on red the next. Ayton and Fischer have theorized that people display positive recency for
the hot-hand fallacy because the fallacy deals with human performance, and that people do not believe that an
inanimate object can become "hot."[17] Human performance is not perceived as "random," and people are more likely
to continue streaks when they believe that the process generating the results is nonrandom.[18] Usually, when a
person exhibits the gambler's fallacy, they are more likely to exhibit the hot-hand fallacy as well, suggesting that one
construct is responsible for the two fallacies.[19]
The difference between the two fallacies is also represented in economic decision-making. A study by Huber,
Kirchler, and Stockl (2010) examined how the hot hand and the gambler's fallacy are exhibited in the financial
market. The researchers gave their participants a choice: they could either bet on the outcome of a series of coin
tosses, use an "expert" opinion to sway their decision, or choose a risk-free alternative instead for a smaller financial
reward. Participants turned to the "expert" opinion to make their decision 24% of the time based on their past
experience of success, which exemplifies the hot-hand. If the expert was correct, 78% of the participants chose the
expert's opinion again, as opposed to 57% doing so when the expert was wrong. The participants also exhibited the
gambler's fallacy, with their selection of either heads or tails decreasing after noticing a streak of that outcome. This
experiment helped bolster Ayton and Fischer's theory that people put more faith in human performance than they do
in seemingly random processes.[20]
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Gambler's fallacy
Neurophysiology
While the representativeness heuristic and other cognitive biases are the most commonly cited cause of the gambler's
fallacy, research suggests that there may be a neurological component to it as well. Functional magnetic resonance
imaging has revealed that, after losing a bet or gamble ("riskloss"), the frontoparietal network of the brain is
activated, resulting in more risk-taking behavior. In contrast, there is decreased activity in the amygdala, caudate and
ventral striatum after a riskloss. Activation in the amygdala is negatively correlated with gambler's fallacy - the more
activity exhibited in the amygdala, the less likely an individual is to fall prey to the gambler's fallacy. These results
suggest that gambler's fallacy relies more on the prefrontal cortex (responsible for executive, goal-directed
processes) and less on the brain areas that control affective decision-making.
The desire to continue gambling or betting is controlled by the striatum, which supports a choice-outcome
contingency learning method. The striatum processes the errors in prediction and the behavior changes accordingly.
After a win, the positive behavior is reinforced and after a loss, the behavior is conditioned to be avoided. In
individuals exhibiting the gambler's fallacy, this choice-outcome contingency method is impaired, and they continue
to make risks after a series of losses.[21]
Possible solutions
The gambler's fallacy is a deep-seated cognitive bias and therefore very difficult to eliminate. For the most part,
educating individuals about the nature of randomness has not proven effective in reducing or eliminating any
manifestation of the gambler's fallacy. Participants in an early study by Beach and Swensson (1967) were shown a
shuffled deck of index cards with shapes on them, and were told to guess which shape would come next in a
sequence. The experimental group of participants was informed about the nature and existence of the gambler's
fallacy, and were explicitly instructed not to rely on "run dependency" to make their guesses. The control group was
not given this information. Even so, the response styles of the two groups were similar, indicating that the
experimental group still based their choices on the length of the run sequence. Clearly, instructing individuals about
randomness is not sufficient in lessening the gambler's fallacy.[22]
It does appear, however, that an individual's susceptibility to the gambler's fallacy decreases with age. Fischbein and
Schnarch (1997) administered a questionnaire to five groups: students in grades 5, 7, 9, 11, and college students
specializing in teaching mathematics. None of the participants had received any prior education regarding
probability. The question was, "Ronni flipped a coin three times and in all cases heads came up. Ronni intends to flip
the coin again. What is the chance of getting heads the fourth time?" The results indicated that as the older the
students got, the less likely they were to answer with "smaller than the chance of getting tails," which would indicate
a negative recency effect. 35% of the 5th graders, 35% of the 7th graders, and 20% of the 9th graders exhibited the
negative recency effect. Only 10% of the 11th graders answered this way, however, and none of the college students
did. Fischbein and Schnarch therefore theorized that an individual's tendency to rely on the representativeness
heuristic and other cognitive biases can be overcome with age.[23]
Another possible solution that could be seen as more proactive comes from Roney and Trick, Gestalt psychologists
who suggest that the fallacy may be eliminated as a result of grouping. When a future event (ex: a coin toss) is
described as part of a sequence, no matter how arbitrarily, a person will automatically consider the event as it relates
to the past events, resulting in the gambler's fallacy. When a person considers every event as independent, however,
the fallacy can be greatly reduced.[24]
In their experiment, Roney and Trick told participants that they were betting on either two blocks of six coin tosses,
or on two blocks of seven coin tosses. The fourth, fifth, and sixth tosses all had the same outcome, either three heads
or three tails. The seventh toss was grouped with either the end of one block, or the beginning of the next block.
Participants exhibited the strongest gambler's fallacy when the seventh trial was part of the first block, directly after
the sequence of three heads or tails. Additionally, the researchers pointed out how insidious the fallacy can be - the
participants that did not show the gambler's fallacy showed less confidence in their bets and bet fewer times than the
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Gambler's fallacy
participants who picked "with" the gambler's fallacy. However, when the seventh trial was grouped with the second
block (and was therefore perceived as not being part of a streak), the gambler's fallacy did not occur.
Roney and Trick argue that a solution to gambler's fallacy could be, instead of teaching individuals about the nature
of randomness, training people to treat each event as if it is a beginning and not a continuation of previous events.
This would prevent people from gambling when they are losing in the vain hope that their chances of winning are
due to increase.
References
[1]
[2]
[3]
[4]
[5]
Lehrer, Jonah (2009). How We Decide. New York: Houghton Mifflin Harcourt. p. 66. ISBN 978-0-618-62011-1.
Blog - "Fallacy Files" (http:/ / www. fallacyfiles. org/ gamblers. html) What happened at Monte Carlo in 1913.
Martin Gardner, Entertaining Mathematical Puzzles, Dover Publications, 69-70.
Barron, G. and Leider, S. (2010). The role of experience in the gambler's fallacy. Journal of Behavioral Decision Making, 23, 117-129.
O'Neill, B. and Puza, B.D. (2004) Dice have no memories but I do: A defence of the reverse gambler's belief. (http:/ / cbe. anu. edu. au/
research/ papers/ pdf/ STAT0004WP. pdf). Reprinted in abridged form as O'Neill, B. and Puza, B.D. (2005) In defence of the reverse
gambler's belief. The Mathematical Scientist 30(1), pp. 13–16.
[6] Burns, B.D. and Corpus, B. (2004). Randomness and inductions from streaks: "Gambler's fallacy" versus "hot hand." Psychonomic Bulletin
and Review. 11, 179-184
[7] Tversky, Amos; Daniel Kahneman (1974). "Judgment under uncertainty: Heuristics and biases". Science 185 (4157): 1124–1131.
doi:10.1126/science.185.4157.1124. PMID 17835457.
[8] Tversky, Amos; Daniel Kahneman (1971). "Belief in the law of small numbers". Psychological Bulletin 76 (2): 105–110.
doi:10.1037/h0031322.
[9] Tversky & Kahneman, 1974.
[10] Tune, G.S. (1964). "Response preferences: A review of some relevant literature". Psychological Bulletin 61 (4): 286–302.
doi:10.1037/h0048618. PMID 14140335.
[11] Tversky & Kahneman, 1971.
[12] Gilovich, Thomas (1991). How we know what isn't so. New York: The Free Press. pp. 16–19. ISBN 0-02-911706-2.
[13] Rogers, P. (1998). The cognitive psychology of lottery gambling: A theoretical review. Journal of Gambling Studies, 14, 111-134
[14] Sundali, J. and Croson, R. (2006). Biases in casino betting: The hot hand and the gambler's fallacy. Judgment and Decision Making, 1, 1-12.
[15] Keren, G. and Lewis, C. (1994). The two fallacies of gamblers: Type I and Type II. Organizational Behavior and Human Decision
Processes, 60, 75-89.
[16] Oppenheimer, D.M. and Monin, B. (2009). The retrospective gambler's fallacy: Unlikely events, constructing the past, and multiple
universes. Judgment and Decision Making, 4, 326-334.
[17] Ayton, P.; Fischer, I. (2004). "The hot hand fallacy and the gambler's fallacy: Two faces of subjective randomness?". Memory and Cognition
32: 1369–1378.
[18] Burns, B.D. and Corpus, B. (2004). Randomness and inductions from streaks: "Gambler's fallacy" versus "hot hand." Psychonomic Bulletin
and Review. 11, 179-184
[19] Sundali, J.; Croson, R. (2006). "Biases in casino betting: The hot hand and the gambler's fallacy". Judgment and Decision Making 1: 1–12.
[20] Huber, J.; Kirchler, M.; Stockl, T. (2010). "The hot hand belief and the gambler's fallacy in investment decisions under risk". Theory and
Decision 68: 445–462.
[21] Xue, G.; Lu, Z.; Levin, I.P.; Bechara, A. (2011). "An fMRI study of risk-taking following wins and losses: Implications for the gambler's
fallacy". Human Brain Mapping 32: 271–281.
[22] Beach, L.R.; Swensson, R.G. (1967). "Instructions about randomness and run dependency in two-choice learning". Journal of Experimental
Psychology 75: 279–282.
[23] Fischbein, E.; Schnarch, D. (1997). "The evolution with age of probabilistic, intuitively based misconceptions". Journal for Research in
Mathematics Education 28: 96–105.
[24] Roney, C.J.; Trick, L.M. (2003). "Grouping and gambling: A gestalt approach to understanding the gambler's fallacy". Canadian Journal of
Experimental Psychology 57: 69–75.
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Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all
but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are
important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not
universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all
mathematics is impossible, giving a negative answer to Hilbert's second problem.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an
"effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths
about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about
the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an
extension of the first, shows that such a system cannot demonstrate its own consistency.
Background
Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a
formal proof from a finite set of axioms is valid. This task, known as automatic proof verification, is closely related
to automated theorem proving. The difference is that instead of constructing a new proof, the proof verifier simply
checks that a provided formal proof (or, in instructions that can be followed to create a formal proof) is correct. This
process is not merely hypothetical; systems such as Isabelle or Coq are used today to formalize proofs and then
check their validity.
Many theories of interest include an infinite set of axioms, however. To verify a formal proof when the set of axioms
is infinite, it must be possible to determine whether a statement that is claimed to be an axiom is actually an axiom.
This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical
induction is expressed as an infinite set of axioms (an axiom schema).
A formal theory is said to be effectively generated if its set of axioms is a recursively enumerable set. This means
that there is a computer program that, in principle, could enumerate all the axioms of the theory without listing any
statements that are not axioms. This is equivalent to the existence of a program that enumerates all the theorems of
the theory without enumerating any statements that are not theorems. Examples of effectively generated theories
with infinite sets of axioms include Peano arithmetic and Zermelo–Fraenkel set theory.
In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any
incorrect results. A set of axioms is complete if, for any statement in the axioms' language, either that statement or its
negation is provable from the axioms. A set of axioms is (simply) consistent if there is no statement such that both
the statement and its negation are provable from the axioms. In the standard system of first-order logic, an
inconsistent set of axioms will prove every statement in its language (this is sometimes called the principle of
explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however,
proves a maximal set of non-contradictory theorems. Gödel's incompleteness theorems show that in certain cases it is
not possible to obtain an effectively generated, complete, consistent theory.
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First incompleteness theorem
Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper On Formally
Undecidable Propositions in Principia Mathematica and Related Systems I.
The formal theorem is written in highly technical language. The broadly accepted natural language statement of the
theorem is:
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and
complete. In particular, for any consistent, effectively generated formal theory that proves certain basic
arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967,
p. 250).
The true but unprovable statement referred to by the theorem is often referred to as "the Gödel sentence" for the
theory. The proof constructs a specific Gödel sentence for each effectively generated theory, but there are infinitely
many statements in the language of the theory that share the property of being true but unprovable. For example, the
conjunction of the Gödel sentence and any logically valid sentence will have this property.
For each consistent formal theory T having the required small amount of number theory, the corresponding Gödel
sentence G asserts: "G cannot be proved within the theory T". This interpretation of G leads to the following
informal analysis. If G were provable under the axioms and rules of inference of T, then T would have a theorem, G,
which effectively contradicts itself, and thus the theory T would be inconsistent. This means that if the theory T is
consistent then G cannot be proved within it, and so the theory T is incomplete. Moreover, the claim G makes about
its own unprovability is correct. In this sense G is not only unprovable but true, and provability-within-the-theory-T
is not the same as truth. This informal analysis can be formalized to make a rigorous proof of the incompleteness
theorem, as described in the section "Proof sketch for the first theorem" below. The formal proof reveals exactly the
hypotheses required for the theory T in order for the self-contradictory nature of G to lead to a genuine contradiction.
Each effectively generated theory has its own Gödel statement. It is possible to define a larger theory T’ that contains
the whole of T, plus G as an additional axiom. This will not result in a complete theory, because Gödel's theorem
will also apply to T’, and thus T’ cannot be complete. In this case, G is indeed a theorem in T’, because it is an axiom.
Since G states only that it is not provable in T, no contradiction is presented by its provability in T’. However,
because the incompleteness theorem applies to T’: there will be a new Gödel statement G’ for T’, showing that T’ is
also incomplete. G’ will differ from G in that G’ will refer to T’, rather than T.
To prove the first incompleteness theorem, Gödel represented statements by numbers. Then the theory at hand,
which is assumed to prove certain facts about numbers, also proves facts about its own statements, provided that it is
effectively generated. Questions about the provability of statements are represented as questions about the properties
of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states
that no natural number exists with a certain, strange property. A number with this property would encode a proof of
the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the
consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number.
Meaning of the first incompleteness theorem
Gödel's first incompleteness theorem shows that any consistent effective formal system that includes enough of the
theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable.
Thus no formal system (satisfying the hypotheses of the theorem) that aims to characterize the natural numbers can
actually do so, as there will be true number-theoretical statements which that system cannot prove. This fact is
sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and
Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob
Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply
equally to second order logic as they do to arithmetic. They argue that only those who believe that the natural
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numbers are to be defined in terms of first order logic have this problem.
The existence of an incomplete formal system is, in itself, not particularly surprising. A system may be incomplete
simply because not all the necessary axioms have been discovered. For example, Euclidean geometry without the
parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining
axioms.
Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent
finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program.
Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with
the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system
inconsistent.
There are complete and consistent lists of axioms for arithmetic that cannot be enumerated by a computer program.
For example, one might take all true statements about the natural numbers to be axioms (and no false statements),
which gives the theory known as "true arithmetic". The difficulty is that there is no mechanical way to decide, given
a statement about the natural numbers, whether it is an axiom of this theory, and thus there is no effective way to
verify a formal proof in this theory.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem,
which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is
often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the
status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem").
Relation to the liar paradox
The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true
(for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a theory T makes a
similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the theory
T." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar
sentence.
It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel
number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's
undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the
incompleteness theorem) and by Alfred Tarski.
Extensions of Gödel's original result
Gödel demonstrated the incompleteness of the theory of Principia Mathematica, a particular theory of arithmetic, but
a parallel demonstration could be given for any effective theory of a certain expressiveness. Gödel commented on
this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern
statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for
the incompleteness theorem, so that it is not limited to any particular formal theory. The terminology used to state
these conditions was not yet developed in 1931 when Gödel published his results.
Gödel's original statement and proof of the incompleteness theorem requires the assumption that the theory is not just
consistent but ω-consistent. A theory is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a
predicate P such that for every specific natural number n the theory proves ~P(n), and yet the theory also proves that
there exists a natural number n such that P(n). That is, the theory says that a number with property P exists while
denying that it has any specific value. The ω-consistency of a theory implies its consistency, but consistency does
not imply ω-consistency. J. Barkley Rosser (1936) strengthened the incompleteness theorem by finding a variation
of the proof (Rosser's trick) that only requires the theory to be consistent, rather than ω-consistent. This is mostly of
technical interest, since all true formal theories of arithmetic (theories whose axioms are all true statements about
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natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger
version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly
known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.
Second incompleteness theorem
Gödel's second incompleteness theorem first appeared as "Theorem XI" in Gödel's 1931 paper On Formally
Undecidable Propositions in Principia Mathematica and Related Systems I.
The formal theorem is written in highly technical language. The broadly accepted natural language statement of the
theorem is:
For any formal effectively generated theory T including basic arithmetical truths and also certain truths about
formal provability, if T includes a statement of its own consistency then T is inconsistent.
This strengthens the first incompleteness theorem, because the statement constructed in the first incompleteness
theorem does not directly express the consistency of the theory. The proof of the second incompleteness theorem is
obtained by formalizing the proof of the first incompleteness theorem within the theory itself.
A technical subtlety in the second incompleteness theorem is how to express the consistency of T as a formula in the
language of T. There are many ways to do this, and not all of them lead to the same result. In particular, different
formalizations of the claim that T is consistent may be inequivalent in T, and some may even be provable. For
example, first-order Peano arithmetic (PA) can prove that the largest consistent subset of PA is consistent. But since
PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent".
What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest
consistent subset of PA" is technically ambiguous, but what is meant here is the largest consistent initial segment of
the axioms of PA ordered according to specific criteria; i.e., by "Gödel numbers", the numbers encoding the axioms
as per the scheme used by Gödel mentioned above).
For Peano arithmetic, or any familiar explicitly axiomatized theory T, it is possible to canonically define a formula
Con(T) expressing the consistency of T; this formula expresses the property that "there does not exist a natural
number coding a sequence of formulas, such that each formula is either of the axioms of T, a logical axiom, or an
immediate consequence of preceding formulas according to the rules of inference of first-order logic, and such that
the last formula is a contradiction".
The formalization of Con(T) depends on two factors: formalizing the notion of a sentence being derivable from a set
of sentences and formalizing the notion of being an axiom of T. Formalizing derivability can be done in canonical
fashion: given an arithmetical formula A(x) defining a set of axioms, one can canonically form a predicate ProvA(P)
which expresses that P is provable from the set of axioms defined by A(x).
In addition, the standard proof of the second incompleteness theorem assumes that ProvA(P) satisfies that
Hilbert–Bernays provability conditions. Letting #(P) represent the Gödel number of a formula P, the derivability
conditions say:
1. If T proves P, then T proves ProvA(#(P)).
2. T proves 1.; that is, T proves that if T proves P, then T proves ProvA(#(P)). In other words, T proves that
ProvA(#(P)) implies ProvA(#(ProvA(#(P)))).
3. T proves that if T proves that (P → Q) and T proves P then T proves Q. In other words, T proves that ProvA(#(P
→ Q)) and ProvA(#(P)) imply ProvA(#(Q)).
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Implications for consistency proofs
Gödel's second incompleteness theorem also implies that a theory T1 satisfying the technical conditions outlined
above cannot prove the consistency of any theory T2 which proves the consistency of T1. This is because such a
theory T1 can prove that if T2 proves the consistency of T1, then T1 is in fact consistent. For the claim that T1 is
consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction
in T1". If T1 were in fact inconsistent, then T2 would prove for some n that n is the code of a contradiction in T1. But
if T2 also proved that T1 is consistent (that is, that there is no such n), then it would itself be inconsistent. This
reasoning can be formalized in T1 to show that if T2 is consistent, then T1 is consistent. Since, by second
incompleteness theorem, T1 does not prove its consistency, it cannot prove the consistency of T2 either.
This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the
consistency of Peano arithmetic using any finitistic means that can be formalized in a theory the consistency of
which is provable in Peano arithmetic. For example, the theory of primitive recursive arithmetic (PRA), which is
widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA
cannot prove the consistency of PA. This fact is generally seen to imply that Hilbert's program, which aimed to
justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical
statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out.
The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually
provide no interesting information if a theory T proved its consistency. This is because inconsistent theories prove
everything, including their consistency. Thus a consistency proof of T in T would give us no clue as to whether T
really is consistent; no doubts about the consistency of T would be resolved by such a consistency proof. The interest
in consistency proofs lies in the possibility of proving the consistency of a theory T in some theory T’ which is in
some sense less doubtful than T itself, for example weaker than T. For many naturally occurring theories T and T’,
such as T = Zermelo–Fraenkel set theory and T’ = primitive recursive arithmetic, the consistency of T’ is provable in
T, and thus T’ can't prove the consistency of T by the above corollary of the second incompleteness theorem.
The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that
could be formalized in the theory that is proved consistent. For example, Gerhard Gentzen proved the consistency of
Peano arithmetic (PA) in a different theory which includes an axiom asserting that the ordinal called ε0 is
wellfounded; see Gentzen's consistency proof. Gentzen's theorem spurred the development of ordinal analysis in
proof theory.
Examples of undecidable statements
There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is
the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor
refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to
computability theory and applies not to statements but to decision problems, which are countably infinite sets of
questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable
function that correctly answers every question in the problem set (see undecidable problem).
Because of the two meanings of the word undecidable, the term independent is sometimes used instead of
undecidable for the "neither provable nor refutable" sense. The usage of "independent" is also ambiguous, however.
Some use it to mean just "not provable", leaving open whether an independent statement might be refuted.
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of
whether the truth value of the statement is well-defined, or whether it can be determined by other means.
Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity
of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be
known or is ill-specified, is a controversial point in the philosophy of mathematics.
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The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the
first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard
axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC
axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940
that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither
is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.
In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in
standard set theory.
Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another
incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any theory that can represent
enough arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have
Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is
related to Berry's paradox.
Undecidable statements provable in larger systems
These are natural mathematical equivalents of the Godel "true but undecidable" sentence. They can be proved in a
larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system
such as Peano Arithmetic.
In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is
undecidable in the first-order axiomatization of arithmetic called Peano arithmetic, but can be proven in the larger
system of second-order arithmetic. Kirby and Paris later showed Goodstein's theorem, a statement about sequences
of natural numbers somewhat simpler than the Paris-Harrington principle, to be undecidable in Peano arithmetic.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but
provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system
codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more
general graph minor theorem (2003) has consequences for computational complexity theory.
Limitations of Gödel's theorems
The conclusions of Gödel's theorems are only proven for the formal theories that satisfy the necessary hypotheses.
Not all axiom systems satisfy these hypotheses, even when these systems have models that include the natural
numbers as a subset. For example, there are first-order axiomatizations of Euclidean geometry, of real closed fields,
and of arithmetic in which multiplication is not provably total; none of these meet the hypotheses of Gödel's
theorems. The key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or
develop basic properties of the natural numbers. Regarding the third example, Dan E. Willard (Willard 2001) has
studied many weak systems of arithmetic which do not satisfy the hypotheses of the second incompleteness theorem,
and which are consistent and capable of proving their own consistency (see self-verifying theories).
Gödel's theorems only apply to effectively generated (that is, recursively enumerable) theories. If all true statements
about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano
arithmetic (called true arithmetic) for which none of Gödel's theorems apply in a meaningful way, because this
theory is not recursively enumerable.
The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the
axioms of those theories themselves. It does not show that the consistency cannot be proved from other (consistent)
axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo–Fraenkel set theory (ZFC),
or in theories of arithmetic augmented with transfinite induction, as in Gentzen's consistency proof.
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Relationship with computability
The incompleteness theorem is closely related to several results about undecidable sets in recursion theory.
Stephen Cole Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of
computability theory. One such result shows that the halting problem is undecidable: there is no computer program
that can correctly determine, given a program P as input, whether P eventually halts when run with a particular given
input. Kleene showed that the existence of a complete effective theory of arithmetic with certain consistency
properties would force the halting problem to be decidable, a contradiction. This method of proof has also been
presented by Shoenfield (1967, p. 132); Charlesworth (1980); and Hopcroft and Ullman (1979).
Franzén (2005, p. 73) explains how Matiyasevich's solution to Hilbert's 10th problem can be used to obtain a proof
to Gödel's first incompleteness theorem. Matiyasevich proved that there is no algorithm that, given a multivariate
polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p
= 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language
of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any
sufficiently strong theory of arithmetic T will prove this. Moreover, if the theory T is ω-consistent, then it will never
prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T
were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation
has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in
contradiction to Matiyasevich's theorem. Moreover, for each consistent effectively generated theory T, it is possible
to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions
over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980).
Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first
incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially
undecidable (see Kleene 1967, p. 274).
Chaitin's incompleteness theorem gives a different method of producing independent sentences, based on
Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only
applies to theories with the additional property that all their axioms are true in the standard model of the natural
numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless
include statements that are false in the standard model; these theories are known as ω-inconsistent.
Proof sketch for the first theorem
The proof by contradiction has three essential parts. To begin, choose a formal system that meets the proposed
criteria:
1. Statements in the system can be represented by natural numbers (known as Gödel numbers). The significance of
this is that properties of statements—such as their truth and falsehood—will be equivalent to determining whether
their Gödel numbers have certain properties, and that properties of the statements can therefore be demonstrated
by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that
"statement S is provable in the system" (which can be applied to any statement "S" in the system).
2. In the formal system it is possible to construct a number whose matching statement, when interpreted, is
self-referential and essentially says that it (i.e. the statement itself) is unprovable. This is done using a technique
called "diagonalization" (so-called because of its origins as Cantor's diagonal argument).
3. Within the formal system this statement permits a demonstration that it is neither provable nor disprovable in the
system, and therefore the system cannot in fact be ω-consistent. Hence the original assumption that the proposed
system met the criteria is false.
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Arithmetization of syntax
The main problem in fleshing out the proof described above is that it seems at first that to construct a statement p
that is equivalent to "p cannot be proved", p would somehow have to contain a reference to p, which could easily
give rise to an infinite regress. Gödel's ingenious technique is to show that statements can be matched with numbers
(often called the arithmetization of syntax) in such a way that "proving a statement" can be replaced with "testing
whether a number has a given property". This allows a self-referential formula to be constructed in a way that avoids
any infinite regress of definitions. The same technique was later used by Alan Turing in his work on the
Entscheidungsproblem.
In simple terms, a method can be devised so that every formula or statement that can be formulated in the system
gets a unique number, called its Gödel number, in such a way that it is possible to mechanically convert back and
forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of
digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is the way
in which English is stored as a sequence of numbers in computers using ASCII or Unicode:
• The word HELLO is represented by 72-69-76-76-79 using decimal ASCII, ie the number 7269767679.
• The logical statement x=y => y=x is represented by 120-061-121-032-061-062-032-121-061-120 using
octal ASCII, ie the number 120061121032061062032121061120.
In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the
statement does or doesn't have a given property. Because the formal system is strong enough to support reasoning
about numbers in general, it can support reasoning about numbers which represent formulae and statements as well.
Crucially, because the system can support reasoning about properties of numbers, the results are equivalent to
reasoning about provability of their equivalent statements.
Construction of a statement about "provability"
Having shown that in principle the system can indirectly make statements about provability, by analyzing properties
of those numbers representing statements it is now possible to show how to create a statement that actually does this.
A formula F(x) that contains exactly one free variable x is called a statement form or class-sign. As soon as x is
replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in
the system, or not. For certain formulas one can show that for every natural number n, F(n) is true if and only if it
can be proven (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In
particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as
"2×3=6".
Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement
form F(x) can be assigned a Gödel number denoted by G(F). The choice of the free variable used in the form F(x) is
not relevant to the assignment of the Gödel number G(F).
Now comes the trick: The notion of provability itself can also be encoded by Gödel numbers, in the following way.
Since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for
every statement p, one may ask whether a number x is the Gödel number of its proof. The relation between the Gödel
number of p and x, the potential Gödel number of its proof, is an arithmetical relation between two numbers.
Therefore there is a statement form Bew(y) that uses this arithmetical relation to state that a Gödel number of a proof
of y exists:
Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula
encoded by y).
The name Bew is short for beweisbar, the German word for "provable"; this name was originally used by Gödel to
denote the provability formula just described. Note that "Bew(y)" is merely an abbreviation that represents a
particular, very long, formula in the original language of T; the string "Bew" itself is not claimed to be part of this
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language.
An important feature of the formula Bew(y) is that if a statement p is provable in the system then Bew(G(p)) is also
provable. This is because any proof of p would have a corresponding Gödel number, the existence of which causes
Bew(G(p)) to be satisfied.
Diagonalization
The next step in the proof is to obtain a statement that says it is unprovable. Although Gödel constructed this
statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for
any sufficiently strong formal system and any statement form F there is a statement p such that the system proves
p ↔ F(G(p)).
By letting F be the negation of Bew(x), we obtain the theorem
p ↔ ~Bew(G(p))
and the p defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.
The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the
resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting
Gödel number turns out to be the Gödel number of p itself. This is similar to the following sentence in English:
", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable.
This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is
obtained as a result, and thus this sentence asserts its own unprovability. The proof of the diagonal lemma employs a
similar method.
Proof of independence
Now assume that the formal system is ω-consistent. Let p be the statement obtained in the previous section.
If p were provable, then Bew(G(p)) would be provable, as argued above. But p asserts the negation of Bew(G(p)).
Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows that p
cannot be provable.
If the negation of p were provable, then Bew(G(p)) would be provable (because p was constructed to be equivalent
to the negation of Bew(G(p))). However, for each specific number x, x cannot be the Gödel number of the proof of p,
because p is not provable (from the previous paragraph). Thus on one hand the system supports construction of a
number with a certain property (that it is the Gödel number of the proof of p), but on the other hand, for every
specific number x, it can be proved that the number does not have this property. This is impossible in an ω-consistent
system. Thus the negation of p is not provable.
Thus the statement p is undecidable: it can neither be proved nor disproved within the chosen system. So the chosen
system is either inconsistent or incomplete. This logic can be applied to any formal system meeting the criteria. The
conclusion is that all formal systems meeting the criteria are either inconsistent or incomplete. It should be noted that
p is not provable (and thus true) in every consistent system. The assumption of ω-consistency is only required for the
negation of p to be not provable. So:
• In an ω-consistent formal system, neither p nor its negation can be proved, and so p is undecidable.
• In a consistent formal system either the same situation occurs, or the negation of p can be proved; In the later
case, a statement ("not p") is false but provable.
Note that if one tries to fix this by "adding the missing axioms" to avoid the undecidability of the system, then one
has to add either p or "not p" as axioms. But this then creates a new formal system2 (old system + p), to which
exactly the same process can be applied, creating a new statement form Bew2(x) for this new system. When the
diagonal lemma is applied to this new form Bew2, a new statement p2 is obtained; this statement will be different
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from the previous one, and this new statement will be undecidable in the new system if it is ω-consistent, thus
showing that system2 is equally inconsistent. So adding extra axioms cannot fix the problem.
Proof via Berry's paradox
George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox
rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently
discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably
enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the
first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a
"different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388).
Formalized proofs
Formalized proofs of versions of the incompleteness theorem have been developed by Natarajan Shankar in 1986
using Nqthm (Shankar 1994) and by Russell O'Connor in 2003 using Coq (O'Connor 2005).
Proof sketch for the second theorem
The main difficulty in proving the second incompleteness theorem is to show that various facts about provability
used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate
for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the
first incompleteness theorem within the system itself.
Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be
proven from within the system itself. The demonstration above shows that if the system is consistent, then p is not
provable. The proof of this implication can be formalized within the system, and therefore the statement "p is not
provable", or "not P(p)" can be proven in the system.
But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can be proven
in the system. This contradiction shows that the system must be inconsistent.
Discussion and implications
The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a
single system formal logic to define their principles. One can paraphrase the first theorem as saying the following:
An all-encompassing axiomatic system can never be found that is able to prove all mathematical truths, but no
falsehoods.
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it
presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to
each formal system.
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:
If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.
Therefore, to establish the consistency of a system S, one needs to use some other system T, but a proof in T is not
completely convincing unless T's consistency has already been established without using S.
Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are
called essentially undecidable or essentially incomplete.
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Minds and machines
Authors including J. R. Lucas have debated what, if anything, Gödel's incompleteness theorems imply about human
intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the
Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness
theorems would apply to it.
Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes
and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general.
Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing
machine.
Avi Wigderson (2010) has proposed that the concept of mathematical "knowability" should be based on
computational complexity rather than logical decidability. He writes that "when knowability is interpreted by modern
standards, namely via computational complexity, the Gödel phenomena are very much with us."
Paraconsistent logic
Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of
paraconsistent logic and of inherently contradictory statements (dialetheia). Graham Priest (1984, 2006) argues that
replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show
that naive mathematics is inconsistent, and uses this as evidence for dialetheism. The cause of this inconsistency is
the inclusion of a truth predicate for a theory within the language of the theory (Priest 2006:47). Stewart Shapiro
(2002) gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism. Carl Hewitt (2008) has
proposed that (inconsistent) paraconsistent logics that prove their own Gödel sentences may have applications in
software engineering.
Appeals to the incompleteness theorems in other fields
Appeals and analogies are sometimes made to the incompleteness theorems in support of arguments that go beyond
mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including
Torkel Franzén (2005); Alan Sokal and Jean Bricmont (1999); and Ophelia Benson and Jeremy Stangroom (2006).
Bricmont and Stangroom (2006, p. 10), for example, quote from Rebecca Goldstein's comments on the disparity
between Gödel's avowed Platonism and the anti-realist uses to which his ideas are sometimes put. Sokal and
Bricmont (1999, p. 187) criticize Régis Debray's invocation of the theorem in the context of sociology; Debray has
defended this use as metaphorical (ibid.).
The role of self-reference
Torkel Franzén (2005, p. 46) observes:
Gödel's proof of the first incompleteness theorem and Rosser's strengthened version have given many
the impression that the theorem can only be proved by constructing self-referential statements [...] or
even that only strange self-referential statements are known to be undecidable in elementary arithmetic.
To counteract such impressions, we need only introduce a different kind of proof of the first
incompleteness theorem.
He then proposes the proofs based on computability, or on information theory, as described earlier in this article, as
examples of proofs that should "counteract such impressions".
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History
After Gödel published his proof of the completeness theorem as his doctoral thesis in 1929, he turned to a second
problem for his habilitation. His original goal was to obtain a positive solution to Hilbert's second problem (Dawson
1997, p. 63). At the time, theories of the natural numbers and real numbers similar to second-order arithmetic were
known as "analysis", while theories of the natural numbers alone were known as "arithmetic".
Gödel was not the only person working on the consistency problem. Ackermann had published a flawed consistency
proof for analysis in 1925, in which he attempted to use the method of ε-substitution originally developed by Hilbert.
Later that year, von Neumann was able to correct the proof for a theory of arithmetic without any axioms of
induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to
announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistency proof of
analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's
modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was
unsound (Zach 2006, p. 418, Zach 2003, p. 33).
In the course of his research, Gödel discovered that although a sentence which asserts its own falsehood leads to
paradox, a sentence that asserts its own non-provability does not. In particular, Gödel was aware of the result now
called Tarski's indefinability theorem, although he never published it. Gödel announced his first incompleteness
theorem to Carnap, Feigel and Waismann on August 26, 1930; all four would attend a key conference in Königsberg
the following week.
Announcement
The 1930 Königsberg conference was a joint meeting of three academic societies, with many of the key logicians of
the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical
philosophies of logicism, intuitionism, and formalism, respectively (Dawson 1996, p. 69). The conference also
included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the
speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,
For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either. ... The
true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no
unsolvable problem. In contrast to the foolish Ignoramibus, our credo avers: We must know. We shall know!
This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "Wir
müssen wissen. Wir werden wissen!", were used as Hilbert's epitaph in 1943). Although Gödel was likely in
attendance for Hilbert's address, the two never met face to face (Dawson 1996, p. 72).
Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the
conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for
conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von
Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated
November 20, 1930 (Dawson 1996, p. 70). Gödel had independently obtained the second incompleteness theorem
and included it in his submitted manuscript, which was received by Monatshefte für Mathematik on November 17,
1930.
Gödel's paper was published in the Monatshefte in 1931 under the title Über formal unentscheidbare Sätze der
Principia Mathematica und verwandter Systeme I (On Formally Undecidable Propositions in Principia Mathematica
and Related Systems I). As the title implies, Gödel originally planned to publish a second part of the paper; it was
never written.
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Generalization and acceptance
Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church,
Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the theory
must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of
ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency, if the
Gödel sentence was changed in an appropriate way. These developments left the incompleteness theorems in
essentially their modern form.
Gentzen published his consistency proof for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary"
although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is
being proved consistent.
The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof
of the incompleteness theorems in the second volume of Grundlagen der Mathematik (1939), along with additional
results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first
full published proof of the second incompleteness theorem.
Criticisms
Finsler
Paul Finsler (1926) used a version of Richard's paradox to construct an expression that was false but unprovable in a
particular, informal framework he had developed. Gödel was unaware of this paper when he proved the
incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote Gödel in 1931 to inform him about this
paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized
provability, and had only a superficial resemblance to Gödel's work (van Heijenoort 1967:328). Gödel read the paper
but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization
(Dawson:89). Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the
remainder of his career.
Zermelo
In September 1931, Ernst Zermelo wrote Gödel to announce what he described as an "essential gap" in Gödel's
argument (Dawson:76). In October, Gödel replied with a 10-page letter (Dawson:76, Grattan-Guinness:512-513).
But Zermelo did not relent and published his criticisms in print with "a rather scathing paragraph on his young
competitor" (Grattan-Guinness:513). Gödel decided that to pursue the matter further was pointless, and Carnap
agreed (Dawson:77). Much of Zermelo's subsequent work was related to logics stronger than first-order logic, with
which he hoped to show both the consistency and categoricity of mathematical theories.
Wittgenstein
Ludwig Wittgenstein wrote several passages about the incompleteness theorems that were published posthumously
in his 1953 Remarks on the Foundations of Mathematics. Gödel was a member of the Vienna Circle during the
period in which Wittgenstein's early ideal language philosophy and Tractatus Logico-Philosophicus dominated the
circle's thinking. Writings in Gödel's Nachlass express the belief that Wittgenstein deliberately misread his ideas.
Multiple commentators have read Wittgenstein as misunderstanding Gödel (Rodych 2003), although Juliet Floyd and
Hilary Putnam (2000), as well as Graham Priest (2004) have provided textual readings arguing that most
commentary misunderstands Wittgenstein. On their release, Bernays, Dummett, and Kreisel wrote separate reviews
on Wittgenstein's remarks, all of which were extremely negative (Berto 2009:208). The unanimity of this criticism
caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In
1972, Gödel, stated: "Has Wittgenstein lost his mind? Does he mean it seriously?" (Wang 1996:197) And wrote to
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Karl Menger that Wittgenstein's comments demonstrate a willful misunderstanding of the incompleteness theorems
writing:
"It is clear from the passages you cite that Wittgenstein did "not" understand [the first incompleteness
theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just
the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics
(finitary number theory or combinatorics)." (Wang 1996:197)
Since the publication of Wittgenstein's Nachlass in 2000, a series of papers in philosophy have sought to evaluate
whether the original criticism of Wittgenstein's remarks was justified. Floyd and Putnam (2000) argue that
Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They
are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent theory as actually saying
"I am not provable", since the theory has no models in which the provability predicate corresponds to actual
provability. Rodych (2003) argues that their interpretation of Wittgenstein is not historically justified, while Bays
(2004) argues against Floyd and Putnam's philosophical analysis of the provability predicate. Berto (2009) explores
the relationship between Wittgenstein's writing and theories of paraconsistent logic.
Notes
[1] The word "true" is used disquotationally here: the Gödel sentence is true in this sense because it "asserts its own unprovability and it is indeed
unprovable" (Smoryński 1977 p. 825; also see Franzén 2005 pp. 28–33). It is also possible to read "GT is true" in the formal sense that
primitive recursive arithmetic proves the implication Con(T)→GT, where Con(T) is a canonical sentence asserting the consistency of T
(Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403)
References
Articles by Gödel
• 1931, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für
Mathematik und Physik 38: 173-98.
• 1931, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. and On formally
undecidable propositions of Principia Mathematica and related systems I in Solomon Feferman, ed., 1986. Kurt
Gödel Collected works, Vol. I. Oxford University Press: 144-195. The original German with a facing English
translation, preceded by a very illuminating introductory note by Kleene.
• Hirzel, Martin, 2000, On formally undecidable propositions of Principia Mathematica and related systems I.
(http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf). A modern translation by
Hirzel.
• 1951, Some basic theorems on the foundations of mathematics and their implications in Solomon Feferman, ed.,
1995. Kurt Gödel Collected works, Vol. III. Oxford University Press: 304-23.
Translations, during his lifetime, of Gödel's paper into English
None of the following agree in all translated words and in typography. The typography is a serious matter, because
Gödel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense
before . . ."(van Heijenoort 1967:595). Three translations exist. Of the first John Dawson states that: "The Meltzer
translation was seriously deficient and received a devastating review in the Journal of Symbolic Logic; "Gödel also
complained about Braithwaite's commentary (Dawson 1997:216). "Fortunately, the Meltzer translation was soon
supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology The Undecidable . . . he
found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its
publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was
marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that
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Gödel favored was that by Jean van Heijenoort"(ibid). For the serious student another version exists as a set of
lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for
Advanced Study during the spring of 1934" (cf commentary by Davis 1965:39 and beginning on p. 41); this version
is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication:
• B. Meltzer (translation) and R. B. Braithwaite (Introduction), 1962. On Formally Undecidable Propositions of
Principia Mathematica and Related Systems, Dover Publications, New York (Dover edition 1992), ISBN
0-486-66980-7 (pbk.) This contains a useful translation of Gödel's German abbreviations on pp. 33–34. As noted
above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all
its suspect content by
• Stephen Hawking editor, 2005. God Created the Integers: The Mathematical Breakthroughs That Changed
History, Running Press, Philadelphia, ISBN 0-7624-1922-9. Gödel's paper appears starting on p. 1097, with
Hawking's commentary starting on p. 1089.
• Martin Davis editor, 1965. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems
and Computable Functions, Raven Press, New York, no ISBN. Gödel's paper begins on page 5, preceded by one
page of commentary.
• Jean van Heijenoort editor, 1967, 3rd edition 1967. From Frege to Gödel: A Source Book in Mathematical Logic,
1979-1931, Harvard University Press, Cambridge Mass., ISBN 0-674-32449-8 (pbk). van Heijenoort did the
translation. He states that "Professor Gödel approved the translation, which in many places was accommodated to
his wishes."(p. 595). Gödel's paper begins on p. 595; van Heijenoort's commentary begins on p. 592.
• Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with
Gödel's corrections of errata and Gödel's added notes begins on page 41, preceded by two pages of Davis's
commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes.
Articles by others
• George Boolos, 1989, "A New Proof of the Gödel Incompleteness Theorem", Notices of the American
Mathematical Society v. 36, pp. 388–390 and p. 676, reprinted in Boolos, 1998, Logic, Logic, and Logic, Harvard
Univ. Press. ISBN 0-674-53766-1
• Arthur Charlesworth, 1980, "A Proof of Godel's Theorem in Terms of Computer Programs," Mathematics
Magazine, v. 54 n. 3, pp. 109–121. JStor (http://links.jstor.org/
sici?sici=0025-570X(198105)54:3<109:APOGTI>2.0.CO;2-1&size=LARGE&origin=JSTOR-enlargePage)
• Martin Davis, " The Incompleteness Theorem (http://www.ams.org/notices/200604/fea-davis.pdf)", in
Notices of the AMS vol. 53 no. 4 (April 2006), p. 414.
• Jean van Heijenoort, 1963. "Gödel's Theorem" in Edwards, Paul, ed., Encyclopedia of Philosophy, Vol. 3.
Macmillan: 348-57.
• Geoffrey Hellman, How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism. Noûs, Vol.
15, No. 4, Special Issue on Philosophy of Mathematics. (Nov., 1981), pp. 451–468.
• David Hilbert, 1900, " Mathematical Problems. (http://aleph0.clarku.edu/~djoyce/hilbert/problems.
html#prob2)" English translation of a lecture delivered before the International Congress of Mathematicians at
Paris, containing Hilbert's statement of his Second Problem.
• Kikuchi, Makoto; Tanaka, Kazuyuki (1994), "On formalization of model-theoretic proofs of Gödel's theorems",
Notre Dame Journal of Formal Logic 35 (3): 403–412, doi:10.1305/ndjfl/1040511346, ISSN 0029-4527,
MR1326122
• Stephen Cole Kleene, 1943, "Recursive predicates and quantifiers," reprinted from Transactions of the American
Mathematical Society, v. 53 n. 1, pp. 41–73 in Martin Davis 1965, The Undecidable (loc. cit.) pp. 255–287.
• John Barkley Rosser, 1936, "Extensions of some theorems of Gödel and Church," reprinted from the Journal of
Symbolic Logic vol. 1 (1936) pp. 87–91, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 230–235.
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• John Barkley Rosser, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem",
Reprinted from the Journal of Symbolic Logic, vol. 4 (1939) pp. 53–60, in Martin Davis 1965, The Undecidable
(loc. cit.) pp. 223–230
• C. Smoryński, "The incompleteness theorems", in J. Barwise, ed., Handbook of Mathematical Logic,
North-Holland 1982 ISBN 978-0-444-86388-1, pp. 821–866.
• Dan E. Willard (2001), " Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection
Principles (http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/
1183746459)", Journal of Symbolic Logic, v. 66 n. 2, pp. 536–596. doi:10.2307/2695030
• Zach, Richard (2003), "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program"
(http://www.ucalgary.ca/~rzach/static/conprf.pdf), Synthese (Berlin, New York: Springer-Verlag) 137 (1):
211–259, doi:10.1023/A:1026247421383, ISSN 0039-7857
• Richard Zach, 2005, "Paper on the incompleteness theorems" in Grattan-Guinness, I., ed., Landmark Writings in
Western Mathematics. Elsevier: 917-25.
Books about the theorems
• Francesco Berto. There's Something about Gödel: The Complete Guide to the Incompleteness Theorem John
Wiley and Sons. 2010.
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1-56881-238-8 MR2007d:03001
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1999 reprint: ISBN 0-465-02656-7. MR80j:03009
• Douglas Hofstadter, 2007. I Am a Strange Loop. Basic Books. ISBN 978-0-465-03078-1. ISBN 0-465-03078-5.
MR2008g:00004
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verb=Display&handle=euclid.lnl/1235416274), Lecture Notes in Logic v. 10.
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0-8147-5816-9. MR2002i:03001
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Press. MR84d:03012
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Press. MathSciNet (http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&
co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&s4=Smith,
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computer science. ISBN 0-521-58533-3
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Miscellaneous references
• Francesco Berto. "The Gödel Paradox and Wittgenstein's Reasons" Philosophia Mathematica (III) 17. 2009.
• John W. Dawson, Jr., 1997. Logical Dilemmas: The Life and Work of Kurt Gödel, A.K. Peters, Wellesley Mass,
ISBN 1-56881-256-6.
• Goldstein, Rebecca, 2005, Incompleteness: the Proof and Paradox of Kurt Gödel, W. W. Norton & Company.
ISBN 0-393-05169-2
• Juliet Floyd and Hilary Putnam, 2000, "A Note on Wittgenstein's 'Notorious Paragraph' About the Gödel
Theorem", Journal of Philosophy v. 97 n. 11, pp. 624–632.
• Carl Hewitt, 2008, "Large-scale Organizational Computing requires Unstratified Reflection and Strong
Paraconsistency", Coordination, Organizations, Institutions, and Norms in Agent Systems III, Springer-Verlag.
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• John Hopcroft and Jeffrey Ullman 1979, Introduction to Automata theory, Addison-Wesley, ISBN
0-201-02988-X
• James P. Jones, Undecidable Diophantine Equations (http://www.ams.org/bull/1980-03-02/
S0273-0979-1980-14832-6/S0273-0979-1980-14832-6.pdf), Bulletin of the American Mathematical Society v. 3
n. 2, 1980, pp. 859–862.
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0505034)", Lecture Notes in Computer Science v. 3603, pp. 245–260.
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0-19-926329-9
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significance, Psychology Press, pp. 207–227.
• Graham Priest, 1984, "Logic of Paradox Revisited", Journal of Philosophical Logic, v. 13,` n. 2, pp. 153–179
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Wittgenstein", Dialectica v. 57 n. 3, pp. 279–313. doi:10.1111/j.1746-8361.2003.tb00272.x
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Picador. ISBN 0-312-20407-8
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Logic, 2001. ISBN 978-1-56881-135-2
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2003. ISBN 978-0-521-75373-9
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0-262-23189-1.
• Richard Zach, 2006, "Hilbert's program then and now" (http://www.ucalgary.ca/~rzach/static/hptn.pdf), in
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Gödel's incompleteness theorems
External links
• Godel's Incompleteness Theorems (http://www.bbc.co.uk/programmes/b00dshx3) on In Our Time at the
BBC. ( listen now (http://www.bbc.co.uk/iplayer/console/b00dshx3/
In_Our_Time_Godel's_Incompleteness_Theorems))
• Stanford Encyclopedia of Philosophy: " Kurt Gödel (http://plato.stanford.edu/entries/goedel/)" — by Juliette
Kennedy.
• MacTutor biographies:
• Kurt Gödel. (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html)
• Gerhard Gentzen. (http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html)
• What is Mathematics:Gödel's Theorem and Around (http://podnieks.id.lv/gt.html) by Karlis Podnieks. An
online free book.
• World's shortest explanation of Gödel's theorem (http://blog.plover.com/math/Gdl-Smullyan.html) using a
printing machine as an example.
• October 2011 RadioLab episode (http://www.radiolab.org/2011/oct/04/break-cycle/) about/including
Gödel's Incompleteness theorem
• Hazewinkel, Michiel, ed. (2001), "Gödel incompleteness theorem" (http://www.encyclopediaofmath.org/index.
php?title=p/g044530), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Interesting number paradox
The interesting number paradox is a semi-humorous paradox that arises from attempting to classify natural
numbers as "interesting" or "dull". The paradox states that all natural numbers are interesting. The "proof" is by
contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting
number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number,
producing a contradiction.
Proof
Claim: There is no such thing as an uninteresting natural number.
Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting. Due to the
well-ordered property of the natural numbers, there must be some smallest number in the set of uninteresting
numbers. Being the smallest number of a set one may consider uninteresting makes that number interesting after all:
a contradiction.
Paradoxical nature
Attempting to classify all numbers this way leads to a paradox or an antinomy of definition. Any hypothetical
partition of natural numbers into interesting and dull sets seems to fail. Since the definition of interesting is usually a
subjective, intuitive notion of "interesting", it should be understood as a half-humorous application of self-reference
in order to obtain a paradox. (The paradox is alleviated if "interesting" is instead defined objectively: for example,
the smallest integer that does not, as of November 2011, appear in an entry of the On-Line Encyclopedia of Integer
Sequences was 12407.[1], but as of April 2012, was 13794) Depending on the sources used for the list of interesting
numbers, a variety of other numbers can be characterized as uninteresting in the same way.[2]
However, as there are many significant results in mathematics that make use of self-reference (such as Gödel's
Incompleteness Theorem), the paradox illustrates some of the power of self-reference, and thus touches on serious
issues in many fields of study.
205
Interesting number paradox
This version of the paradox applies only to well-ordered sets with a natural order, such as the natural numbers; the
argument would not apply to the real numbers.
One proposed resolution of the paradox asserts that only the first uninteresting number is made interesting by that
fact. For example, if 39 and 41 were the first two uninteresting numbers, then 39 would become interesting as a
result, but 41 would not since it is not the first uninteresting number.[3] However, this resolution is invalid, since the
paradox is proved by contradiction: assuming that there is any uninteresting number, we arrive to the fact that that
same number is interesting, hence no number can be uninteresting; its aim is not in particular to identify the
interesting or uninteresting numbers, but to speculate whether any number can in fact exhibit such properties.
An obvious weakness in the proof is that what qualifies as "interesting" is not defined. However, assuming this
predicate is defined with a finite, definite list of "interesting properties of positive integers", and is defined
self-referentially to include the smallest number not in such a list, a paradox arises. The Berry paradox is closely
related, since it arises from a similar self-referential definition. As the paradox lies in the definition of "interesting",
it applies only to persons with particular opinions on numbers: if one's view is that all numbers are boring, and one
finds uninteresting the observation that 0 is the smallest boring number, there is no paradox.
Notes
[1] Johnston, N. (June 12, 2009). "11630 is the First Uninteresting Number" (http:/ / www. nathanieljohnston. com/ index. php/ 2009/ 06/
11630-is-the-first-uninteresting-number/ ). . Retrieved November 12, 2011.
[2] Charles R Greathouse IV. "Uninteresting Numbers" (http:/ / math. crg4. com/ uninteresting. html). . Retrieved 2011-08-28.
[3] Clark, M., 2007, Paradoxes from A to Z, Routledge, ISBN 0-521-46168-5.
Further reading
• Gardner, Martin (1959). Mathematical Puzzles and Diversions. ISBN 0-226-28253-8.
External links
• Most Boring Day in History (http://timesofindia.indiatimes.com/world/uk/
April-11-1954-was-most-boring-day-in-history/articleshow/6994947.cms)
206
KleeneRosser paradox
Kleene–Rosser paradox
In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are
inconsistent, in particular the version of Curry's combinatory logic introduced in 1930, and Church's original lambda
calculus, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited
by Stephen Kleene and J. B. Rosser in 1935.
The paradox
Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total,
definable number-theoretic functions, which enabled them to construct a term that essentially replicates the Richard
paradox in formal language.
Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox,
and used this to construct a much simpler paradox, now known as Curry's paradox.
References
• Andrea Cantini, "The inconsistency of certain formal logics [1]", in the Paradoxes and Contemporary Logic entry
of Stanford Encyclopedia of Philosophy (2007).
• Kleene, S. C. & Rosser, J. B. (1935). "The inconsistency of certain formal logics". Annals of Mathematics 36 (3):
630–636. doi:10.2307/1968646.
References
[1] http:/ / plato. stanford. edu/ entries/ paradoxes-contemporary-logic/ #IncCerForLog
207
Lindley's paradox
208
Lindley's paradox
Lindley's paradox is a counterintuitive situation in statistics in which the Bayesian and frequentist approaches to a
hypothesis testing problem give different results for certain choices of the prior distribution. The problem of the
disagreement between the two approaches was discussed in Harold Jeffreys' textbook;[1] it became known as
Lindley's paradox after Dennis Lindley called the disagreement a paradox in a 1957 paper.[2]
Although referred to as a paradox, the differing results from the Bayesian and Frequentist approaches can be
explained as using them to answer fundamentally different questions, rather than actual disagreement between the
two methods.
Description of the paradox
Consider the result
of some experiment, with two possible explanations, hypotheses
and
representing uncertainty as to which hypothesis is more accurate before taking into account
, and some prior
.
Lindley's paradox occurs when
1. The result
is "significant" by a frequentist test of
5% level, and
2. The posterior probability of
than
given
, indicating sufficient evidence to reject
is high, indicating strong evidence that
, say, at the
is in better agreement with
.
These results can occur at the same time when
is very specific,
more diffuse, and the prior distribution does
not strongly favor one or the other, as seen below.
Numerical example
We can illustrate Lindley's paradox with a numerical example. Imagine a certain city where 49,581 boys and 48,870
girls have been born over a certain time period. The observed proportion of male births is thus 49,581/98,451 ≈
0.5036. We assume the number of male births is a binomial variable with parameter . We are interested in testing
whether
is 0.5 or some other value. That is, our null hypothesis is
and the alternative is
.
Frequentist approach
The frequentist approach to testing
least as large as
assuming
approximation
for
the
is to compute a p-value, the probability of observing a fraction of boys at
is true. Because the number of births is very large, we can use a normal
fraction
of
male
births
,
with
and
, to compute
.
We would have been equally surprised if we had seen 49,581 female births, i.e.
would usually perform a two-sided test, for which the p-value would be
cases, the p-value is lower than the significance level of 5%, so the frequentist approach rejects
with the observed data.
, so a frequentist
. In both
as disagreeing
Lindley's paradox
209
Bayesian approach
Assuming no reason to favor one hypothesis over the other, the Bayesian approach would be to assign prior
probabilities
, and then to compute the posterior probability of
using Bayes'
theorem,
After observing
boys out of
births, we can compute the posterior probability of each
hypothesis using the probability mass function for a binomial variable,
where
is the Beta function.
From these values, we find the posterior probability of
, which strongly favors
over
.
The two approaches—the Bayesian and the frequentist—appear to be in conflict, and this is the "paradox".
The lack of an actual paradox
The apparent disagreement between the two approaches is caused by a combination of factors. First, the frequentist
approach above tests
without reference to
. The Bayesian approach evaluates
as an alternative to
,
and finds the first to be in better agreement with the observations. This is because the latter hypothesis is much more
diffuse, as can be anywhere in
, which results in it having a very low posterior probability. To understand
why, it is helpful to consider the two hypotheses as generators of the observations:
• Under
, we choose
, and ask how likely it is to see 49,581 boys in 98,451 births.
• Under
, we choose randomly from anywhere within 0 to 1, and ask the same question.
Most of the possible values for
under
are very poorly supported by the observations. In essence, the apparent
disagreement between the methods is not a disagreement at all, but rather two different statements about how the
hypotheses relate to the data:
• The Frequentist finds that
• The Bayesian finds that
is a poor explanation for the observation.
is a far better explanation for the observation than
.
For practical purposes (and particularly in the numerical example above), it could also be said that disagreement is
rooted in the poor choice of prior probabilities in the Bayesian approach. This becomes clear if the region of
is examined.
For example, this choice of hypotheses and prior probabilities implies the statement: "if
then the prior probability of
being exactly 0.5 is 0.50/0.51
, it is easy to see why the Bayesian approach favors
value of
lies
> 0.49 and
< 0.51,
98%." Given such a strong preference for
in the face of
, even though the observed
away from 0.5. The deviation of over 2 sigma from
is considered significant in the
frequentist approach, but its significance is overruled by the prior in the Bayesian approach.
Looking at it another way, we can see that the prior is essentially flat with a delta function at
. Clearly this
is dubious. In fact if you were to picture real numbers as being continuous, then it would be more logical to assume
that it would impossible for any given number to be exactly the parameter value, i.e., we should assume P(theta =
0.5) = 0.
A more realistic distribution for
. For example, if we replace
posterior probability of
in the alternative hypothesis produces a less surprising result for the posterior of
with
, i.e., the maximum likelihood estimate for
would be only 0.07 compared to 0.93 for
.
, the
Lindley's paradox
210
Reconciling the Bayesian and Frequentist approaches
If one uses an uninformative prior and tests a hypothesis more similar to that in the Frequentist approach, the
paradox disappears.
For example, if we calculate the posterior distribution
, using a uniform prior on
(i.e.,
), we find
If we use this to check the probability that more boys are born than girls, i.e.,
, we find
In other words, it is very likely that the proportion of male births is above 0.5.
Neither analysis gives an estimate of the effect size, directly, but both could be used to determine, for instance, if the
fraction of boy births is likely to be above some particular threshold.
Notes
[1] Jeffreys, Harold (1939). Theory of Probability. Oxford University Press. MR924.
[2] Lindley, D.V. (1957). "A Statistical Paradox". Biometrika 44 (1–2): 187–192. doi:10.1093/biomet/44.1-2.187. JSTOR 2333251.
References
• Shafer, Glenn (1982). "Lindley's paradox". Journal of the American Statistical Association 77 (378): 325–334.
doi:10.2307/2287244. JSTOR 2287244. MR664677.
Low birth weight paradox
The low birth-weight paradox is an apparently paradoxical observation relating to the birth weights and mortality
of children born to tobacco smoking mothers. Low birth-weight children born to smoking mothers have a lower
infant mortality rate than the low birth weight children of non-smokers. The same is true of children born to poor
parents, and of children born at high altitude; these are all examples of Simpson's paradox.
History
Traditionally, babies weighing less than a certain amount (which varies between countries) have been classified as
having low birth weight. In a given population, low birth weight babies have a significantly higher mortality rate
than others; thus, populations with a higher rate of low birth weights typically also have higher rates of child
mortality than other populations.
Based on prior research, the children of smoking mothers are more likely to be of low birth weight than children of
non-smoking mothers. Thus, by extension the child mortality rate should be higher among children of smoking
mothers. So it is a surprising real-world observation that low birth weight babies of smoking mothers have a lower
child mortality than low birth weight babies of non-smokers.
Low birth weight paradox
Explanation
At first sight these findings seemed to suggest that, at least for some babies, having a smoking mother might be
beneficial to one's health. However the paradox can be explained statistically by uncovering a lurking variable
between smoking and the two key variables: birth weight and risk of mortality. Both are acted on independently
when the mother of the child smokes — birth weight is lowered and the risk of mortality increases.
The birth weight distribution for children of smoking mothers is shifted to lower weights by their mothers' actions.
Therefore, otherwise healthy babies (who would weigh more if it were not for the fact their mother smoked) are born
underweight. They have a lower mortality rate than children who have other medical reasons why they are born
underweight, regardless of the fact their mother does not smoke.
In short, smoking may be harmful in that it contributes to low birth weight, but other causes of low birth
weight are generally more harmful only with regard to their weight.
Evidence
If one corrects and adjusts for the confounding by smoking, via stratification or multivariable regression modelling
to statistical control for smoking, then one finds that the association between birth weight and mortality may be
attenuated towards the null. Nevertheless, most epidemiologic studies of birth weight and mortality have controlled
for maternal smoking, and the adjusted results, although attenuated after adjusting for smoking, still indicated a
significant association.
Additional support for the hypothesis that birth weight and mortality can be acted on independently came from the
analysis of birth data from Colorado: compared with the birth weight distribution in the US as a whole, the
distribution curve in Colorado is also shifted to lower weights. The overall child mortality of Colorado children is the
same as that for US children however, and if one corrects for the lower weights as above, one finds that babies of a
given (corrected) weight are just as likely to die, whether they are from Colorado or not. The likely explanation here
is that the higher altitude of Colorado affects birth weight, but not mortality.
References
• Wilcox, Allen (2001). "On the importance — and the unimportance — of birthweight [1]". International Journal
of Epidemiology. 30:1233–1241.
• Wilcox, Allen (2006). "The Perils of Birth Weight — A Lesson from Directed Acyclic Graphs [2]". American
Journal of Epidemiology. 164(11):1121–1123.
External links
• The Analysis of Birthweight [3], by Allen Wilcox
References
[1] http:/ / eb. niehs. nih. gov/ bwt/ V0M3QDQU. pdf
[2] http:/ / aje. oxfordjournals. org/ cgi/ content/ abstract/ 164/ 11/ 1121
[3] http:/ / eb. niehs. nih. gov/ bwt/ index. htm
211
Missing square puzzle
Missing square puzzle
The missing square puzzle is an optical illusion used in mathematics classes to help students reason about
geometrical figures. It depicts two arrangements of shapes, each of which apparently forms a 13×5 right-angled
triangle, but one of which has a 1×1 hole in it.
Solution
The key to the puzzle is the fact that
neither of the 13×5 "triangles" is truly
a triangle, because what would be the
hypotenuse is bent. In other words, the
hypotenuse does not maintain a
consistent slope, even though it may
appear that way to the human eye. A
true 13×5 triangle cannot be created
from the given component parts.
The four figures (the yellow, red, blue
and green shapes) total 32 units of
area. The apparent triangles formed
from the figures are 13 units wide and
5 units tall, so it appears that the area
should be
units.
However the blue triangle has a ratio
of 5:2 (=2.5:1), while the red triangle
has the ratio 8:3 (≈2.667:1), so the
apparent combined hypotenuse in each
figure is actually bent.
The amount of bending is around 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this
puzzle. Note the grid point where the red and blue hypotenuses meet, and compare it to the same point on the other
figure; the edge is slightly over or under the mark. Overlaying the hypotenuses from both figures results in a very
thin parallelogram with the area of exactly one grid square, the same area "missing" from the second figure.
According to Martin Gardner,[1] this particular puzzle was invented by a New York City amateur magician, Paul
Curry, in 1953. The principle of a dissection paradox has however been known since the start of the 16th century.
The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive Fibonacci numbers. Many other
geometric dissection puzzles are based on a few simple properties of the famous Fibonacci sequence.[2]
212
Missing square puzzle
213
Similar puzzles
A different puzzle of the same kind (depicted in the
animation) uses four congruent quadrilaterals and a small
square, which form a larger square. When the quadrilaterals
are rotated about their centers they fill the space of the small
square, although the total area of the figure seems unchanged.
The apparent paradox is explained by the fact that the side of
the new large square is a little smaller than the original one. If
a is the side of the large square and θ is the angle between two
opposing sides in each quadrilateral, then the quotient
between the two areas is given by sec2θ − 1. For θ = 5°, this is
approximately 1.00765, which corresponds to a difference of
about 0.8%.
References
[1] Martin, Gardner (1956). Mathematics Magic and Mystery. Dover.
pp. 139–150.
[2] Weisstein, Eric. "Cassini's Identity" (http:/ / mathworld. wolfram. com/
CassinisIdentity. html). .
External links
• A printable Missing Square variant (http://www.
archimedes-lab.org/workshop13skulls.html) with a video
demonstration.
Sam Loyd's paradoxial dissection. In the "larger"
rearrangement, the gaps between the figures have
a combined unit square more area than their
square gaps counterparts, creating an illusion that
the figures there take up more space than those in
the square figure. In the "smaller" rearrangement,
the gaps take up one fewer unit squares than in
the square.
• Curry's Paradox: How Is It Possible? (http://www.
cut-the-knot.org/Curriculum/Fallacies/CurryParadox.
shtml) at cut-the-knot
• Triangles and Paradoxes (http://www.archimedes-lab.
org/page3b.html) at archimedes-lab.org
• The Triangle Problem or What's Wrong with the Obvious
Truth (http://www.marktaw.com/blog/
TheTriangleProblem.html)
• Jigsaw Paradox (http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html)
• The Eleven Holes Puzzle (http://www.slideshare.net/sualeh/the-eleven-holes-puzzle)
• Very nice animated Excel workbook of the Missing Square Puzzle (http://www.excelhero.com/blog/2010/09/
excel-optical-illusions-week-30.html)
• A video explaining Curry's Paradox and Area (http://www.youtube.com/watch?v=eFw0878Ig-A&
feature=related) by James Stanton
Paradoxes of set theory
Paradoxes of set theory
This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally
reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern
axiomatic set theory.
Basics
Cardinal numbers
Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved
from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the
existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the same
size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the
even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. These sets have in common
the cardinal number |N| =
(aleph-nought), a number greater than every natural number.
Cardinal numbers can be defined as follows. Define two sets to have the same size by: there exists a bijection
between the two sets (a one-to-one correspondence between the elements). Then a cardinal number is, by definition,
a class consisting of all sets of the same size. To have the same size is an equivalence relation, and the cardinal
numbers are the equivalence classes.
Ordinal numbers
Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of
choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements
such that every nonempty subset has a first element with respect to that order. The order of a well-ordered set is
described by an ordinal number. For instance, 3 is the ordinal number of the set {0, 1, 2} with the usual order 0 < 1 <
2; and ω is the ordinal number of the set of all natural numbers ordered the usual way. Neglecting the order, we are
left with the cardinal number |N| = |ω| = .
Ordinal numbers can be defined with the same method used for cardinal numbers. Define two well-ordered sets to
have the same order type by: there exists a bijection between the two sets respecting the order: smaller elements are
mapped to smaller elements. Then an ordinal number is, by definition, a class consisting of all well-ordered sets of
the same order type. To have the same order type is an equivalence relation on the class of well-ordered sets, and the
ordinal numbers are the equivalence classes.
Two sets of the same order type have the same cardinality. The converse is not true in general for infinite sets: it is
possible to impose different well-orderings on the set of natural numbers that give rise to different ordinal numbers.
There is a natural ordering on the ordinals, which is itself a well-ordering. Given any ordinal α, one can consider the
set of all ordinals less than α. This set turns out to have ordinal number α. This observation is used for a different
way of introducing the ordinals, in which an ordinal is equated with the set of all smaller ordinals. This form of
ordinal number is thus a canonical representative of the earlier form of equivalence class.
214
Paradoxes of set theory
Power sets
By forming all subsets of a set S (all possible choices of its elements), we obtain the power set P(S). Georg Cantor
proved that the power set is always larger than the set, i.e., |P(S)| > |S|. A special case of Cantor's theorem proves that
the set of all real numbers R cannot be enumerated by natural numbers. R is uncountable: |R| > |N|.
Paradoxes of the infinite set
Instead of relying on ambiguous descriptions such as "that which cannot be enlarged" or "increasing without bound",
set theory provides definitions for the term infinite set to give an unambiguous meaning to phrases such as "the set of
all natural numbers is infinite". Just as for finite sets, the theory makes further definitions which allow us to
consistently compare two infinite sets as regards whether one set is "larger than", "smaller than", or "the same size
as" the other. But not every intuition regarding the size of finite sets applies to the size of infinite sets, leading to
various apparently paradoxical results regarding enumeration, size, measure and order.
Paradoxes of enumeration
Before set theory was introduced, the notion of the size of a set had been problematic. It had been discussed by
Galileo Galilei and Bernard Bolzano, among others. Are there as many natural numbers as squares of natural
numbers when measured by the method of enumeration?
• The answer is yes, because for every natural number n there is a square number n2, and likewise the other way
around.
• The answer is no, because the squares are a proper subset of the naturals: every square is a natural number but
there are natural numbers, like 2, which are not squares of natural numbers.
By defining the notion of the size of a set in terms of its cardinality, the issue can be settled. Since there is a bijection
between the two sets involved, this follows in fact directly from the definition of the cardinality of a set.
See Hilbert's paradox of the Grand Hotel for more on paradoxes of enumeration.
Je le vois, mais je ne crois pas
"I see it but I can't believe it", Cantor wrote to Richard Dedekind, after proving that the set of points of a square has
the same cardinality as that of the points on just a side of the square: the cardinality of the continuum.
This demonstrates that the "size" of sets as defined by cardinality alone is not the only useful way of comparing sets.
Measure theory provides a more nuanced theory of size that conforms to our intuition that length and area are
incompatible measures of size.
Paradoxes of well-ordering
In 1904 Ernst Zermelo proved by means of the axiom of choice (which was introduced for this reason) that every set
can be well-ordered. In 1963 Paul J. Cohen showed that using the axiom of choice is essential to well-ordering the
real numbers; no weaker assumption suffices.
However, the ability to well order any set allows certain constructions to be performed that have been called
paradoxical. One example is the Banach–Tarski paradox, a theorem widely considered to be nonintuitive. It states
that it is possible to decompose a ball of a fixed radius into a finite number of pieces and then move and reassemble
those pieces by ordinary translations and rotations (with no scaling) to obtain two copies from the one original copy.
The construction of these pieces requires the axiom of choice; the pieces are not simple regions of the ball, but
complicated subsets.
215
Paradoxes of set theory
Paradoxes of the Supertask
In set theory, an infinite set is not considered to be created by some mathematical process such as "adding one
element" that is then carried out "an infinite number of times". Instead, a particular infinite set (such as the set of all
natural numbers) is said to already exist, "by fiat", as an assumption or an axiom. Given this infinite set, other
infinite sets are then proven to exist as well, as a logical consequence. But it is still a natural philosophical question
to contemplate some physical action that actually completes after an infinite number of discrete steps; and the
interpretation of this question using set theory gives rise to the paradoxes of the supertask.
The diary of Tristram Shandy
Tristram Shandy, the hero of a novel by Laurence Sterne, writes his autobiography so conscientiously that it takes
him one year to lay down the events of one day. If he is mortal he can never terminate; but if he lived forever then no
part of his diary would remain unwritten, for to each day of his life a year devoted to that day's description would
correspond.
The Ross-Littlewood paradox
An increased version of this type of paradox shifts the infinitely remote finish to a finite time. Fill a huge reservoir
with balls enumerated by numbers 1 to 10 and take off ball number 1. Then add the balls enumerated by numbers 11
to 20 and take off number 2. Continue to add balls enumerated by numbers 10n - 9 to 10n and to remove ball number
n for all natural numbers n = 3, 4, 5, .... Let the first transaction last half an hour, let the second transaction last
quarter an hour, and so on, so that all transactions are finished after one hour. Obviously the set of balls in the
reservoir increases without bound. Nevertheless, after one hour the reservoir is empty because for every ball the time
of removal is known.
The paradox is further increased by the significance of the removal sequence. If the balls are not removed in the
sequence 1, 2, 3, ... but in the sequence 1, 11, 21, ... after one hour infinitely many balls populate the reservoir,
although the same amount of material as before has been moved.
Paradoxes of proof and definability
For all its usefulness in resolving questions regarding infinite sets, naive set theory has some fatal flaws. In
particular, it is prey to logical paradoxes such as those exposed by Russell's paradox. The discovery of these
paradoxes revealed that not all sets which can be described in the language of naive set theory can actually be said to
exist without creating a contradiction. The 20th century saw a resolution to these paradoxes in the development of
the various axiomatizations of set theories such as ZFC and NBG in common use today. However, the gap between
the very formalized and symbolic language of these theories and our typical informal use of mathematical language
results in various paradoxical situations, as well as the philosophical question of exactly what it is that such formal
systems actually propose to be talking about.
216
Paradoxes of set theory
Early paradoxes: the set of all sets
In 1897 the Italian mathematician Cesare Burali-Forti discovered that there is no set containing all ordinal numbers.
As every ordinal number is defined by a set of smaller ordinal numbers, the well-ordered set Ω of all ordinal
numbers (if it exists) fits the definition and is itself an ordinal. On the other hand, no ordinal number can contain
itself, so Ω cannot be an ordinal. Therefore, the set of all ordinal numbers cannot exist.
By the end of the 19th century Cantor was aware of the non-existence of the set of all cardinal numbers and the set of
all ordinal numbers. In letters to David Hilbert and Richard Dedekind he wrote about inconsistent sets, the elements
of which cannot be thought of as being all together, and he used this result to prove that every consistent set has a
cardinal number.
After all this, the version of the "set of all sets" paradox conceived by Bertrand Russell in 1903 led to a serious crisis
in set theory. Russell recognized that the statement x = x is true for every set, and thus the set of all sets is defined by
{x | x = x}. In 1906 he constructed several paradox sets, the most famous of which is the set of all sets which do not
contain themselves. Russell himself explained this abstract idea by means of some very concrete pictures. One
example, known as the Barber paradox, states: The male barber who shaves all and only men who don't shave
themselves has to shave himself only if he does not shave himself.
There are close similarities between Russell's paradox in set theory and the Grelling–Nelson paradox, which
demonstrates a paradox in natural language.
Paradoxes by change of language
König's paradox
In 1905, the Hungarian mathematician Julius König published a paradox based on the fact that there are only
countably many finite definitions. If we imagine the real numbers as a well-ordered set, those real numbers which
can be finitely defined form a subset. Hence in this well-order there should be a first real number that is not finitely
definable. This is paradoxical, because this real number has just been finitely defined by the last sentence. This leads
to a contradiction in naive set theory.
This paradox is avoided in axiomatic set theory. Although it is possible to represent a proposition about a set as a set,
by a system of codes known as Gödel numbers, there is no formula
in the language of set theory which
holds exactly when a is a code for a finite description of a set and this description is a true description of the set x.
This result is known as Tarski's indefinability theorem; it applies to a wide class of formal systems including all
commonly studied axiomatizations of set theory.
Richard's paradox
In the same year the French mathematician Jules Richard used a variant of Cantor's diagonal method to obtain
another contradiction in naive set theory. Consider the set A of all finite agglomerations of words. The set E of all
finite definitions of real numbers is a subset of A. As A is countable, so is E. Let p be the nth decimal of the nth real
number defined by the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal if p
is not equal either to 8 or 9, and unity if p is equal to 8 or 9. This number N is not defined by the set E because it
differs from any finitely defined real number, namely from the nth number by the nth digit. But N has been defined
by a finite number of words in this paragraph. It should therefore be in the set E. That is a contradiction.
As with König's paradox, this paradox cannot be formalized in axiomatic set theory because it requires the ability to
tell whether a description applies to a particular set (or, equivalently, to tell whether a formula is actually the
definition of a single set).
217
Paradoxes of set theory
Paradox of Löwenheim and Skolem
Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem
showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most
countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming
paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in
which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable
in a larger model (because the bijections that establish countability are in the larger model but not the smaller one).
References
• G. Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, E. Zermelo (Ed.), Olms,
Hildesheim 1966.
• H. Meschkowski, W. Nilson: Georg Cantor - Briefe, Springer, Berlin 1991.
• A. Fraenkel: Einleitung in die Mengenlehre, Springer, Berlin 1923.
• A. A. Fraenkel, A. Levy: Abstract Set Theory, North Holland, Amsterdam 1976.
• F. Hausdorff: Grundzüge der Mengenlehre, Chelsea, New York 1965.
• B. Russell: The principles of mathematics I, Cambridge 1903.
• B. Russell: On some difficulties in the theory of transfinite numbers and order types, Proc. London Math. Soc. (2)
4 (1907) 29-53.
• P. J. Cohen: Set Theory and the Continuum Hypothesis, Benjamin, New York 1966.
• S. Wagon: The Banach-Tarski-Paradox, Cambridge University Press, Cambridge 1985.
• A. N. Whitehead, B. Russell: Principia Mathematica I, Cambridge Univ. Press, Cambridge 1910, p. 64.
• E. Zermelo: Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908) p. 107-128.
External links
• [1] PDF
• [2]
• Definability paradoxes [3] by Timothy Gowers
References
[1] http:/ / www. hti. umich. edu/ cgi/ t/ text/ pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=AAT3201. 0001.
001;didno=AAT3201. 0001. 001;view=pdf;seq=00000086
[2] http:/ / dz-srv1. sub. uni-goettingen. de/ sub/ digbib/ loader?ht=VIEW& did=D38183& p=125
[3] http:/ / www. dpmms. cam. ac. uk/ ~wtg10/ richardsparadox. html
218
Parrondo's paradox
Parrondo's paradox
Parrondo's paradox, a paradox in game theory, has been described as: A combination of losing strategies becomes
a winning strategy. It is named after its creator, Juan Parrondo, who discovered the paradox in 1996. A more
explanatory description is:
There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to
construct a winning strategy by playing the games alternately.
Parrondo devised the paradox in connection with his analysis of the Brownian ratchet, a thought experiment about a
machine that can purportedly extract energy from random heat motions popularized by physicist Richard Feynman.
However, the paradox disappears when rigorously analyzed.
Illustrative examples
The saw-tooth example
Consider an example in which there are two points A and B having the
same altitude, as shown in Figure 1. In the first case, we have a flat
profile connecting them. Here, if we leave some round marbles in the
middle that move back and forth in a random fashion, they will roll
around randomly but towards both ends with an equal probability. Now
Figure 1
consider the second case where we have a saw-tooth-like region
between them. Here also, the marbles will roll towards either ends with
equal probability. Now if we tilt the whole profile towards the right, as shown in Figure 2, it is quite clear that both
these cases will become biased towards B.
Now consider the game in which we alternate the two profiles while judiciously choosing the time between
alternating from one profile to the other.
When we leave a few marbles on the first profile at point E, they
distribute themselves on the plane showing preferential movements
towards point B. However, if we apply the second profile when some
of the marbles have crossed the point C, but none have crossed point
D, we will end up having most marbles back at point E (where we
Figure 2
started from initially) but some also in the valley towards point A
given sufficient time for the marbles to roll to the valley. Then we
again apply the first profile and repeat the steps (points C, D and E now shifted one step to refer to the final valley
closest to A). If no marbles cross point C before the first marble crosses point D, we must apply the second profile
shortly before the first marble crosses point D, to start over.
It easily follows that eventually we will have marbles at point A, but none at point B. Hence for a problem defined
with having marbles at point A being a win and having marbles at point B a loss, we clearly win by playing two
losing games.
219
Parrondo's paradox
220
The coin-tossing example
A second example of Parrondo's paradox is drawn from the field of gambling. Consider playing two games, Game A
and Game B with the following rules. For convenience, define
to be our capital at time t, immediately before we
play a game.
1. Winning a game earns us $1 and losing requires us to surrender $1. It follows that
step t and
if we lose at step t.
2. In Game A, we toss a biased coin, Coin 1, with probability of winning
clearly a losing game in the long run.
3. In Game B, we first determine if our capital is a multiple of some integer
2, with probability of winning
probability of winning
if we win at
. If
, this is
. If it is, we toss a biased coin, Coin
. If it is not, we toss another biased coin, Coin 3, with
. The role of modulo
provides the periodicity as in the ratchet
teeth.
It is clear that by playing Game A, we will almost surely lose in the long run. Harmer and Abbott[1] show via
simulation that if
and
Game B is an almost surely losing game as well. In fact, Game B is a
Markov chain, and an analysis of its state transition matrix (again with M=3) shows that the steady state probability
of using coin 2 is 0.3836, and that of using coin 3 is 0.6164.[2] As coin 2 is selected nearly 40% of the time, it has a
disproportionate influence on the payoff from Game B, and results in it being a losing game.
However, when these two losing games are played in some alternating sequence - e.g. two games of A followed by
two games of B (AABBAABB...), the combination of the two games is, paradoxically, a winning game. Not all
alternating sequences of A and B result in winning games. For example, one game of A followed by one game of B
(ABABAB...) is a losing game, while one game of A followed by two games of B (ABBABB...) is a winning game.
This coin-tossing example has become the canonical illustration of Parrondo's paradox – two games, both losing
when played individually, become a winning game when played in a particular alternating sequence. The apparent
paradox has been explained using a number of sophisticated approaches, including Markov chains,[3] flashing
ratchets,[4] Simulated Annealing[5] and information theory.[6] One way to explain the apparent paradox is as follows:
• While Game B is a losing game under the probability distribution that results for
played individually (
modulo
is the remainder when
is divided by
modulo
when it is
), it can be a winning game
under other distributions, as there is at least one state in which its expectation is positive.
• As the distribution of outcomes of Game B depend on the player's capital, the two games cannot be independent.
If they were, playing them in any sequence would lose as well.
The role of
now comes into sharp focus. It serves solely to induce a dependence between Games A and B, so
that a player is more likely to enter states in which Game B has a positive expectation, allowing it to overcome the
losses from Game A. With this understanding, the paradox resolves itself: The individual games are losing only
under a distribution that differs from that which is actually encountered when playing the compound game. In
summary, Parrondo's paradox is an example of how dependence can wreak havoc with probabilistic computations
made under a naive assumption of independence. A more detailed exposition of this point, along with several related
examples, can be found in Philips and Feldman.[7]
Parrondo's paradox
A simplified example
For a simpler example of how and why the paradox works, again consider two games Game A and Game B, this
time with the following rules:
1. In Game A, you lose 100% of the time, losing $3 each time you play
2. In Game B, you count how much money you have left. If it's a multiple of 5, then you win $8. If it's not, then you
lose $6.
Obviously playing Game A exclusively is a losing proposition, since you lose every time. Playing Game B
exclusively is also a losing strategy, since you will lose three out of every four times you play, thus losing $18 for
every $8 you win, for an average net loss of $2.50 per game.
However if Games A and B are played in alternating sequence of one game of A followed by one game of B
(ABABAB), then you will win in the long run, because the first time you play Game B with a remaining balance that
is a multiple of 5, you will win $8, then play Game A and lose $3, and play Game B again with exactly $5 more,
which guarantees you will continue to win Game B indefinitely, netting $5 each time ($8 - $3). Thus, even though
Game B would be a losing proposition if played alone, because the results of Game B are affected by Game A, the
sequence in which the games are played can affect how often Game B is won, and subsequently produce different
results than if either game is played by itself.
Application
Parrondo's paradox is used extensively in game theory, and its application in engineering, population dynamics,[8]
financial risk, etc., are also being looked into as demonstrated by the reading lists below. Parrondo's games are of
little practical use such as for investing in stock markets[9] as the original games require the payoff from at least one
of the interacting games to depend on the player's capital. However, the games need not be restricted to their original
form and work continues in generalizing the phenomenon. Similarities to volatility pumping and the two-envelope
problem[10] have been pointed out. Simple finance textbook models of security returns have been used to prove that
individual investments with negative median long-term returns may be easily combined into diversified portfolios
with positive median long-term returns.[11] Similarly, a model that is often used to illustrate optimal betting rules has
been used to prove that splitting bets between multiple games can turn a negative median long-term return into a
positive one.[12]
Name
In the early literature on Parrondo's paradox, it was debated whether the word 'paradox' is an appropriate description
given that the Parrondo effect can be understood in mathematical terms. The 'paradoxical' effect can be
mathematically explained in terms of a convex linear combination.
However, Derek Abbott, a leading Parrondo's paradox researcher provides the following answer regarding the use of
the word 'paradox' in this context:
Is Parrondo's paradox really a "paradox"? This question is sometimes asked by mathematicians, whereas
physicists usually don't worry about such things. The first thing to point out is that "Parrondo's paradox" is just
a name, just like the "Braess paradox" or "Simpson's paradox." Secondly, as is the case with most of these
named paradoxes they are all really apparent paradoxes. People drop the word "apparent" in these cases as it is
a mouthful, and it is obvious anyway. So no one claims these are paradoxes in the strict sense. In the wide
sense, a paradox is simply something that is counterintuitive. Parrondo's games certainly are
counterintuitive—at least until you have intensively studied them for a few months. The truth is we still keep
finding new surprising things to delight us, as we research these games. I have had one mathematician
complain that the games always were obvious to him and hence we should not use the word "paradox." He is
either a genius or never really understood it in the first place. In either case, it is not worth arguing with people
221
Parrondo's paradox
like that.
Parrondo's paradox does not seem that paradoxical if one notes that it is actually a combination of three simple
games: two of which have losing probabilities and one of which has a high probability of winning. To suggest that
one can create a winning strategy with three such games is neither counterintuitive nor paradoxical.
Further reading
• John Allen Paulos, A Mathematician Plays the Stock Market [13], Basic Books, 2004, ISBN 0-465-05481-1.
• Neil F. Johnson, Paul Jefferies, Pak Ming Hui, Financial Market Complexity [14], Oxford University Press, 2003,
ISBN 0-19-852665-2.
• Ning Zhong and Jiming Liu, Intelligent Agent Technology: Research and Development, [15] World Scientific,
2001, ISBN 981-02-4706-0.
• Elka Korutcheva and Rodolfo Cuerno, Advances in Condensed Matter and Statistical Physics [16], Nova
Publishers, 2004, ISBN 1-59033-899-5.
• Maria Carla Galavotti, Roberto Scazzieri, and Patrick Suppes, Reasoning, Rationality, and Probability [17],
Center for the Study of Language and Information, 2008, ISBN 1-57586-557-2.
• Derek Abbott and Laszlo B. Kish, Unsolved Problems of Noise and Fluctuations [18], American Institute of
Physics, 2000, ISBN 1-56396-826-6.
• Visarath In, Patrick Longhini, and Antonio Palacios, Applications of Nonlinear Dynamics: Model and Design of
Complex Systems [19], Springer, 2009, ISBN 3-540-85631-5.
• Marc Moore, Constance van Eeden, Sorana Froda, and Christian Léger, Mathematical Statistics and Applications:
Festschrift for Constance van Eeden [20], IMS, 2003, ISBN 0-940600-57-9.
• Ehrhard Behrends, Fünf Minuten Mathematik: 100 Beiträge der Mathematik-Kolumne der Zeitung Die Welt [21],
Vieweg+Teubner Verlag, 2006, ISBN 3-8348-0082-1.
• Lutz Schimansky-Geier, Noise in Complex Systems and Stochastic Dynamics [22], SPIE, 2003, ISBN
0-8194-4974-1.
• Susan Shannon, Artificial Intelligence and Computer Science [23], Nova Science Publishers, 2005, ISBN
1-59454-411-5.
• Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics [24], CRC Press, 2003, ISBN 1-58488-347-2.
• David Reguera, José M. G. Vilar, and José-Miguel Rubí, Statistical Mechanics of Biocomplexity [25], Springer,
1999, ISBN 3-540-66245-6.
• Sergey M. Bezrukov, Unsolved Problems of Noise and Fluctuations [26], Springer, 2003, ISBN 0-7354-0127-6.
• Julián Chela Flores, Tobias C. Owen, and F. Raulin, First Steps in the Origin of Life in the Universe [27],
Springer, 2001, ISBN 1-4020-0077-4.
• Tönu Puu and Irina Sushko, Business Cycle Dynamics: Models and Tools [28], Springer, 2006, ISBN
3-540-32167-5.
• Andrzej S. Nowak and Krzysztof Szajowski, Advances in Dynamic Games: Applications to Economics, Finance,
Optimization, and Stochastic Control [29], Birkhäuser, 2005, ISBN 0-8176-4362-1.
• Cristel Chandre, Xavier Leoncini, and George M. Zaslavsky, Chaos, Complexity and Transport: Theory and
Applications [30], World Scientific, 2008, ISBN 981-281-879-0.
• Richard A. Epstein, The Theory of Gambling and Statistical Logic (Second edition), Academic Press, 2009, ISBN
0-12-374940-9.
• Clifford A. Pickover, The Math Book, [31] Sterling, 2009, ISBN 1-4027-5796-4.
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Parrondo's paradox
References
[1]
[2]
[3]
[4]
G. P. Harmer and D. Abbott, "Losing strategies can win by Parrondo's paradox", Nature 402 (1999), 864
D. Minor, "Parrondo's Paradox - Hope for Losers!", The College Mathematics Journal 34(1) (2003) 15-20
G. P. Harmer and D. Abbott, "Parrondo's paradox", Statistical Science 14 (1999) 206-213
G. P. Harmer, D. Abbott, P. G. Taylor, and J. M. R. Parrondo, in Proc. 2nd Int. Conf. Unsolved Problems of Noise and Fluctuations, D.
Abbott, and L. B. Kish, eds., American Institute of Physics, 2000
[5] G. P. Harmer, D. Abbott, and P. G. Taylor, The Paradox of Parrondo's games, Proc. Royal Society of London A 456 (2000), 1-13
[6] G. P. Harmer, D. Abbott, P. G. Taylor, C. E. M. Pearce and J. M. R. Parrondo, Information entropy and Parrondo's discrete-time ratchet, in
Proc. Stochastic and Chaotic Dynamics in the Lakes, Ambleside, U.K., P. V. E. McClintock, ed., American Institute of Physics, 2000
[7] Thomas K. Philips and Andrew B. Feldman, Parrondo's Paradox is not Paradoxical (http:/ / papers. ssrn. com/ sol3/ papers.
cfm?abstract_id=581521), Social Science Research Network (SSRN) Working Papers, August 2004
[8] V. A. A. Jansen and J. Yoshimura "Populations can persist in an environment consisting of sink habitats only". Proceedings of the National
Academy of Sciences USA, 95(1998), 3696-3698 .
[9] R. Iyengar and R. Kohli, "Why Parrondo's paradox is irrelevant for utility theory, stock buying, and the emergence of life," Complexity, 9(1),
pp. 23-27, 2004
[10] Winning While Losing: New Strategy Solves'Two-Envelope' Paradox (http:/ / www. physorg. com/ pdf169811689. pdf) at Physorg.com
[11] M. Stutzer, The Paradox of Diversification, The Journal of Investing, Vol. 19, No.1, 2010.
[12] M. Stutzer, "A Simple Parrondo Paradox", Mathematical Scientist, V.35, 2010.
[13] http:/ / books. google. com. au/ books?id=FUGI7KDTkTUC
[14] http:/ / books. google. com. au/ books?id=8jfV6nntNPkC& pg=PA74& dq=parrondo*
[15] http:/ / books. google. com. au/ books?id=eZ6YCz5NamsC& pg=PA150
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
http:/ / books. google. com. au/ books?id=lIoZeb_domwC& pg=PA103
http:/ / books. google. com. au/ books?id=ZuMQAQAAIAAJ& q=parrondo*
http:/ / books. google. com. au/ books?id=ePoaAQAAIAAJ
http:/ / books. google. com. au/ books?id=FidKZcUqdIQC& pg=PA307
http:/ / books. google. com. au/ books?id=SJsDHpgsVgsC& pg=PA185
http:/ / books. google. com. au/ books?id=liNP2CpsU8EC& pg=PA10
http:/ / books. google. com. au/ books?id=WgJTAAAAMAAJ& q=parrondo*
http:/ / books. google. com. au/ books?id=PGtGAAAAYAAJ& q=parrondo*
http:/ / books. google. com. au/ books?id=UDk8QARabpwC& pg=PA2152& dq=parrondo*
http:/ / books. google. com. au/ books?id=0oMp60wubKIC& pg=PA95
http:/ / books. google. com. au/ books?id=soGS-YcwvxsC& pg=PA82
http:/ / books. google. com. au/ books?id=q8JwN_1p78UC& pg=PA17& dq=parrondo*
http:/ / books. google. com. au/ books?id=cTfwjzihuiIC& pg=PA148& dq=parrondo*
http:/ / books. google. com. au/ books?id=l5W20mVBeT4C& pg=PA650& dq=parrondo*
http:/ / books. google. com. au/ books?id=md092lhGSOQC& pg=PA107& dq=parrondo*
http:/ / sprott. physics. wisc. edu/ pickover/ math-book. html
External links
• J. M. R. Parrondo, Parrondo's paradoxical games (http://seneca.fis.ucm.es/parr/GAMES/index.htm)
• Google Scholar profiling of Parrondo's paradox (http://scholar.google.com.au/citations?hl=en&
user=aeNdbrUAAAAJ)
• Nature news article on Parrondo's paradox (http://www.nature.com/news/1999/991223/full/news991223-13.
html)
• Alternate game play ratchets up winnings: It's the law (http://www.eleceng.adelaide.edu.au/Groups/
parrondo/articles/sandiego.html)
• Official Parrondo's paradox page (http://www.eleceng.adelaide.edu.au/Groups/parrondo)
• Parrondo's Paradox - A Simulation (http://www.cut-the-knot.org/ctk/Parrondo.shtml)
• The Wizard of Odds on Parrondo's Paradox (http://wizardofodds.com/askthewizard/149)
• Parrondo's Paradox at Wolfram (http://mathworld.wolfram.com/ParrondosParadox.html)
• Online Parrondo simulator (http://hampshire.edu/lspector/parrondo/parrondo.html)
• Parrondo's paradox at Maplesoft (http://www.maplesoft.com/applications/view.aspx?SID=1761)
• Donald Catlin on Parrondo's paradox (http://catlin.casinocitytimes.com/article/parrondos-paradox-46851)
223
Parrondo's paradox
• Parrondo's paradox and poker (http://emergentfool.com/2008/02/16/parrondos-paradox-and-poker/)
• Parrondo's paradox and epistemology (http://www.fil.lu.se/HommageaWlodek/site/papper/
StjernbergFredrik.pdf)
• A Parrondo's paradox resource (http://pagesperso-orange.fr/jean-paul.davalan/proba/parr/index-en.html)
• Optimal adaptive strategies and Parrondo (http://www.molgen.mpg.de/~rahmann/parrondo/parrondo.shtml)
• Behrends on Parrondo (http://www.math.uni-potsdam.de/~roelly/WorkshopCDFAPotsdam09/Behrends.pdf)
• God doesn't shoot craps (http://www.goddoesntshootcraps.com/paradox.html)
• Parrondo's paradox in chemistry (http://www.fasebj.org/cgi/content/meeting_abstract/23/
1_MeetingAbstracts/514.1)
• Parrondo's paradox in genetics (http://www.genetics.org/cgi/content/full/176/3/1923)
• Parrondo effect in quantum mechanics (http://www.ingentaconnect.com/content/els/03784371/2003/
00000324/00000001/art01909)
• Financial diversification and Parrondo (http://leeds.colorado.edu/uploadedFiles/Centers_of_Excellence/
Burridge_Center/Working_Papers/ParadoxOfDiversification.pdf)
Russell's paradox
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand
Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. The same
paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known
only to Hilbert, Husserl and other members of the University of Göttingen.
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of
themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that
are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a
member of itself by the same definition. This contradiction is Russell's paradox. Symbolically:
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first
constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited
set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).[1]
Informal presentation
Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all
geometrical squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is
"normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a
square and so should be one of its own members. It is "abnormal".
Now we consider the set of all normal sets, R. Determining whether R is normal or abnormal is impossible: If R were
a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if R were
abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the
conclusion that R is neither normal nor abnormal: Russell's paradox.
224
Russell's paradox
225
Formal presentation
Define Naive Set Theory (NST) as the theory of predicate logic with a binary predicate
schema of unrestricted comprehension:
for any formula P with only the variable x free. Substitute
for
and the following axiom
. Then by existential instantiation
(reusing the symbol y) and universal instantiation we have
a contradiction. Therefore NST is inconsistent.
Set-theoretic responses
In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by
replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation
(Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf
Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely
accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic
set theory down to the present day.
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts
that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be
constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called
proper classes. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a
background type theory.
In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of
themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows
that no set contains everything.
Through the work of Zermelo and others, especially John von Neumann, the structure of what some see as the
"natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe, V,
built up from the empty set by transfinitely iterating the power set operation. It is thus now possible again to reason
about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the
elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of
view on the philosophy of mathematics.
Other resolutions to Russell's paradox, more in the spirit of type theory, include the axiomatic set theories New
Foundations and Scott-Potter set theory.
History
Russell discovered the paradox in May or June 1901.[2] By his own admission in his 1919 Introduction to
Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal".[3]
In a 1902 letter,[4] he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and
framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function;
in the following, p. 17 refers to a page in the original Begriffsschrift, and page 23 refers to the same page in van
Heijenoort 1967:
There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function
too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me
because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of
itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that
Russell's paradox
w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do
not belong to themselves. From this I conclude that under certain circumstances a definable collection
[Menge] does not form a totality.[5]
Russell would go to cover it at length in his 1903 The Principles of Mathematics where he repeats his first encounter
with the paradox:[6]
Before taking leave of fundamental questions, it is necessary to examine more in detail the singular
contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that
I was led to it in the endeavour to reconcile Cantor's proof...."
Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der
Arithmetik.[7] Frege did not waste time responding to Russell, his letter dated 22 June 1902 appears, with van
Heijenoort's commentary in Heijenoort 1967:126–127. Frege then wrote an appendix admitting to the paradox,[8]
and proposed a solution that Russell would endorse in his Principles of Mathematics,[9] but was later considered by
some unsatisfactory.[10] For his part, Russell had his work at the printers and he added an appendix on the doctrine of
types.[11]
Ernst Zermelo in his (1908) A new proof of the possibility of a well-ordering (published at the same time he
published "the first axiomatic set theory")[12] laid claim to prior discovery of the antinomy in Cantor's naive set
theory. He states: "And yet, even the elementary form that Russell9 gave to the set-theoretic antinomies could have
persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the
surrender of well-ordering but only in a suitable restriction of the notion of set".[13] Footnote 9 is where he stakes his
claim:
9
1903, pp. 366–368. I had, however, discovered this antinomy myself, independently of Russell, and had
communicated it prior to 1903 to Professor Hilbert among others.[14]
A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl.[15]
It is also known that unpublished discussions of set theoretical paradoxes took place in the mathematical community
at the turn of the century. van Heijenoort in his commentary before Russell's 1902 Letter to Frege states that
Zermelo "had discovered the paradox independently of Russell and communicated it to Hilbert, among others, prior
to its publication by Russell".[16]
In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:
The reason why a function cannot be its own argument is that the sign for a function already contains the
prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be
its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and
the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer
one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself
signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'.
That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)
Russell and Alfred North Whitehead wrote their three-volume Principia Mathematica (PM) hoping to achieve what
Frege had been unable to do. They sought to banish the paradoxes of naive set theory by employing a theory of types
they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that
they did so by purely logical means. While PM avoided the known paradoxes and allows the derivation of a great
deal of mathematics, its system gave rise to new problems.
In any event, Kurt Gödel in 1930–31 proved that while the logic of much of PM, now known as first-order logic, is
complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely – though not universally
– regarded as having shown the logicist program of Frege to be impossible to complete.
226
Russell's paradox
227
Applied versions
There are some versions of this paradox that are closer to real-life situations and may be easier to understand for
non-logicians. For example, the Barber paradox supposes a barber who shaves all men who do not shave themselves
and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not,
the paradox begins to emerge.
As another example, consider five lists of encyclopedia entries within the same encyclopedia:
List of articles about
people:
List of articles starting with the
letter L:
List of articles about
places:
List of articles about
Japan:
List of all lists that do not contain
themselves:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Ptolemy VII of
Egypt
Hermann Hesse
Don Nix
Don Knotts
Nikola Tesla
Sherlock Holmes
Emperor Kōnin
L
L!VE TV
L&H
Leivonmäki
Katase River
Enoshima
...
•
•
•
Emperor Showa
Katase River
Enoshima
List of articles about Japan
List of articles about places
List of articles about people
...
List of articles starting with
the letter K
List of articles starting with
the letter L
List of articles starting with
the letter M
...
•
•
List of articles starting with the
letter K
List of articles starting with the
letter M
...
•
List of all lists that do not contain
themselves?
If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be
removed. However, if it does not list itself, then it should be added to itself.
While appealing, these layman's versions of the paradox share a drawback: an easy refutation of the Barber paradox
seems to be that such a barber does not exist, or at least does not shave (a variant of which is that the barber is a
woman). The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of
the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does
not exist" and "it is an empty set". It is like the difference between saying, "There is no bucket", and saying, "The
bucket is empty".
A notable exception to the above may be the Grelling–Nelson paradox, in which words and meaning are the
elements of the scenario rather than people and hair-cutting. Though it is easy to refute the Barber's paradox by
saying that such a barber does not (and cannot) exist, it is impossible to say something similar about a meaningfully
defined word.
One way that the paradox has been dramatised is as follows:
Suppose that every public library has to compile a catalog of all its books. Since the catalog is itself one of the
library's books, some librarians include it in the catalog for completeness; while others leave it out as it being
one of the library's books is self-evident.
Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their
listings, others do not. The national librarian compiles two master catalogs – one of all the catalogs that list
themselves, and one of all those that don't.
The question is: should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no
problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do
include themselves. If he does include it, it remains a true catalog of those that list themselves.
However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the
second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it
in its own listing, because then it would include itself. But in that case, it should belong to the other catalog,
that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete.
Russell's paradox
Either way, it can never be a true catalog of catalogs that do not list themselves.
Applications and related topics
Russell-like paradoxes
As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:
• A transitive verb <V>, that can be applied to its substantive form.
Form the sentence:
The <V>er that <V>s all (and only those) who don't <V> themselves,
Sometimes the "all" is replaced by "all <V>ers".
An example would be "paint":
The painter that paints all (and only those) that don't paint themselves.
or "elect"
The elector (representative), that elects all that don't elect themselves.
Paradoxes that fall in this scheme include:
• The barber with "shave".
• The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain
themselves.
• The Grelling–Nelson paradox with "describer": The describer (word) that describes all words, that don't describe
themselves.
• Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote
themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all
denoters (numbers) that don't denote themselves" is here called Richardian.)
Related paradoxes
• The liar paradox and Epimenides paradox, whose origins are ancient
• The Kleene–Rosser paradox, showing that the original lambda calculus is inconsistent, by means of a
self-negating statement
• Curry's paradox (named after Haskell Curry), which does not require negation
• The smallest uninteresting integer paradox
Notes
[1] Set theory paradoxes (http:/ / www. suitcaseofdreams. net/ Set_theory_Paradox. htm)
[2] Godehard Link (2004), One hundred years of Russell's paradox (http:/ / books. google. com/ ?id=Xg6QpedPpcsC& pg=PA350), p. 350,
ISBN 978-3-11-017438-0,
[3] Russell 1920:136
[4] Gottlob Frege, Michael Beaney (1997), The Frege reader (http:/ / books. google. com/ ?id=4ktC0UrG4V8C& pg=PA253), p. 253,
ISBN 978-0-631-19445-3, . Also van Heijenoort 1967:124–125
[5] Remarkably, this letter was unpublished until van Heijenoort 1967 – it appears with van Heijenoort's commentary at van Heijenoort
1967:124–125.
[6] Russell 1903:101
[7] cf van Heijenoort's commentary before Frege's Letter to Russell in van Heijenoort 1967:126.
[8] van Heijenoort's commentary, cf van Heijenoort 1967:126 ; Frege starts his analysis by this exceptionally honest comment : "Hardly anything
more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the
position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion" (Appendix of
Grundgesetze der Arithmetik, vol. II, in The Frege Reader, p.279, translation by Michael Beaney
228
Russell's paradox
[9] cf van Heijenoort's commentary, cf van Heijenoort 1967:126. The added text reads as follows: " Note. The second volume of Gg., which
appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction (pp. 253–265), suggesting that the
solution is to be found by denying that two propositional functions that determine equal classes must be equivalent. As it seems very likely
that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (Russell 1903:522); The
abbreviation Gg. stands for Frege's Grundgezetze der Arithmetik. Begriffsschriftlich abgeleitet. Vol. I. Jena, 1893. Vol. II. 1903.
[10] Livio states that "While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion
appeared to be disastrous...." Livio 2009:188. But van Heijenoort in his commentary before Frege's (1902) Letter to Russell describes Frege's
proposed "way out" in some detail – the matter has to do with the " 'transformation of the generalization of an equality into an equality of
courses-of-values. For Frege a function is something incomplete, 'unsaturated' "; this seems to contradict the contemporary notion of a
"function in extension"; see Frege's wording at page 128: "Incidentally, it seems to me that the expession 'a predicate is predicated of itself' is
not exact. ...Therefore I would prefer to say that 'a concept is predicated of its own extension' [etc]". But he waffles at the end of his suggestion
that a function-as-concept-in-extension can be written as predicated of its function. van Heijenoort cites Quine: "For a late and thorough study
of Frege's "way out", see Quine 1955": "On Frege's way out", Mind 64, 145–159; reprinted in Quine 1955b: Appendix. Completeness of
quantification theory. Loewenheim's theorem, enclosed as a pamphlet with part of the third printing (1955) of Quine 1950 and incorporated in
the revised edition (1959), 253—260" (cf REFERENCES in van Heijenoort 1967:649)
[11] Russell mentions this fact to Frege, cf van Heijenoort's commentary before Frege's (1902) Letter to Russell in van Heijenoort 1967:126
[12] van Heijenoort's commentary before Zermelo (1908a) Investigations in the foundations of set theory I in van Heijenoort 1967:199
[13] van Heijenoort 1967:190–191. In the section before this he objects strenuously to the notion of impredicativity as defined by Poincaré (and
soon to be taken by Russell, too, in his 1908 Mathematical logic as based on the theory of types cf van Heijenoort 1967:150–182).
[14] Ernst Zermelo (1908) A new proof of the possibility of a well-ordering in van Heijenoort 1967:183–198. Livio 2009:191 reports that
Zermelo "discovered Russell's paradox independently as early as 1900"; Livio in turn cites Ewald 1996 and van Heijenoort 1967 (cf Livio
2009:268).
[15] B. Rang and W. Thomas, "Zermelo's discovery of the 'Russell Paradox'", Historia Mathematica, v. 8 n. 1, 1981, pp. 15–22.
doi:10.1016/0315-0860(81)90002-1
[16] van Heijenoort 1967:124
References
• Potter, Michael (15 January 2004), Set Theory and its Philosophy, Clarendon Press (Oxford University Press),
ISBN 978-0-19-926973-0
• van Heijenoort, Jean (1967, third printing 1976), From Frege to Gödel: A Source Book in Mathematical Logic,
1979-1931, Cambridge, Massachusetts: Harvard University Press, ISBN 0-674-32449-8
• Livio, Mario (6 January 2009), Is God a Mathematician?, New York: Simon & Schuster,
ISBN 978-0-7432-9405-8
External links
• Russell's Paradox (http://www.cut-the-knot.org/selfreference/russell.shtml) at Cut-the-Knot
• Stanford Encyclopedia of Philosophy: " Russell's Paradox (http://plato.stanford.edu/entries/russell-paradox/)"
– by A. D. Irvine.
229
Simpson's paradox
230
Simpson's paradox
In probability and statistics, Simpson's paradox (or the
Yule–Simpson effect) is a paradox in which a trend that appears in
different groups of data disappears when these groups are combined,
and the reverse trend appears for the aggregate data. This result is often
encountered in social-science and medical-science statistics,[1] and is
particularly confounding when frequency data are unduly given causal
interpretations.[2] Simpson's Paradox disappears when causal relations
are brought into consideration.
Though it is mostly unknown to laypeople, Simpson's Paradox is well
known to statisticians, and it is described in a few introductory
statistics books.[3][4] Many statisticians believe that the mainstream
public should be informed of the counter-intuitive results in statistics
such as Simpson's paradox.[5][6]
Simpson's paradox for continuous data: a positive
trend appears for two separate groups (blue and
red), a negative trend (black, dashed) appears
when the data are combined.
Edward H. Simpson first described this phenomenon in a technical paper in 1951,[7] but the statisticians Karl
Pearson, et al., in 1899,[8] and Udny Yule, in 1903, had mentioned similar effects earlier.[9] The name Simpson's
paradox was introduced by Colin R. Blyth in 1972.[10] Since Edward Simpson did not actually discover this
statistical paradox (an instance of Stigler's law of eponymy), some writers, instead, have used the impersonal names
reversal paradox and amalgamation paradox in referring to what is now called Simpson's Paradox and the
Yule-Simpson effect.[11]
Examples
This section provides one fictional simple example and three real-life examples.
Teacher's performance
This is a fictional example meant to explain the paradox in relatable terms. Suppose we are trying to measure the
effectiveness of a teacher (say, Mrs. Brown), from looking at the change in exam scores of students she taught over
several years.
During Year 1, say that she teaches 10 disadvantaged students (define any way you like) and 90 regular students.
Suppose that at the end of term, the average standardized test score for the disadvantaged students is 400 and the
standardized test score for the regular students is 600.
These are relatively good scores, so the school decides to assign Mrs. Brown more disadvantaged students. During
Year 2, say Mrs. Brown teaches 50 disadvantaged students and 50 regular students. Suppose that at the end of term,
the average standardized test score for the disadvantaged students is 450 and the standardized test score for the
regular students is 650.
Note that within both groups (the disadvantaged students and the regular students), the standardized scores
INCREASED. This teacher is getting better!
However, because bureaucratic people have very little time, they just look at the average scores of Mrs. Brown's
class. Year 1, the average is .1*400+.9*600=580. During year 2, the average is .5*450+.5*650=550. Thus, the
average scores DECREASED by 30 points! They conclude that Mrs. Brown is doing worse.
Simpson's paradox
231
avg. score for disadvantaged students avg. score for regular students avg. score for entire group
Year 1
400
600
580
Year 2
450
650
550
Simpson's paradox is that when several groups of data are combined (the disadvantaged and regular students' test
scores), the combined data may show the reverse trend as each of the individual groups. We can see why this is the
case above: the second year's students are much more skewed towards those who are disadvantaged and hence have
lower scores.[12]
Kidney stone treatment
This is a real-life example from a medical study[13] comparing the success rates of two treatments for kidney
stones.[14]
The table shows the success rates and numbers of treatments for treatments involving both small and large kidney
stones, where Treatment A includes all open procedures and Treatment B is percutaneous nephrolithotomy:
Treatment A
Treatment B
Small Stones
Group 1
93% (81/87)
Group 2
87% (234/270)
Large Stones
Group 3
73% (192/263)
Group 4
69% (55/80)
Both
78% (273/350) 83% (289/350)
The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on
large stones, yet treatment B is more effective when considering both sizes at the same time. In this example the
"lurking" variable (or confounding variable) of the stone size was not previously known to be important until its
effects were included.
Which treatment is considered better is determined by an inequality between two ratios (successes/total). The
reversal of the inequality between the ratios, which creates Simpson's paradox, happens because two effects occur
together:
1. The sizes of the groups, which are combined when the lurking variable is ignored, are very different. Doctors
tend to give the severe cases (large stones) the better treatment (A), and the milder cases (small stones) the
inferior treatment (B). Therefore, the totals are dominated by groups three and two, and not by the two much
smaller groups one and four.
2. The lurking variable has a large effect on the ratios, i.e. the success rate is more strongly influenced by the
severity of the case than by the choice of treatment. Therefore, the group of patients with large stones using
treatment A (group three) does worse than the group with small stones, even if the latter used the inferior
treatment B (group two).
Simpson's paradox
232
Berkeley gender bias case
One of the best known real life examples of Simpson's paradox occurred when the University of California, Berkeley
was sued for bias against women who had applied for admission to graduate schools there. The admission figures for
the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so
large that it was unlikely to be due to chance.[3][15]
Applicants Admitted
Men
8442
44%
Women 4321
35%
But when examining the individual departments, it appeared that no department was significantly biased against
women. In fact, most departments had a "small but statistically significant bias in favor of women."[15] The data
from the six largest departments are listed below.
Department
Men
Women
Applicants Admitted Applicants Admitted
A
825
62%
108
82%
B
560
63%
25
68%
C
325
37%
593
34%
D
417
33%
375
35%
E
191
28%
393
24%
F
272
6%
341
7%
The research paper by Bickel, et al.[15] concluded that women tended to apply to competitive departments with low
rates of admission even among qualified applicants (such as in the English Department), whereas men tended to
apply to less-competitive departments with high rates of admission among the qualified applicants (such as in
engineering and chemistry). The conditions under which the admissions' frequency data from specific departments
constitute a proper defense against charges of discrimination are formulated in the book Causality by Pearl.[2]
Low birth weight paradox
The low birth weight paradox is an apparently paradoxical observation relating to the birth weights and mortality of
children born to tobacco smoking mothers. As a usual practice, babies weighing less than a certain amount (which
varies between different countries) have been classified as having low birth weight. In a given population, babies
with low birth weights have had a significantly higher infant mortality rate than others. However, it has been
observed that babies of low birth weights born to smoking mothers have a lower mortality rate than the babies of low
birth weights of non-smokers.[16]
Batting averages
A common example of Simpson's Paradox involves the batting averages of players in professional baseball. It is
possible for one player to hit for a higher batting average than another player during a given year, and to do so again
during the next year, but to have a lower batting average when the two years are combined. This phenomenon can
occur when there are large differences in the number of at-bats between the years. (The same situation applies to
calculating batting averages for the first half of the baseball season, and during the second half, and then combining
all of the data for the season's batting average.)
A real-life example is provided by Ken Ross[17] and involves the batting average of two baseball players, Derek Jeter
and David Justice, during the baseball years 1995 and 1996:[18]
Simpson's paradox
233
1995
Derek Jeter
12/48
1996
Combined
.250 183/582 .314 195/630 .310
David Justice 104/411 .253 45/140
.321 149/551 .270
In both 1995 and 1996, Justice had a higher batting average (in bold type) than Jeter did. However, when the two
baseball seasons are combined, Jeter shows a higher batting average than Justice. According to Ross, this
phenomenon would be observed about once per year among the possible pairs of interesting baseball players. In this
particular case, the Simpson's Paradox can still be observed if the year 1997 is also taken into account:
1995
Derek Jeter
12/48
1996
1997
Combined
.250 183/582 .314 190/654 .291 385/1284 .300
David Justice 104/411 .253 45/140
.321 163/495 .329 312/1046 .298
The Jeter and Justice example of Simpson's paradox was referred to in the "Conspiracy Theory" episode of the
television series Numb3rs, though a chart shown omitted some of the data, and listed the 1996 averages as 1995.
Description
Suppose two people, Lisa and Bart, each edit document
articles for two weeks. In the first week, Lisa improves
0 of the 3 articles she edited, and Bart improves 1 of
the 7 articles he edited. In the second week, Lisa
improves 5 of 7 articles she edited, while Bart improves
all 3 of the articles he edited.
Illustration of Simpson's Paradox; The first graph (on the top)
represents Lisa's contribution, the second one Bart's. The blue bars
represent the first week, the red bars the second week; the triangles
indicate the combined percentage of good contributions (weighted
average). While Bart's bars both show a higher rate of success than
Lisa's, Lisa's combined rate is higher because basically she improved
a greater ratio relative to the quantity edited.
Simpson's paradox
234
Week 1 Week 2 Total
Lisa
0/3
5/7
5/10
Bart
1/7
3/3
4/10
Both times Bart improved a higher percentage of articles than Lisa, but the actual number of articles each edited (the
bottom number of their ratios, also known as the sample size) were not the same for both of them either week. When
the totals for the two weeks are added together, Bart and Lisa's work can be judged from an equal sample size, i.e.
the same number of articles edited by each. Looked at in this more accurate manner, Lisa's ratio is higher and,
therefore, so is her percentage. Also when the two tests are combined using a weighted average, overall, Lisa has
improved a much higher percentage than Bart because the quality modifier had a significantly higher percentage.
Therefore, like other paradoxes, it only appears to be a paradox because of incorrect assumptions, incomplete or
misguided information, or a lack of understanding a particular concept.
Week 1 quantity Week 2 quantity Total quantity and weighted quality
Lisa
0%
71.4%
50%
Bart
14.2%
100%
40%
This imagined paradox is caused when the percentage is provided but not the ratio. In this example, if only the
14.2% in the first week for Bart was provided but not the ratio (1:7), it would distort the information causing the
imagined paradox. Even though Bart's percentage is higher for the first and second week, when two weeks of articles
is combined, overall Lisa had improved a greater proportion, 50% of the 10 total articles. Lisa's proportional total of
articles improved exceeds Bart's total.
Here are some notations:
• In the first week
•
•
— Lisa improved 0% of the articles she edited.
— Bart had a 14.2% success rate during that time.
Success is associated with Bart.
• In the second week
•
•
— Lisa managed 71.4% in her busy life.
— Bart achieved a 100% success rate.
Success is associated with Bart.
On both occasions Bart's edits were more successful than Lisa's. But if we combine the two sets, we see that Lisa and
Bart both edited 10 articles, and:
•
— Lisa improved 5 articles.
•
•
— Bart improved only 4.
— Success is now associated with Lisa.
Bart is better for each set but worse overall.
The paradox stems from the intuition that Bart could not possibly be a better editor on each set but worse overall.
Pearl proved how this is possible, when "better editor" is taken in the counterfactual sense: "Were Bart to edit all
items in a set he would do better than Lisa would, on those same items".[2] Clearly, frequency data cannot support
this sense of "better editor," because it does not tell us how Bart would perform on items edited by Lisa, and vice
versa. In the back of our mind, though, we assume that the articles were assigned at random to Bart and Lisa, an
assumption which (for a large sample) would support the counterfactual interpretation of "better editor." However,
under random assignment conditions, the data given in this example are unlikely, which accounts for our surprise
Simpson's paradox
235
when confronting the rate reversal.
The arithmetical basis of the paradox is uncontroversial. If
must be greater than
and
we feel that
. However if different weights are used to form the overall score for each person then
this feeling may be disappointed. Here the first test is weighted
for Lisa and
for Bart while the weights are
reversed on the second test.
•
•
Lisa is a better editor on average, as her overall success rate is higher. But it is possible to have told the story in a
way which would make it appear obvious that Bart is more diligent.
Simpson's paradox shows us an extreme example of the importance of including data about possible confounding
variables when attempting to calculate causal relations. Precise criteria for selecting a set of "confounding variables,"
(i.e., variables that yield correct causal relationships if included in the analysis), is given in Pearl[2] using causal
graphs.
While Simpson's paradox often refers to the analysis of count tables, as shown in this example, it also occurs with
continuous data:[19] for example, if one fits separated regression lines through two sets of data, the two regression
lines may show a positive trend, while a regression line fitted through all data together will show a negative trend, as
shown on the picture above.
Vector interpretation
Simpson's paradox can also be illustrated using the 2-dimensional
vector space.[20] A success rate of
can be represented by a vector
, with a slope of
. If two rates
and
are combined, as in the examples given above, the result can be
represented by the sum of the vectors
and
, which
according to the parallelogram rule is the vector
with slope
.
Simpson's paradox says that even if a vector
has a smaller slope than another vector
smaller slope than
,
(in blue in the figure)
(in red), and
has a
, the sum of the two vectors
Vector interpretation of Simpson's paradox
(indicated by "+" in the figure) can still have a larger slope than the
sum of the two vectors
, as shown in the example.
Implications for decision making
The practical significance of Simpson's paradox surfaces in decision making situations where it poses the following
dilemma: Which data should we consult in choosing an action, the aggregated or the partitioned? In the Kidney
Stone example above, it is clear that if one is diagnosed with "Small Stones" or "Large Stones" the data for the
respective subpopulation should be consulted and Treatment A would be preferred to Treatment B. But what if a
patient is not diagnosed, and the size of the stone is not known; would it be appropriate to consult the aggregated
data and administer Treatment B? This would stand contrary to common sense; a treatment that is preferred both
under one condition and under its negation should also be preferred when the condition is unknown.
On the other hand, if the partitioned data is to be preferred a priori, what prevents one from partitioning the data into
arbitrary sub-categories (say based on eye color or post-treatment pain) artificially constructed to yield wrong
Simpson's paradox
choices of treatments? Pearl[2] shows that, indeed, in many cases it is the aggregated, not the partitioned data that
gives the correct choice of action. Worse yet, given the same table, one should sometimes follow the partitioned and
sometimes the aggregated data, depending on the story behind the data; with each story dictating its own choice.
Pearl[2] considers this to be the real paradox behind Simpson's reversal.
As to why and how a story, not data, should dictate choices, the answer is that it is the story which encodes the
causal relationships among the variables. Once we extract these relationships and represent them in a graph called a
causal Bayesian network we can test algorithmically whether a given partition, representing confounding variables,
gives the correct answer. The test, called "back-door," requires that we check whether the nodes corresponding to the
confounding variables intercept certain paths in the graph. This reduces Simpson's Paradox to an exercise in graph
theory.
Psychology
Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first,
offended by the idea that a treatment could benefit both males and females and harm the population as a whole. The
question is where people get this strong intuition from, and how it is encoded in the mind. Simpson's paradox
demonstrates that this intuition cannot be supported by probability calculus alone, and thus led philosophers to
speculate that it is supported by an innate causal logic that guides people in reasoning about actions and their
consequences. Savage's "sure thing principle"[10] is an example of what such logic may entail. A qualified version of
Savage's sure thing principle can indeed be derived from Pearl's do-calculus[2] and reads: "An action A that increases
the probability of an event B in each subpopulation Ci of C must also increase the probability of B in the population
as a whole, provided that the action does not change the distribution of the subpopulations." This suggests that
knowledge about actions and consequences is stored in a form resembling Causal Bayesian Networks.
Probability
If a 2 × 2 × 2 table is selected at random, the probability is approximately 1/60 that Simpson's paradox will occur
purely by chance.[21]
Related concepts
• Ecological fallacy (and ecological correlation)
• Modifiable areal unit problem
• Prosecutor's fallacy
References
[1] Clifford H. Wagner (February 1982). "Simpson's Paradox in Real Life". The American Statistician 36 (1): 46–48. doi:10.2307/2684093.
JSTOR 2684093.
[2] Judea Pearl. Causality: Models, Reasoning, and Inference, Cambridge University Press (2000, 2nd edition 2009). ISBN 0-521-77362-8.
[3] David Freedman, Robert Pisani and Roger Purves. Statistics (4th edition). W.W. Norton, 2007, p. 19. ISBN 978-0393929720.
[4] David S. Moore and D.S. George P. McCabe (February 2005). "Introduction to the Practice of Statistics" (5th edition). W.H. Freeman &
Company. ISBN 0-7167-6282-X.
[5] Robert L. Wardrop (February 1995). "Simpson's Paradox and the Hot Hand in Basketball". The American Statistician, 49 (1): pp. 24–28.
[6] Alan Agresti (2002). "Categorical Data Analysis" (Second edition). John Wiley and Sons ISBN 0-471-36093-7
[7] Simpson, Edward H. (1951). "The Interpretation of Interaction in Contingency Tables". Journal of the Royal Statistical Society, Ser. B 13:
238–241.
[8] Pearson, Karl; Lee, A.; Bramley-Moore, L. (1899). "Genetic (reproductive) selection: Inheritance of fertility in man". Philosophical
Translations of the Royal Statistical Society, Ser. A 173: 534–539.
[9] G. U. Yule (1903). "Notes on the Theory of Association of Attributes in Statistics". Biometrika 2 (2): 121–134. doi:10.1093/biomet/2.2.121.
[10] Colin R. Blyth (June 1972). "On Simpson's Paradox and the Sure-Thing Principle". Journal of the American Statistical Association 67 (338):
364–366. doi:10.2307/2284382. JSTOR 2284382.
236
Simpson's paradox
[11] I. J. Good, Y. Mittal (June 1987). "The Amalgamation and Geometry of Two-by-Two Contingency Tables". The Annals of Statistics 15 (2):
694–711. doi:10.1214/aos/1176350369. ISSN 0090-5364. JSTOR 2241334.
[12] "Mathematics: What are the most basic mathematical proofs or facts that the majority of people would benefit from?" (http:/ / www. quora.
com/ Mathematics/ What-are-the-most-basic-mathematical-proofs-or-facts-that-the-majority-of-people-would-benefit-from?srid=3aft& st=ns).
Quora. . Retrieved 29 November 2012.
[13] C. R. Charig, D. R. Webb, S. R. Payne, J. E. Wickham (29 March 1986). "Comparison of treatment of renal calculi by open surgery,
percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy". Br Med J (Clin Res Ed) 292 (6524): 879–882.
doi:10.1136/bmj.292.6524.879. PMC 1339981. PMID 3083922.
[14] Steven A. Julious and Mark A. Mullee (12/03/1994). "Confounding and Simpson's paradox" (http:/ / bmj. bmjjournals. com/ cgi/ content/
full/ 309/ 6967/ 1480). BMJ 309 (6967): 1480–1481. PMC 2541623. PMID 7804052. .
[15] P.J. Bickel, E.A. Hammel and J.W. O'Connell (1975). "Sex Bias in Graduate Admissions: Data From Berkeley" (http:/ / www. sciencemag.
org/ cgi/ content/ abstract/ 187/ 4175/ 398). Science 187 (4175): 398–404. doi:10.1126/science.187.4175.398. PMID 17835295. ..
[16] Wilcox Allen (2006). "The Perils of Birth Weight — A Lesson from Directed Acyclic Graphs" (http:/ / aje. oxfordjournals. org/ cgi/ content/
abstract/ 164/ 11/ 1121). American Journal of Epidemiology 164 (11): 1121–1123. doi:10.1093/aje/kwj276. PMID 16931545. .
[17] Ken Ross. "A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (Paperback)" Pi Press, 2004. ISBN 0-13-147990-3.
12–13
[18] Statistics available from http:/ / www. baseball-reference. com/ : Data for Derek Jeter (http:/ / www. baseball-reference. com/ j/ jeterde01.
shtml), Data for David Justice (http:/ / www. baseball-reference. com/ j/ justida01. shtml).
[19] John Fox (1997). "Applied Regression Analysis, Linear Models, and Related Methods". Sage Publications. ISBN 0-8039-4540-X. 136–137
[20] Kocik Jerzy (2001). "Proofs without Words: Simpson's Paradox" (http:/ / www. math. siu. edu/ kocik/ papers/ simpson2. pdf) (PDF).
Mathematics Magazine 74 (5): 399. .
[21] Marios G. Pavlides and Michael D. Perlman (August 2009). "How Likely is Simpson's Paradox?". The American Statistician 63 (3):
226–233. doi:10.1198/tast.2009.09007.
External links
• Stanford Encyclopedia of Philosophy: " Simpson's Paradox (http://plato.stanford.edu/entries/
paradox-simpson/)" – by Gary Malinas.
• Earliest known uses of some of the words of mathematics: S (http://jeff560.tripod.com/s.html)
• For a brief history of the origins of the paradox see the entries "Simpson's Paradox" and "Spurious Correlation"
• Pearl, Judea, " "The Art and Science of Cause and Effect. (http://bayes.cs.ucla.edu/LECTURE/lecture_sec1.
htm)" A slide show and tutorial lecture.
• Pearl, Judea, "Simpson's Paradox: An Anatomy" (http://bayes.cs.ucla.edu/R264.pdf) (PDF)
• Short articles by Alexander Bogomolny at cut-the-knot:
• " Mediant Fractions. (http://www.cut-the-knot.org/blue/Mediant.shtml)"
• " Simpson's Paradox. (http://www.cut-the-knot.org/Curriculum/Algebra/SimpsonParadox.shtml)"
• The Wall Street Journal column "The Numbers Guy" (http://online.wsj.com/article/SB125970744553071829.
html) for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's
paradox in the comparison of unemployment rates of the 2009 recession with the 1983 recession. by Cari Tuna
(substituting for regular columnist Carl Bialik)
237
Skolem's paradox
Skolem's paradox
In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward
Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of
the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not
an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a
"paradoxical state of affairs" by Skolem (1922: p. 295).
Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a
model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a
sentence which intuitively says (or which precisely says in the standard model of the theory) that there exist sets that
are not countable. Thus the seeming contradiction is that a model which is itself countable, and which contains only
countable sets, satisfies the first order sentence that intuitively states "there are uncountable sets".
A mathematical explanation of the paradox, showing that it is not a contradiction in mathematics, was given by
Skolem (1922). Skolem's work was harshly received by Ernst Zermelo, who argued against the limitations of
first-order logic, but the result quickly came to be accepted by the mathematical community.
The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions
whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets". This line of
thought can be extended to question whether any set is uncountable in an absolute sense. More recently, the paper
"Models and Reality" by Hilary Putnam, and responses to it, led to renewed interest in the philosophical aspects of
Skolem's result.
Background
One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets,
such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set X is
countable if there is a function that gives a one-to-one correspondence between X and the natural numbers, and is
uncountable if there is no such correspondence function. When Zermelo proposed his axioms for set theory in 1908,
he proved Cantor's theorem from them to demonstrate their strength.
Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim–Skolem theorem. The downward form of this
theorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure, then the same
axioms are satisfied by some countable structure. In particular, this implies that if the first order versions of
Zermelo's axioms of set theory are satisfiable, they are satisfiable in some countable model. The same is true of any
consistent first order axiomatisation of set theory.
The paradoxical result and its mathematical implications
Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand,
which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which
states that uncountable sets exist, and which is provable from Zermelo's axioms. "So far as I know," Skolem writes,
"no one has called attention to this paradoxical state of affairs. By virtue of the axioms we can prove the existence of
higher cardinalities... How can it be, then, that the entire domain B [a countable model of Zermelo's axioms] can
already be enumerated by means of finite positive integers?" (Skolem 1922, p. 295, translation by
Bauer-Mengelberg)
More specifically, let B be a countable model of Zermelo's axioms. Then there is some set u in B such that B satisfies
the first-order formula saying that u is uncountable. For example, u could be taken as the set of real numbers in B.
Now, because B is countable, there are only countably many elements c such that c ∈ u according to B, because there
are only countably many elements c in B to begin with. Thus it appears that u should be countable. This is Skolem's
238
Skolem's paradox
paradox.
Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term
"set" does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of
countability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible to
recognize that a particular set u is countable, but not countable in a particular model of set theory, because there is no
set in the model that gives a one-to-one correspondence between u and the natural numbers in that model.
Skolem used the term "relative" to describe this state of affairs, where the same set is included in two models of set
theory, is countable in one model, and is not countable in the other model. He described this as the "most important"
result in his paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitive
model as absolute. From their point of view, Skolem's paradox simply shows that countability is not an absolute
property in first order logic. (Kunen 1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406).
Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a
foundational system:
"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of
mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent
times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the
ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique."
(Ebbinghaus and van Dalen, 2000, p. 147)
Reception by the mathematical community
A central goal of early research into set theory was to find a first order axiomatisation for set theory which was
categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed
this is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time for
the theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem's result;
no resolution of the paradox was widely accepted during the 1920s. Fraenkel (1928) still described the result as an
antinomy:
"Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible
solution yet been reached." (van Dalen and Ebbinghaus, 2000, p. 147).
In 1925, von Neumann presented a novel axiomatization of set theory, which developed into NBG set theory. Very
much aware of Skolem's 1922 paper, von Neumann investigated countable models of his axioms in detail. In his
concluding remarks, Von Neumann comments that there is no categorical axiomatization of set theory, or any other
theory with an infinite model. Speaking of the impact of Skolem's paradox, he wrote,
"At present we can do no more than note that we have one more reason here to entertain reservations about set
theory and that for the time being no way of rehabilitating this theory is known."(Ebbinghaus and van Dalen,
2000, p. 148)
Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.), and spoke
against it starting in 1929. Skolem's result applies only to what is now called first-order logic, but Zermelo argued
against the finitary metamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued that
his axioms should instead be studied in second-order logic, a setting in which Skolem's result does not apply.
Zermelo published a second-order axiomatization in 1930 and proved several categoricity results in that context.
Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative
hierarchy and formalization of infinitary logic (van Dalen and Ebbinghaus, 2000, note 11).
Fraenkel et al. (1973, pp. 303–304) explain why Skolem's result was so surprising to set theorists in the 1920s. At
the time, Gödel's completeness theorem and the compactness theorem had not yet been proved. These theorems
illuminated the way that first-order logic behaves and established its finitary nature. The method of Henkin models,
239
Skolem's paradox
now a standard technique for constructing countable models of a consistent first-order theory, was not developed
until 1950. Thus, in 1922, the particular properties of first-order logic that permit Skolem's paradox to go through
were not yet understood. It is now known that Skolem's paradox is unique to first-order logic; if set theory is
formalized using higher-order logic then it does not have any countable models.
Contemporary mathematical opinion
Contemporary mathematical logicians do not view Skolem's paradox as any sort of fatal flaw in set theory. Kleene
(1967, p. 324) describes the result as "not a paradox in the sense of outright contradiction, but rather a kind of
anomaly". After surveying Skolem's argument that the result is not contradictory, Kleene concludes "there is no
absolute notion of countability." Hunter (1971, p. 208) describes the contradiction as "hardly even a paradox".
Fraenkel et al. (1973, p. 304) explain that contemporary mathematicians are no more bothered by the lack of
categoricity of first-order theories than they are bothered by the conclusion of Gödel's incompleteness theorem that
no consistent, effective, and sufficiently strong set of first-order axioms is complete.
Countable models of ZF have become common tools in the study of set theory. Forcing, for example, is often
explained in terms of countable models. The fact that these countable models of ZF still satisfy the theorem that
there are uncountable sets is not considered a pathology; van Heijenoort (1967) describes it as "a novel and
unexpected feature of formal systems." (van Heijenoort 1967, p. 290)
Although mathematicians no longer consider Skolem's result paradoxical, the result is often discussed by
philosophers. In the setting of philosophy, a merely mathematical resolution of the paradox may be less than
satisfactory.
References
• Barwise, Jon (1977), "An introduction to first-order logic", in Barwise, Jon, ed. (1982), Handbook of
Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland,
ISBN 978-0-444-86388-1
• Van Dalen, Dirk and Heinz-Dieter Ebbinghaus, "Zermelo and the Skolem Paradox", The Bulletin of Symbolic
Logic Volume 6, Number 2, June 2000.
• Dragalin, A.G. (2001), "S/s085750" [1], in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer,
ISBN 978-1-55608-010-4
• Abraham Fraenkel, Yehoshua Bar-Hillel, Azriel Levy, Dirk van Dalen (1973), Foundations of Set Theory,
North-Holland.
• Henkin, L. (1950), "Completeness in the theory of types", Journal of Symbolic Logic (The Journal of Symbolic
Logic, Vol. 15, No. 2) 15 (2): 81–91, doi:10.2307/2266967, JSTOR 2266967.
• Kanamori, Akihiro (2004), "Zermelo and set theory" [2], The Bulletin of Symbolic Logic 10 (4): 487–553,
doi:10.2178/bsl/1102083759, ISSN 1079-8986, MR2136635
• Stephen Cole Kleene, (1952, 1971 with emendations, 1991 10th printing), Introduction to Metamathematics,
North-Holland Publishing Company, Amsterdam NY. ISBN 0-444-10088-1. cf pages 420-432: § 75. Axiom
systems, Skolem's paradox, the natural number sequence.
• Stephen Cole Kleene, (1967). Mathematical Logic.
• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Amsterdam: North-Holland,
ISBN 978-0-444-85401-8
• (1915), "Über Möglichkeiten im Relativkalkül", Mathematische Annalen 76 (4): 447–470,
doi:10.1007/BF01458217, ISSN 0025-5831
• Moore, A.W., "Set Theory, Skolem's Paradox and the Tractatus", Analysis 1985: 45.
• Hilary Putnam, "Models and Reality", The Journal of Symbolic Logic, Vol. 45, No. 3 (Sep., 1980), pp. 464–482
240
Skolem's paradox
241
• Rautenberg, Wolfgang (2010), A Concise Introduction to Mathematical Logic [3] (3rd ed.), New York: Springer
Science+Business Media, doi:10.1007/978-1-4419-1221-3, ISBN 978-1-4419-1220-6
• Skolem, Thoralf (1922). "Axiomatized set theory". Reprinted in From Frege to Gödel, van Heijenoort, 1967, in
English translation by Stefan Bauer-Mengelberg, pp. 291–301.
External links
• Bays's Ph.D. thesis on the paradox [4]
• Vaughan Pratt's celebration of his academic ancestor Skolem's 120th birthday [5]
• Extract from Moore's discussion of the paradox(broken link) [6]
References
[1]
[2]
[3]
[4]
[5]
[6]
http:/ / www. encyclopediaofmath. org/ index. php?title=S/ s085750
http:/ / www. math. ucla. edu/ ~asl/ bsl/ 1004-toc. htm
http:/ / www. springerlink. com/ content/ 978-1-4419-1220-6/
http:/ / www. nd. edu/ ~tbays/ papers/ pthesis. pdf
http:/ / boole. stanford. edu/ skolem
http:/ / www. webcitation. org/ query?url=http:/ / uk. geocities. com/ frege%40btinternet. com/ cantor/ skolem_moore. htm&
date=2009-10-25+ 04:16:47
Smale's paradox
In differential topology, Smale's paradox states that it is possible to
turn a sphere inside out in a three-dimensional space with possible
self-intersections but without creating any crease, a process often
called sphere eversion (eversion means "to turn inside out"). This is
surprising, and is hence deemed a veridical paradox. More precisely,
let
be the standard embedding; then there is a regular homotopy of
immersions
such that ƒ0 = ƒ and ƒ1 = −ƒ.
A Morin surface seen from "above"
History
This 'paradox' was discovered by Stephen Smale (1958). It is difficult to visualize a particular example of such a
turning, although some digital animations have been produced that make it somewhat easier. The first example was
exhibited through the efforts of several mathematicians, including Arnold Shapiro and Bernard Morin who was
blind. On the other hand, it is much easier to prove that such a "turning" exists and that is what Smale did.
Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong (Levy 1995). His
reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that
there is no such turning of S1in R2. But the degree of the Gauss map for the embeddings f, −f in R3 are both equal to
1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of a
2-sphere in R3 is 1; so there is no obstacle.
See h-principle for further generalizations.
Smale's paradox
Proof
Smale's original proof was indirect: he identified (regular homotopy) classes of immersions of spheres with a
homotopy group of the Stiefel manifold. Since the homotopy group that corresponds to immersions of
in
vanishes, the standard embedding and the inside-out one must be regular homotopic. In principle the proof can be
unwound to produce an explicit regular homotopy, but this is not easy to do.
There are several ways of producing explicit examples and mathematical visualization:
• the method of half-way models: these consist of very special homotopies. This is the original method, first done
by Shapiro and Phillips via Boy's surface, later refined by many others. A more recent and definitive refinement
(1980s) is minimax eversions, which is a variational method, and consist of special homotopies (they are shortest
paths with respect to Willmore energy). The original half-way model homotopies were constructed by hand, and
worked topologically but weren't minimal.
• Thurston's corrugations: this is a topological method and generic; it takes a homotopy and perturbs it so that it
becomes a regular homotopy.
References
• Francis, George K. (2007), A topological picturebook, Berlin, New York: Springer-Verlag,
ISBN 978-0-387-34542-0, MR2265679
• Levy, Silvio (1995), "A brief history of sphere eversions" [1], Making waves, Wellesley, MA: A K Peters Ltd.,
ISBN 978-1-56881-049-2, MR1357900
• Nelson Max, "Turning a Sphere Inside Out", International Film Bureau, Chicago, 1977 (video)
• Anthony Phillips, "Turning a surface inside out, Scientific American, May 1966, pp. 112–120.
• Smale, Stephen (1958), "A classification of immersions of the two-sphere", Transactions of the American
Mathematical Society 90: 281–290, ISSN 0002-9947, JSTOR 1993205, MR0104227
External links
•
•
•
•
•
Outside In [2], full video (short clip here [3])
Optiverse video [4], portions available online
A History of Sphere Eversions [5]
"Turning a Sphere Inside Out" [6]
Software for visualizing sphere eversion [7]
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
http:/ / www. geom. uiuc. edu/ docs/ outreach/ oi/ history. html
http:/ / video. google. com/ videoplay?docid=-6626464599825291409
http:/ / www. th. physik. uni-bonn. de/ th/ People/ netah/ cy/ movies/ sphere. mpg
http:/ / new. math. uiuc. edu/ optiverse/
http:/ / torus. math. uiuc. edu/ jms/ Papers/ isama/ color/ opt2. htm
http:/ / www. cs. berkeley. edu/ ~sequin/ SCULPTS/ SnowSculpt04/ eversion. html
http:/ / www. dgp. utoronto. ca/ ~mjmcguff/ eversion/
242
Thomson's lamp
243
Thomson's lamp
Thomson's lamp is a philosophical puzzle that is a variation on Zeno's paradoxes. It was devised in 1954 by British
philosopher James F. Thomson, who also coined the term supertask.
Time State
0.000
On
1.000
Off
1.500
On
1.750
Off
1.875
On
...
...
2.000
?
Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp
off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At
the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another
quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking
the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of all
these progressively smaller times is exactly two minutes.
The following questions are then considered:
• Is the lamp switch on or off after exactly two minutes?
• Would the final state be different if the lamp had started out being on, instead of off?
Thomson wasn't interested in actually answering these questions, because he believed these questions had no
answers. This is because Thomson used this thought experiment to argue against the possibility of supertasks, which
is the completion of an infinite number of tasks. To be specific, Thomson argued that if supertasks are possible, then
the scenario of having flicked the lamp on and off infinitely many times should be possible too (at least logically,
even if not necessarily physically). But, Thomson reasoned, the possibility of the completion of the supertask of
flicking a lamp on and off infinitely many times creates a contradiction. The lamp is either on or off at the 2-minute
mark. If the lamp is on, then there must have been some last time, right before the 2-minute mark, at which it was
flicked on. But, such an action must have been followed by a flicking off action since, after all, every action of
flicking the lamp on before the 2-minute mark is followed by one at which it is flicked off between that time and the
2-minute mark. So, the lamp cannot be on. Analogously, one can also reason that the lamp cannot be off at the
2-minute mark. So, the lamp cannot be either on or off. So, we have a contradiction. By reductio ad absurdum, the
assumption that supertasks are possible must therefore be rejected: supertasks are logically impossible.
Discussion
The status of the lamp and the switch is known for all times strictly less than two minutes. However the question
does not state how the sequence finishes, and so the status of the switch at exactly two minutes is indeterminate.
Though acceptance of this indeterminacy is resolution enough for some, problems do continue to present themselves
under the intuitive assumption that one should be able to determine the status of the lamp and the switch at any time,
given full knowledge of all previous statuses and actions taken.
Another interesting issue is that measuring two minutes exactly is a supertask in the sense that it requires measuring
time with infinite precision.
Thomson's lamp
244
Mathematical series analogy
The question is similar to determining the value of Grandi's series, i.e. the limit as n tends to infinity of
For even values of n, the above finite series sums to 1; for odd values, it sums to 0. In other words, as n takes the
values of each of the non-negative integers 0, 1, 2, 3, ... in turn, the series generates the sequence {1, -1, 1, -1, ...},
representing the changing state of the lamp. The sequence does not converge as n tends to infinity, so neither does
the infinite series.
Another way of illustrating this problem is to let the series look like this:
The series can be rearranged as:
The unending series in the brackets is exactly the same as the original series S. This means S = 1 - S which implies S
= ½. In fact, this manipulation can be rigorously justified: there are generalized definitions for the sums of series that
do assign Grandi's series the value ½. On the other hand, according to other definitions for the sum of a series this
series has no defined sum (the limit does not exist).
One of Thomson's objectives in his original 1954 paper is to differentiate supertasks from their series analogies. He
writes of the lamp and Grandi's series,
"Then the question whether the lamp is on or off… is the question: What is the sum of the infinite divergent
sequence
+1, −1, +1, …?
"Now mathematicians do say that this sequence has a sum; they say that its sum is 1⁄2. And this answer does
not help us, since we attach no sense here to saying that the lamp is half-on. I take this to mean that there is no
established method for deciding what is done when a super-task is done. … We cannot be expected to pick up
this idea, just because we have the idea of a task or tasks having been performed and because we are
acquainted with transfinite numbers."[1]
Later, he claims that even the divergence of a series does not provide information about its supertask: "The
impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be-associated arithmetical
sequence is convergent or divergent."[2]
References
[1] Thomson p.6. For the mathematics and its history he cites Hardy and Waismann's books, for which see History of Grandi's series.
[2] Thomson p.7
• Thomson, James F. (October 1954). "Tasks and Super-Tasks". Analysis (Analysis, Vol. 15, No. 1) 15 (1): 1–13.
doi:10.2307/3326643. JSTOR 3326643.
Two envelopes problem
Two envelopes problem
The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle or paradox in logic,
philosophy, probability, and recreational mathematics, of special interest in decision theory and for the Bayesian
interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.
A statement of the problem starts with:
Let us say you are given two indistinguishable envelopes, each of which contains a positive sum of money.
One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it
contains. You pick one envelope at random but before you open it you are offered the possibility to take the
other envelope instead.
It is possible to give arguments that show that it will be to your advantage to swap envelopes by showing that your
expected return on swapping exceeds the sum in your envelope. This leads to the absurdity that it is beneficial to
continue to swap envelopes indefinitely.
A large number of different solutions have been proposed. The usual scenario is that one writer proposes a solution
that solves the problem as stated, but then some other writer discovers that by altering the problem a little the
paradox is brought back to life again. In this way a family of closely related formulations of the problem is created
which are then discussed in the literature.
There is not yet any one proposed solution that is widely accepted as being the correct one. [1] Despite this it is
common for authors to claim that the solution to the problem is easy, even elementary.[2] However, when
investigating these elementary solutions they often differ from one author to the next. During the last two decades
several new papers have been published every year. [3]
The problem
The basic setup: You are given two indistinguishable envelopes, each
of which contains a positive sum of money. One envelope contains
twice as much as the other. You may pick one envelope and keep
whatever amount it contains. You pick one envelope at random but
before you open it you are offered the possibility to take the other
envelope instead.[4]
The switching argument: Now suppose you reason as follows:
1. I denote by A the amount in my selected envelope.
2.
3.
4.
5.
6.
7.
The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
The other envelope may contain either 2A or A/2.
If A is the smaller amount, then the other envelope contains 2A.
If A is the larger amount, then the other envelope contains A/2.
Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.
So the expected value of the money in the other envelope is
8. This is greater than A, so I gain on average by swapping.
9. After the switch, I can denote that content by B and reason in exactly the same manner as above.
10. I will conclude that the most rational thing to do is to swap back again.
11. To be rational, I will thus end up swapping envelopes indefinitely.
12. As it seems more rational to open just any envelope than to swap indefinitely, we have a contradiction.
245
Two envelopes problem
The puzzle: The puzzle is to find the flaw in the very compelling line of reasoning above.
A common resolution
A common way to resolve the paradox, both in popular literature and in the academic literature in philosophy, is to
observe that A stands for different things at different places in the expected value calculation, step 7 above.[5] In the
first term A is the smaller amount while in the second term A is the larger amount. To mix different instances of a
variable in the same formula like this is said to be illegitimate, so step 7 is incorrect, and this is the cause of the
paradox.
According to this analysis, a correct alternative argument would have run on the following lines. Assume that there
are only two possible sums that might be in the envelope. Denoting the lower of the two amounts by X, we can
rewrite the expected value calculation as
Here X stands for the same thing in every term of the equation. We learn that 1.5X is the average expected value in
either of the envelopes, hence no reason to swap envelopes according to this calculation.
Mathematical details
Let us rewrite the preceding calculations in a more detailed notation which explicitly distinguishes random from
not-random quantities (that is a different distinction from the usual distinction in ordinary, deterministic,
mathematics between variables and constants). This is useful in order to compare with the next, alternative,
resolution. So far we were thinking of the two amounts of money in the two envelopes as being fixed; the only
randomness lies in which one goes into which envelope. We called the smaller amount X, let us denote the larger
amount by Y. Given the values x and y of X and Y, where y = 2x and x > 0, the problem description tells us (whether
or not x and y are known)
for all possible values x of the smaller amount X; there is a corresponding definition of the probability distribution of
B given X and Y. In our resolution of the paradox, we guessed that in Step 7 the writer was trying to compute the
expected value of B given X=x. Splitting the calculation over the two possibilities for which envelope contains the
smaller amount, it is certainly correct to write
At this point the writer correctly substitutes the value 1/2 for both of the conditional probabilities on the right hand
side of this equation (Step 2). At the same time he correctly substitutes the random variable B inside the first
conditional expectation for 2A, when taking its expectation value given B > A and X = x, and he similarly correctly
substitutes the random variable B for A/2 when taking its expectation value given B < A and X = x (Steps 4 and 5).
He would then arrive at the completely correct equation
However he now proceeds, in the first of the two terms on the right hand side, to replace the expectation value of A
given that Envelope A contains the smaller amount and given that the amounts are x and 2x, by the random quantity
A itself. Similarly, in the second term on the right hand side he replaces the expectation value of A given now that
Envelope A contains the larger amount and given that the amounts are x and 2x, also by the random quantity A itself.
The correct substitutions would have been, of course, x and 2x respectively, leading to a correct conclusion
246
Two envelopes problem
247
.
Naturally this coincides with the expectation value of A given X=x.
Indeed, in the two contexts in which the random variable A appears on the right hand side, it is standing for two
different things, since its distribution has been conditioned on different events. Obviously, A tends to be larger, when
we know that it is greater than B and when the two amounts are fixed, and it tends to be smaller, when we know that
it is smaller than B and the two amounts are fixed, cf. Schwitzgebel and Dever (2007, 2008). In fact, it is exactly
twice as large in the first situation as in the second situation.
The preceding resolution was first noted by Bruss in 1996.[6] A concise exposition is given by Falk in 2009.[7]
Alternative interpretation
The first solution above doesn't explain what's wrong if the player is allowed to open the first envelope before being
offered the option to switch. In this case, A stands for the value which is seen then, throughout all subsequent
calculations. The mathematical variable A stands for any particular amount he might see there (it is a mathematical
variable, a generic possible value of a random variable). The reasoning appears to show that whatever amount he
would see there, he would decide to switch. Hence, he does not need to look in the envelope at all: he knows that if
he would look, and go through the calculations, they would tell him to switch, whatever he saw in the envelope.
In this case, at Steps 6, 7 and 8 of the reasoning, A is any fixed possible value of the amount of money in the first
envelope.
Thus, the proposed "common resolution" above breaks down and another explanation is needed.
This interpretation of the two envelopes problem appears in the first publications in which the paradox was
introduced, Gardner (1989) and Nalebuff (1989). It is common in the more mathematical literature on the problem.
The "common resolution" above depends on a particular interpretation of what the writer of the argument is trying to
calculate: namely, it assumes he is after the (unconditional) expectation value of what's in Envelope B. In the
mathematical literature on Two Envelopes Problem (and in particular, in the literature where it was first introduced
to the world), another interpretation is more common, involving the conditional expectation value (conditional on
what might be in Envelope A). In order to solve this and related interpretations or versions of the problem most
authors utilize the Bayesian interpretation of probability.
Introduction to resolutions based on Bayesian probability theory
Here the ways in which the paradox can be resolved depend to a large degree on the assumptions that are made about
the things that are not made clear in the setup and the proposed argument for switching.[8] The most usual
assumption about the way the envelopes are set up is that a sum of money is put in one envelope and twice that sum
is put in another envelope and then one of the two envelopes is selected randomly, called "Envelope A", and given to
the player. It is not made clear exactly how the first mentioned sum of money (the smaller of the two sums) is
determined and what values it could possibly take and, in particular, whether there is a maximum sum which it might
contain.[9][10] It is also not specified whether the player can look in Envelope A before deciding whether or not to
switch. A further ambiguity in the paradox is that it is not made clear in the proposed argument whether the amount
A in Envelope A is intended to be a constant, a random variable, or some other quantity.
If it assumed that there is a maximum sum that can be put in the first envelope then a very simple and
mathematically sound resolution is possible within the second interpretation. Step 6 in the proposed line of reasoning
is not always true, since if the player holds more than the maximum sum that can be put into the first envelope they
must hold the envelope containing the larger sum and are thus certain to lose by switching. Although this may not
occur often, when it does the heavy loss incurred by the player means that, on average, there is no advantage in
switching. This resolves all practical cases of the problem, whether or not the player looks in their envelope.[11]
Two envelopes problem
It can be envisaged, however, that the sums in the two envelopes are not limited. This requires a more careful
mathematical analysis, and also uncovers other possible interpretations of the problem. If, for example, the smaller
of the two sums of money is considered to be equally likely to be one of infinitely many positive integers, thus
without upper limit, it means that the probability that it will be any given number is always zero. This absurd
situation is an example of what is known as an improper prior and this is generally considered to resolve the paradox
in this case.
It is possible to devise a distribution for the sums possible in the first envelope such that the maximum value is
unlimited, computation of the expectation of what's in B given what's in A seems to dictate you should switch, and
the distribution constitutes a proper prior.[12] In these cases it can be shown that the expected sum in both envelopes
is infinite. There is no gain, on average, in swapping.
The first two resolutions we present correspond, technically speaking, first to A being a random variable, and
secondly to it being a possible value of a random variable (and the expectation being computed is a conditional
expectation). At the same time, in the first resolution the two original amounts of money seem to be thought of as
being fixed, while in the second they are also thought of as varying. Thus there are two main interpretations of the
problem, and two main resolutions.
Proposed resolutions to the alternative interpretation
Nalebuff (1989), Christensen and Utts (1992), Falk and Konold (1992), Blachman, Christensen and Utts (1996),[13]
Nickerson and Falk (2006), pointed out that if the amounts of money in the two envelopes have any proper
probability distribution representing the player's prior beliefs about the amounts of money in the two envelopes, then
it is impossible that whatever the amount A=a in the first envelope might be, it would be equally likely, according to
these prior beliefs, that the second contains a/2 or 2a. Thus step 6 of the argument which leads to always switching is
a non-sequitur.
Mathematical details
According to this interpretation, the writer is carrying out the following computation, where he is conditioning now
on the value of A, the amount in Envelope A, not on the pair amounts in the two envelopes X and Y:
Completely correctly, and according to Step 5, the two conditional expectation values are evaluated as
However in Step 6 the writer is invoking Steps 2 and 3 to get the two conditional probabilities, and effectively
replacing the two conditional probabilities of Envelope A containing the smaller and larger amount, respectively,
given the amount actually in that envelope, both by the unconditional probability 1/2: he makes the substitutions
But intuitively we would expect that the larger the amount in A, the more likely it is to be the larger of the two, and
vice-versa. And it is a mathematical fact, as we will see in a moment, that it is impossible that both of these
conditional probabilities are equal to 1/2 for all possible values of a. In fact, in order for step 6 to be true, whatever a
might be, the smaller amount of money in the two envelopes must be equally likely to be between 1 and 2, as
between 2 and 4, as between 4 and 8, ... ad infinitum. But there is no way to divide total probability 1 into an infinite
number of pieces which are not only all equal to one another, but also all larger than zero. Yet the smaller amount of
money in the two envelopes must have probability larger than zero to be in at least one of the just mentioned ranges.
248
Two envelopes problem
249
To see this, suppose that the chance that the smaller of the two envelopes contains an amount between 2n and 2n+1 is
p(n), where n is any whole number, positive or negative, and for definiteness we include the lower limit but exclude
the upper in each interval. It follows that the conditional probability that the envelope in our hands contains the
smaller amount of money of the two, given that its contents are between 2n and 2n+1, is
If this is equal to 1/2, it follows by simple algebra that
or p(n)=p(n-1). This has to be true for all n, an impossibility.
A new variant
Though Bayesian probability theory can resolve the alternative interpretation of the paradox above, it turns out that
examples can be found of proper probability distributions, such that the expected value of the amount in the second
envelope given that in the first does exceed the amount in the first, whatever it might be. The first such example was
already given by Nalebuff (1989). See also Christensen and Utts (1992)[14]
Denote again the amount of money in the first envelope by A and that in the second by B. We think of these as
random. Let X be the smaller of the two amounts and Y=2X be the larger. Notice that once we have fixed a
probability distribution for X then the joint probability distribution of A,B is fixed, since A,B = X,Y or Y,X each with
probability 1/2, independently of X,Y.
The bad step 6 in the "always switching" argument led us to the finding
for all a, and hence to
the recommendation to switch, whether or not we know a. Now, it turns out that one can quite easily invent proper
probability distributions for X, the smaller of the two amounts of money, such that this bad conclusion is still true!
One example is analysed in more detail, in a moment.
It cannot be true that whatever a, given A=a, B is equally likely to be a/2 or 2a, but it can be true that whatever a,
given A=a, B is larger in expected value than a.
Suppose for example (Broome, 1995)[15] that the envelope with the smaller amount actually contains 2n dollars with
probability 2n/3n+1 where n = 0, 1, 2,… These probabilities sum to 1, hence the distribution is a proper prior (for
subjectivists) and a completely decent probability law also for frequentists.
Imagine what might be in the first envelope. A sensible strategy would certainly be to swap when the first envelope
contains 1, as the other must then contain 2. Suppose on the other hand the first envelope contains 2. In that case
there are two possibilities: the envelope pair in front of us is either {1, 2} or {2, 4}. All other pairs are impossible.
The conditional probability that we are dealing with the {1, 2} pair, given that the first envelope contains 2, is
and consequently the probability it's the {2, 4} pair is 2/5, since these are the only two possibilities. In this
derivation,
is the probability that the envelope pair is the pair 1 and 2, and Envelope A happens to
contain 2;
is the probability that the envelope pair is the pair 2 and 4, and (again) Envelope A
happens to contain 2. Those are the only two ways in which Envelope A can end up containing the amount 2.
It turns out that these proportions hold in general unless the first envelope contains 1. Denote by a the amount we
imagine finding in Envelope A, if we were to open that envelope, and suppose that a = 2n for some n ≥ 1. In that case
the other envelope contains a/2 with probability 3/5 and 2a with probability 2/5.
Two envelopes problem
250
So either the first envelope contains 1, in which case the conditional expected amount in the other envelope is 2, or
the first envelope contains a > 1, and though the second envelope is more likely to be smaller than larger, its
conditionally expected amount is larger: the conditionally expected amount in Envelope B is
which is more than a. This means that the player who looks in Envelope A would decide to switch whatever he saw
there. Hence there is no need to look in Envelope A in order to make that decision.
This conclusion is just as clearly wrong as it was in the preceding interpretations of the Two Envelopes Problem. But
now the flaws noted above don't apply; the a in the expected value calculation is a constant and the conditional
probabilities in the formula are obtained from a specified and proper prior distribution.
Proposed resolutions
Some writers think that the new paradox can be defused.[16] Suppose
for all a. As remarked
before, this is possible for some probability distributions of X (the smaller amount of money in the two envelopes).
Averaging over a, it follows either that
, or alternatively that
. But A and
B have the same probability distribution, and hence the same expectation value, by symmetry (each envelope is
equally likely to be the smaller of the two). Thus both have infinite expectation values, and hence so must X too.
Thus if we switch for the second envelope because its conditional expected value is larger than what actually is in
the first, whatever that might be, we are exchanging an unknown amount of money whose expectation value is
infinite for another unknown amount of money with the same distribution and the same infinite expected value. The
average amount of money in both envelopes is infinite. Exchanging one for the other simply exchanges an average of
infinity with an average of infinity.
Probability theory therefore tells us why and when the paradox can occur and explains to us where the sequence of
apparently logical steps breaks down. In this situation, Steps 6 and Steps 7 of the standard Two Envelopes argument
can be replaced by correct calculations of the conditional probabilities that the other envelope contains half or twice
what's in A, and a correct calculation of the conditional expectation of what's in B given what's in A. Indeed, that
conditional expected value is larger than what's in A. But because the unconditional expected amount in A is infinite,
this does not provide a reason to switch, because it does not guarantee that on average you'll be better off after
switching. One only has this mathematical guarantee in the situation that the unconditional expectation value of
what's in A is finite. But then the reason for switching without looking in the envelope,
for all
a, simply cannot arise.
Many economists prefer to argue that in a real-life situation, the expectation of the amount of money in an envelope
cannot be infinity, for instance, because the total amount of money in the world is bounded; therefore any probability
distribution describing the real world would have to assign probability 0 to the amount being larger than the total
amount of money on the world. Therefore the expectation of the amount of money under this distribution cannot be
infinity. The resolution of the second paradox, for such writers, is that the postulated probability distributions cannot
arise in a real-life situation. These are similar arguments as used to explain the St. Petersburg Paradox.
We certainly have a counter-intuitive situation. If the two envelopes are set up exactly as the Broome recipe requires
(not with real money, but just with numbers), then it is a fact that in many, many repetitions, the average of the
number written on the piece of paper in Envelope B, taken over all those occasions where the number in envelope A
is, say, 4, is definitely larger than 4. And the same thing holds with B and A exchanged! And the same thing holds
with the number 4 replaced by any other possible number! This might be hard to imagine, but we have now shown
how it can be arranged. There is no logical contradiction because we have seen why, logically, there is not
implication that we should switch envelopes. There does remain a conflict with our intuition.
The Broome paradox can be resolved at a purely formal level by showing where the error in the sequence of
apparently logical deductions occurs. But one still is left with a strange situation which simply does not feel right.
Two envelopes problem
One can try to soften the blow by giving real-world reasons why this counter-intuitive situation could not occur in
reality (with real money). As far as practical economics is concerned, we need not worry about the insult to our
intuition.
Foundations of mathematical economics
In mathematical economics and the theory of utility, which explains economic behaviour in terms of expected utility,
there remains a problem to be resolved.[17] In the real world we presumably wouldn't indefinitely exchange one
envelope for the other (and probability theory, as just discussed, explains quite well why calculations of conditional
expectations might mislead us). Yet the expected utility based theory of economic behaviour says (or assumes) that
people do (or should) make economic decisions by maximizing expected utility, conditional on present knowledge,
and hence predicts that people would (or should) switch indefinitely.
Fortunately for mathematical economics and the theory of utility, it is generally agreed that as an amount of money
increases, its utility to the owner increases less and less, and ultimately there is a finite upper bound to the utility of
all possible amounts of money. We can pretend that the amount of money in the whole world is as large as we like,
yet the owner of all that money will not have more and more use of it, the more is in his possession. For decision
theory and utility theory, the two envelope paradox illustrates that unbounded utility does not exist in the real world,
so fortunately there is no need to build a decision theory which allows unbounded utility, let alone utility of infinite
expectation.
Controversy among philosophers
As mentioned above, any distribution producing this variant of the paradox must have an infinite mean. So before
the player opens an envelope the expected gain from switching is "∞ − ∞", which is not defined. In the words of
Chalmers this is "just another example of a familiar phenomenon, the strange behaviour of infinity".[18] Chalmers
suggests that decision theory generally breaks down when confronted with games having a diverging expectation,
and compares it with the situation generated by the classical St. Petersburg paradox.
However, Clark and Shackel argue that this blaming it all on "the strange behaviour of infinity" doesn't resolve the
paradox at all; neither in the single case nor the averaged case. They provide a simple example of a pair of random
variables both having infinite mean but where it is clearly sensible to prefer one to the other, both conditionally and
on average.[19] They argue that decision theory should be extended so as to allow infinite expectation values in some
situations. Most mathematical economists are happy to exclude infinite expected utility by assumption, hence
excluding the paradox altogether. Some try to generalise some of the existing theory to allow infinite expectations.
They have to come up with clever ways to get around the paradoxical example just given.
Non-probabilistic variant
The logician Raymond Smullyan questioned if the paradox has anything to do with probabilities at all. He did this by
expressing the problem in a way which doesn't involve probabilities. The following plainly logical arguments lead to
conflicting conclusions:
1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the
potential gain is strictly greater than the potential loss.
2. Let the amounts in the envelopes be X and 2X. Now by swapping, the player may gain X or lose X. So the
potential gain is equal to the potential loss.
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Two envelopes problem
Proposed resolutions
A number of solutions have been put forward. Careful analyses have been made by some logicians. Though solutions
differ, they all pinpoint semantic issues concerned with counterfactual reasoning. We want to compare the amount
that we would gain by switching if we would gain by switching, with the amount we would lose by switching if we
would indeed lose by switching. However, we cannot both gain and lose by switching at the same time. We are
asked to compare two incompatible situations. Only one of them can factually occur, the other will be a
counterfactual situation, somehow imaginary. In order to compare them at all, we must somehow "align" the two
situations, we must give them some definite points in common.
James Chase (2002) argues that the second argument is correct because it does correspond to the way to align two
situations (one in which we gain, the other in which we lose) which is preferably indicated by the problem
description.[20] Also Bernard Katz and Doris Olin (2007) argue this point of view.[21] In the second argument, we
consider the amounts of money in the two envelopes as being fixed; what varies is which one is first given to the
player. Because that was an arbitrary and physical choice, the counterfactual world in which the player,
counterfactually, got the other envelope to the one he was actually (factually) given is a highly meaningful
counterfactual world and hence the comparison between gains and losses in the two worlds is meaningful. This
comparison is uniquely indicated by the problem description, in which two amounts of money are put in the two
envelopes first, and only after that is one chosen arbitrarily and given to the player. In the first argument, however,
we consider the amount of money in the envelope first given to the player as fixed and consider the situations where
the second envelope contains either half or twice that amount. This would only be a reasonable counterfactual world
if in reality the envelopes had been filled as follows: first, some amount of money is placed in the specific envelope
which will be given to the player; and secondly, by some arbitrary process, the other envelope is filled (arbitrarily or
randomly) either with double or with half of that amount of money.
Byeong-Uk Yi (2009), on the other hand, argues that comparing the amount you would gain if you would gain by
switching with the amount you would lose if you would lose by switching is a meaningless exercise from the
outset.[22] According to his analysis, all three implications (switch, indifferent, don't switch) are incorrect. He
analyses Smullyan's arguments in detail, showing that intermediate steps are being taken, and pinpointing exactly
where an incorrect inference is made according to his formalization of counterfactual inference. An important
difference with Chase's analysis is that he does not take account of the part of the story where we are told that which
envelope is called Envelope A is decided completely at random. Thus Chase puts probability back into the problem
description in order to conclude that arguments 1 and 3 are incorrect, argument 2 is correct, while Yi keeps "two
envelope problem without probability" completely free of probability, and comes to the conclusion that there are no
reasons to prefer any action. This corresponds to the view of Albers et al., that without probability ingredient, there is
no way to argue that one action is better than another, anyway.
In perhaps the most recent paper on the subject, Bliss argues that the source of the paradox is that when one
mistakenly believes in the possibility of a larger payoff that does not, in actuality, exist, one is mistaken by a larger
margin than when one believes in the possibility of a smaller payoff that does not actually exist.[23] If, for example,
the envelopes contained $5.00 and $10.00 respectively, a player who opened the $10.00 envelope would expect the
possibility of a $20.00 payout that simply does not exist. Were that player to open the $5.00 envelope instead, he
would believe in the possibility of a $2.50 payout, which constitutes a smaller deviation from the true value.
Albers, Kooi, and Schaafsma (2005) consider that without adding probability (or other) ingredients to the problem,
Smullyan's arguments do not give any reason to swap or not to swap, in any case. Thus there is no paradox. This
dismissive attitude is common among writers from probability and economics: Smullyan's paradox arises precisely
because he takes no account whatever of probability or utility.
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Two envelopes problem
253
Extensions to the Problem
Since the two envelopes problem became popular, many authors have studied the problem in depth in the situation in
which the player has a prior probability distribution of the values in the two envelopes, and does look in Envelope A.
One of the most recent such publications is by McDonnell and Douglas (2009), who also consider some further
generalizations.[24]
If a priori we know that the amount in the smaller envelope is a whole number of some currency units, then the
problem is determined, as far as probability theory is concerned, by the probability mass function
describing
our prior beliefs that the smaller amount is any number x = 1,2, ... ; the summation over all values of x being equal to
1. It follows that given the amount a in Envelope A, the amount in Envelope B is certainly 2a if a is an odd number.
However, if a is even, then the amount in Envelope B is 2a with probability
, and a/2
with probability
. If one would like to switch envelopes if the expectation value of
what is in the other is larger than what we have in ours, then a simple calculation shows that one should switch if
, keep to Envelope A if
.
If on the other hand the smaller amount of money can vary continuously, and we represent our prior beliefs about it
with a probability density
, thus a function which integrates to one when we integrate over x running from
zero to infinity, then given the amount a in Envelope A, the other envelope contains 2a with probability
, and a/2 with probability
. If again we decide to
switch or not according to the expectation value of what's in the other envelope, the criterion for switching now
becomes
.
The difference between the results for discrete and continuous variables may surprise many readers. Speaking
intuitively, this is explained as follows. Let h be a small quantity and imagine that the amount of money we see when
we look in Envelope A is rounded off in such a way that differences smaller than h are not noticeable, even though
actually it varies continuously. The probability that the smaller amount of money is in an interval around a of length
h, and Envelope A contains the smaller amount is approximately
. The probability that the larger
amount of money is in an interval around a of length h corresponds to the smaller amount being in an interval of
length h/2 around a/2. Hence the probability that the larger amount of money is in a small interval around a of length
h and Envelope A contains the larger amount is approximately
. Thus, given Envelope A
contains an amount about equal to a, the probability it is the smaller of the two is roughly
.
If the player only wants to end up with the larger amount of money, and does not care about expected amounts, then
in the discrete case he should switch if a is an odd number, or if a is even and
. In the continuous
case he should switch if
.
Some authors prefer to think of probability in a frequentist sense. If the player knows the probability distribution
used by the organizer to determine the smaller of the two values, then the analysis would proceed just as in the case
when p or f represents subjective prior beliefs. However, what if we take a frequentist point of view, but the player
does not know what probability distribution is used by the organiser to fix the amounts of money in any one
instance? Thinking of the arranger of the game and the player as two parties in a two person game, puts the problem
into the range of game theory. The arranger's strategy consists of a choice of a probability distribution of x, the
smaller of the two amounts. Allowing the player also to use randomness in making his decision, his strategy is
determined by his choosing a probability of switching
for each possible amount of money a he might see in
Envelope A. In this section we so far only discussed fixed strategies, that is strategies for which q only takes the
values 0 and 1, and we saw that the player is fine with a fixed strategy, if he knows the strategy of the organizer. In
the next section we will see that randomized strategies can be useful when the organizer's strategy is not known.
Two envelopes problem
Randomized solutions
Suppose as in the previous section that the player is allowed to look in the first envelope before deciding whether to
switch or to stay. We'll think of the contents of the two envelopes as being two positive numbers, not necessarily two
amounts of money. The player is allowed either to keep the number in Envelope A, or to switch and take the number
in Envelope B. We'll drop the assumption that one number is exactly twice the other, we'll just suppose that they are
different and positive. On the other hand, instead of trying to maximize expectation values, we'll just try to maximize
the chance that we end up with the larger number.
In this section we ask the question, is it possible for the player to make his choice in such a way that he goes home
with the larger number with probability strictly greater than half, however the organizer has filled the two envelopes?
We are given no information at all about the two numbers in the two envelopes, except that they are different, and
strictly greater than zero. The numbers were written down on slips of paper by the organiser, put into the two
envelopes. The envelopes were then shuffled, the player picks one, calls it Envelope A, and opens it.
We are not told any joint probability distribution of the two numbers. We are not asking for a subjectivist solution.
We must think of the two numbers in the envelopes as chosen by the arranger of the game according to some
possibly random procedure, completely unknown to us, and fixed. Think of each envelope as simply containing a
positive number and such that the two numbers are not the same. The job of the player is to end up with the envelope
with the larger number. This variant of the problem, as well as its solution, is attributed by McDonnell and Abbott,
and by earlier authors, to information theorist Thomas M. Cover.[25]
Counter-intuitive though it might seem, there is a way that the player can decide whether to switch or to stay so that
he has a larger chance than 1/2 of finishing with the bigger number, however the two numbers are chosen by the
arranger of the game. However, it is only possible with a so-called randomized algorithm, that means to say, the
player needs himself to be able to generate random numbers. Suppose he is able to think up a random number, let's
call it Z, such that the probability that Z is larger than any particular quantity z is exp(-z). Note that exp(-z) starts off
equal to 1 at z=0 and decreases strictly and continuously as z increases, tending to zero as z tends to infinity. So the
chance is 0 that Z is exactly equal to any particular number, and there is a positive probability that Z lies between any
two particular different numbers. The player compares his Z with the number in Envelope A. If Z is smaller he keeps
the envelope. If Z is larger he switches to the other envelope.
Think of the two numbers in the envelopes as fixed (though of course unknown to the player). Think of the player's
random Z as a probe with which he decides whether the number in Envelope A is small or large. If it is small
compared to Z he will switch, if it is large compared to Z he will stay.
If both numbers are smaller than the player's Z then his strategy does not help him, he ends up with the Envelope B,
which is equally likely to be the larger or the smaller of the two. If both numbers are larger than Z his strategy does
not help him either, he ends up with the first Envelope A, which again is equally likely to be the larger or the smaller
of the two. However if Z happens to be in between the two numbers, then his strategy leads him correctly to keep
Envelope A if its contents are larger than those of B, but to switch to Envelope B if A has smaller contents than B.
Altogether, this means that he ends up with the envelope with the larger number with probability strictly larger than
1/2. To be precise, the probability that he ends with the "winning envelope" is 1/2 + P(Z falls between the two
numbers)/2.
In practice, the number Z we have described could be determined to the necessary degree of accuracy as follows.
Toss a fair coin many times, and convert the sequence of heads and tails into the binary representation of a number U
between 0 and 1: for instance, HTHHTH... becomes the binary representation of u=0.101101.. . In this way, we
generate a random number U, uniformly distributed between 0 and 1. Then define Z = - ln (U) where "ln" stands for
natural logarithm, i.e., logarithm to base e. Note that we just need to toss the coin long enough to be able to see for
sure whether Z is smaller or larger than the number a in the first envelope, we do not need to go on for ever. We will
only need to toss the coin a finite (though random) number of times: at some point we can be sure that the outcomes
254
Two envelopes problem
of further coin tosses is not going to change the outcome of the comparison.
The particular probability law (the so-called standard exponential distribution) used to generate the random number
Z in this problem is not crucial. Any probability distribution over the positive real numbers which assigns positive
probability to any interval of positive length will do the job.
This problem can be considered from the point of view of game theory, where we make the game a two-person
zero-sum game with outcomes win or lose, depending on whether the player ends up with the higher or lower
amount of money. The organiser chooses the joint distribution of the amounts of money in both envelopes, and the
player chooses the distribution of Z. The game does not have a "solution" (or saddle point) in the sense of game
theory. This is an infinite game and von Neumann's minimax theorem does not apply.[26]
History of the paradox
The envelope paradox dates back at least to 1953, when Belgian mathematician Maurice Kraitchik proposed a puzzle
in his book Recreational Mathematics concerning two equally rich men who meet and compare their beautiful
neckties, presents from their wives, wondering which tie actually cost more money. It is also mentioned in a 1953
book on elementary mathematics and mathematical puzzles by the mathematician John Edensor Littlewood, who
credited it to the physicist Erwin Schroedinger. Martin Gardner popularized Kraitchik's puzzle in his 1982 book Aha!
Gotcha, in the form of a wallet game:
Two people, equally rich, meet to compare the contents of their wallets. Each is ignorant of the contents of the
two wallets. The game is as follows: whoever has the least money receives the contents of the wallet of the
other (in the case where the amounts are equal, nothing happens). One of the two men can reason: "I have the
amount A in my wallet. That's the maximum that I could lose. If I win (probability 0.5), the amount that I'll
have in my possession at the end of the game will be more than 2A. Therefore the game is favourable to me."
The other man can reason in exactly the same way. In fact, by symmetry, the game is fair. Where is the
mistake in the reasoning of each man?
In 1988 and 1989, Barry Nalebuff presented two different two-envelope problems, each with one envelope
containing twice what's in the other, and each with computation of the expectation value 5A/4. The first paper just
presents the two problems, the second paper discusses many solutions to both of them. The second of his two
problems is the one which is nowadays the most common and which is presented in this article. According to this
version, the two envelopes are filled first, then one is chosen at random and called Envelope A. Martin Gardner
independently mentioned this same version in his 1989 book Penrose Tiles to Trapdoor Ciphers and the Return of
Dr Matrix. Barry Nalebuff's asymmetric variant, often known as the Ali Baba problem, has one envelope filled first,
called Envelope A, and given to Ali. Then a coin is tossed to decide whether Envelope B should contain half or twice
that amount, and only then given to Baba.
In the Ali-Baba problem, it is a priori clear that (even if they don't look in their envelopes) Ali should want to switch,
while Baba should want to keep what he has been given. The Ali-Baba paradox comes about by imagining Baba
working through the steps of the two-envelopes problem argument (second interpretation), and wrongly coming to
the conclusion that he too wants to switch, just like Ali.
255
Two envelopes problem
Notes and references
[1] Markosian, Ned (2011). "A Simple Solution to the Two Envelope Problem". Logos & Episteme II (3): 347-357.
[2] McDonnell, Mark D; Grant, Alex J.; Land, Ingmar; Vellambi, Badri N.; Abbott, Derek; Lever, Ken (2011). "Gain from the two-envelope
problem via information asymmetry: on the suboptimality of randomized switching". Proceedings of the Royal Society, A to appear.
doi:10.1098/rspa.2010.0541.
[3] A complete list of published and unpublished sources in chronological order can be found here
[4] Falk, Ruma (2008). "The Unrelenting Exchange Paradox". Teaching Statistics 30 (3): 86–88. doi:10.1111/j.1467-9639.2008.00318.x.
[5] Eckhardt, William (2013). "The Two-Envelopes Problem". Paradoxes in Probability Theory. Springer. pp. 47–48.
[6] Bruss, F.T. (1996). "The Fallacy of the Two Envelopes Problem". The Mathematical Scientist 21 (2): 112–119..
[7] Falk, Ruma (2009). "An inside look at the two envelope paradox". Teaching Statistics 31 (2): 39–41..
[8] Casper Albers, Trying to resolve the two-envelope problem, Chapter 2 of his thesis Distributional Inference: The Limits of Reason, March
2003. (Has also appeared as Albers, Casper J.; Kooi, Barteld P. and Schaafsma, Willem (2005), Trying to resolve the two-envelope problem,
Synthese, 145(1): 89–109 p91)
[9] Ruma Falk, Raymond Nickerson, An inside look at the two envelopes paradox, Teaching Statistics 31(2): 39-41.
[10] Jeff Chen, The Puzzle of the Two-Envelope Puzzle—a Logical Approach, published online p274
[11] Barry Nalebuff, Puzzles: The Other Person’s Envelope is Always Greener, Journal of Economic Perspectives 3(1): 171–181.
[12] John Broome, The Two-envelope Paradox, Analysis 55(1): 6–11.
[13] Blachman, N. M.; Christensen, R.; Utts, J. (1996). The American Statistician 50 (1): 98–99.
[14] Christensen, R.; Utts, J. (1992). The American Statistician 46 (4): 274–276., the letters to editor and responses by Christensen and Utts by
D.A. Binder (1993; vol. 47, nr. 2, p. 160) and Ross (1994; vol. 48, nr. 3, p. 267), and a letter with corrections to the original article by N.M.
Blachman, R. Christensen and J.M. Utts (1996; vol. 50, nr. 1, pp. 98-99)
[15] Here we present some details of a famous example due to John Broome of a proper probability distribution of the amounts of money in the
two envelopes, for which
for all a. Broome, John (1995). "The Two-envelope Paradox". Analysis 55 (1): 6–11.
doi:10.1093/analys/55.1.6.
[16] Binder, D. A. (1993). The American Statistician 47 (2): 160. (letters to the editor, comment on Christensen and Utts (1992)
[17] Fallis, D. (2009). "Taking the Two Envelope Paradox to the Limit". Southwest Philosophy Review 25 (2).
[18] Chalmers, David J. (2002). "The St. Petersburg Two-Envelope Paradox". Analysis 62 (2): 155–157. doi:10.1093/analys/62.2.155.
[19] Clark, M.; Shackel, N. (2000). "The Two-Envelope Paradox". Mind 109 (435): 415–442. doi:10.1093/mind/109.435.415.
[20] Chase, James (2002). "The Non-Probabilistic Two Envelope Paradox". Analysis 62 (2): 157–160. doi:10.1093/analys/62.2.157.
[21] Katz, Bernard; Olin, Doris (2007). "A tale of two envelopes". Mind 116 (464): 903–926. doi:10.1093/mind/fzm903.
[22] Byeong-Uk Yi (2009). The Two-envelope Paradox With No Probability (http:/ / philosophy. utoronto. ca/ people/ linked-documents-people/
c two envelope with no probability. pdf). .
[23] Bliss (2012). A Concise Resolution to the Two Envelope Paradox (http:/ / arxiv. org/ abs/ 1202. 4669). .
[24] McDonnell, M. D.; Abott, D. (2009). "Randomized switching in the two-envelope problem". Proceedings of the Royal Society A 465 (2111):
3309–3322. doi:10.1098/rspa.2009.0312.
[25] Cover, Thomas M.. "Pick the largest number". In Cover, T.; Gopinath, B.. Open Problems in Communication and Computation.
Springer-Verlag.
[26] http:/ / www. mit. edu/ ~emin/ writings/ envelopes. html
256
Von Neumann paradox
Von Neumann paradox
In mathematics, the von Neumann paradox, named after John von Neumann, is the idea that one can break a planar
figure such as the unit square into sets of points and subject each set to an area-preserving affine transformation such
that the result is two planar figures of the same size as the original. This was proved in 1929 by John von Neumann,
assuming the axiom of choice. It is based on the earlier Banach–Tarski paradox which is in turn based on the
Hausdorff paradox.
Banach and Tarski had proved that, using isometric transformations, the result of taking apart and reassembling a
two-dimensional figure would necessarily have the same area as the original. This would make creating two unit
squares out of one impossible. But von Neumann realized that the trick of such so-called paradoxical decompositions
was the use of a group of transformations which include as a subgroup a free group with two generators. The group
of area preserving transformations (whether the special linear group or the special affine group) contains such
subgroups, and this opens the possibility of performing paradoxical decompositions using them.
Sketch of the method
The following is an informal description of the method found by von Neumann. Assume that we have a free group H
of area-preserving linear transformations generated by two transformations, σ and τ, which are not far from the
identity element. Being a free group means that all the its elements can be expressed uniquely in the form
for some n, where the 's and 's are all non-zero integers, except possiblly the first
u and the last v. We can divide this group into two parts: those that start on the left with σ to some non-zero power
(let's call this set A) and those that start with τ to some power (that is,
is zero—let's call this set B, and it includes
the identity).
If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that
point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the
cardinality of the continuum. According to the axiom of choice, we can choose one point from each orbit and call the
set of these points M. We exclude the origin, which is a fixed point in H. If we then operate on M by all the elements
of H, we generate each point of the plane (except the origin) exactly once. If we operate on M by all the elements of
A or of B, we get two disjoint sets whose union is all points but the origin.
Now let us take some figure such as the unit square or the unit disk. We then choose another figure totally inside it,
such as a smaller square, centred at the origin. We can cover the big figure with several copies of the small figure,
albeit with some points covered by two or more copies. We can then assign each point of the big figure to one of the
copies of the small figure. Let us call the sets corresponding to each copy
. We shall now make a
one-to-one mapping of each point in the big figure to a point in its interior, using only area-preserving
transformations! We take the points belonging to
and translate them so that the centre of the
square is at the
origin. We then take those points in it which are in the set A defined above and operate on them by the
area-preserving operation σ τ. Note that this puts them into set B. We then take the points belonging to B and operate
on them with σ2. They will now still be in B, but the set of these points will be disjoint from the previous set. We
proceed in this manner, using σ3τ on the A points from C2 (after centring it) and σ4 on its B points, and so on. In this
way, we have mapped all points from the big figure (except some fixed points) in a one-to-one manner to B type
points not too far from the centre, and within the big figure. We can then make a second mapping to A type points.
At this point we can apply the method of the Cantor-Bernstein-Schroeder theorem. This theorem tells us that if we
have an injection from set D to set E (such as from the big figure to the A type points in it), and an injection from E
to D (such as the identity mapping from the A type points in the figure to themselves), then there is a one-to-one
correspondence between D and E. In other words, having a mapping from the big figure to a subset of the A points in
it, we can make a mapping (a bijection) from the big figure to all the A points in it. (In some regions points are
257
Von Neumann paradox
mapped to themselves, in others they are mapped using the mapping described in the previous paragraph.) Likewise
we can make a mapping from the big figure to all the B points in it. So looking at this the other way round, we can
separate the figure into its A and B points, and then map each of these back into the whole figure (that is, containing
both kinds of points)!
We have glossed over some things, like how to handle fixed points. It turns out that we have to use more mappings
and more sets to work around this.
Consequences
The paradox for the square can be strengthened as follows:
Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect
to the area-preserving affine maps.
This has consequences concerning the problem of measure. As von Neumann notes,
"Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das
Maß 1 hat), dass [sic] gegenüber allen Abbildungen von A2 invariant wäre."[1]
"In accordance with this, already in the plane there is no nonnegative additive measure (for which the unit
square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of
area-preserving affine transformations]."
To explain this a bit more, the question of whether a finitely additive measure exists, that is preserved under certain
transformations, depends on what transformations are allowed. The Banach measure of sets in the plane, which is
preserved by translations and rotations, is not preserved by non-isometric transformations even when they do
preserve the area of polygons. As explained above, the points of the plane (other than the origin) can be divided into
two dense sets which we may call A and B. If the A points of a given polygon are transformed by a certain
area-preserving transformation and the B points by another, both sets can become subsets of the B points in two new
polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the
same measure as before (since they contain only part of the B points), and therefore there is no measure that "works".
The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be
very important for many areas of mathematics: these are amenable groups, or groups with an invariant mean, and
include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used
for equivalences in the definition of equidecomposability is not amenable.
Recent progress
Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with
respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a
decomposition exists.[2] More precisely, let A be the family of all bounded subsets of the plane with non-empty
interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of
finitely many translates under some elements of SL(2,R) contains a punctured neighbourhood of the origin. Then all
sets in the family A are SL(2,R)-equidecomposable, and likewise for the sets in B. It follows that both families
consist of paradoxical sets.
258
Von Neumann paradox
References
[1] On p. 85 of: von Neumann, J. (1929), "Zur allgemeinen Theorie des Masses" (http:/ / matwbn. icm. edu. pl/ ksiazki/ fm/ fm13/ fm1316. pdf),
Fundamenta Mathematica 13: 73–116,
[2] Laczkovich, Miklós (1999), "Paradoxical sets under SL2[R]", Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42: 141–145
259
260
Miscellaneous
Bracketing paradox
In linguistic morphology, the term bracketing paradox refers to morphologically complex words which apparently
have more than one incompatible analysis, or bracketing, simultaneously.
One type of a bracketing paradox found in English is exemplified by words like unhappier or uneasier.[1] The
synthetic comparative suffix -er generally occurs with monosyllabic adjectives and a small class of disyllabic
adjectives with the primary (and only) stress on the first syllable. Other adjectives take the analytic comparative
more. Thus, we have older and grumpier, but more correct and more restrictive. This suggests that a word like
uneasier must be formed by combining the suffix er with the adjective easy, since uneasy is a three syllable word:
However, uneasier means "more uneasy", not "more difficult". Thus, from a semantic perspective, uneasier must be
a combination of er with the adjective uneasy:
however violates the morphophonological rules for the suffix -er. Phenomena such as this have been argued to
represent a mismatch between different levels of grammatical structure.[2]
Another type of English bracketing paradox is found in compound words that are a name for a professional of a
particular discipline, preceded by a modifier that narrows that discipline: nuclear physicist, historical linguist,
political scientist, etc.[3][4] Taking nuclear physicist as an example, we see that there are at least two reasonable ways
that the compound word can be bracketed (ignoring the fact that nuclear itself is morphologically complex):
1.
- one who studies physics, and who happens also to be nuclear
2.
- one who studies nuclear physics, a subfield of physics that deals with nuclear
phenomena
What is interesting to many morphologists about this type of bracketing paradox in English is that the correct
bracketing 2 (correct in the sense that this is the way that a native speaker would understand it) does not follow the
usual bracketing pattern 1 typical for most compound words in English.
References
[1] Pesetsky, D. 1985. "Morphology and logical form." Linguistic Inquiry 16:193-246.
[2] Sproat, R. 1988. "Bracketing paradoxes, cliticization, and other topics: The mapping between syntactic and phonological structure." In
Everaert et al. (eds), Morphology and Modularity. Amsterdam: North-Holland.
[3] Williams, E. 1981. "On the notions 'lexically related' and 'head of a word.'" Linguistic Inquiry 12:245-274.
[4] Spencer, A. 1988. "Bracketing paradoxes and the English lexicon." Language 64:663-682.
Buridan's ass
261
Buridan's ass
Buridan's ass is an illustration of a paradox in philosophy in the
conception of free will.
It refers to a hypothetical situation wherein an ass that is equally
hungry and thirsty is placed precisely midway between a stack of hay
and a pail of water. Since the paradox assumes the ass will always go
to whichever is closer, it will die of both hunger and thirst since it
cannot make any rational decision to choose one over the other.[1] The
paradox is named after the 14th century French philosopher Jean
Buridan, whose philosophy of moral determinism it satirizes. A
common variant of the paradox substitutes two identical piles of hay
for the hay and water; the ass, unable to choose between the two, dies
of hunger.
History
Political cartoon ca. 1900, showing the United
States Congress as Buridan's ass, hesitating
between a Panama route or a Nicaragua route for
an Atlantic-Pacific canal.
The paradox predates Buridan; it dates to antiquity, being found in Aristotle's On the Heavens.[2] Aristotle, in
ridiculing the Sophist idea that the Earth is stationary simply because it is circular and any forces on it must be equal
in all directions, says that is as ridiculous as saying that[2]
...a man, being just as hungry as thirsty, and placed in between food and drink, must necessarily remain where
he is and starve to death.
— Aristotle, On the Heavens, ca.350 BCE
However, the Greeks only used this paradox as an analogy in the context of discussions of the equilibrium of
physical forces.[2]
The 12th century Persian Islamic scholar and philosopher Al-Ghazali discusses the application of this paradox to
human decision making, asking whether it is possible to make a choice between equally good courses without
grounds for preference.[2] He takes the attitude that free will can break the stalemate.
Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them
both. Surely he will take one of them, through a quality in him, the nature of which is to differentiate between
two similar things.
— Abu Hamid al-Ghazali,The Incoherence of the Philosophers 1100[3]
Moorish Islamic philosopher Averroes (1126-1198), in commentary on Ghazali, takes the opposite view.[2]
Although Buridan nowhere discusses this specific problem, its relevance is that he did advocate a moral determinism
whereby, save for ignorance or impediment, a human faced by alternative courses of action must always choose the
greater good. In the face of equally good alternatives Buridan believed a rational choice could not be made
Should two courses be judged equal, then the will cannot break the deadlock, all it can do is to suspend
judgement until the circumstances change, and the right course of action is clear.
— Jean Buridan, 1340
Later writers satirised this view in terms of an ass which, confronted by both food and water must necessarily die of
both hunger and thirst while pondering a decision.
Buridan's ass
262
Discussion
Some proponents of hard determinism have granted the unpleasantness of the scenario, but have denied that it
illustrates a true paradox, since one does not contradict oneself in suggesting that a man might die between two
equally plausible routes of action. For example, Baruch Spinoza in his Ethics, suggests that a person who sees two
options as truly equally compelling cannot be fully rational:
[I]t may be objected, if man does not act from free will, what will happen if the incentives to action are equally
balanced, as in the case of Buridan's ass? [In reply,] I am quite ready to admit, that a man placed in the
equilibrium described (namely, as perceiving nothing but hunger and thirst, a certain food and a certain drink,
each equally distant from him) would die of hunger and thirst. If I am asked, whether such a one should not
rather be considered an ass than a man; I answer, that I do not know, neither do I know how a man should be
considered, who hangs himself, or how we should consider children, fools, madmen, &c.
— Baruch Spinoza, Ethics, Book 2, Scholium, 1677
Other writers have opted to deny the validity of the illustration. A typical counter-argument is that rationality as
described in the paradox is so limited as to be a straw man version of the real thing, which does allow the
consideration of meta-arguments. In other words, it is entirely rational to recognize that both choices are equally
good and arbitrarily (randomly) pick one instead of starving. This counter-argument is sometimes used as an
attempted justification for faith or intuitivity (called by Aristotle noetic or noesis). The argument is that, like the
starving ass, we must make a choice in order to avoid being frozen in endless doubt. Other counter-arguments exist.
Buridan's principle
The situation of Buridan's ass was given a mathematical basis in a 1984 paper by American computer scientist Leslie
Lamport, in which Lamport presents an argument that, given certain assumptions about continuity in a simple
mathematical model of the Buridan's ass problem, there will always be some starting conditions under which the ass
will starve to death, no matter what strategy it takes.
Lamport calls this result Buridan’s principle, and states it as:
A discrete decision based upon an input having a continuous range of values cannot be made within a bounded
length of time.[4]
Application to digital logic: metastability
A version of Buridan's principle actually occurs in electrical engineering. Specifically, the input to a digital logic
gate must convert a continuous voltage value into either a 0 or a 1 which is typically sampled and then processed. If
the input is changing and at an intermediate value when sampled, the input stage acts like a comparator. The voltage
value can then be likened to the position of the ass, and the values 0 and 1 represent the bales of hay. Like the
situation of the starving ass, there exists an input on which the converter cannot make a proper decision, resulting in
a metastable state. Having the converter make an arbitrary choice in ambiguous situations does not solve the
problem, as the boundary between ambiguous values and unambiguous values introduces another binary decision
with its own metastable state.
Buridan's ass
References
[1] "Buridan's ass: Oxford Companion to Phrase and Fable" (http:/ / www. encyclopedia. com/ doc/ 1O214-Buridansass. html).
Encyclopedia.com. . Retrieved 2009-12-15.
[2] Rescher, Nicholas (2005). Cosmos and Logos: Studies in Greek Philosophy (http:/ / books. google. com/ books?id=qU3MdvVlvpAC&
pg=PA89& lpg=PA89& dq=aristotle+ "buridan's+ ass"& source=bl& ots=4PONJwhdOt& sig=5CqJklNpF8cLbfBWEjN1DxWbUNY&
hl=en& sa=X& ei=7Ek_UPeVBYODiwLG-ICICA& ved=0CEcQ6AEwBA#v=onepage& q=aristotle "buridan's ass"& f=false). Ontos Verlag.
pp. 93-99. ISBN 393720265X. .
[3] Kane, Robert (2005). A Contemporary Introduction to Free Will. New York: Oxford. pp. 37.
[4] Leslie Lamport (December 1984). "Buridan's Principle" (http:/ / research. microsoft. com/ users/ lamport/ pubs/ buridan. pdf). . Retrieved
2010-07-09.
Bibliography
•
•
•
•
The Columbia Encyclopedia (6th ed.). 2006.
Knowles, Elizabeth (2006). The Oxford Dictionary of Phrase and Fable.
Mawson, T.J. (2005). Belief in God. New York, NY: Oxford University (Clarendon) Press. p. 201.
Rescher, Nicholas (1959/60). "Choice Without Preference: A Study of the History and of the Logic of the
Problem of “Buridan’s Ass”". Kant-Studien 51: 142–75.
• Zupko, Jack (2003). John Buridan: Portrait of a Fourteenth-Century Arts Master. Notre Dame, Indiana:
University of Notre Dame Press. pp. 258, 400n71.
• E. Ullmann-Margalit and S. Morgenbesser, "Picking and Choos- ing," Social Research, XLIV (1977), 757-785.
External links
• Vassiliy Lubchenko (August 2008). "Competing interactions create functionality through frustration" (http://
www.pnas.org/content/105/31/10635.full). Proc. Natl. Acad. Sci. U.S.A. 105 (31): 10635–6.
doi:10.1073/pnas.0805716105. PMC 2504771. PMID 18669666.
• Definition of term at wordsmith.org (http://wordsmith.org/words/buridans_ass.html)
263
Buttered cat paradox
264
Buttered cat paradox
The buttered cat paradox is a common joke based on the tongue-in-cheek
combination of two adages:
• Cats always land on their feet.
• Buttered toast always lands buttered side down.
The paradox arises when one considers what would happen if one attached a
piece of buttered toast (butter side up) to the back of a cat, then dropped the cat
from a large height. The buttered cat paradox, submitted by artist John Frazee
of Kingston, New York, won a 1993 OMNI magazine competition about
paradoxes.[1]
Thought experiments
Some people jokingly maintain that the experiment will produce an anti-gravity
effect. They propose that as the cat falls towards the ground, it will slow down
and start to rotate, eventually reaching a steady state of hovering a short
distance from the ground while rotating at high speed as both the buttered side
of the toast and the cat’s feet attempt to land on the ground.[2] In June 2003,
Kimberly Miner won a Student Academy Award for her film Perpetual
Motion.[3] Miner based her film on a paper written by a high-school friend that
explored the potential implications of the cat and buttered toast idea.[4][5]
In humor
A cartoon illustration of the thought
experiment.
The faux paradox has captured the imagination of science-oriented humorists.
Testing the theory is the main theme in an episode of the comic book strip Jack
B. Quick, the title character seeks to test this theory, leading to the cat hovering above the ground, with the cat's
wagging tail providing propulsion. The March 31, 2005, strip of the webcomic Bunny also explored the idea in the
guise of a plan for a "Perpetual Motion MoggieToast 5k Power Generator", based on Sod's Law.[6] In Science Askew,
Donald E. Simanek comments on this phenomenon.[7]
The idea appeared on the British panel game QI, where the idea was discussed. As well as talking about the idea,
they also brought up other questions regarding the paradox. These included "Would it still work if you used
margarine?", "Would it still work if you used I Can't Believe It's Not Butter?", and "What if the toast was covered in
something that was not butter, but the cat thought it was butter?", the idea being that it would act like a placebo.[8]
The paradox also appeared in the episode "Gravitational Anarchy" of the scientific podcast RadioLab.[9] Later, a
humoristic explainer animation[10] was put together by the animated production company Barq,[11] based on an
extracted audio clip from this very RadioLab episode.
Brazilian energy drink brand Flying Horse has released a commercial that simulates the recreation of this
phenomenon.[12]
Buttered cat paradox
Cat righting reflex
In reality, cats do possess the ability to turn themselves right side up in mid-air if they should fall upside-down,
known as the cat righting reflex, which enables them to land on their feet if dropped from sufficient height (about
30cm). A similar ability has not been reported for buttered toast.
References
[1]
[2]
[3]
[4]
[5]
Morris, Scot (July, 1993). "I have a theory..." (http:/ / www. aps. org/ publications/ apsnews/ 200111/ letters. cfm). Omni 15 (9): 96. .
"UoWaikato newsletter" (http:/ / www. waikato. ac. nz/ fmd/ newsletter/ Newsletter_No14. pdf) (PDF). . Retrieved 2012-06-20.
Available at http:/ / www. kminer. net/ 2011/ 07/ perpetual-motion/
PG Klein. University of Leeds. Perpetual Motion (http:/ / www. physics. leeds. ac. uk/ pages/ PerpetualMotion).
"Oscar Nominated Animated Shorts 2003" (http:/ / www. scifidimensions. com/ Apr04/ oscaranimation. htm). Scifidimensions.com. .
Retrieved 2012-06-20.
[6] "Feline cunning and sods law" (http:/ / www. bunny. frozenreality. co. uk/ 211. html). Bunny.frozenreality.co.uk. . Retrieved 2012-06-20.
[7] Simanek, Donald E.; Holden, John C. (2002). Science askew: a light-hearted look at the scientific world (http:/ / books. google. com/
?id=ldX0FkgurzoC& pg=PA201). CRC Press. p. 201. ISBN 978-0-7503-0714-7. . Retrieved 30 June 2010.
[8] "Hypothetical" (http:/ / www. comedy. co. uk/ guide/ tv/ qi/ episodes/ 8/ 8/ ). QI. episode 8. series H. 5 November 2010. BBC. BBC One. .
[9] "The RadioLab podcast" (http:/ / itunes. apple. com/ podcast/ wnycs-radiolab/ id152249110). Itunes.apple.com. . Retrieved 2012-06-20.
[10] "The Cat and Jelly Toast Experiment aka the Buttered Cat Paradox explainer" (http:/ / www. youtube. com/ watch?v=y7Is22CmY_Y).
Youtube.com. 2012-04-25. . Retrieved 2012-06-20.
[11] www.barqvideo.com (http:/ / www. barqvideo. com)
[12] http:/ / www. youtube. com/ watch?feature=player_embedded& v=Z8yW5cyXXRc
External links
• Frazee Fine Arts (http://www.frazeefinearts.com) Website of Teresa and John Frazee.
• "Feedback" (http://www.newscientist.com/article/mg15220568.300). New Scientist (2056). 16 November
1996.
• Loopholes for the paradox (http://www.xs4all.nl/~jcdverha/scijokes/2_21.html#subindex)
265
Lombard's Paradox
Lombard's Paradox
Lombard's Paradox describes a paradoxical muscular contraction in humans. When rising to stand from a sitting or
squatting position, both the hamstrings and quadriceps contract at the same time, despite their being antagonists to
each other.
The rectus femoris biarticular muscle acting over the hip has a smaller hip moment arm than the hamstrings.
However, the rectus femoris moment arm is greater over the knee than the hamstring knee moment. This means that
contraction from both rectus femoris and hamstrings will result in hip and knee extension. Hip extension also adds a
passive stretch component to rectus femoris, which results in a knee extension force. This paradox allows for
efficient movement, especially during gait.
References
• Lombard, W.P., & Abbott, F.M. (1907). The mechanical effects produced by the contraction of individual
muscles of the thigh of the frog. American Journal of Physiology, 20, 1-60.
• http://moon.ouhsc.edu/dthompso/namics/lombard.htm
External links
• Andrews JG (1987). "The functional roles of the hamstrings and quadriceps during cycling: Lombard's Paradox
revisited". J Biomech 20 (6): 565–75. doi:10.1016/0021-9290(87)90278-8. PMID 3611133.
• Gregor RJ, Cavanagh PR, LaFortune M (1985). "Knee flexor moments during propulsion in cycling--a creative
solution to Lombard's Paradox". J Biomech 18 (5): 307–16. doi:10.1016/0021-9290(85)90286-6. PMID 4008501.
Mere addition paradox
The mere addition paradox is a problem in ethics, identified by Derek Parfit, and appearing in his book, Reasons
and Persons (1984). The paradox identifies apparent inconsistency between three seemingly true beliefs about
population ethics by arguing that utilitarianism leads to an apparent overpopulated world with minimal individual
happiness.
The paradox
The paradox arises from consideration of four different possibilities. The following diagrams show different
situations, with each bar representing a population. The group's size is represented by column width, and the group's
happiness represented by column height. For simplicity, in each group of people represented, everyone in the group
has exactly the same level of happiness, though Parfit did not consider this essential to the argument.
266
Mere addition paradox
In situation A, everyone is happy.
In situation A+, there are the extra people. There is the same population as in A, and another population of the same
size, which is less happy, but whose lives are nevertheless worth living. The two populations are entirely separate,
that is, they cannot communicate and are not even aware of each other. Parfit gives the example of A+ being a
possible state of the world before the Atlantic was crossed, and says that A, in that case, represents an alternative
history in which the Americas had never been inhabited by any humans.
In situation B-, there are again two separate populations, of the same size as before, but now of equal happiness. The
increase in happiness of the right-hand population is greater than the decrease in happiness of the left-hand
population. Therefore, average happiness in B- is higher.
Finally in situation B, there is a single population whose size is the sum of the two populations in situation B-, and at
the same level of happiness.
Going from A to B
The difference between the situations A and A+ is only in the existence of extra people at a lower level of happiness.
If the two populations were known to each other, and were aware of the inequality, this would arguably constitute
social injustice. However, as they do not, Parfit says this represents a Mere Addition, and it seems implausible to him
that it would be worse for the extra people to exist. Thus, he argues, A+ is not worse than A.
Furthermore, there are the same numbers of people on both sides of the divide in situation B- as there are in situation
A+. The average happiness in B- is higher than A+ (though lower than A). Since A+ and B- have the same number
of people, and because there is a greater level of equality and average happiness in B-, it seems that, all things
considered, B- is better than A+. But the situations B- and B are the same, except the communication gap is
removed. It seems that B is at least as good as B-.
267
Mere addition paradox
The Repugnant Conclusion
We can then repeat the argument, and imagine another divide, and ask if it would be better for more extra people to
exist, unknown to the people in B, and so on, as before. We then arrive at a situation C, in which the population is
even larger, and less happy, though still with lives worth living. And if we agree that B is not worse than A, then we
would conclude in the same fashion that C is not worse than B. But then we could repeat that argument again, finally
arriving at a situation Z, in which there are an enormous number of people whose lives are worth living, but just
barely. Parfit calls this the Repugnant Conclusion, and says that Z is in fact worse than A, and that it goes against
what he believes about overpopulation to say otherwise. This is a contradiction, but it is not clear how to avoid it.
Criticisms and responses
Some scholars, such as Larry Temkin, argue that the paradox is resolved by the conclusion that the "better than"
relation is not transitive, meaning that our assertion that B- is better than A by way of A+ is not justified—it could
very well be the case that B- is better than A+, and A+ is better than A, and yet A is better than B-. The paradox is
defeated, it is argued, by asserting that A+ is actually worse than A, in other words, that adding people of
less-than-average happiness into the world makes the overall situation worse. This is the conclusion of "average
utilitarianism", which aims at maximizing average happiness. However, this solution may commit one to the position
that it is actually bad for people of less than average happiness to be born, even if their lives are worth living.
Another position argues for some threshold above the level at which lives become worth living, but below which
additional lives would nonetheless make the situation worse. Parfit argues that for this position to be plausible, such
a threshold would be so low as to apply only to lives that are "gravely deficient" and which, "though worth living ...
must be crimped and mean." Parfit calls this hypothetical threshold the "bad level," and argues that its existence
would not resolve the paradox because population A would still be better than an enormous population with all
members having lives at the "bad level."
Torbjörn Tännsjö argues that we have a false intuition of the moral weight of billions upon billions of lives "barely
worth living". He argues that we must consider that life in Z would not be terrible, and that in our actual world, most
lives are actually not far above, and often fall below, the level of "not worth living". The Repugnant Conclusion
therefore is not repugnant.
Another criticism is that the paradox only considers utility at one point in time, instead of taking into account the
population's sum utility over time. While mere addition may increase the present value of net utility, it may damage
the sum-over-time of utility, such as by consuming extra resources that will be unavailable for future generations.
268
Mere addition paradox
Also, it has been argued that the conclusion is an illusion: since it is much easier for us to empathise with different
individuals than with groups, we naturally focus on the decrease in average utility for each individual, rather than the
fact there are many more individuals to enjoy it.
Self-criticism
Parfit also considers an objection where the comparison between A+ and B- is attacked. The comparison between A
and A+ was partly dependent on their separation. Thus A+ and B- might simply be incomparable. Parfit gives the
Rich and Poor example, in which two people live in separate societies, and are unknown to each other, but are both
known to you, and you have to make a choice between helping one or the other. Thus, despite their separation, it is
meaningful to ask whether A+ is better than B- or not. One could deny that B- is better than A+, and therefore
neither is B. But this rejection implies that what is most important is the happiness of the happiest people, and
commits one to the view that a smaller decrease in the happiness of the happiest people outweighs a bigger increase
in the happiness of less happy people, at least in some cases. Parfit calls this the Elitist view.
External links
• The Repugnant Conclusion [1] (Stanford Encyclopedia of Philosophy)
References
• Parfit, Derek. Reasons and Persons, ch. 17 and 19. Oxford University Press 1986.
• Temkin, Larry. Intransitivity and the Mere Addition Paradox [2], Philosophy and Public Affairs, Vol. 16 No. 2
(Spring 1987) pp. 138–187
• Tännsjö, Torbjörn. Hedonistic Utilitarianism [3]. Edinburgh University Press 1988.
• Contestabile, Bruno. On the Buddhist Truths and the Paradoxes in Population Ethics [4], Contemporary
Buddhism, Vol. 11 Issue 1, pp. 103–113, Routledge 2010
References
[1]
[2]
[3]
[4]
http:/ / plato. stanford. edu/ entries/ repugnant-conclusion/
http:/ / www. jstor. org/ pss/ 2265425
http:/ / www. amazon. com/ dp/ 0748610421
http:/ / www. socrethics. com/ Folder2/ Population. htm
269
Navigation paradox
Navigation paradox
The Navigation paradox states that increased navigational precision may result in increased collision risk. In the
case of ships and aircraft, the advent of Global Positioning System (GPS) navigation has enabled craft to follow
navigational paths with such greater precision (often on the order of plus or minus 2 meters), that, without better
distribution of routes, coordination between neighboring craft and collision avoidance procedures, the likelihood of
two craft occupying the same space on the shortest distance line between two navigational points has increased.
Research
Robert E. Machol[1] attributes the term "navigation paradox" to Peter G. Reich, writing in 1964,[2] and 1966,[3] who
recognized that "in some cases (see below) increases in navigational precision increase collision risk." In the "below"
explanation, Machol noted "that if vertical station-keeping is sloppy, then if longitudinal and lateral separation are
lost, the planes will probably pass above and below each other. This is the ‘navigation paradox’ mentioned earlier."
Russ Paielli wrote a mid-air collision simulating computer model 500 sq mi (1,300 km2) centered on Denver,
Colorado[4] In Table 3 Paielli[4] notes that aircraft cruising at random altitudes have five times fewer collisions than
those obeying with only 25 ft (7.6 m) RMS of vertical error discrete cruising altitude rule, such as the internationally
required hemispherical cruising altitude rules. At the same vertical error, the prototype linear cruising altitude rule
tested produced 33.8 fewer mid-air collisions than the hemispherical cruising altitude rules.
Paielli’s 2000 model corroborated an earlier 1997 model by Patlovany[5] showing in Figure 1 that zero altitude error
by pilots obeying the hemispherical cruising altitude rules resulted in six times more mid-air collisions than random
cruising altitude non compliance. Similarly, Patlovany’s computer model test of the Altimeter-Compass Cruising
Altitude Rule (ACCAR) with zero piloting altitude error (a linear cruising altitude rule similar to the one
recommended by Paielli), resulted in about 60% of the mid-air collisions counted from random altitude non
compliance, or 10 times fewer collisions than the internationally accepted hemispherical cruising altitude rules. In
other words, Patlovany’s ACCAR alternative and Paielli’s linear cruising altitude rule would reduce cruising midair
collisions between 10 and 33 times, compared to the currently recognized, and internationally required,
hemispherical cruising altitude rules, which institutionalize the navigation paradox on a world wide basis.
The ACCAR alternative to the hemispherical cruising altitude rules, if adopted in 1997, could have eliminated the
navigation paradox at all altitudes, and could have saved 342 lives in over 30 midair collisions (up to November
2006) since the Risk Analysis proof that the current regulations multiply midair collision risk in direct proportion to
pilot accuracy in compliance.[6] The Namibian collision in 1997, the Japanese near-miss in 2001, the Überlingen
collision in Germany in 2002, and the Amazon collision in 2006,[7] are all examples where human or hardware errors
doomed altitude-accurate pilots killed by the navigation paradox designed into the current cruising altitude rules. The
current system as described by Paielli noted as examples that nuclear power plants and elevators are designed to be
passively safe and fault tolerant. Reactivity control rods fall into the reactor to cause a shutdown on loss of electrical
power, and elevator fall-arresting brakes are released by torque from support cable tension. The navigation paradox
describes a midair collision safety system that by design cannot tolerate a single failure in human performance or
electronic hardware.
To mitigate the described problem, many recommend, as legally allowed in very limited authorized airspace, that
planes fly one or two miles offset from the center of the airway (to the right side) thus eliminating the problem only
in the head-on collision scenario. The International Civil Aviation Organization's (ICAO) "Procedures for Air
Navigation--Air Traffic Management Manual," authorizes lateral offset only in oceanic and remote airspace
worldwide.[8] However, this workaround for the particular case of a head-on collision threat on a common assigned
airway fails to address the navigation paradox in general, and it fails to specifically address the inherent system
safety fault intolerance inadvertently designed into international air traffic safety regulations.[4] To be specific, in the
270
Navigation paradox
cases of intersecting flight paths where either aircraft is not on an airway (for example, flying under a "direct"
clearance, or a temporary diversion clearance for weather threats), or where intersecting aircraft flights are on
deliberately intersecting airways, these more general threats receive no protection from flying one or two miles to the
right of the center of the airway. Intersecting flight paths must still intersect somewhere. As with the midair collision
over Germany, an offset to the right of an airway would have simply changed the impact point by a mile or two away
from where the intersection actually did occur. Of the 342 deaths since 1997 so far encouraged by the lack of a linear
cruising altitude rule (like ACCAR) improvement to the fault intolerance of the hemispherical cruising altitude rules,
only the head-on collision over the Amazon could have been prevented if either pilot had been flying an offset to the
right of the airway centerline. In contrast, ACCAR systematically separates conflicting traffic in all airspace at all
altitudes on any heading, whether over the middle of the ocean or over high density multinational-interface
continental airspace. Nothing about Reduced Vertical Separation Minima (RVSM) system design addresses the
inherent vulnerability of the air traffic system to expected faults in hardware and human performance, as experienced
in the Namibian, German, Amazon and Japanese accidents.[5]
References
[1] Machol, Robert E., Interfaces 25:5, September–October 1995 (151-172), page 154.
[2] Reich, Peter G., "A theory of safe separation standards for air traffic control," RAE Technical Reports Nos. 64041, 64042, 64043, Royal
Aircraft Establishment, Farnborough, United Kingdom.
[3] Reich, Peter G., "Analysis of long-range air traffic systems: Separation standards—I, II, and III," Journal of Navigation, Vol. 19, No. 1, pp.
88-96; No. 2, pp. 169-176; No. 3, pp. 331-338.
[4] Paielli, Russ A., "A Linear Altitude Rule for Safer and More Efficient Enroute Air Traffic," Air Traffic Control Quarterly, Vol. 8, No. 3, Fall
2000.
[5] Patlovany, Robert W., "U.S. Aviation Regulations Increase Probability of Midair Collisions," Risk Analysis: An International Journal, April
1997, Volume 17, No. 2, Pages 237-248.
[6] Patlovany, Robert, W., "Preventable Midair Collisions Since 26 June 1997 Request Denied for Notice of Proposed Rulemaking (NPRM)
28996 Altimeter-Compass Cruising Altitude Rule (ACCAR)," Preventable Midair Collisions Since 26 June 1997 Request Denied for Notice
of Proposed Rulemaking (NPRM) 28996 Altimeter-Compass Cruising Altitude Rule (ACCAR) (http:/ / web. archive. org/ web/
20091027124508/ http:/ / www. geocities. com/ rpatlovany/ PreventableMidairs. html)
[7] Langwiesche, William, "The Devil at 37,000 Feet", Vanity Fair, January 2009 (http:/ / www. vanityfair. com/ magazine/ 2009/ 01/
air_crash200901)
[8] Werfelman, Linda, "Sidestepping the Airway," AeroSafety World March 2007, pages 40-45, Flight Safety Foundation (http:/ / flightsafety.
org/ asw/ mar07/ asw_mar07_p40-45. pdf).
271
Paradox of the plankton
272
Paradox of the plankton
In aquatic biology, the paradox of the plankton describes the
situation in which a limited range of resources (light, nutrients)
supports a much wider range of planktonic organisms. The paradox
results from the competitive exclusion principle (sometimes referred to
as Gause's Law), which suggests that when two species compete for
the same resource, ultimately only one will persist and the other will be
driven to extinction. Phytoplankton life is diverse at all phylogenetic
levels despite the limited range of resources (e.g. light, nitrate,
phosphate, silicic acid, iron) for which they compete amongst
themselves.
A batch of marine diatoms, just some of the many
organisms to gain energy from the Sun
The paradox was originally described in 1961 by limnologist G. Evelyn Hutchinson, who proposed that the paradox
could be resolved by factors such as vertical gradients of light or turbulence, symbiosis or commensalism,
differential predation, or constantly changing environmental conditions.[1] More recent work has proposed that the
paradox can be resolved by factors such as: size-selective grazing;[2] spatio-temporal heterogeneity;[3] and
environmental fluctuations.[4] More generally, some researchers suggest that ecological and environmental factors
continually interact such that the planktonic habitat never reaches an equilibrium for which a single species is
favoured.[5]
References
[1] Hutchinson, G. E. (1961) The paradox of the plankton (http:/ / cmore. soest. hawaii. edu/ agouron/ 2007/ documents/
paradox_of_the_plankton. pdf). American Naturalist 95, 137-145.
[2] Wiggert, J.D., Haskell, A.G.E., Paffenhofer, G.A., Hofmann, E.E. and Klinck, J.M. (2005) The role of feeding behavior in sustaining copepod
populations in the tropical ocean (http:/ / plankt. oxfordjournals. org/ cgi/ content/ short/ fbi090v1). Journal of Plankton Research 27,
1013-1031.
[3] Miyazaki, T., Tainaka, K., Togashi, T., Suzuki, T. and Yoshimura, J. (2006) Spatial coexistence of phytoplankton species in ecological
timescale (http:/ / sciencelinks. jp/ j-east/ article/ 200611/ 000020061106A0324715. php). Population Ecology 48(2), 107-112.
[4] Descamps-Julien, B. and Gonzalez, A. (2005) Stable coexistence in a fluctuating environment: An experimental demonstration (http:/ / www.
biologie. ens. fr/ bioemco/ biodiversite/ descamps/ ecology05. pdf). Ecology 86, 2815-2824.
[5] Scheffer, M., Rinaldi, S., Huisman, J. and Weissing, F.J. (2003) Why plankton communities have no equilibrium: solutions to the paradox
(http:/ / www. springerlink. com/ content/ vn768133l633114x/ ). Hydrobiologia 491, 9-18.
External links
• The Paradox of the Plankton (http://knol.google.com/k/klaus-rohde/the-paradox-of-the-plankton/
xk923bc3gp4/40#)
Temporal paradox
273
Temporal paradox
The temporal paradox, or time problem is a controversial issue in the evolutionary relationships of birds. It was
described by paleornithologist Alan Feduccia[1][2] in 1994.
Objection to consensus
The concept of a "temporal paradox" is based on the following facts.
The consensus view is that birds evolved from dinosaurs, but the most
bird-like dinosaurs, including almost all of the feathered dinosaurs and
those believed to be most closely related to birds are known mostly
from the Cretaceous, by which time birds had already evolved and
diversified. If bird-like dinosaurs are the ancestors of birds they should,
then, be older than birds, but Archaeopteryx is 155 million years old,
while the very bird-like Deinonychus is 35 million years younger. This
idea is sometimes summarized as "you can't be your own
grandmother". As Dodson pointed out:
Diagram illustrating the determined age of four
different prehistoric bird genera, compared to the
age of some birdlike dinosaurs and feathered
dinosaurs.
I hasten to add that none of the known small theropods,
including Deinonychus, Dromaeosaurus, Velociraptor,
Unenlagia, nor Sinosauropteryx, Protarchaeopteryx, nor Caudipteryx is itself relevant to the origin of
birds; these are all Cretaceous fossils ... and as such can at best represent only structural stages through
which an avian ancestor may be hypothezised to have passed.[3]
Problems
Numerous researchers have discredited the idea of the temporal paradox. Witmer (2002) summarized this critical
literature by pointing out that there are at least three lines of evidence that contradict it.
First, no one has proposed that maniraptoran dinosaurs of the Cretaceous are the ancestors of birds. They have
merely found that dinosaurs like dromaeosaurs, troodontids and oviraptorosaurs are close relatives of birds. The true
ancestors are thought to be older than Archaeopteryx, perhaps Early Jurassic or even older. The scarcity of
maniraptoran fossils from then is not surprising since fossilization is a rare event requiring special circumstances,
and we may never find fossils of animals in sediments from ages that they actually inhabited.
Second, fragmentary remains of maniraptoran dinosaurs actually had been known from Jurassic deposits in China,
North America, and Europe for many years. For example, the femur of a tiny maniraptoran from the Late Jurassic of
Colorado was reported by Padian and Jensen in 1989 [4] while teeth of dromaeosaurids and troodontids are known
from Jurassic England.[5] Complete skeletons of Middle-Late Jurassic maniraptorans were subsequently described
from China. The known diversity of pre-Tithonian (and thus pre-Archaeopteryx) non-avian maniraptorans includes
Ornitholestes, the possible therizinosaur Eshanosaurus, the troodontids Anchiornis[6] and the as yet unnamed
Morrison WDC DML 001, the scansoriopterygids Epidexipteryx and Epidendrosaurus [7] and the basal alvarezsaur
Haplocheirus.
Third, if the temporal paradox would indicate that birds should not have evolved from dinosaurs, then this raises the
question of what animals would be more likely ancestors considering their age. Brochu and Norell (2001) [8]
analyzed this question using six [9] of the other archosaurs that have been proposed as bird ancestors, and found that
all of them create temporal paradoxes — long stretches between the ancestor and Archaeopteryx where there are no
intermediate fossils — that are actually worse. The MSM value for the theropod option was 0.438 - 0.466 [9].
However, because of computational limitations, six taxa considered in SCI and SMIG calculations (Compsognathus,
Temporal paradox
Eoraptor, Herrerasauridae, Marasuchus, Pseudolagosuchus, and Choristodera) were not included in calculation of
MSM. Brochu and Norell (2001) [10] Thus, even if one used the logic of the temporal paradox, one should still prefer
dinosaurs as the ancestors to birds.[11] Pol and Norell [12] (2006) calculated MSM* values for the same proposed bird
ancestors and obtained the same relative results. The MSM* value for the theropod option was 0.31 - 0.40 [13] .
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Feduccia, Alan (1994) "The Great Dinosaur Debate" Living Bird. 13:29-33.
Feduccia, Alan (1996) "The Origin and Evolution of Birds." Yale University Press. New Haven, Conn. USA.
Dodson P., "Origin of birds: the final solution?", American zoologist 40: 505-506, 2000.
Jensen, James A. & Padian, Kevin. (1989) "Small pterosaurs and dinosaurs from the Uncompahgre fauna (Brushy Basin member, Morrison
Formation: ?Tithonian), Late Jurassic, western Colorado" Journal of Paleontology Vol. 63 no. 3 pg. 364 - 373
Witmer, L.M. (2002). “The Debate on Avian Ancestry; Phylogeny, Function and Fossils”, “Mesozoic Birds: Above the Heads of Dinosaurs”
pp.3-30. ISBN 0-520-20094-2
Hu, D.; Hou, L.; Zhang, L.; Xu, X. (2009). "A pre-Archaeopteryx troodontid theropod from China with long feathers on the metatarsus".
Nature 461 (7264): 640–3. Bibcode 2009Natur.461..640H. doi:10.1038/nature08322. PMID 19794491.
Zhang, F.; Zhou, Z.; Xu, X.; Wang, X.; Sullivan, C. (2008). "A bizarre Jurassic maniraptoran from China with elongate ribbon-like feathers".
Nature 455 (7216): 1105–8. Bibcode 2008Natur.455.1105Z. doi:10.1038/nature07447. PMID 18948955.
http:/ / www. jstor. org/ pss/ 4524078
http:/ / www. bioone. org/ action/ showFullPopup?doi=10. 1671%2F0272-4634%282000%29020%5B0197%3ATCATOO%5D2. 0.
CO%3B2& id=i0272-4634-20-1-197-t02
[10] http:/ / www. bioone. org/ action/ showFullPopup?doi=10. 1671%2F0272-4634%282000%29020%5B0197%3ATCATOO%5D2. 0.
CO%3B2& id=i0272-4634-20-1-197-f01
[11] Brochu, Christopher A. Norell, Mark A. (2001) "Time and trees: A quantitative assessment of temporal congruence in the bird origins
debate" pp.511-535 in "New Perspectives on the Origin and Early Evolution of Birds" Gauthier&Gall, ed. Yale Peabody Museum. New
Haven, Conn., USA.
[12] http:/ / sysbio. oxfordjournals. org/ cgi/ content-nw/ full/ 55/ 3/ 512/
[13] http:/ / sysbio. oxfordjournals. org/ cgi/ content-nw/ full/ 55/ 3/ 512/ FIG6
274
Tritone paradox
Tritone paradox
The tritone paradox is an auditory illusion in which a
sequentially played pair of Shepard tones [1] separated by an
interval of a tritone, or half octave, is heard as ascending by
some people and as descending by others.[2] Different
populations tend to favor one of a limited set of different
spots around the chromatic circle as central to the set of
"higher" tones. The tritone paradox was first reported by
psychology of music researcher Diana Deutsch in 1986.[3]
Each Shepard tone consists of a set of octave related
sinusoids, whose amplitudes are scaled by a fixed bell
shaped spectral envelope based on a log frequency scale.
For example, one tone might consist of a sinusoid at 440
Hz, accompanied by sinusoid at the higher octaves (880 Hz,
1760 Hz, etc.) and lower octaves (220 Hz, 110 Hz, etc.).
The other tone might consist of a 311 Hz sinusoid, again
accompanied by higher and lower octaves (622 Hz, 155.5
Hz, etc.). The amplitudes of the sinusoids of both
complexes are determined by the same fixed amplitude
envelope - for example the envelope might be centered at
370 Hz and span a 6 octave range.
Shepard predicted that the two tones would constitute a
bistable figure, the auditory equivalent of the Necker cube,
that could be heard ascending or descending, but never both
at the same time. Diana Deutsch later found that perception of which tone was higher depended on the absolute
frequencies involved: an individual will usually find the same tone to be higher, and this is determined by the tones'
absolute pitches. This is consistently done by a large portion of the population, despite the fact that responding
differently to different tones must involve the ability to hear absolute pitch, which was thought to be extremely rare.
This finding has been used to argue that latent absolute-pitch ability is present in a large proportion of the
population. In addition, Deutsch found that subjects from the south of England and from California resolved the
ambiguity the opposite way. Also, Deutsch, Henthorn and Dolson found that native speakers of Vietnamese, a tonal
language, heard the tritone paradox differently from Californians who were native speakers of English.
Notes
[1] R.N. Shepard. Circularity in judgments of relative pitch. Journal of the Acoustical Society of America, 36(12):2346–2353, 1964.
[2] Deutsch's Musical Illusions (http:/ / deutsch. ucsd. edu/ psychology/ deutsch_research6. php)
[3] Deutsch (1986).
References
• Deutsch, D. (1986). "A musical paradox". Music Perception 3: 275–280.. Weblink (http://psycnet.apa.org/
?fa=main.doiLanding&uid=1987-27127-001) PDF Document (http://philomel.com/pdf/MP-1986_3_275-280.
pdf)
• Deutsch, D. (1986). "An auditory paradox". Journal of the Acoustical Society of America 80: s93.
doi:10.1121/1.2024050. Weblink (http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ASADL&
275
Tritone paradox
•
•
•
•
•
smode=strresults&sort=chron&maxdisp=25&threshold=0&pjournals=journals&pjournals=JASMAN&
pjournals=ARLOFJ&pjournals=NOCOAN&pjournals=SOUCAU&possible1=An auditory paradox. &
possible1zone=title&OUTLOG=NO&viewabs=JASMAN&key=DISPLAY&docID=1&page=1&chapter=0)
Deutsch, D. (1987). "The tritone paradox: Effects of spectral variables". Perception & Psychophysics 41 (6):
563–575. doi:10.3758/BF03210490. PMID 3615152. PDF Document (http://philomel.com/pdf/P&
P-1987_41_563-575.pdf)
Deutsch, D., North, T. and Ray, L. (1990). "The tritone paradox: Correlate with the listener's vocal range for
speech". Music Perception 7: 371–384. PDF Document (http://philomel.com/pdf/MP-1990_7_371-384.pdf)
Deutsch, D. (1991). "The tritone paradox: An influence of language on music perception". Music Perception 8:
335–347. PDF Document (http://philomel.com/pdf/MP-1991_8_335-347.pdf)
Deutsch, D. (1992). "Paradoxes of musical pitch". Scientific American 267 (2): 88–95.
doi:10.1038/scientificamerican0892-88. PMID 1641627. PDF Document (http://philomel.com/pdf/
Sci_Am-1992-Aug_267_88_95.pdf)
Deutsch, D. (1992). "Some new pitch paradoxes and their implications. In Auditory Processing of Complex
Sounds". Philosophical Transactions of the Royal Society, Series B 336 (1278): 391–397.
doi:10.1098/rstb.1992.0073. PMID 1354379. PDF Document (http://philomel.com/pdf/
Proc_Royal_Soc-1992_336_391-397.pdf)
• Deutsch, D. (1997). "The tritone paradox: A link between music and speech". Current Directions in
Psychological Science 6 (6): 174–180. doi:10.1111/1467-8721.ep10772951. PDF Document (http://philomel.
com/pdf/Curr_Dir-1997_6_174-180.pdf)
• Deutsch, D., Henthorn T. and Dolson, M. (2004). "Speech patterns heard early in life influence later perception of
the tritone paradox". Music Perception 21 (3): 357–372. doi:10.1525/mp.2004.21.3.357. PDF Document (http://
philomel.com/pdf/MP-2004-21_357-372.pdf)
• Deutsch, D. (2007). "Mothers and their offspring perceive the tritone paradox in closely similar ways". Archives
of Acoustics 32: 3–14. PDF Document (http://philomel.com/pdf/archives_of_acoustics-2007_32_3-14.pdf)
External links
• Audio example (requires Java) (http://www.cs.ubc.ca/nest/imager/contributions/flinn/Illusions/TT/tt.html)
• Diana Deutsch's page on auditory illusions (http://deutsch.ucsd.edu/psychology/deutsch_research1.php)
• Sound example of the tritone paradox (http://philomel.com/musical_illusions/play.
php?fname=Tritone_paradox)
276
Voting paradox
277
Voting paradox
The voting paradox (also known as Condorcet's paradox or the paradox of voting) is a situation noted by the
Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i.e. not transitive), even
if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in
conflict with each other. When this occurs, it is because the conflicting majorities are each made up of different
groups of individuals.
For example, suppose we have three candidates, A, B, and C, and that there are three voters with preferences as
follows (candidates being listed in decreasing order of preference):
Voter
First preference Second preference Third preference
Voter 1
A
B
C
Voter 2
B
C
A
Voter 3
C
A
B
If C is chosen as the winner, it can be argued that B should win instead, since two voters (1 and 2) prefer B to C and
only one voter (3) prefers C to B. However, by the same argument A is preferred to B, and C is preferred to A, by a
margin of two to one on each occasion. The requirement of majority rule then provides no clear winner.
Also, if an election were held with the above three voters as the only participants, nobody would win under majority
rule, as it would result in a three way tie with each candidate getting one vote. However, Condorcet's paradox
illustrates that the person who can reduce alternatives can essentially guide the election. For example, if Voter 1 and
Voter 2 choose their preferred candidates (A and B respectively), and if Voter 3 was willing to drop his vote for C,
then Voter 3 can choose between either A or B - and become the agenda-setter.
When a Condorcet method is used to determine an election, a voting paradox among the ballots can mean that the
election has no Condorcet winner. The several variants of the Condorcet method differ on how they resolve such
ambiguities when they arise to determine a winner. Note that there is no fair and deterministic resolution to this
trivial example because each candidate is in an exactly symmetrical situation.
278
Philosophy
Fitch's paradox of knowability
Fitch's paradox of knowability is one of the fundamental puzzles of epistemic logic. It provides a challenge to the
knowability thesis, which states that any truth is, in principle, knowable. The paradox is that this assumption implies
the omniscience principle, which asserts that any truth is known. Essentially, Fitch's paradox asserts that the
existence of an unknown truth is unknowable. So if all truths were knowable, it would follow that all truths are in
fact known.
The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very
plausible, but the omniscience principle is very implausible.
The paradox appeared as a minor theorem in a 1963 paper by Frederic Fitch, "A Logical Analysis of Some Value
Concepts". Other than the knowability thesis, his proof makes only modest assumptions on the modal nature of
knowledge and of possibility. He also generalised the proof to different modalities. It resurfaced in 1979 when W.D.
Hart wrote that Fitch's proof was an "unjustly neglected logical gem".
Proof
Suppose p is a sentence which is an unknown truth; that is, the sentence p is true, but it is not known that p is true. In
such a case, the sentence "the sentence p is an unknown truth" is true; and, if all truths are knowable, it should be
possible to know that "p is an unknown truth". But this isn't possible, because as soon as we know "p is an unknown
truth", we know that p is true, rendering p no longer an unknown truth, so the statement "p is an unknown truth"
becomes a falsity. Hence, the statement "p is an unknown truth" cannot be both known and true at the same time.
Therefore, if all truths are knowable, the set of "all truths" must not include any of the form "something is an
unknown truth"; thus there must be no unknown truths, and thus all truths must be known.
This can be formalised with modal logic. K and L will stand for known and possible, respectively. Thus LK means
possibly known, in other words, knowable. The modality rules used are:
(A) Kp → p
- knowledge implies truth.
(B) K(p & q) → (Kp
& Kq)
- knowing a conjunction implies knowing each conjunct.
(C) p → LKp
- all truths are knowable.
(D) from ¬p, deduce
¬Lp
- if p can be proven false without assumptions, then p is impossible (which is similar to the rule of necessitation: if p can
be proven true without assumptions, then p is necessarily true).
The proof proceeds:
Fitch's paradox of knowability
279
1. Suppose K(p & ¬Kp)
2. Kp & K¬Kp
from line 1 by rule (B)
3. Kp
from line 2 by conjunction elimination
4. K¬Kp
from line 2 by conjunction elimination
5. ¬Kp
from line 4 by rule (A)
6. ¬K(p & ¬Kp)
from lines 3 and 5 by reductio ad absurdum, discharging assumption 1
7. ¬LK(p & ¬Kp)
from line 6 by rule (D)
8. Suppose p & ¬Kp
9. LK(p & ¬Kp)
from line 8 by rule (C)
10. ¬(p & ¬Kp)
from lines 7 and 9 by reductio ad absurdum, discharging assumption 8.
11. p → Kp
from line 10 by a classical tautology
The last line states that if p is true then it is known. Since nothing else about p was assumed, it means that every truth
is known.
Generalisations
The proof uses minimal assumptions about the nature of K and L, so other modalities can be substituted for
"known". Salerno gives the example of "caused by God": rule (C) becomes that every true fact could have been
caused by God, and the conclusion is that every true fact was caused by God. Rule (A) can also be weakened to
include modalities which don't imply truth. For instance instead of "known" we could have the doxastic modality
"believed by a rational person" (represented by B). Rule (A) is replaced with:
(E) Bp → BBp
- rational belief is transparent; if p is rationally believed, then it is rationally believed that p is rationally believed.
(F) ¬(Bp & B¬p) - rational beliefs are consistent
This time the proof proceeds:
1. Suppose B(p & ¬Bp)
2. Bp & B¬Bp
from line 1 by rule (B)
3. Bp
from line 2 by conjunction elimination
4. BBp
from line 3 by rule (E)
5. B¬Bp
from line 2 by conjunction elimination
6. BBp & B¬Bp
from lines 4 and 5 by conjunction introduction
7. ¬(BBp & B¬Bp)
by rule (F)
8. ¬B(p & ¬Bp)
from lines 6 and 7 by reductio ad absurdum, discharging assumption 1
The last line matches line 6 in the previous proof, and the remainder goes as before. So if any true sentence could
possibly be believed by a rational person, then that person does believe all true sentences.
Some anti-realists advocate the use of intuitionistic logic; however, except for the very last line which moves from
there are no unknown truths to all truths are known, the proof is, in fact, intuitionistically valid.
Fitch's paradox of knowability
The knowability thesis
Rule (C) is generally held to be at fault rather than any of the other logical principles employed. It may be contended
that this rule does not faithfully translate the idea that all truths are knowable, and that rule (C) should not apply
unrestrictedly. Kvanvig contends that this represents an illicit substitution into a modal context.
Berit Brogaard and Joseph Salerno offer a criticism of Kvanvig's proposal and then defend a new proposal according
to which quantified expressions play a special role in modal contexts. On the account of this special role articulated
by Stanley and Szabo, they propose a solution to the knowability paradoxes. Another way to resolve the paradox is
to restrict the paradox only to atomic sentences. Brogaard and Salerno have argued against this strategy in several
papers that have appeared in journals such as Analysis and American Philosophical Quarterly.
External links
• Fitch's Paradox of Knowability [1]. Article at the Stanford Encyclopedia of Philosophy, by Berit Brogaard and Joe
Salerno.
• Not Every Truth Can Be Known: at least, not all at once [2]. Discussion page on an article of the same name by
Greg Restall to appear in Salerno's book
• Joe Salerno [3]
References
[1] http:/ / plato. stanford. edu/ entries/ fitch-paradox/
[2] http:/ / consequently. org/ writing/ notevery/
[3] http:/ / sites. google. com/ site/ knowability/ joesalerno
• Frederick Fitch, " A Logical Analysis of Some Value Concepts (http://www.jstor.org/pss/2271594)". Journal
of Symbolic Logic Vol. 28, No. 2 (Jun., 1963), pp. 135–142
• W. D. Hart. "The Epistemology of Abstract Objects", Proceedings of the Aristotelian Society, suppl. vol. 53,
1979, pp. 153–65.
• Johnathan Kvanvig. The Knowability Paradox (http://books.google.ca/books?id=nhRZqgREEQMC). Oxford
University Press, 2006.
• Joe Salerno, ed. New essays on the knowability paradox (http://knowability.googlepages.com/home). Oxford
University Press, to appear.
External links
• Fitch's paradox of knowability (http://plato.stanford.edu/entries/fitch-paradox/) entry in the Stanford
Encyclopedia of Philosophy
• Knowability (http://philpapers.org/browse/knowability) at PhilPapers
• Fitch's paradox of knowability (https://inpho.cogs.indiana.edu/idea/998) at the Indiana Philosophy Ontology
Project
280
Grandfather paradox
Grandfather paradox
The grandfather paradox is a proposed paradox of time travel first described (in this exact form) by the science
fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent (Future Times Three).[1] The paradox is
described as following: the time traveller went back in time to the time when his or her grandfather had not married
yet. At that time, the time traveller kills his or her grandfather, and therefore, the time traveller is never born when he
or she was meant to be.
Despite the name, the grandfather paradox does not exclusively regard the impossibility of one's own birth. Rather, it
regards any action that makes impossible the ability to travel back in time in the first place. The paradox's namesake
example is merely the most commonly thought of when one considers the whole range of possible actions. Another
example would be using scientific knowledge to invent a time machine, then going back in time and (whether
through murder or otherwise) impeding a scientist's work that would eventually lead to the very information that you
used to invent the time machine. An equivalent paradox is known (in philosophy) as autoinfanticide, going back in
time and killing oneself as a baby.[2]
The grandfather paradox has been used to argue that backwards time travel must be impossible. However, a number
of hypotheses have been postulated to avoid the paradox, such as the idea that the past is unchangeable, so the
grandfather must have already survived the attempted killing (as stated earlier); or the time traveller creates - or joins
- an alternate time line[3] in which the traveller was never born.
Scientific theories
Novikov self-consistency principle
The Novikov self-consistency principle and Kip S. Thorne expresses one view on how backwards time travel could
be possible without a danger of paradoxes. According to this hypothesis, the only possible time lines are those
entirely self-consistent—so anything a time traveler does in the past must have been part of history all along, and the
time traveler can never do anything to prevent the trip back in time from happening, since this would represent an
inconsistency. Nicholas J. J. Smith argues, for example, that if some time traveler killed the child who lived in his
old address, this would ipso facto necessitate that the child was not the time traveler's younger self, nor the younger
self of anyone alive in the time frame that the time traveler came from. This could be extrapolated further into the
possibility that the child's death led to the family moving away, which in turn led to the time traveller's family
moving into the house guaranteeing that the house later became the home the time traveller would then grow up in,
forming a predestination paradox.
Seth Lloyd and other researchers at MIT have proposed an expanded version of the Novikov principle, according to
which probability bends to prevent paradoxes from occurring. Outcomes would become stranger as one approaches a
forbidden act, as the universe must favor improbable events to prevent impossible ones.[4][5]
Huggins Displacement Theory
The Huggins Displacement Theory allows for backwards time travel, but only if there is an equal space
displacement. A Time Traveler who went back 1 year in time would also be displaced by 1 light year. The Traveler
would be prevented by the Theory of Relativity from doing anything that would affect his past. In the parlance of
Relativity, you can only travel back in time along the edge of your Past Light Cone. Or more precisely, along your
Past Light Cones. You may not travel back to the left, wait and then travel back to the right. Past Light Cones
continue to exist and you may only travel their edges.
281
Grandfather paradox
Parallel universes
There could be "an ensemble of parallel universes" such that when the traveller kills the grandfather, the act took
place in (or resulted in the creation of) a parallel universe where the traveler's counterpart never exists as a result.
However, his prior existence in the original universe is unaltered. Succinctly, this explanation states that: if time
travel is possible, then multiple versions of the future exist in parallel universes. This theory would also apply if a
person went back in time to shoot himself, because in the past he would be dead as in the future he would be alive
and well.
Examples of parallel universes postulated in physics are:
• In quantum mechanics, the many-worlds interpretation suggests that every seemingly random quantum event with
a non-zero probability actually occurs in all possible ways in different "worlds", so that history is constantly
branching into different alternatives. The physicist David Deutsch has argued that if backwards time travel is
possible, it should result in the traveler ending up in a different branch of history than the one he departed from.[6]
See also quantum suicide and immortality.
• M-theory is put forward as a hypothetical master theory that unifies the six superstring theories, although at
present it is largely incomplete. One possible consequence of ideas drawn from M-theory is that multiple
universes in the form of 3-dimensional membranes known as branes could exist side-by-side in a fourth large
spatial dimension (which is distinct from the concept of time as a fourth dimension) - see Brane cosmology.
However, there is currently no argument from physics that there would be one brane for each physically possible
version of history as in the many-worlds interpretation, nor is there any argument that time travel would take one
to a different brane.
Nonexistence theory
According to this theory, if one were to do something in the past that would cause their nonexistence, upon returning
to the future, they would find themselves in a world where the effects of (and chain reactions thereof) their actions
are not present, as the person never existed. Through this theory, they would still exist, though. A famous example of
this theory is It's A Wonderful Life.
Theories in science fiction
Parallel universes resolution
The idea of preventing paradoxes by supposing that the time traveler is taken to a parallel universe while his original
history remains intact, which is discussed above in the context of science, is also common in science fiction—see
Time travel as a means of creating historical divergences.
Restricted action resolution
Another resolution, of which the Novikov self-consistency principle can be taken as an example, holds that if one
were to travel back in time, the laws of nature (or other intervening cause) would simply forbid the traveler from
doing anything that could later result in their time travel not occurring. For example, a shot fired at the traveler's
grandfather misses, or the gun jams or misfires, or the grandfather is injured but not killed, or the person killed turns
out to be not the real grandfather—or some other event prevents the attempt from succeeding. No action the traveler
takes to affect or change history can ever succeed, as some form of "bad luck" or coincidence always prevents the
outcome. In effect, the traveler cannot change history. Often in fiction, the time traveler does not merely fail to
prevent the actions, but in fact precipitates them (see predestination paradox), usually by accident.
This theory might lead to concerns about the existence of free will (in this model, free will may be an illusion, or at
least not unlimited). This theory also assumes that causality must be constant: i.e. that nothing can occur in the
282
Grandfather paradox
absence of cause, whereas some theories hold that an event may remain constant even if its initial cause was
subsequently eliminated.
Closely related but distinct is the notion of the time line as self-healing. The time-traveller's actions are like throwing
a stone in a large lake; the ripples spread, but are soon swamped by the effect of the existing waves. For instance, a
time traveller could assassinate a politician who led his country into a disastrous war, but the politician's followers
would then use his murder as a pretext for the war, and the emotional effect of that would cancel out the loss of the
politician's charisma. Or the traveller could prevent a car crash from killing a loved one, only to have the loved one
killed by a mugger, or fall down the stairs, choke on a meal, killed by a stray bullet, etc. In the 2002 film The Time
Machine, this scenario is shown where the main character builds a time machine to save his fiance from being killed
by a mugger, only for her to die in a car crash instead; as he learns from a trip to the future, he cannot save her with
the machine or he would never have been inspired to build the machine so that he could go back and save her in the
first place. In some stories it is only the event that precipitated the time traveler's decision to travel back in time that
cannot be substantially changed, in others all attempted changes "heal" in this way, and in still others the universe
can heal most changes but not sufficiently drastic ones. This is also the explanation advanced by the Doctor Who
role-playing game, which supposes that Time is like a stream; you can dam it, divert it, or block it, but the overall
direction resumes after a period of conflict.
It also may not be clear whether the time traveler altered the past or precipitated the future he remembers, such as a
time traveler who goes back in time to persuade an artist— whose single surviving work is famous— to hide the rest
of the works to protect them. If, on returning to his time, he finds that these works are now well-known, he knows he
has changed the past. On the other hand, he may return to a future exactly as he remembers, except that a week after
his return, the works are found. Were they actually destroyed, as he believed when he traveled in time, and has he
preserved them? Or was their disappearance occasioned by the artist's hiding them at his urging, and the skill with
which they were hidden, and so the long time to find them, stemmed from his urgency?
Destruction resolution
Some science fiction stories suggest that any paradox would destroy the universe, or at least the parts of space and
time affected by the paradox. The plots of such stories tend to revolve around preventing paradoxes, such as the final
episode of Star Trek: The Next Generation.
A less destructive alternative of this theory suggests the death of the time traveller whether the history is altered or
not; an example would be in the first part of the Back to the Future trilogy, where the lead character's alteration of
history results in a risk of his own disappearance, and he has to fix the alteration to preserve his own existence. In
this theory, killing one's grandfather would result in the disappearance of oneself, history would erase all traces of
the person's existence, and the death of the grandfather would be caused by another means (say, another existing
person firing the gun); thus, the paradox would never occur from a historical viewpoint.
Temporal Modification Negation Theory
While stating that if time travel is possible it would be impossible to violate the grandfather paradox, it goes further
to state that any action taken that itself negates the time travel event cannot occur. The consequences of such an
event would in some way negate that event, be it by either voiding the memory of what one is doing before doing it,
by preventing the action in some way, or even by destroying the universe among other possible consequences. It
states therefore that to successfully change the past one must do so incidentally.
For example, if one tried to stop the murder of one's parents, he would fail. On the other hand, if one traveled back
and did something else that as a result prevented the death of someone else's parents, then such an event would be
successful, because the reason for the journey and therefore the journey itself remains unchanged preventing a
paradox.
283
Grandfather paradox
284
In addition, if this event had some colossal change in the history of mankind, and such an event would not void the
ability or purpose of the journey back, it would occur, and would hold. In such a case, the memory of the event
would immediately be modified in the mind of the time traveler.
An example of this would be for someone to travel back to observe life in Austria in 1887 and while there shoot five
people, one of which was one of Hitler's parents. Hitler would therefore never have existed, but since this would not
prevent the invention of the means for time travel, or the purpose of the trip, then such a change would hold. But for
it to hold, every element that influenced the trip must remain unchanged. The Third Reich would not exist and the
world we know today would be completely different. This would void someone convincing another party to travel
back to kill the people without knowing who they are and making the time line stick, because by being successful,
they would void the first party's influence and therefore the second party's actions.
These issues are treated humorously in an episode of Futurama in which Fry travels back in time and inadvertently
causes his grandfather Enid's death before Enid marries his grandmother. Fry's distraught grandmother then seduces
him, and on returning to his own time, Fry learns that he is his own grandfather.
Other considerations
Consideration of the grandfather paradox has led some to the idea that time travel is by its very nature paradoxical
and therefore logically impossible, on the same order as round squares. For example, the philosopher Bradley
Dowden made this sort of argument in the textbook Logical Reasoning, where he wrote:
Nobody has ever built a time machine that could take a person back to an earlier time. Nobody should be seriously trying to build one, either,
because a good argument exists for why the machine can never be built. The argument goes like this: suppose you did have a time machine
right now, and you could step into it and travel back to some earlier time. Your actions in that time might then prevent your grandparents from
ever having met one another. This would make you not born, and thus not step into the time machine. So, the claim that there could be a time
machine is self-contradictory.
“
”
However, some philosophers and scientists believe that time travel into the past need not be logically impossible
provided that there is no possibility of changing the past, as suggested, for example, by the Novikov self-consistency
principle. Bradley Dowden himself revised the view above after being convinced of this in an exchange with the
philosopher Norman Swartz.[7]
Consideration of the possibility of backwards time travel in a hypothetical universe described by a Gödel metric led
famed logician Kurt Gödel to assert that time might itself be a sort of illusion.[8][9] He seems to have been suggesting
something along the lines of the block time view in which time does not really "flow" but is just another dimension
like space, with all events at all times being fixed within this 4-dimensional "block".
References
[1] Barjavel, René (1943). Le voyageur imprudent ("The imprudent traveller").; actually, the book refers to an ancestor of the time traveler not
his grandfather.
[2] Horwich, Paul (1987). Asymmetries in Time. Cambridge, MIT Press. pp. 116.
When the term was coined by Paul Horwich, he used the term autofanticide.
[3] See also Alfred Bester, The Men Who Murdered Mohammed, published in 1958, just the year following Everett's Ph.D thesis
[4] Laura Sanders, "Physicists Tame Time Travel by Forbidding You to Kill Your Grandfather", Wired, 20 July 2010. "But this dictum against
paradoxical events causes possible but unlikely events to happen more frequently. 'If you make a slight change in the initial conditions, the
paradoxical situation won’t happen. That looks like a good thing, but what it means is that if you’re very near the paradoxical condition, then
slight differences will be extremely amplified,' says Charles Bennett of IBM’s Watson Research Center in Yorktown Heights, New York."
[5] Seth Lloyd et. al., " The quantum mechanics of time travel through post-selected teleportation (http:/ / arxiv. org/ abs/ 1007. 2615)",
arXiv.org, submitted 15 July 2010, revised 19 July 2010.
[6] Deutsch, David (1991). "Quantum mechanics near closed timelike curves". Physical Review D 44 (10): 3197–3217.
Bibcode 1991PhRvD..44.3197D. doi:10.1103/PhysRevD.44.3197.
[7] "Dowden-Swartz Exchange" (http:/ / www. sfu. ca/ philosophy/ swartz/ time_travel1. htm). .
[8] Yourgrau, Palle (2004). A World Without Time: The Forgotten Legacy Of Godel And Einstein. Basic Books. ISBN 0-465-09293-4.
Grandfather paradox
285
[9] Holt, Jim (2005-02-21). "Time Bandits" (http:/ / www. newyorker. com/ printables/ critics/ 050228crat_atlarge). The New Yorker. . Retrieved
2006-10-19.
Liberal paradox
The liberal paradox is a logical paradox advanced by Amartya Sen, building on the work of Kenneth Arrow and his
impossibility theorem, which showed that within a system of menu-independent social choice, it is impossible to
have both a commitment to "Minimal Liberty", which was defined as the ability to order tuples of choices, and
Pareto optimality.
Since this theorem was advanced in 1970, it has attracted a wide body of commentary from philosophers such as
James M. Buchanan and Robert Nozick.
The most contentious aspect is, on one hand, to contradict the libertarian notion that the market mechanism is
sufficient to produce a Pareto-optimal society—and on the other hand, argue that degrees of choice and freedom,
rather than welfare economics, should be the defining trait of that market mechanism. As a result it attracts
commentary from both the left and the right of the political spectrum.
The theorem
The formal statement of the theorem is as follows.
Suppose there is a set of social outcomes
people each with individual preferences
with at least two alternatives and that there is a group of at least two
over
.
A benign social planner has to choose a single outcome from the set using the information about the individuals'
preferences. The planner uses a social choice function, which selects a choice
for every possible set of
preferences.
There are two desirable properties for this social choice function:
1. A social choice function respects the Paretian principle (also called Pareto optimality) if it never selects an
outcome when there is an alternative that everyone strictly prefers. So if there are two choices,
such that
for all individuals, then the social choice function does not select .
2. A social choice function respects minimal liberalism if there are at least two individuals, each of whom has at
least one pair of alternatives over which he is decisive. For example, there is a pair
such that if he prefers
to
, then the society should also prefer
to
.
The Paretian principle assumption is that there exists an individual
and a pair of alternatives
strictly prefers
and vice-versa.
to
, then the social choice function cannot chose
Similarly there must be another individual called
pair of alternatives
. If
such that if
whose preferences can veto a choice over a (possibly different)
then the social choice function cannot select
.
The Minimal Liberty assumption is that there exist at least two individuals who may each independently decide at
least one thing (for instance, to decide whether to sleep on his belly or back.) This is a very weak form of liberalism in reality, almost everyone can decide whether to sleep on his belly or back without society's input. However, even
under this weak-form assumption, the social choice cannot reach a Pareto efficient outcome.
Sen's impossibility theorem establishes that it is impossible for the social planner to satisfy both conditions. In other
words, for every social choice function there is at least one set of preferences (there exists at least one situation in
the social preferences sets) that forces the planner to violate either condition (1) or condition (2).
Liberal paradox
286
Sen's example
The following simple example involving two agents and three alternatives was put forward by Sen.[1]
There is a copy of a certain book, say Lady Chatterly's Lover, which is viewed differently by individuals
1 and 2. The three alternatives are: that individual 1 reads it ( ), that individual 2 reads it ( ), that
no one reads it ( ). Person 1, who is a prude, prefers most that no one reads it, but given the choice
between either of the two reading it, he would prefer that he read it himself rather than exposing the
gullible Mr. 2 to the influences of Lawrence. (Prudes, I am told, tend to prefer to be censors than being
censored.) In decreasing order of preference, his ranking is
. Person 2, however, prefers that
either of them should read it rather than neither. Furthermore he takes delight in the thought that prudish
Mr. 1 may have to read Lawrence, and his first preference is that person 1 should read it, next best that
he himself should read it, and worst that neither should. His ranking is, therefore,
.
Suppose that we give each individual the right to decide whether they want or don't want to read the book. Then it's
impossible to find a social choice function without violating "Minimal liberalism" or the "Paretian principle".
"Minimal liberalism" requires that Mr. 1 not be forced to read the book, so cannot be chosen. It also requires that
Mr. 2 not be forbidden from reading the book, so cannot be chosen. But alternative cannot be chosen either
because of the Paretian principle. Both Mr. 1 and Mr. 2 agree that that they prefer Mr. 1 to read the book (
Mr. 2 ( ).
) than
Since we have ruled out any possible solutions, we must conclude that it's impossible to find a social choice function.
Another example
Suppose Alice and Bob have to decide whether to go to the cinema to see a 'chick flick', and that each has the liberty
to decide whether to go themselves. If the personal preferences are based on Alice first wanting to be with Bob, then
thinking it is a good film, and on Bob first wanting Alice to see it but then not wanting to go himself, then the
personal preference orders might be:
• Alice wants: both to go > neither to go > Alice to go > Bob to go
• Bob wants: Alice to go > both to go > neither to go > Bob to go
There are two Pareto efficient solutions: either Alice goes alone or they both go. Clearly Bob will not go on his own:
he would not set off alone, but if he did then Alice would follow, and Alice's personal liberty means the joint
preference must have both to go > Bob to go. However, since Alice also has personal liberty if Bob does not go, the
joint preference must have neither to go > Alice to go. But Bob has personal liberty too, so the joint preference must
have Alice to go > both to go and neither to go > Bob to go. Combining these gives
• Joint preference: neither to go > Alice to go > both to go > Bob to go
and in particular neither to go > both to go. So the result of these individual preferences and personal liberty is that
neither go to see the film.
But this is Pareto inefficient given that Alice and Bob each think both to go > neither to go.
Liberal paradox
287
Bob
Goes
Alice
Goes
4,3
Doesn't
→
↑
Doesn't
1,1
2,4
↓
→
3,2
The diagram shows the strategy graphically. The numbers represent ranks in Alice and Bob's personal preferences,
relevant for Pareto efficiency (thus, either 4,3 or 2,4 is better than 1,1 and 4,3 is better than 3,2 – making 4,3 and 2,4
the two solutions). The arrows represent transitions suggested by the individual preferences over which each has
liberty, clearly leading to the solution for neither to go.
Liberalism and externalities
The example shows that liberalism and Pareto-efficiency cannot always be attained at the same time. Hence, if
liberalism exists in just a rather constrained way,[2] then Pareto-inefficiency could arise. Note that this is not always
the case. For instance if one individual makes use of her liberal right to decide between two alternatives, chooses one
of them and society would also prefer this alternative, no problem arises.
Nevertheless, the general case will be that there are some externalities. For instance, one individual is free to go to
work by car or by bicycle. If the individual takes the car and drives to work, whereas society wants him to go to
work by bicycle there will be an externality. However, no one can force the other to prefer cycling. So, one
implication of Sen's paradox is that these externalities will exist wherever liberalism exists.
Ways out of the paradox
There are several ways to resolve the paradox.
• First, the way Sen preferred, the individuals may decide simply to "respect" each other's choice by constraining
their own choice. Assume that individual A orders three alternatives (x, y, z) according to x P y P z and individual
B orders the same alternative according to z P x P y: according to the above reasoning, it will be impossible to
achieve a Pareto-efficient outcome. But, if A refuses to decide over z and B refuses to decide over x, then for A
follows x P y (x is chosen), and for B z P y (z is chosen). Hence A chooses x and respects that B chooses z; B
chooses z and respects that A chooses x. So, the Pareto-efficient solution can be reached, if A and B constrain
themselves and accept the freedom of the other player.
• A second way out of the paradox draws from game theory by assuming that individuals A and B pursue
self-interested actions, when they decide over alternatives or pairs of alternatives. Hence, the collective outcome
will be Pareto-inferior as the prisoner's dilemma predicts. The way out (except Tit for tat) will be to sign a
contract, so trading away one's right to act selfishly and get the other's right to act selfishly in return.
• A third possibility starts with assuming that again A and B have different preferences towards four states of the
world, w, x, y, and z. A's preferences are given by w P x P y P z; B's preferences are given by y P z P w P x. Now,
liberalism implies that each individual is a dictator in a least one social area. Hence, A and B should be allowed to
decide at least over one pair of alternatives. For A, the "best" pair will be (w,z), because w is most preferred and z
is least preferred. Hence A can decide that w is chosen and at the same time make sure that z is not chosen. For B,
the same applies and implies, that B would most preferably decide between y and x. Furthermore assume that A is
not free to decide (w,z), but has to choose between y and x. Then A will choose x. Conversely, B is just allowed
to choose between w and z and eventually will rest with z. The collective outcome will be (x,z), which is
Pareto-inferior. Hence again A and B can make each other better off by employing a contract and trading away
their right to decide over (x,y) and (w,z). The contract makes sure that A decides between w and z and chooses w.
B decides between (x,y) and chooses y. The collective outcome will be (w,y), the Pareto-optimal result.
Liberal paradox
• A fourth possibility is to dispute the paradox's very existence, as the concept of demonstrated preference, as
explained by Austrian economist Murray Rothbard, would mean the preferences that other people do certain
things are incapable of being shown in action.
And we are not interested in his opinions about the exchanges made by others, since his preferences are not
demonstrated through action and are therefore irrelevant. How do we know that this hypothetical envious one
loses in utility because of the exchanges of others? Consulting his verbal opinions does not suffice, for his
proclaimed envy might be a joke or a literary game or a deliberate lie.
—Murray Rothbard[3]
References
[1] Amartya, Sen (1970). "The Impossibility of a Paretian Liberal". Journal of Political Economy 78: 152–157. JSTOR 1829633.
[2] Sen, Amartya (1984) [1970]. Collective Choice and Social Welfare.
ch. 6.4 "Critique of Liberal Values"
ch. 6.5, "Critique of the Pareto Principle"
ch. 6*, "The Liberal Paradox"
[3] Rothbard, Murray. "Toward A Reconstruction of Utility and Welfare Economics" (http:/ / mises. org/ rothbard/ toward. pdf). . Retrieved 1
December 2012.
Moore's paradox
Moore's paradox concerns the putative absurdity involved in asserting a first-person present-tense sentence such as
'It's raining but I don't believe that it is raining' or 'It's raining but I believe that it is not raining'. The first author to
note this apparent absurdity was G.E. Moore.[1] These 'Moorean' sentences, as they have become known:
1. can be true,
2. are (logically) consistent, and moreover
3. are not (obviously) contradictions.
The 'paradox' consists in explaining why asserting a Moorean sentence is (or less strongly, strikes us as being) weird,
absurd or nonsensical in some way. The term 'Moore's Paradox' is due to Ludwig Wittgenstein,[2] who considered it
Moore's most important contribution to philosophy.[3] Wittgenstein devoted numerous remarks to the problem in his
later writings, which has brought Moore's Paradox the attention it might otherwise not have received.[4] Subsequent
commentators have further noted that there is an apparent residual absurdity in asserting a first-person future-tense
sentence such as 'It will be raining and I will believe that it is not raining'.[5]
Moore's Paradox has also been connected to many other of the well-known logical paradoxes including, though not
limited to, the liar paradox, the knower paradox, the unexpected hanging paradox, and the Preface paradox.[6]
There is currently no generally accepted explanation of Moore's Paradox in the philosophical literature. However,
while Moore's Paradox has perhaps been seen as a philosophical curiosity by philosophers themselves, Moorean-type
sentences are used by logicians, computer scientists, and those working in the artificial intelligence community, as
examples of cases in which a knowledge, belief or information system is unsuccessful in updating its
knowledge/belief/information store in the light of new or novel information.[7]
288
Moore's paradox
The problem
Since Jaakko Hintikka's seminal treatment of the problem,[8] it has become standard to present Moore's Paradox as
explaining why it is absurd to assert sentences that have the logical form: (OM) P and NOT(I believe that P), or
(COM) P and I believe that NOT-P. Commentators nowadays refer to these, respectively, as the omissive and
commissive versions of Moore's Paradox, a distinction according to the scope of the negation in the apparent
assertion of a lack of belief ('I don't believe that p') or belief that NOT-P.[6] The terms pertain to the kind of doxastic
error (i.e. error of belief) that one is subject to, or guilty of, if one is as the Moorean sentence says one is.
Moore himself presented the problem in two ways.[1][9]
The first more fundamental way of setting the problem up starts from the following three premises:
1. It can be true at a particular time both that P, and that I do not believe that P.
2. I can assert or believe one of the two at a particular time.
3. It is absurd to assert or believe both of them at the same time.
I can assert that it is raining at a particular time. I can assert that I don't believe that it is raining at a particular time.
If I say both at the same time, I am saying or doing something absurd. But the content of what I say—the proposition
the sentence expresses—is perfectly consistent: it may well be raining and I may not believe it. So why can't I assert
that it is so?
Moore presents the problem in a second, distinct, way:
1. It is not absurd to assert the past-tense counterpart, e.g. 'It was raining but I did not believe that it was raining'.
2. It is not absurd to assert the second- or third-person counterparts to Moore's sentences, e.g. 'It is raining but you
do not believe that it is raining', or 'Michael is dead but they do not believe that he is'.
3. It is absurd to assert the present-tense 'It is raining and I don't believe that it is raining'.
I can assert that I was a certain way (e.g. believing it was raining when it wasn't), that you, he, or they, are that way,
but not that I am that way. Why not?
Many commentators—though by no means all—also hold that Moore's Paradox arises not only at the level of
assertion but also at the level of belief. Interestingly imagining someone who believes an instance of a Moorean
sentence is tantamount to considering an agent who is subject to, or engaging in, self-deception (at least on one
standard way of describing it).
Proposed explanations
Philosophical interest in Moore's paradox, since Moore and Wittgenstein, has undergone a resurgence, starting with,
though not limited to, Jaakko Hintikka,[8] continuing with Roy Sorensen,[6] David Rosenthal,[10] Sydney
Shoemaker[11] and the first publication, in 2007, of a collection of articles devoted to the problem.[12]
There have been several proposed constraints on a satisfactory explanation in the literature, including (though not
limited to):
• It should explain the absurdity of both the omissive and the commissive versions.
• It should explain the absurdity of both asserting and believing Moore's sentences.
• It should preserve, and reveal the roots of, the intuition that contradiction (or something contradiction-like) is at
the root of the absurdity.
The first two conditions have generally been the most challenged, while the third appears to be the least
controversial. Some philosophers have claimed that there is, in fact, no problem in believing the content of Moore's
sentences (e.g. David Rosenthal). Others (e.g. Sydney Shoemaker) hold that an explanation of the problem at the
level of belief will automatically provide us with an explanation of the absurdity at the level of assertion via the
linking principle that what can reasonably be asserted is determined by what can reasonably be believed. Some have
also denied (e.g. Rosenthal) that a satisfactory explanation to the problem need be uniform in explaining both the
289
Moore's paradox
omissive AND commissive versions. Most of the explanations offered of Moore's paradox are united in holding that
contradiction is at the heart of the absurdity.
One type of explanation at the level of assertion exploits the view that assertion implies or expresses belief in some
way so that if someone asserts that p they imply or express the belief that p. Several versions of this view exploit
elements of speech act theory, which can be distinguished according to the particular explanation given of the link
between assertion and belief. Whatever version of this view is preferred, whether cast in terms of the Gricean
intentions (see Paul Grice) or in terms of the structure of Searlean illocutionary acts[13] (see speech act), it does not
obviously apply to explaining the absurdity of the commissive version of Moore's paradox. To take one version of
this type of explanation, if someone asserts p and conjoins it with the assertion (or denial) that he does not believe
that p, then he has in that very act contradicted himself, for in effect what the speaker says is: I believe that p and I
do not believe that p. The absurdity of asserting p & I do not believe that p is thus revealed as being of a more
familiar kind. Depending on one's view of the nature of contradiction, one might thus interpret a speaker of the
omissive Moorean sentence as asserting everything (that is, asserting too much) or asserting nothing (that is, not
asserting enough).
An alternative view is that the assertion "I believe that p" often (though not always) functions as an alternative way
of asserting "p", so that the semantic content of the assertion "I believe that p" is just p: it functions as a statement
about the world and not about anyone's state of mind. Accordingly what someone asserts when they assert "p and I
believe that not-p" is just "p and not-p" Asserting the commissive version of Moore's sentences is again assimilated
to the more familiar (putative) impropriety of asserting a contradiction.[14]
At the level of belief, there are two main kinds of explanation. The first, much more popular one, agrees with those
at the level of assertion that contradiction is at the heart of the absurdity. The contradiction is revealed in various
ways, some using the resources of doxastic logic (e.g. Hintikka), others (e.g. Sorensen) principles of rational belief
maintenance and formation, while still others appeal to our putative capacity for self-knowledge and the first-person
authority (e.g. Shoemaker) we have over our states of mind.
Another alternative view, due to Richard Moran,[15] views the existence of Moore's paradox as symptomatic of
creatures who are capable of self-knowledge, capable of thinking for themselves from a deliberative point of view, as
well as about themselves from a theoretical point of view. On this view, anyone who asserted or believed one of
Moore's sentences would be subject to a loss of self-knowledge—in particular, would be one who, with respect to a
particular 'object', broadly construed, e.g. person, apple, the way of the world, would be in a situation which violates,
what Moran calls, the Transparency Condition: if I want to know what I think about X, then I consider/think about
nothing but X itself. Moran's view seems to be that what makes Moore's paradox so distinctive is not some
contradictory-like phenomenon (or at least not in the sense that most commentators on the problem have construed
it), whether it be located at the level of belief or that of assertion. Rather, that the very possibility of Moore's paradox
is a consequence of our status as agents (albeit finite and resource-limited ones) who are capable of knowing (and
changing) their own minds.
290
Moore's paradox
References
[1] Moore, G. E. (1993). "Moore's Paradox". In Baldwin, Thomas. G. E. Moore: Selected Writings. London: Routledge. pp. 207–212.
ISBN 0-415-09853-X.
[2] Wittgenstein, Ludwig (1953). Philosophical Investigations. Section II.x. Blackwell Publishers. p. 190.
[3] Wittgenstein, Ludwig (1974). von Wright, G. H.. ed. Letters to Russell, Keynes and Moore. Oxford: Blackwell Publishers.
[4] Wittgenstein, Ludwig (1980). Anscombe, G. E. M.; von Wright, G. H.. eds. Remarks on the Philosophy of Psychology, Volume I. Translated
by G. E. M. Anscombe. Oxford: Blackwell Publishers. ISBN 0-631-12541-8.
[5] Bovens, Luc (1995). "'P and I Will Believe that not-P': Diachronic Constraints on Rational Belief". Mind 104 (416): 737–760.
doi:10.1093/mind/104.416.737.
[6] Sorensen, Roy A. (1988). Blindspots. New York: Oxford University Press. ISBN 0-19-824981-0.
[7] Philosophical Studies 128. 2006.
[8] Hintikka, Jaakko (1962). Knowledge and Belief: An Introduction to the Logic of the Two Notions. Cornell, NY: Cornell University Press.
[9] Moore, G. E. (1991). "Russell's Theory of Descriptions". In Schilpp, P. A.. The Philosophy of Bertrand Russell. The Library of Living
Philosophers. 5. La Salle, IL: Open Court Publishing. pp. 177–225.
[10] Rosenthal, David (1995). "Moore's Paradox and Consciousness". AI, Connectionism and Philosophical Psychology. Philosophical
Perspectives. 9. Atascadero, CA: Ridgeview. pp. 313–334. ISBN 0-924922-73-7.
[11] Shoemaker, Sydney (1996). "Moore's Paradox and Self-Knowledge". The First-Person Perspective and other essays. New York: Cambridge
University Press. pp. 74–96. ISBN 0-521-56871-4.
[12] Green, Mitchell S.; Williams, John N., eds. (2007). Moore's Paradox: New Essays on Belief, Rationality and the First-Person. New York:
Oxford University Press. ISBN 978-0-19-928279-1.
[13] Searle, John & Vanderveken, Daniel (1985). Foundations of Illocutionary Logic. New York: Cambridge University Press.
ISBN 0-521-26324-7.
[14] Linville, Kent & Ring, Merrill. "Moore's Paradox Revisited". Synthese 87 (2): 295–309. doi:10.1007/BF00485405.
[15] Moran, Richard (2001). Authority & Estrangement: An Essay on Self-knowledge. Princeton: Princeton University Press.
ISBN 0-691-08944-2.
External links
• "Epistemic Paradoxes" (including Moore's) at the Stanford Encyclopedia of Philosophy (http://plato.stanford.
edu/entries/epistemic-paradoxes/#MooPro)
291
Moravec's paradox
Moravec's paradox
Moravec's paradox is the discovery by artificial intelligence and robotics researchers that, contrary to traditional
assumptions, high-level reasoning requires very little computation, but low-level sensorimotor skills require
enormous computational resources. The principle was articulated by Hans Moravec, Rodney Brooks, Marvin Minsky
and others in the 1980s. As Moravec writes: "it is comparatively easy to make computers exhibit adult level
performance on intelligence tests or playing checkers, and difficult or impossible to give them the skills of a
one-year-old when it comes to perception and mobility."[1]
Linguist and cognitive scientist Steven Pinker considers this the most significant discovery uncovered by AI
researchers. In his book The Language Instinct, he writes:
"The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy
problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face,
lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest
engineering problems ever conceived.... As the new generation of intelligent devices appears, it will be
the stock analysts and petrochemical engineers and parole board members who are in danger of being
replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to
come."[2]
Marvin Minsky emphasizes that the most difficult human skills to reverse engineer are those that are unconscious.
"In general, we're least aware of what our minds do best," he writes, and adds "we're more aware of simple processes
that don't work well than of complex ones that work flawlessly."[3]
The biological basis of human skills
One possible explanation of the paradox, offered by Moravec, is based on evolution. All human skills are
implemented biologically, using machinery designed by the process of natural selection. In the course of their
evolution, natural selection has tended to preserve design improvements and optimizations. The older a skill is, the
more time natural selection has had to improve the design. Abstract thought developed only very recently, and
consequently, we should not expect its implementation to be particularly efficient.
As Moravec writes:
“Encoded in the large, highly evolved sensory and motor portions of the human brain is a billion years of
experience about the nature of the world and how to survive in it. The deliberate process we call
reasoning is, I believe, the thinnest veneer of human thought, effective only because it is supported by
this much older and much powerful, though usually unconscious, sensorimotor knowledge. We are all
prodigious olympians in perceptual and motor areas, so good that we make the difficult look easy.
Abstract thought, though, is a new trick, perhaps less than 100 thousand years old. We have not yet
mastered it. It is not all that intrinsically difficult; it just seems so when we do it.”[4]
A compact way to express this argument would be:
• We should expect the difficulty of reverse-engineering any human skill to be roughly proportional to the amount
of time that skill has been evolving in animals.
• The oldest human skills are largely unconscious and so appear to us to be effortless.
• Therefore, we should expect skills that appear effortless to be difficult to reverse-engineer, but skills that require
effort may not necessarily be difficult to engineer at all.
Some examples of skills that have been evolving for millions of years: recognizing a face, moving around in space,
judging people’s motivations, catching a ball, recognizing a voice, setting appropriate goals, paying attention to
things that are interesting; anything to do with perception, attention, visualization, motor skills, social skills and so
on.
292
Moravec's paradox
Some examples of skills that have appeared more recently: mathematics, engineering, human games, logic and much
of what we call science. These are hard for us because they are not what our bodies and brains were primarily
designed to do. These are skills and techniques that were acquired recently, in historical time, and have had at most a
few thousand years to be refined, mostly by cultural evolution.[5]
Historical influence on artificial intelligence
In the early days of artificial intelligence research, leading researchers often predicted that they would be able to
create thinking machines in just a few decades (see history of artificial intelligence). Their optimism stemmed in part
from the fact that they had been successful at writing programs that used logic, solved algebra and geometry
problems and played games like checkers and chess. Logic and algebra are difficult for people and are considered a
sign of intelligence. They assumed that, having (almost) solved the "hard" problems, the "easy" problems of vision
and commonsense reasoning would soon fall into place. They were wrong, and one reason is that these problems are
not easy at all, but incredibly difficult. The fact that they had solved problems like logic and algebra was irrelevant,
because these problems are extremely easy for machines to solve.[6]
Rodney Brooks explains that, according to early AI research, intelligence was "best characterized as the things that
highly educated male scientists found challenging", such as chess, symbolic integration, proving mathematical
theorems and solving complicated word algebra problems. "The things that children of four or five years could do
effortlessly, such as visually distinguishing between a coffee cup and a chair, or walking around on two legs, or
finding their way from their bedroom to the living room were not thought of as activities requiring intelligence."[7]
This would lead Brooks to pursue a new direction in artificial intelligence and robotics research. He decided to build
intelligent machines that had "No cognition. Just sensing and action. That is all I would build and completely leave
out what traditionally was thought of as the intelligence of artificial intelligence."[7] This new direction, which he
called "Nouvelle AI" was highly influential on robotics research and AI.[8]
Notes
[1]
[2]
[3]
[4]
[5]
Moravec 1988, p. 15
Pinker 2007, pp. 190-191
Minsky 1988, p. 29
Moravec 1988, pp. 15–16
Even given that cultural evolution is faster than genetic evolution, the difference in development time between these two kinds of skills is five
or six orders of magnitude, and (Moravec would argue) there hasn't been nearly enough time for us to have "mastered" the new skills.
[6] These are not the only reasons that their predictions did not come true: see the problems
[7] Brooks (2002), quoted in McCorduck (2004, p. 456)
[8] McCorduck 2004, p. 456
References
•
•
•
•
•
•
Brooks, Rodney (1986), Intelligence Without Representation, MIT Artificial Intelligence Laboratory
Brooks, Rodney (2002), Flesh and Machines, Pantheon Books
Campbell, Jeremy (1989), The Improbable Machine, Simon and Schuster, pp. 30–31
Minsky, Marvin (1986), The Society of Mind, Simon and Schuster, p. 29
Moravec, Hans (1988), Mind Children, Harvard University Press
McCorduck, Pamela (2004), Machines Who Think (http://www.pamelamc.com/html/machines_who_think.
html) (2nd ed.), Natick, MA: A. K. Peters, Ltd., ISBN 1-56881-205-1, p. 456.
• Nilsson, Nils (1998), Artificial Intelligence: A New Synthesis, Morgan Kaufmann Publishers,
ISBN 978-1-55860-467-4, pg. 7
• Pinker, Steven (September 4, 2007) [1994], The Language Instinct, Harper Perennial Modern Classics,
ISBN 0-06-133646-7
293
Newcomb's paradox
294
Newcomb's paradox
Newcomb's paradox, also referred to as Newcomb's problem, is a thought experiment involving a game between
two players, one of whom purports to be able to predict the future. Whether the problem is actually a paradox is
disputed.
Newcomb's paradox was created by William Newcomb of the University of California's Lawrence Livermore
Laboratory. However, it was first analyzed and was published in a philosophy paper spread to the philosophical
community by Robert Nozick in 1969, and appeared in Martin Gardner's Scientific American column in 1974. Today
it is a much debated problem in the philosophical branch of decision theory but has received little attention from the
mathematical side.
The problem
A person is playing a game operated by the Predictor, an entity somehow presented as being exceptionally skilled at
predicting people's actions. The exact nature of the Predictor varies between retellings of the paradox. Some assume
that the character always has a reputation for being completely infallible and incapable of error; others assume that
the predictor has a very low error rate. The Predictor can be presented as a psychic, as a superintelligent alien, as a
deity, as a brain-scanning computer, etc. However, the original discussion by Nozick says only that the Predictor's
predictions are "almost certainly" correct, and also specifies that "what you actually decide to do is not part of the
explanation of why he made the prediction he made". With this original version of the problem, some of the
discussion below is inapplicable.
The player of the game is presented with two boxes, one transparent (labeled A) and the other opaque (labeled B).
The player is permitted to take the contents of both boxes, or just the opaque box B. Box A contains a visible $1,000.
The contents of box B, however, are determined as follows: At some point before the start of the game, the Predictor
makes a prediction as to whether the player of the game will take just box B, or both boxes. If the Predictor predicts
that both boxes will be taken, then box B will contain nothing. If the Predictor predicts that only box B will be taken,
then box B will contain $1,000,000.
By the time the game begins, and the player is called upon to choose which boxes to take, the prediction has already
been made, and the contents of box B have already been determined. That is, box B contains either $0 or $1,000,000
before the game begins, and once the game begins even the Predictor is powerless to change the contents of the
boxes. Before the game begins, the player is aware of all the rules of the game, including the two possible contents
of box B, the fact that its contents are based on the Predictor's prediction, and knowledge of the Predictor's
infallibility. The only information withheld from the player is what prediction the Predictor made, and thus what the
contents of box B are.
Predicted choice Actual choice
Payout
A and B
A and B
$1,000
A and B
B only
$0
B only
A and B
$1,001,000
B only
B only
$1,000,000
The problem is called a paradox because two strategies that both sound intuitively logical give conflicting answers to
the question of what choice maximizes the player's payout. The first strategy argues that, regardless of what
prediction the Predictor has made, taking both boxes yields more money. That is, if the prediction is for both A and
B to be taken, then the player's decision becomes a matter of choosing between $1,000 (by taking A and B) and $0
(by taking just B), in which case taking both boxes is obviously preferable. But, even if the prediction is for the
Newcomb's paradox
player to take only B, then taking both boxes yields $1,001,000, and taking only B yields only $1,000,000—taking
both boxes is still better, regardless of which prediction has been made.
The second strategy suggests taking only B. By this strategy, we can ignore the possibilities that return $0 and
$1,001,000, as they both require that the Predictor has made an incorrect prediction, and the problem states that the
Predictor is almost never wrong. Thus, the choice becomes whether to receive $1,000 (both boxes) or to receive
$1,000,000 (only box B)—so taking only box B is better.
In his 1969 article, Nozick noted that "To almost everyone, it is perfectly clear and obvious what should be done.
The difficulty is that these people seem to divide almost evenly on the problem, with large numbers thinking that the
opposing half is just being silly."
The crux of the problem
The crux of the paradox is in the existence of two contradictory arguments, both being seemingly correct.
1. A powerful intuitive belief, that past events cannot be affected. My future action cannot determine the fate of an
event that happened before the action.
2. Newcomb proposes a way of doing precisely this - affecting a past event. The prediction of the Predictor
establishes equivalence between my choice (of renouncing the open box) and the content of the closed box, which
was determined in the past. Since I can affect the future event, I can also affect the past event, which is equivalent
to it.
The use of first person in the formulation of the second argument is essential: only when playing the role of the
chooser I feel that I determine the fate of the past event. Looking from aside at another person participating in the
experiment does not arouse a feeling of contradiction. Their choice and its prediction are part of a causal chain, that
in principle is not problematic.
A solution of the paradox must point out an error in one of the two arguments. Either the intuition is wrong, or there
is something wrong with the way proposed for affecting the past.
The relationship to the idle argument
There is a version of the famous idle argument (see fatalism) that is equivalent to the paradox. It is this:
Suppose that the omniscient predictor predicted the grade I will get in tomorrow's exam, and wrote their prediction
in a note. Since the content of the note was determined a while ago, I cannot change it. Since I believe that it reflects
precisely the grade I will get, I cannot also change my grade. So I can just as well rest, rather than prepare for the
exam (hence the name "the idle argument").
In both situations an equivalence between a past event P and a future event F is used to draw a paradoxical
conclusion, and both use the same argumentation. In Newcomb's paradox the claim is "I can determine F, hence I
can change P", while in the idle argument the claim is "I cannot change P, hence I cannot determine F", which is the
same argument, formulated in reverse direction.
295
Newcomb's paradox
Attempted resolutions
Many argue that the paradox is primarily a matter of conflicting decision making models. Using the expected utility
hypothesis will lead one to believe that one should expect the most utility (or money) from taking only box B.
However if one uses the Dominance principle, one would expect to benefit most from taking both boxes.
More recent work has reformulated the problem as a noncooperative game in which players set the conditional
distributions in a Bayes net. It is straightforward to prove that the two strategies for which boxes to choose make
mutually inconsistent assumptions for the underlying Bayes net. Depending on which Bayes net one assumes, one
can derive either strategy as optimal. In this there is no paradox, only unclear language that hides the fact that one is
making two inconsistent assumptions.[1]
Some argue that Newcomb's Problem is a paradox because it leads logically to self-contradiction. Reverse causation
is defined into the problem and therefore logically there can be no free will. However, free will is also defined in the
problem; otherwise the chooser is not really making a choice.
Other philosophers have proposed many solutions to the problem, many eliminating its seemingly paradoxical
nature:
Some suggest a rational person will choose both boxes, and an irrational person will choose just the one, therefore
rational people fare better, since the Predictor cannot actually exist. Others have suggested that an irrational person
will do better than a rational person and interpret this paradox as showing how people can be punished for making
rational decisions.
Others have suggested that in a world with perfect predictors (or time machines because a time machine could be the
mechanism for making the prediction) causation can go backwards.[2] If a person truly knows the future, and that
knowledge affects their actions, then events in the future will be causing effects in the past. Chooser's choice will
have already caused Predictor's action. Some have concluded that if time machines or perfect predictors can exist,
then there can be no free will and Chooser will do whatever they're fated to do. Others conclude that the paradox
shows that it is impossible to ever know the future. Taken together, the paradox is a restatement of the old contention
that free will and determinism are incompatible, since determinism enables the existence of perfect predictors. Some
philosophers argue this paradox is equivalent to the grandfather paradox. Put another way, the paradox presupposes a
perfect predictor, implying the "chooser" is not free to choose, yet simultaneously presumes a choice can be debated
and decided. This suggests to some that the paradox is an artifact of these contradictory assumptions. Nozick's
exposition specifically excludes backward causation (such as time travel) and requires only that the predictions be of
high accuracy, not that they are absolutely certain to be correct. So the considerations just discussed are irrelevant to
the paradox as seen by Nozick, which focuses on two principles of choice, one probabilistic and the other causal assuming backward causation removes any conflict between these two principles.
Newcomb's paradox can also be related to the question of machine consciousness, specifically if a perfect simulation
of a person's brain will generate the consciousness of that person.[3] Suppose we take the Predictor to be a machine
that arrives at its prediction by simulating the brain of the Chooser when confronted with the problem of which box
to choose. If that simulation generates the consciousness of the Chooser, then the Chooser cannot tell whether they
are standing in front of the boxes in the real world or in the virtual world generated by the simulation in the past. The
"virtual" Chooser would thus tell the Predictor which choice the "real" Chooser is going to make.
296
Newcomb's paradox
Notes
[1] Wolpert, D. H.; Benford, G. (2010). What does Newcomb's paradox teach us?. arXiv:1003.1343.
[2] Craig, William Lane (1988). "Tachyons, Time Travel, and Divine Omniscience". Journal of Philosophy 85 (3): 135–150. JSTOR 2027068.
[3] Neal, R. M. (2006). Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning. arXiv:math.ST/0608592.
References
• Nozick, Robert (1969), "Newcomb's Problem and Two principles of Choice," in Essays in Honor of Carl G.
Hempel, ed. Nicholas Rescher, Synthese Library (Dordrecht, the Netherlands: D. Reidel), p. 114-115.
• Bar-Hillel, Maya & Margalit, Avishai (1972), Newcomb's paradox revisited. British Journal of Philosophy of
Science, 23, 295-304.
• Gardner, Martin (1974), "Mathematical Games," Scientific American, March 1974, p. 102; reprinted with an
addendum and annotated bibliography in his book The Colossal Book of Mathematics (ISBN 0-393-02023-1)
• Campbell, Richmond and Lanning Sowden, ed. (1985), Paradoxes of Rationality and Cooperation: Prisoners'
Dilemma and Newcomb's Problem, Vancouver: University of British Columbia Press. (an anthology discussing
Newcomb's Problem, with an extensive bibliography)
• Levi, Isaac (1982), "A Note on Newcombmania," Journal of Philosophy 79 (1982): 337-42. (a paper discussing
the popularity of Newcomb's Problem)
• John Collins, "Newcomb's Problem", International Encyclopedia of the Social and Behavioral Sciences, Neil
Smelser and Paul Baltes (eds), Elsevier Science (2001) (http://collins.philo.columbia.edu/econphil/newcomb.
pdf) (Requires proper credentials)
External links
•
•
•
•
The cat that is not there (http://losthunderlads.com/2010/04/04/the-cat-that-is-not-there/)
Newcomb's Paradox (http://www.kiekeben.com/newcomb.html) by Franz Kiekeben
Thinking Inside the Boxes (http://www.slate.com/?id=2061419) by Jim Holt, for Slate
Free Will: Two Paradoxes of Choice (http://mises.org/multimedia/mp3/Long/Long-3.mp3) (lecture) by
Roderick T. Long
• Newcomb's Problem (http://w3.ub.uni-konstanz.de/kops/volltexte/2000/524/) by Marion Ledwig
• Newcomb's Problem and Regret of Rationality (http://www.overcomingbias.com/2008/01/newcombs-proble.
html) by Eliezer Yudkowsky
• The Resolution of Newcomb's Paradox (http://www.megasociety.org/noesis/44/newcomb.html) by Chris
Langan
297
Omnipotence paradox
298
Omnipotence paradox
The omnipotence paradox is a family of semantic
paradoxes which address two issues: Is an omnipotent
entity logically possible? and What do we mean by
'omnipotence'?. The paradox states that: if a being can
perform any action, then it should be able to create a
task which this being is unable to perform; hence, this
being cannot perform all actions. Yet, on the other
hand, if this being cannot create a task that it is unable
to perform, then there exists something it cannot do.
One version of the omnipotence paradox is the
so-called paradox of the stone: "Could an omnipotent
being create a stone so heavy that even he could not lift
it?" If he could lift the rock, then it seems that the being
could cease to be omnipotent, as the rock was not
heavy enough; if he could not, it seems that the being
was not omnipotent to begin with.[1]
14th-century depiction of Averroes (detail from Triunfo de Santo
Tomás by Andrea da Firenze), who addressed the omnipotence
paradox in the 12th century
The argument is medieval, dating at least to the 12th century, addressed by Averroës (1126–1198) and later by
Thomas Aquinas.[2] Pseudo-Dionysius the Areopagite (before 532) has a predecessor version of the paradox, asking
whether it is possible for God to "deny himself".
Many answers to the paradox have been proposed.
Overview
A common modern version of the omnipotence paradox is expressed in the question: "Can [an omnipotent being]
create a stone so heavy that it cannot lift it?" This question generates a dilemma. The being can either create a stone
which it cannot lift, or it cannot create a stone which it cannot lift. If the being can create a stone that it cannot lift,
then it seems that it can cease to be omnipotent. If the being cannot create a stone which it cannot lift, then it seems
it is already not omnipotent.[1]
The problem is whether the above question is ad hoc, or, instead, is inherently required by the concept of
omnipotence. If it is ad hoc, then the concept of omnipotence does not include being subject to be exceeded. If it is
inherently required, then there is no way to exclude answering the question in either the affirmative or the negative,
and, thus, no way to determine whether an omnipotent being is logically possible or impossible. But, if the question
is inherently required by the concept of omnipotence, then it seems the logic which allows it to be inherently
required is a paradox since the particular concept of omnipotence which requires it is a paradox. In short, the act of
seeming to find omnipotence to be a contradiction-of-terms is founded on the act of conceiving something against
which to construct the contradiction: prior to any ‘act’, omnipotence is conceived as coherent both with itself and
with the possibility of knowledge (which begs the question of what is the knowledge that constitutes the
identifiability of omnipotence-as-a-paradox?).
But, whether the concept of omnipotence itself is a material paradox, or is simply too obscure to us to preclude being
construed by paradoxical thinking, the central issue of the omnipotence paradox is whether the concept of the
'logically possible' is different for a world in which omnipotence exists from a world in which omnipotence does not
exist. The reason this is the central issue is because our sense of material paradox, and of the logical contradiction of
which material paradox is an expression, are functions of the fact that we presuppose that there must be something
Omnipotence paradox
which exists which is inherently meaningful or logical, that is, which is concretely not a compound of other things or
other concepts. So, for example, in a world in which exists a materially paradoxical omnipotence, its very
paradoxicality seems either to be a material-paradox-of-a-material-paradox, or to be a non-paradox per the
proposition that it exists (i.e., if it exists, then nothing has inherent meaning, including itself). Whereas, a world in
which exists non-paradoxical omnipotence, its own omnipotence is coextensive with whatever is the concrete basis
of our presupposition that something must be inherently meaningful.
The dilemma of omnipotence is similar to another classic paradox, the irresistible force paradox: What happens when
an irresistible force meets an immovable object? One response to this paradox is that if a force is irresistible, then,
by definition, there is no truly immovable object; conversely, if an immovable object were to exist, then no force
could be defined as being truly irresistible. Some claim that the only way out of this paradox is if the irresistible
force and the immovable object never meet. But, this way out is not possible in the omnipotence case, because the
purpose is to ask if the being's own inherent omnipotence makes its own inherent omnipotence impossible.
Moreover, an object cannot in principle be immovable, if there is a force which may move it, regardless of whether
the force and the object never meet. So, while, prior to any task, it is easy to imagine that omnipotence is in state of
coherence with itself, some imaginable tasks are not possible for such a coherent omnipotence to perform without
compromising its coherence.
Types of omnipotence
Peter Geach describes and rejects four levels of omnipotence. He also defines and defends a lesser notion of the
"almightiness" of God.
1. "Y is absolutely omnipotent" means that "Y" can do everything absolutely. Everything that can be expressed in
a string of words even if it can be shown to be self-contradictory, "Y"is not bound in action, as we are in thought
by the laws of logic."[3] This position is advanced by Descartes. It has the theological advantage of making God
prior to the laws of logic. Some claim that it in addition gives rise to the theological disadvantage of making
God's promises suspect. However, this claim is unfounded; for if God could do anything, then he could make it so
all of his promises are genuine, and do anything, even to the contrary, while they remain so. On this account, the
omnipotence paradox is a genuine paradox, but genuine paradoxes might nonetheless be so.
2. "Y is omnipotent" means "Y can do X" is true if and only if X is a logically consistent description of a state of
affairs. This position was once advocated by Thomas Aquinas.[4] This definition of omnipotence solves some of
the paradoxes associated with omnipotence, but some modern formulations of the paradox still work against this
definition. Let X = "to make something that its maker cannot lift". As Mavrodes points out there is nothing
logically contradictory about this; a man could, for example, make a boat which he could not lift.[5] It would be
strange if humans could accomplish this feat, but an omnipotent being could not. Additionally, this definition has
problems when X is morally or physically untenable for a being like God.
3. "Y is omnipotent" means "Y can do X" is true if and only if "Y does X" is logically consistent. Here the idea is
to exclude actions which would be inconsistent for Y to do but might be consistent for others. Again sometimes it
looks as if Aquinas takes this position.[6] Here Mavrodes' worry about X= "to make something its maker cannot
lift" will no longer be a problem because "God does X" is not logically consistent. However, this account may
still have problems with moral issues like X = "tells a lie" or temporal issues like X = "brings it about that Rome
was never founded."[3]
4. "Y is omnipotent" means whenever "Y will bring about X" is logically possible, then "Y can bring about X" is
true. This sense, also does not allow the paradox of omnipotence to arise, and unlike definition #3 avoids any
temporal worries about whether or not an omnipotent being could change the past. However, Geach criticizes
even this sense of omnipotence as misunderstanding the nature of God's promises.[3]
5. "Y is almighty" means that Y is not just more powerful than any creature; no creature can compete with Y in
power, even unsuccessfully.[3] In this account nothing like the omnipotence paradox arises, but perhaps that is
299
Omnipotence paradox
because God is not taken to be in any sense omnipotent. On the other hand, Anselm of Canterbury seems to think
that almightiness is one of the things that makes God count as omnipotent.[7]
St Augustine in his City of God writes "God is called omnipotent on account of His doing what He wills" and thus
proposes the definition that "Y is omnipotent" means "If Y wishes to do X then Y can and does do X".
The notion of omnipotence can also be applied to an entity in different ways. An essentially omnipotent being is an
entity that is necessarily omnipotent. In contrast, an accidentally omnipotent being is an entity that can be
omnipotent for a temporary period of time, and then becomes non-omnipotent. The omnipotence paradox can be
applied differently to each type of being.[8]
Some Philosophers, such as René Descartes, argue that God is absolutely omnipotent.[9] In addition, some
philosophers have considered the assumption that a being is either omnipotent or non-omnipotent to be a false
dilemma, as it neglects the possibility of varying degrees of omnipotence.[10] Some modern approaches to the
problem have involved semantic debates over whether language—and therefore philosophy—can meaningfully
address the concept of omnipotence itself.[11]
Proposed answers
A common response from Christian philosophers, such as Norman Geisler or Richard Swinburne is that the paradox
assumes a wrong definition of omnipotence. Omnipotence, they say, does not mean that God can do anything at all
but, rather, that he can do anything that's possible according to his nature. The distinction is important. God cannot
perform logical absurdities; he can't, for instance, make 1+1=3. Likewise, God cannot make a being greater than
himself because he is, by definition, the greatest possible being. God is limited in his actions to his nature. The Bible
supports this, they assert, in passages such as Hebrews 6:18 which says it is "impossible for God to lie." This raises
the question, similar to the Euthyphro Dilemma, of where this law of logic, which God is bound to obey, comes
from. According to these theologians, this law is not a law above God that he assents to but, rather, logic is an eternal
part of God's nature, like his omniscience or omnibenevolence. God obeys the laws of logic because God is eternally
logical in the same way that God doesn't perform evil actions because God is eternally good. So, God, by nature
logical and unable to violate the laws of logic, cannot make a boulder so heavy he cannot lift it because that would
violate the law of non contradiction by creating an immovable object and an unstoppable force. This is similar to the
Hebrews 6:18 verse, which teaches that God, by nature honest, cannot lie.
Another common response is that since God is supposedly omnipotent, the phrase "could not lift" doesn't make sense
and the paradox is meaningless.[12][13] This may mean that the complexity involved in rightly understanding
omnipotence---contra all the logical details involved in misunderstanding it---is a function of the fact that
omnipotence, like infinity, is perceived at all by contrasting reference to those complex and variable things which it
is not. But, an alternative meaning is that a non-corporeal God cannot lift anything, but can raise it (a linguistic
pedantry) - or to use the beliefs of Christians and Hindus (that there is one God, who can be manifest as several
different beings) that whilst it is possible for God to do all things, it is not possible for all his incarnations to do them.
As such, God could create a stone so heavy that, in one incarnation, he was unable to lift it - but would be able to do
something that an incarnation that could lift it couldn't.
Other responses claim that the question is sophistry, meaning it makes grammatical sense, but has no intelligible
meaning. The lifting a rock paradox (Can God lift a stone larger than he can carry?) uses human characteristics to
cover up the main skeletal structure of the question. With these assumptions made, two arguments can stem from it:
1. Lifting covers up the definition of translation, which means moving something from one point in space to
another. With this in mind, the real question would be, "Can God move a rock from one location in space to
another that is larger than possible?" In order for the rock to not be able to move from one space to another, it
would have to be larger than space itself. However, it is impossible for a rock to be larger than space, as space
will always adjust itself to cover the space of the rock. If the supposed rock was out of space-time dimension, then
the question would not make sense, because it would be impossible to move an object from one location in space
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Omnipotence paradox
to another if there is no space to begin with, meaning the faulting is with the logic of the question and not God's
capabilities.
2. The words, "Lift a Stone", are used instead to substitute capability. With this in mind, essentially the question is
asking if God is incapable, so the real question would be, "Is God capable of being incapable?" If God is capable
of being incapable, it means that He is incapable, because He has the potential to not be able to do something.
Conversely, if God is incapable of being incapable, then the two inabilities cancel each other out, making God
have the capability to do something.
The act of killing oneself is not applicable to an omnipotent being, since, despite that such an act does involve some
power, it also involves a lack of power: the human person who can kill himself is already not indestructible, and, in
fact, every agent constituting his environment is more powerful in some ways than himself. In other words, all
non-omnipotent agents are concretely synthetic: constructed as contingencies of other, smaller, agents, meaning that
they, unlike an omnipotent agent, logically can exist not only in multiple instantiation (by being constructed out of
the more basic agents of which they are made), but are each bound to a differentiated location in space contra
transcendent omnipresence.
Isaac Asimov, a confirmed atheist, answered a variation of this question: what happens when an irresistible force
meets an immovable object? He points out that Albert Einstein demonstrated the equivalence of mass-energy. That
is, according to relativity theory, mass is simply frozen energy, energy is simply liquid mass. In order to be either
"immovable" or "irresistible", the entity must possess the majority of energy in the system. No system can have two
majorities. A universe in which there exists such a thing as an irresistible force is, by definition, a universe which
cannot also contain an immovable object. And a universe which contains an immovable object cannot, by definition,
also contain an irresistible force. So the question is essentially meaningless: either the force is irresistible or the
object is immovable, but not both. Asimov points out that this question is the logical fallacy of the pseudo-question.
Just because we can string words together to form what looks like a coherent sentence does not mean the sentence
really makes any sense.
Thomas Aquinas asserts that the paradox arises from a misunderstanding of omnipotence. He maintains that inherent
contradictions and logical impossibilities do not fall under the omnipotence of God.[14] J. L Cowan sees this paradox
as a reason to reject the concept of 'absolute' omnipotence,[15] while others, such as René Descartes, argue that God
is absolutely omnipotent, despite the problem.[9]
C. S. Lewis argues that when talking about omnipotence, referencing "a rock so heavy that God cannot lift it" is
nonsense just as much as referencing "a square circle"; that it is not logically coherent in terms of power to think that
omnipotence includes the power to do the logically impossible. So asking "Can God create a rock so heavy that even
he cannot lift it?" is just as much nonsense as asking "Can God draw a square circle?" The logical contradiction here
being God's simultaneous ability and disability in lifting the rock: the statement "God can lift this rock" must have a
truth value of either true or false, it cannot possess both. This is justified by observing that in order for the
omnipotent agent to create such a stone, the omnipotent agent must already be more powerful than itself: such a
stone is too heavy for the omnipotent agent to lift, but the omnipotent agent already can create such a stone; If an
omnipotent agent already is more powerful than itself, then it already is just that powerful. Which means that its
power to create a stone that’s too heavy for it to lift is identical to its power to lift that very stone. While this doesn’t
quite make complete sense, Lewis wished to stress its implicit point: that even within the attempt to prove that the
concept of omnipotence is immediately incoherent, one admits that it is immediately coherent, and that the only
difference is that this attempt if forced to admit this despite that the attempt is constituted by a perfectly irrational
route to its own unwilling end, with a perfectly irrational set of 'things' included in that end. In other words, that the
'limit' on what omnipotence 'can' do is not a limit on its actual agency, but an epistemological boundary without
which omnipotence could not be identified (paradoxically or otherwise) in the first place. In fact, this process is
merely a fancier form of the classic Liar ParadoxA: If I say, "I am a liar", then how can it be true if I am telling the
truth therewith, and, if I am telling the truth therewith, then how can I be a liar? So, to think that omnipotence is an
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epistemological paradox is like failing to recognize that, when taking the statement, 'I am a liar' self-referentially, the
statement is reduced to an actual failure to lie. In other words, if one maintains the supposedly 'initial' position that
the necessary conception of omnipotence includes the 'power' to compromise both itself and all other identity, and if
one concludes from this position that omnipotence is epistemologically incoherent, then one implicitly is asserting
that one's own 'initial' position is incoherent. Therefore the question (and therefore the perceived paradox) is
meaningless. Nonsense does not suddenly acquire sense and meaning with the addition of the two words, "God can"
before it.[12] Lewis additionally said that "unless something is self-evident, nothing can be proved", which implies
for the debate on omnipotence that, as in matter, so in the human understanding of truth: it takes no true insight to
destroy a perfectly integrated structure, and the effort to destroy has greater effect than an equal effort to build; so,
a man is thought a fool who assumes its integrity, and thought an abomination who argues for it. It is easier to teach
a fish to swim in outer space than to convince a room full of ignorant fools why it cannot be done.
John Christian Uy said that it is just the same as someone with double-bladed sword (accidentally omnipotent), or
sword and a shield (essentially omnipotent). Therefore, an accidentally omnipotent deity CAN remove its
omnipotence while an essentially omnipotent deity CANNOT do anything that would make it non-omnipotent. Both
however, have no limitations so far other than the essential omnipotent being who cannot do anything which will
make it non-omnipotent like making someone equal with him, lowering or improving himself(for omnipotence is the
highest) etc. It could, however, make someone with a great power, though it cannot be 99% because Omnipotence is
infinite, because that created being is not equal with him. Overall, God in the Christian Bible, is essentially
omnipotent.
William Jennings Bryan said this is roughly the view espoused by Matthew Harrison Brady, a character in the 1955
play Inherit the Wind loosely based upon William Jennings Bryan. In the climactic scene of the 1960 movie version,
Brady argues, "Natural law was born in the mind of the Creator. He can change it—cancel it—use it as he pleases!"
But this solution merely pushes the problem back a step; one may ask whether an omnipotent being can create a
stone so immutable that the being itself cannot later alter it. But a similar response can be offered to respond to this
and any further steps.
In a 1955 article published in the philosophy journal Mind, J. L. Mackie attempted to resolve the paradox by
distinguishing between first-order omnipotence (unlimited power to act) and second-order omnipotence (unlimited
power to determine what powers to act things shall have).[16] An omnipotent being with both first and second-order
omnipotence at a particular time might restrict its own power to act and, henceforth, cease to be omnipotent in either
sense. There has been considerable philosophical dispute since Mackie, as to the best way to formulate the paradox
of omnipotence in formal logic.[17]
Another common response to the omnipotence paradox is to try to define omnipotence to mean something weaker
than absolute omnipotence, such as definition 3 or 4 above. The paradox can be resolved by simply stipulating that
omnipotence does not require the being to have abilities which are logically impossible, but only to be able to do
anything which conforms to the laws of logic. A good example of a modern defender of this line of reasoning is
George Mavrodes.[5] Essentially, Mavrodes argues that it is no limitation on a being's omnipotence to say that it
cannot make a round square. Such a "task" is termed by him a "pseudo-task" as it is self-contradictory and inherently
nonsense. Harry Frankfurt—following from Descartes—has responded to this solution with a proposal of his own:
that God can create a stone impossible to lift and also lift said stone
For why should God not be able to perform the task in question? To be sure, it is a task—the task of
lifting a stone which He cannot lift—whose description is self-contradictory. But if God is supposed
capable of performing one task whose description is self-contradictory—that of creating the problematic
stone in the first place—why should He not be supposed capable of performing another—that of lifting
the stone? After all, is there any greater trick in performing two logically impossible tasks than there is
in performing one?[18]
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Omnipotence paradox
If a being is accidentally omnipotent, then it can resolve the paradox by creating a stone which it cannot lift and
thereby becoming non-omnipotent. Unlike essentially omnipotent entities, it is possible for an accidentally
omnipotent being to be non-omnipotent. This raises the question, however, of whether or not the being was ever
truly omnipotent, or just capable of great power.[8] On the other hand, the ability to voluntarily give up great power
is often thought of as central to the notion of the Christian Incarnation.[19]
If a being is essentially omnipotent, then it can also resolve the paradox (as long as we take omnipotence not to
require absolute omnipotence). The omnipotent being is essentially omnipotent, and therefore it is impossible for it
to be non-omnipotent. Further, the omnipotent being can do what is logically impossible and have no limitations just
like the accidentally omnipotent but the ability to make oneself non-omnipotent. The creation of a stone which the
omnipotent being cannot lift would be an impossibility. The omnipotent being cannot create such a stone because its
power will be equal to him and thus, remove his omnipotence for there can only be one omnipotent being in
existence, but nevertheless retains its omnipotence. This solution works even with definition 2, as long as we also
know the being is essentially omnipotent rather than accidentally so. However, it is possible for non-omnipotent
beings to compromise their own powers, which presents the paradox that non-omnipotent beings can do something
(to themselves) which an essentially omnipotent being cannot do (to itself).
This was essentially the position taken by Augustine of Hippo in his The City of God:
For He is called omnipotent on account of His doing what He wills, not on account of His suffering what He
wills not; for if that should befall Him, He would by no means be omnipotent. Wherefore, He cannot do some
things for the very reason that He is omnipotent.[20]
Thus Augustine argued that God could not do anything or create any situation that would in effect make God not
God.
Some philosophers maintain that the paradox can be resolved if the definition of omnipotence includes Descartes'
view that an omnipotent being can do the logically impossible. In this scenario, the omnipotent being could create a
stone which it cannot lift, but could also then lift the stone anyway. Presumably, such a being could also make the
sum 2 + 2 = 5 become mathematically possible or create a square triangle. This attempt to resolve the paradox is
problematic in that the definition itself forgoes logical consistency. The paradox may be solved, but at the expense of
making the logic a paraconsistent logic. This might not seem like a problem if one is already committed to
dialetheism or some other form of logical transcendence.
St Augustine's definition of omnipotence, i.e. that God can do and does everything that God wishes, resolves all
possible paradoxes, because God, being perfectly rational, never wishes to do something that is paradoxical.
If God can do absolutely anything, then God can remove His own omnipotence. If God can remove His own
omnipotence, then God can create an enormous stone, remove His own omnipotence, then not be able to lift the
stone. This preserves the belief that God is omnipotent because God can create a stone that He couldn't lift.
Therefore, in this theory, God would not be omnipotent while not being able to lift the stone. This is a trivial solution
because, for example, an omnipotent being could create a boulder that the strongest human could not lift (it needn't
do that anyway since such boulders exist) and then give itself the potency of an average human; it would then not be
able to lift the stone. This solves nothing as the entity that is unable to lift the stone is not "God" as understood by the
paradox, but a very average being with the same potency as a human. The solution only produces a reduced-potency
"God"; it does not deal with the matter at hand: God maintaining omnipotence even while performing a task, the
success or failure of which seems to imply impotence.
David Hemlock has proposed an incarnational resolution: "On one small planet, lying in a manger, one incarnate
babe could not lift the rocks He had made. All the rocks of all of the starfields in Him consist, with their whirling
atoms; by Him were and ever-are all things lifted up (Col 1:17; Phil 2:5-8)".[21]
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Omnipotence paradox
Language and omnipotence
The philosopher Ludwig Wittgenstein is often interpreted as arguing that language is not up to the task of describing
the kind of power an omnipotent being would have. In his Tractatus Logico-Philosophicus he stays generally within
the realm of logical positivism, until claim 6.4, but at 6.41 and following the succeeding propositions argue that
ethics and several other issues are "transcendental" subjects which we cannot examine with language. Wittgenstein
also mentions the will, life after death, and God; arguing that "When the answer cannot be put into words, neither
can the question be put into words".[22]
Wittgenstein's work makes the omnipotence paradox a problem in semantics, the study of how symbols are given
meaning. (The retort "That's only semantics" is a way of saying that a statement only concerns the definitions of
words, instead of anything important in the physical world.) According to the Tractatus, then, even attempting to
formulate the omnipotence paradox is futile, since language cannot refer to the entities the paradox considers. The
final proposition of the Tractatus gives Wittgenstein's dictum for these circumstances: "What we cannot speak of, we
must pass over in silence".[23] Wittgenstein's approach to these problems is influential among other 20th century
religious thinkers such as D. Z. Phillips.[24]
But in his later years, Wittgenstein wrote works which are often interpreted as conflicting with his positions in the
Tractatus,[25] and indeed the later Wittgenstein is mainly seen as the leading critic of the early Wittgenstein.
Other versions of the paradox
In the 6th century, Pseudo-Dionysius claims that a version of the omnipotence paradox constituted the dispute
between St. Paul and Elmyas the Magician mentioned in Acts 13:8, but it is phrased in terms of a debate as to
whether or not God can "deny himself" ala 2 Tim 2:13.[26] In the 11th century, St. Anselm argues that there are many
things that God cannot do, but that nonetheless he counts as omnipotent.[27]
Thomas Aquinas advanced a version of the omnipotence paradox by asking whether God could create a triangle with
internal angles that did not add up to 180 degrees. As Aquinas put it in Summa contra Gentiles:
Since the principles of certain sciences, such as logic, geometry and arithmetic are taken only from the formal
principles of things, on which the essence of the thing depends, it follows that God could not make things
contrary to these principles. For example, that a genus was not predicable of the species, or that lines drawn
from the centre to the circumference were not equal, or that a triangle did not have three angles equal to two
right angles.[28]
This can be done on a sphere, and not on a flat surface. The later invention of non-Euclidean geometry does not
resolve this question; for one might as well ask, "If given the axioms of Riemannian geometry, can an omnipotent
being create a triangle whose angles do not add up to more than 180 degrees?" In either case, the real question is
whether or not an omnipotent being would have the ability to evade the consequences which follow logically from a
system of axioms that the being created.
A version of the paradox can also be seen in non-theological contexts. A similar problem occurs when accessing
legislative or parliamentary sovereignty, which holds a specific legal institution to be omnipotent in legal power, and
in particular such an institution's ability to regulate itself.[29]
In a sense, the classic statement of the omnipotence paradox — a rock so heavy that its omnipotent creator cannot
lift it — is grounded in Aristotelian science. After all, if one considers the stone's position relative to the sun around
which the planet orbits, one could hold that the stone is constantly being lifted—strained though that interpretation
would be in the present context. Modern physics indicates that the choice of phrasing about lifting stones should
relate to acceleration; however, this does not in itself of course invalidate the fundamental concept of the generalized
omnipotence paradox. However, one could easily modify the classic statement as follows: "An omnipotent being
creates a universe which follows the laws of Aristotelian physics. Within this universe, can the omnipotent being
create a stone so heavy that the being cannot lift it?"
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Omnipotence paradox
Ethan Allen's Reason addresses the topics of original sin, theodicy and several others in classic Enlightenment
fashion.[30] In Chapter 3, section IV, he notes that "omnipotence itself" could not exempt animal life from mortality,
since change and death are defining attributes of such life. He argues, "the one cannot be without the other, any more
than there could be a compact number of mountains without valleys, or that I could exist and not exist at the same
time, or that God should effect any other contradiction in nature." Labeled by his friends a Deist, Allen accepted the
notion of a divine being, though throughout Reason he argues that even a divine being must be circumscribed by
logic.
In Principles of Philosophy, Descartes tried refuting the existence of atoms with a variation of this argument,
claiming God could not create things so indivisible that he could not divide them.
It is even in popular culture. In an episode of The Simpsons, Homer asks Ned Flanders the question "Could Jesus
microwave a burrito so hot that He Himself could not eat it?" In one strip of the webcomic Saturday Morning
Breakfast Cereal, a child is seen asking a priest "Could God make an argument so circular that even He couldn't
believe it?"
In the Marvel Comics Runaways, Victor Mancha, the technorganic android created by Ultron, is shown as unable to
process correctly paradoxes: as such, it's known that a small number of well known paradoxes may force his logic in
a permanent loop, shutting his functions down until someone steps in to give Victor the proper solution. As such, his
peers stop him once by asking "Could God make a sandwich so big that even he couldn't finish it?", and reboot his
mind by explaining him a simplified version of the God as essentially omnipotent solution ("Yes. God could make a
sandwich so big that even he couldn't finish it, and eat it all").
In the book' 'Bart Simpson's guide to Life this question is phrased as if God can do anything, could he create a hot
dog so big that even he couldn't eat it.
Notes
[1]
[2]
[3]
[4]
[5]
Savage, C. Wade. "The Paradox of the Stone" Philosophical Review, Vol. 76, No. 1 (Jan., 1967), pp. 74–79 doi:10.2307/2182966
Averroës, Tahafut al-Tahafut (The Incoherence of the Incoherence) trans. Simon Van Der Bergh, Luzac & Company 1969, sections 529–536
Geach, P. T. "Omnipotence" 1973 in Philosophy of Religion: Selected Readings, Oxford University Press, 1998, pp. 63–75
Aquinas, Thomas Summa Theologica Book 1 Question 25 article 3
Mavrodes, George. " Some Puzzles Concerning Omnipotence (http:/ / spot. colorado. edu/ ~kaufmad/ courses/ Mavrodes. pdf)" first
published 1963 now in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University
Press 1978 pp. 131–34
[6] Aquinas Summa Theologica Book 1 Question 25 article 4 response #3
[7] Anselm of Canterbury Proslogion Chap VII in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass Walton
eds. Oxford University Press 1978 pp. 35–36
[8] Hoffman, Joshua, Rosenkrantz, Gary. "Omnipotence" (http:/ / plato. stanford. edu/ archives/ sum2002/ entries/ omnipotence/ ) The Stanford
Encyclopedia of Philosophy (Summer 2002 Edition). Edward N. Zalta (ed.). (Accessed on 19 April 2006)
[9] Descartes, Rene, 1641. Meditations on First Philosophy. Cottingham, J., trans., 1996. Cambridge University Press. Latin original. Alternative
English title: Metaphysical Meditations. Includes six Objections and Replies. A second edition published the following year, includes an
additional ‘’Objection and Reply’’ and a Letter to Dinet
[10] Haeckel, Ernst. The Riddle of the Universe. Harper and Brothers, 1900.
[11] Wittgenstein, Ludwig. Tractatus Logico-Philosophicus (6.41 and following)
[12] The Problem of Pain, Clive Staples Lewis, 1944 MacMillan
[13] Loving Wisdom: Christian Philosophy of Religion by Paul Copan, Chalice Press, 2007 page 46
[14] "Summa Theologica" (http:/ / www. ccel. org/ a/ aquinas/ summa/ FP/ FP025. html#FPQ25A3THEP1). Ccel.org. . Retrieved 2012-05-10.
[15] Cowan, J. L. "The Paradox of Omnipotence" first published 1962, in The Power of God: Readings on Omnipotence and Evil. Linwood
Urban and Douglass Walton eds. Oxford University Press 1978 pp. 144–52
[16] Mackie, J. L., "Evil and Omnipotence." Mind LXIV, No, 254 (April 1955).
[17] The Power of God: Readings on Omnipotence and Evil. Linwood Urban and Douglass Walton eds. Oxford University Press 1978. Keene
and Mayo disagree p. 145, Savage provides 3 formalizations p. 138–41, Cowan has a different strategy p. 147, and Walton uses a whole
separate strategy p. 153–63
[18] Frankfurt, Harry. "The Logic of Omnipotence" first published in 1964 in Philosophical Review and now in Necessity, Volition, and Love.
Cambridge University Press November 28, 1998 pp.1–2
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Omnipotence paradox
[19] Gore, Charles, "A Kenotic Theory of Incarnation" first published 1891, in The Power of God: readings on Omnipotence and Evil. Linwood
Urban and Douglass Walton eds. Oxford University Press 1978 pp. 165–68
[20] City of God, Book 5, Chapter 10 (http:/ / www. ccel. org/ ccel/ schaff/ npnf102. iv. V. 10. html)
[21] "Can God Make a Stone He Cannot Lift? | katachriston" (http:/ / katachriston. wordpress. com/ 2011/ 04/ 01/
can-god-make-a-stone-he-cannot-lift/ ). Katachriston.wordpress.com. 2011-04-01. . Retrieved 2012-05-10.
[22] Wittgenstein, Ludwig. proposition 6.5
[23] Wittgenstein, Ludwig. proposition 7
[24] D. Z. Phillips "Philosophy, Theology and the Reality of God" in Philosophy of Religion: Selected Readings. William Rowe and William
Wainwright eds. 3rd ed. 1998 Oxford University Press
[25] Hacker, P.M.S. Wittgenstein's Place in Twentieth-Century Analytic Philosophy. 1996 Blackwell
[26] Pseudo-Dionysius, "Divine Names" 893B in Pseudo-Dionysius: The Complete Works. trans Colm Luibheid Paulist Press. 1987. ISBN
0-8091-2838-1
[27] Anselm of Canterbury Proslogion Chap. VII, in The Power of God: readings on Omnipotence and Evil. Linwood Urban and Douglass
Walton eds. Oxford University Press 1978 pp. 35–36
[28] "Cum principia quarundam scientiarum, ut logicae, geometriae et arithmeticae, sumantur ex solis principiis formalibus rerum, ex quibus
essentia rei dependet, sequitur quod contraria horum principiorum Deus facere non possit: sicut quod genus non sit praedicabile de specie; vel
quod lineae ductae a centro ad circumferentiam non sint aequales; aut quod triangulus rectilineus non habeat tres angulos aequales duobus
rectis". Aquinas, T. Summa Contra Gentiles, Book 2, Section 25. trans. Edward Buckner
[29] Suber, P. (1990) The Paradox of Self-Amendment: A Study of Law, Logic, Omnipotence, and Change (http:/ / www. earlham. edu/ ~peters/
writing/ psa/ sec01. htm#C). Peter Lang Publishing
[30] Allen, Ethan. Reason: The Only Oracle of Man. (http:/ / libertyonline. hypermall. com/ allen-reason. html) J.P. Mendum, Cornill; 1854.
Originally published 1784. (Accessed on 19 April 2006)
References
• Allen, Ethan. Reason: The Only Oracle of Man. (http://libertyonline.hypermall.com/allen-reason.html) J.P.
Mendum, Cornill; 1854. Originally published 1784. (Accessed on 19 April 2006)
• Augustine. City of God and Christian Doctrine. (http://www.ccel.org/ccel/schaff/npnf102.html) The
Christian Literature Publishing Co., 1890. (Accessed on 26 September 2006)
• Burke, James. The Day the Universe Changed. Little, Brown; 1995 (paperback edition). ISBN 0-316-11704-8.
• Gleick, James. Genius. Pantheon, 1992. ISBN 0-679-40836-3.
• Haeckel, Ernst. The Riddle of the Universe. Harper and Brothers, 1900.
• Hoffman, Joshua, Rosenkrantz, Gary. "Omnipotence" (http://plato.stanford.edu/archives/sum2002/entries/
omnipotence/) The Stanford Encyclopedia of Philosophy (Summer 2002 Edition). Edward N. Zalta (ed.).
(Accessed on 19 April 2006)
• Mackie, J. L., "Evil and Omnipotence." Mind LXIV, No, 254 (April 1955).
• Wierenga, Edward. "Omnipotence" The Nature of God: An Inquiry into Divine Attributes (http://www.courses.
rochester.edu/wierenga/REL111/omnipch.html). Cornell University Press, 1989. (Accessed on 19 April 2006)
• Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. Available online (http://www.gutenberg.org/etext/
5740) via Project Gutenberg. Accessed 19 April 2006.
306
Paradox of hedonism
Paradox of hedonism
The paradox of hedonism, also called the pleasure paradox, is the idea in the study of ethics which points out that
pleasure and happiness are strange phenomena that do not obey normal principles. First explicitly noted by the
philosopher Henry Sidgwick in The Methods of Ethics, the paradox of hedonism points out that pleasure cannot be
acquired directly, it can only be acquired indirectly.
Overview
It is often said that we fail to attain pleasures if we deliberately seek them. This has been described variously, by
many:
• John Stuart Mill, the utilitarian philosopher, in his autobiography:
But I now thought that this end [one's happiness] was only to be attained by not making it the direct end.
Those only are happy (I thought) who have their minds fixed on some object other than their own
happiness[....] Aiming thus at something else, they find happiness along the way[....] Ask yourself
whether you are happy, and you cease to be so.[1]
• Viktor Frankl in Man's Search for Meaning:
Happiness cannot be pursued; it must ensue, and it only does so as the unintended side effect of one's
personal dedication to a cause greater than oneself or as the by-product of one's surrender to a person
other than oneself.
The more a man tries to demonstrate his sexual potency or a woman her ability to experience orgasm,
the less they are able to succeed. Pleasure is, and must remain, a side-effect or by-product, and is
destroyed and spoiled to the degree to which it is made a goal in itself.[2]
• Philosopher Søren Kierkegaard in Either/Or:
Most men pursue pleasure with such breathless haste that they hurry past it.[3]
• Philosopher Friedrich Nietzsche in The Antichrist (1895) and The Will to Power (1901):
What is good? Everything that heightens the feeling of power in man, the will to power, power itself.
What is bad? Everything that is born of weakness.
What is happiness? The feeling that power increases — that a resistance is overcome.[4]
[...] it is significantly enlightening to substitute for the individual 'happiness' (for which every living
being is supposed to strive) power [...] joy is only a symptom of the feeling of attained power [...] (one
does not strive for joy [...] joy accompanies; joy does not move)[5]
• Psychologist Alfred Adler in The Neurotic Constitution (1912):
Nietzsche's "will to power" and "will to seem" embrace many of our views, which again resemble in
some respects the views of Féré and the older writers, according to whom the sensation of pleasure
originates in a feeling of power, that of pain in a feeling of feebleness.[6]
• Poet and satirist Edward Young:
The love of praise, howe'er concealed by art,
Reigns more or less supreme in every heart;
The Proud to gain it, toils on toils endure;
The modest shun it, but to make it sure![7]
• Politician William Bennett:
Happiness is like a cat, If you try to coax it or call it, it will avoid you; it will never come. But if you pay
no attention to it and go about your business, you'll find it rubbing against your legs and jumping into
307
Paradox of hedonism
your lap.
• Novelist João Guimarães Rosa:
Happiness is found only in little moments of inattention.[8]
Example
Suppose Paul likes to collect stamps. According to most models of behavior, including not only utilitarianism, but
most economic, psychological and social conceptions of behavior, it is believed that Paul likes collecting stamps
because he gets pleasure from collecting stamps. Stamp collecting is an avenue towards acquiring pleasure.
However, if you tell Paul this, he will likely disagree. He does get pleasure from collecting stamps, but this is not the
process that explains why he collects stamps. It is not as though he says, "I must collect stamps so I, Paul, can obtain
pleasure". Collecting stamps is not just a means toward pleasure. He just likes collecting stamps.
This paradox is often spun around backwards, to illustrate that pleasure and happiness cannot be reverse-engineered.
If for example you heard that collecting stamps was very pleasurable, and began a stamp collection as a means
towards this happiness, it would inevitably be in vain. To achieve happiness, you must not seek happiness directly,
you must strangely motivate yourself towards things unrelated to happiness, like the collection of stamps.
The hedonistic paradox would probably mean that if one sets the goal to please oneself too highly then the
mechanism would in fact jam itself.
Suggested explanations
Happiness is often imprecisely equated with pleasure. If, for whatever reason, one does equate happiness with
pleasure, then the paradox of hedonism arises. When one aims solely towards pleasure itself, one's aim is frustrated.
Henry Sidgwick comments on such frustration after a discussion of self-love in the above-mentioned work:
I should not, however, infer from this that the pursuit of pleasure is necessarily self-defeating and futile;
but merely that the principle of Egoistic Hedonism, when applied with a due knowledge of the laws of
human nature, is practically self-limiting; i.e., that a rational method of attaining the end at which it aims
requires that we should to some extent put it out of sight and not directly aim at it.[9]
While not addressing the paradox directly, Aristotle commented on the futility of pursuing pleasure. Human beings
are actors whose endeavors bring about consequences, and among these is pleasure. Aristotle then argues as follows:
How, then, is it that no one is continuously pleased? Is it that we grow weary? Certainly all human
things are incapable of continuous activity. Therefore pleasure also is not continuous; for it accompanies
activity.[10]
Sooner or later, finite beings will be unable to acquire and expend the resources necessary to maintain their sole goal
of pleasure; thus, they find themselves in the company of misery. Evolutionary theory explains that humans evolved
through natural selection and follow genetic imperatives that seek to maximize reproduction[11], not happiness. As a
result of these selection pressures, the extent of human happiness is limited biologically. David Pearce argues in his
treatise The Hedonistic Imperative that humans might be able to use genetic engineering, nanotechnology, and
neuroscience to eliminate suffering in all sentient life.
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Paradox of hedonism
References
[1] John Stuart Mill, Autobiography in The Harvard Classics, Vol. 25, Charles Eliot Norton, ed. (New York: P. F. Collier & Son Company, 1909
(p. 94)
[2] Viktor Frankl. Man's Search for Meaning.
[3] Søren Kierkegaard. Either/Or. Diapsalmata
[4] The Antichrist, § 2
[5] The Will to Power, § 688
[6] Adler, Alfred (1912). The Neurotic Constitution (http:/ / www. archive. org/ details/ neuroticconstitu00adle). New York: Moffat, Yard and
Company. pp. ix. .
[7] Geoffrey Brennan. The Esteem Engine: A Resource for Institutional Design (http:/ / www. assa. edu. au/ publications/ occasional_papers/
2005_No1. php)
[8] Rosa, Guimarães. Tutaméia – Terceiras Estórias (8.a ed.). Rio de Janeiro: Ed. Nova Fronteira, 2001, p. 60.
[9] Henry Sidgwick. The Methods of Ethics. BookSurge Publishing (1 Mar 2001) (p. 3)
[10] Aristotle. Nicomachean Ethics, (Written 350 B.C.E) Book X, page 4 (http:/ / classics. mit. edu/ Aristotle/ nicomachaen. 10. x. html)
[11] Raymond Bohlin. "Sociobiology: Evolution, Genes and Morality" (http:/ / www. leaderu. com/ orgs/ probe/ docs/ sociobio. html). .
Retrieved 2007-01-03.
Further reading
• Aristotle, Nicomachean Ethics 1175, 3-6 in The Basic Works of Aristotle, Richard McKeon ed. (New York:
Random House, 1941)
• John Stuart Mill, Autobiography in The Harvard Classics, Vol. 25, Charles Eliot Norton, ed. (New York: P. F.
Collier & Son Company, 1909)
• Henry Sidgwick, The Methods of Ethics (London: Macmillan & Co. Ltd., 1874/1963)
Paradox of nihilism
Paradox of nihilism is the name of several paradoxes.
Meaning
According to Hegarty, the paradox of nihilism is "that the absence of meaning seems to be some sort of meaning".[1]
Truth
Niklas Luhmann construes the paradox as stating "that consequently, only the untrue could be the truth".[2] In a
footnote in his PhD thesis, Slocombe equates nihilism with the liar paradox.[3]
Religion
Rivas locates the paradox in the "conservative attitude of Roman Catholicism" developed in reaction to Nietzschean
nihilism, in that it "betrays a form of nihilism, that is, the forced oblivion of the real ambiguity and the paradox that
inform the distinction between the secular and the sacred".[4]
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Paradox of nihilism
Critical legal theory
In Critical Legal Studies (CLS) theory, the arguments used to criticize the centrist position also undermine the
position of CLS.[5]
Ethics
According to Jonna Bornemark, "the paradox of nihilism is the choice to continue one's own life while at the same
time stating that it is not worth more than any other life".[6] Richard Ian Wright sees relativism as the root of the
paradox.[7]
References
[1] Hegarty, Paul (2006). "Noise Music" (http:/ / scirus. com/ srsapp/ sciruslink?src=web& url=http:/ / www. chass. toronto. edu/ epc/ srb/
vol%2016. 1. pdf) (PDF). The Semiotic Review of Books (955 Oliver Road, Thunder Bay, Ontario, Canada P7B 5E1: Department of
Sociology, Lakehead University) 16 (1-2): 2. ISSN 0847-1622. . Retrieved 4 April 2010. "Failure/impossibility: noise is only ever defined
against something else, operating in the absence of meaning, but caught in the paradox of nihilism – that the absence of meaning seems to be
some sort of meaning."
[2] Luhmann, Niklas (1 February 2002). "2 The Modernity of Science" (http:/ / books. google. com/ books?id=L0G2bwV9VMMC& pg=PA64).
In Rasch, William; Bal, Mieke; de Vries, Hent. Theories of Distinction: Redescribing the Descriptions of Modernity. Cultural memory in the
present. William Rasch (introduction), translations by Joseph O'Neil, Elliott Schreiber, Kerstin Behnke, William Whobrey. Stanford
University Press. p. 64. ISBN 978-0-8047-4123-1. OCLC 318368737. . Retrieved 4 April 2010. "... the loss of reference had to appear as a
loss of truth, resulting in the paradox of “nihilism,” which states that consequently only the untrue could be truth."
The quoted chapter was originally published as Luhmann, Niklas; Behnke, Kerstin (1994). Oppenheimer, Andrew. ed. "The Modernity of
Science". New German Critique (New York: TELOS Press) (61 Special Issue on Niklas Luhmann (Winter 1994)): 9–23. ISSN 1558-1462.
JSTOR 488618. OCLC 50709608.
There is a 2002 review of the book (http:/ / www. cjsonline. ca/ reviews/ luhmann. html) in Canadian Journal of Sociology Online.
[3] Slocombe, Will (September 2003). Postmodern Nihilism: Theory and Literature. PhD theses from Aberystwyth University. University of
Wales Aberystwyth. pp. 154. hdl:2160/267. "'There is no truth' is not inherently paradoxical. If it is considered true, then it creates a paradox
because it is therefore false. However, if it is considered false, then no such paradox exists. Therefore, it is only when considered true that it
creates a paradox, in much the same way as critics suggest that nihilism must be invalid for this very reason. Having now introduced this
stronger formulation of nihilism, from this point on nihilism can be considered equivalent to the statement that 'This sentence is not true'.
(footnote 110)"
A revised version was published as Slocombe, Will (2006). Nihilism and the Sublime Postmodern : The (Hi)Story of a Difficult Relationship
From Romanticism to Postmodernism (http:/ / books. google. com/ books?id=VLx7QgAACAAJ). Routledge. ISBN 978-0-415-97529-2.
OCLC 62281496. . Retrieved 5 April 2010.
[4] Rivas, Virgilio Aquino (2008). "The Role of the Church in the Politics of Social Transformation: The Paradox of Nihilism" (http:/ / www.
politicsandreligionjournal. com/ PDF/ broj 4/ rivas. pdf) (PDF). Политикологија религије (Politics and Religion) (11000 Beograd (Belgrade,
Serbia): Centar za proučavanje religije i versku toleranciju, 27.marta 95 (Center for Studies of Religion and Religious Tolerance)) II (2):
53–77. ISSN 1820-659X. . Retrieved 4 April 2010.
[5] Belliotti, Raymond A. (1987). "critical legal studies: the paradoxes of indeterminacy and nihilism" (http:/ / psc. sagepub. com/ cgi/ reprint/
13/ 2/ 145. pdf) (PDF). Philosophy & Social Criticism 13 (2): 145–154. doi:10.1177/019145378701300203. . Retrieved 4 April 2010. "...
Critical Legal Studies Movement (CLS) ... CLS' view generates a "paradox of nihilism" which CLS has recognized and tried unsuccessfully to
resolve." (subscription required)
Belliotti, Raymond A. (25 January 1994). Justifying Law: The Debate Over Foundations, Goals, and Methods (http:/ / books. google. com/
books?id=YK0VEi-S8m4C& pg=PA169). Temple University Press. p. 169. ISBN 978-1-56639-203-7. . Retrieved 4 April 2010. "The
argument supporting CLS' attack on centrist ideology, adhering as it does to social contingency, jurisprudential indeterminacy, and pervasive
conditionality flowing from the fundamental contradiction, seems to preclude CLS from establishing a normative justification for its own
vision. CLS' critical attack seems to cut the heart from all efforts to provide non-question-begging adjudication of epistemological and moral
truth claims. This nihilistic paradox, in which CLS' critical attack is so extreme that it prohibits CLS from constructing persuasively its own
alternative vision, ..."
[6] Bornemark, Jonna (2006). "Limit-situation: Antinomies and Transcendence in Karl Jaspers’ Philosophy" (http:/ / cjas. dk/ index. php/ sats/
article/ viewArticle/ 703) (PDF). sats - Nordic Journal of Philosophy 7 (2). ISSN 1600-1974. . Retrieved 5 April 2010. (subscription
required)
[7] Wright, Richard Ian (April 1994) (PDF). The Dream of Enlightenment: An Essay on the Promise of Reason and Freedom in Modernity
(https:/ / circle. ubc. ca/ handle/ 2429/ 9168). University of British Columbia. . Retrieved 5 April 2010. "But essentially these values can be
negated by extending the same critical methods which Marx uses to negate earlier philosophical idealism and liberal bourgeois ideology. In
other words, from a Nietzschean perspective, Marx's foundational principles are not sufficient to defend his humanistic values. Thus it can be
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argued that they still maintain the residue of the Christian ethics and Platonic metaphysics which have permeated western thought for several
thousand years and which continue to provide modem thinkers with many of their illusory presuppositions. Nevertheless, one is justified in
asking: Without such presuppositions, does not the critique of law, politics, or "this earth" lose its ultimate justification or meaning? How can
one critique laws without holding on to a sense of justice? And herein lies the crux of the paradox of nihilism. If nihilism is the basis of human
existence then all values are relative, and as such, particular values can only be maintained through a "will to power." (page 97)"
Paradox of tolerance
The tolerance paradox arises from a problem that a tolerant person might be antagonistic toward intolerance, hence
intolerant of it. The tolerant individual would then be by definition intolerant of intolerance.
Karl Popper[1] and John Rawls,[2] have discussed this paradox.
References
[1] Karl Popper, The Open Society and Its Enemies, Vol. 1, Notes to the Chapters: Ch. 7, Note 4
[2] J. Rawls, A Theory of Justice, Harvard University Press, 1971, p. 216 (http:/ / books. google. com/ books?id=TdvHKizvuTAC& pg=PA216&
lpg=PA216#v=onepage& q=& f=false)
External links
• The Concept of Toleration and its Paradoxes (http://plato.stanford.edu/entries/toleration/#ConTolPar), in
Stanford Encyclopedia of Philosophy.
• The Paradoxes of Tolerance (http://www.eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/
detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=EJ707855&
ERICExtSearch_SearchType_0=no&accno=EJ707855), Barbara Pasamonik in Social Studies, v95 n5 p206
Sep-Oct 2004.
• "Puzzles and Paradoxes of Tolerance" (http://books.google.com/books?id=S1j8KXp20qwC&pg=PA9&
lpg=PA9&dq#v=onepage&q=&f=false), Hans Oberdiek, 2001.
• Tolerating the Intolerant (http://www.michaeltotten.com/archives/2006/07/tolerating-the-intolerant.php),
Michael Totten.
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Predestination paradox
Predestination paradox
A predestination paradox (also called causal loop, causality loop, and, less frequently, closed loop or closed time
loop) is a paradox of time travel that is often used as a convention in science fiction. It exists when a time traveler is
caught in a loop of events that "predestines" or "predates" them to travel back in time. Because of the possibility of
influencing the past while time traveling, one way of explaining why history does not change is by saying that
whatever has happened must happen. A time traveler attempting to alter the past in this model, intentionally or not,
would only be fulfilling their role in creating history as we know it, not changing it. Or that the time-traveler's
personal knowledge of history already includes their future travels to their own experience of the past (for the
Novikov self-consistency principle).
In layman's terms, it means this: the time traveller is in the past, which means they were in the past before.
Therefore, their presence is vital to the future, and they do something that causes the future to occur in the same way
that their knowledge of the future has already happened. It is very closely related to the ontological paradox and
usually occurs at the same time.
Examples
A dual example of a predestination paradox is depicted in the classic Ancient Greek play 'Oedipus':
Laius hears a prophecy that his son will kill him and marry his wife. Fearing the prophecy, Laius
pierces newborn Oedipus' feet and leaves him out to die, but a herdsman finds him and takes him away
from Thebes. Oedipus, not knowing he was adopted, leaves home in fear of the same prophecy that he
would kill his father and marry his mother. Laius, meanwhile, ventures out to find a solution to the
Sphinx's riddle. As prophesied, Oedipus crossed paths with a wealthy man leading to a fight in which
Oedipus slays him. Unbeknownst to Oedipus the man is Laius. Oedipus then defeats the Sphinx by
solving a mysterious riddle to become king. He marries the widow queen Jocasta not knowing she is his
mother.
A typical example of a predestination paradox (used in The Twilight Zone episode "No Time Like the Past") is as
follows:
A man travels back in time. While trying to prevent a school fire he had read about in a historical
account he had brought with him, he accidentally causes it.
An example of a predestination paradox in the television show Family Guy (Season 9, Episode 16):
Stewie and Brian travel back in time using Stewie's time machine. They are warped outside the space-time
continuum, before the Big Bang. To return home, Stewie overloads the return pad and they are boosted back into the
space-time continuum by an explosion. Stewie later studies the radiation footprints of the Big Bang and the
explosion of his return pad. He discovers that they match, and he concludes that he is actually the creator of the
universe. He explains his theory to Brian, who replies with "That doesn't make any sense; you were born into the
universe. How could you create it?" Stewie explains that it is a temporal causality loop, which is an example of a
predestination paradox.
A variation on the predestination paradoxes which involves information, rather than objects, traveling through time
is similar to the self-fulfilling prophecy:
A man receives information about his own future, telling him that he will die from a heart attack. He
resolves to get fit so as to avoid that fate, but in doing so overexerts himself, causing him to suffer the
heart attack that kills him.
Here is a peculiar example from Barry Dainton's Time and Space:
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Many years from now, a transgalactic civilization has discovered time travel. A deep-thinking temporal
engineer wonders what would happen if a time machine were sent back to the singularity from which the
big bang emerged. His calculations yield an interesting result: the singularity would be destabilized,
producing an explosion resembling the big bang. Needless to say, a time machine was quickly sent on its
way.[1]
In all five examples, causality is turned on its head, as the flanking events are both causes and effects of each other,
and this is where the paradox lies. In the third example, the paradox lies in the temporal causality loop. So, if Stewie
had never traveled back in time, the universe would not exist. Since it would not have existed, it could not have
created Stewie, so Stewie would not have existed.
One example of a predestination paradox that is not simultaneously an ontological paradox is:
In 1850, Bob's horse was spooked by something, and almost took Bob over a cliff, had it not been for a
strange man stopping the horse. This strange man was later honored by having a statue of him erected.
Two hundred years later, Bob goes back in time to sight-see, and sees someone's horse about to go over
a cliff. He rushes to his aid and saves his life.
In The Big Loop the Big Bang owes its causation to the temporal engineers. Interestingly enough, it seems the
engineers could have chosen not to send the time machine back (after all, they knew what the result would be),
thereby failing to cause the Big Bang. But the Big Bang failing to happen is obviously impossible because the
universe does exist, so perhaps in the situation where the engineers decide not to send a time machine to the Big
Bang's singularity, some other cause will turn out to have been responsible.
In another example, on the show Mucha Lucha in the episode "Woulda Coulda Hasbeena", Senior Hasbeena goes
back in time to stop a flash from blinding him in an important wrestling match, when the three main protagonists try
to stop him due to dangerous possible outcomes he unleashes a disco ball move thereby blinding himself in the past
causing the future he knows to that day.
Another example is in "The Legend of Zelda: Ocarina of Time", when the player travels to the future and meets a
man in a windmill, who tells him about a mean Ocarina kid who played a song that sped up his windmill and dried
up the well. He then teaches Link the song, who plays it in the past, causing him to learn the song in the future. This
also an example of a Bootstrap Paradox, as the song itself was never written, but taught back and forth between Link
and the man in the windmill.
In most examples of the predestination paradox, the person travels back in time and ends up fulfilling their role in an
event that has already occurred. In a self-fulfilling prophecy, the person is fulfilling their role in an event that has yet
to occur, and it is usually information that travels in time (for example, in the form of a prophecy) rather than a
person. In either situation, the attempts to avert the course of past or future history both fail.
Examples from fiction
Time travel
Many fictional works have dealt with various circumstances that can logically arise from time travel, usually dealing
with paradoxes. The predestination paradox is a common literary device in such fiction.
• In Robert Heinlein's "—All You Zombies—", a young man (later revealed to be intersex) is taken back in time and
tricked into impregnating his younger, female self (before he underwent a sex change); he then turns out to be the
offspring of that union, with the paradoxical result that he is his own mother and father. As the story unfolds, all
the major characters are revealed to be the same person, at different stages of her/his life. In another of his stories,
"By His Bootstraps", the protagonist in a series of twists, interacts with future versions of himself.
• The Man Who Folded Himself is a 1973 science fiction novel by David Gerrold that deals with time travel and the
predestination paradox, much like Heinlein's. The protagonist, Daniel Eakins, inherits a time belt from his "uncle"
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that allows him to travel in time. This results in a series of time paradoxes, which are only resolved by the
existence of multiple universes and multiple histories. Eakins, who repeatedly encounters alternate versions of
himself, finds himself in progressively more bizarre situations. The character spends much of his own contorted
lifetime at an extended party with dozens of versions of himself at different ages, before understanding the true
nature of the gathering, and his true identity. Much of the book deals with the psychological, physical, and
personal challenges that manifest when time travel is possible for a single individual at the touch of a button.
Eakins repeatedly meets himself; has sex with himself; and ultimately cohabitates with an opposite-sex version of
himself. Eventually, that relationship ends up with a male child who he finally realizes is him, and he is now his
own "uncle".
• In the SpongeBob SquarePants episode SB-129m, Squidward, inspired by 'jellyfishing', teaches prehistoric
SpongeBob and Patrick to catch a jelly in a net. This means that Squidward invented Jellyfishing.
• In the video game Timesplitters: Future Perfect the main protagonist, Sergeant Cortez, often helps himself solve
puzzles, and protects himself during hard situations.
• In Flatterland, Vikki Line and the Space Hopper fall into a black hole, are rescued by future versions of
themselves, and then go back in time to rescue themselves.
• In the American Dad! episode "Fart-Break Hotel", Steve becomes drawn to a Patrick Nagel painting of a woman.
A hotel concierge explains to Steve that the painting was made in 1981, meaning Steve would have to travel back
in time to meet the woman. After successfully traveling back in time, Steve meets Nagel, who drugs his
champagne, causing Steve to pass out. When he wakes up, he finds himself naked on a bed and sees the painting
of the woman. However, Nagel explains that he painted Steve, meaning Steve was the woman in the painting he
had become attracted to.
• In the film 12 Monkeys, James Cole travels into the past to stop an attack attributed to the elusive "Army of the
Twelve Monkeys", which leads indirectly to the formation of the group. The fatal shooting at the end of the movie
is witnessed by his childhood version and leads to the nightmares that haunt him throughout his life.
• In The Twilight Zone 2002-2003 revival, in the episode, Cradle of Darkness, Andrea (played by Katherine Heigl)
goes back in time to assassinate Adolf Hitler while he is a baby. She kills the baby (whom she presumes to be
actual Adolf Hitler, though the viewer might note it seems like a very normal baby, perhaps not very dark hair),
but the nanny (discovering the death) replaces the baby with a street gypsy's baby (the mother being a very crazy
looking woman who has black hair resembling the Hitler we know), and she presents this baby to the father as his
own. The father proceeds to introduce this son to his guests as "Adolf", presumably the Adolf Hitler known to
history in the first place.
• In Bill and Ted's Bogus Journey the antagonist, unhappy with the future, sends evil robots back in time to kill Bill
and Ted. When his robots are defeated, he goes back himself and takes control of the world's satellites so the
whole world can see them defeated. Instead, the whole world watches them play their music, cementing their
place in history. In Bill and Ted's Excellent Adventure we see that the band could not have formed if not for Rufus
appearing from the future to help them with their history project.
• The episode "Roswell That Ends Well" of the animated television series Futurama puts a more humorous spin on
the paradox. In the episode, the main characters go back in time to 1947 in Roswell, New Mexico, sparking the
Roswell UFO Incident. Meanwhile, Fry, told that the death of his grandfather Enos would nullify his own
existence, becomes obsessed with protecting the man. He shuts Enos in a deserted house in the desert in order to
protect him, failing to realize that the house is in a nuclear testing site. The resulting atomic test kills Enos, but
Fry does not disappear. Fry later comforts Enos' fiancée, no longer believing her to be his future grandmother. He
has sex with her, only to realize afterward that she is his grandmother and therefore he is his own grandfather.
• The video game Prince of Persia: Warrior Within, the Prince is chased by the Dahaka, whose purpose is to
preserve the time-line by erasing the Prince from it. Unable to fight the monster, the Prince travels to the Island of
Time to kill the Empress of Time, who created the time-manipulating sands from the first game. He hopes to
prevent the sands from being created, since it was the sands that put him in his current predicament. However, the
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Prince realizes too late that killing the Empress is what creates the sands, and hence he becomes the architect of
his own fate. A secondary paradox is the Sand Wraith, who seems to stalk the Prince throughout the first half of
the game, even trying to kill him at one point. The wraith is killed by the Dahaka shortly before the Prince kills
the Empress. After killing the Empress, the Prince realizes that he can change his fate by using the Mask of the
Wraith, which transforms him into the Sand Wraith and sends him back in time a short distance. He learns that the
wraith (who he now understands to be his future self) was trying to protect him, rather than attack him. Upon
reaching the point at which the Dahaka is supposed to kill him, the Prince uses his knowledge of the encounter to
have his younger self die instead, ending the mask's power and creating a grandfather paradox as well.
• The film Donnie Darko incorporates an example of fictional predestination paradox. Donnie avoids death by a jet
engine that appears out of nowhere, only to later, because of information he has learned since, send the engine
back in time himself so that he may die by it. He thereby negates all activity that occurred between the appearance
of the engine and him sending it back, including his learning of the reason that he must die. This is explained
through use of a tangent universe and a physical and temporal theory.
• In Harry Potter and The Prisoner of Azkaban, Harry is saved from the Dementors by a stag patronus. At that
time, he thought it was his dead father's spirit of some sort watching over him. After traveling back in time, he
realizes he was the one who produced the patronus- after watching himself being attacked and seeing that no one
had produced the stag patronus- he himself casts the spell, producing the stag patronus he had seen earlier.[2]
Similarly, in the film, Harry and his friends are alerted to the presence of the Minister for Magic when a rock hits
Harry in the head; but after traveling back in time, Hermione recognizes the same rock and throws it at Harry
herself.
• In the Legacy of Kain video game series, more specifically Soul Reaver, Soul Reaver 2, and Defiance, the
predestination is evident in the Soul Reaver as well as Raziel, whose soul is contained inside. Through the
storyline of the 3 games it is learned that Raziel's soul must become part of the Reaver, despite the fact that it has
been a part of the weapon the whole time. Defiance ends in Raziel being stabbed by the Reaver, allowing his soul
to be transferred to it, however because of the purification his soul had gone through earlier the cycle is broken
rather than beginning again.
• In the Terminator films, Skynet, a computer program that controls nearly the whole world in the future, sends a
machine to the past in order to kill John Connor, the future leader of the human resistance, at different points of
his life: once before he is conceived (by killing his mother, Sarah Connor), again when he is 10 years old (in
Terminator 2: Judgment Day) and a final time a few days before Judgment Day happens (Terminator 3: Rise of
the Machines). In the second film Dr. Dyson (Joe Morton), the lead scientist for the Skynet project, explains that
the surviving arm and CPU chip of the original Terminator was analyzed and found that the technology was so
advanced, they (humans) would have never invented the technology themselves and was used to create Skynet in
the first place. However, all the components and research were destroyed in an attempt to prevent Skynet, but in
(Terminator 3: Rise of the Machines) Skynet is built anyway without any information or components from the
future, implying that it was inevitable. In a not yet made movie, the humans somehow successfully invaded the
complex in which the time machine is placed, manage to send someone else to the past so that the Connors can be
protected, which is what starts the series. In The Terminator, the machines send the T-800 and the humans send
Kyle Reese: Kyle will be John Connor's father (that is, if Skynet had not have happened, Kyle Reese would have
no reason to go back in time to protect Sarah, and thus John Connor would not have been born).
• In the episode "He's Our You" of the television series Lost, several characters travel back into the 1970s. One of
them, Sayid Jarrah, encounters the younger version of Benjamin Linus, the leader of the Others, and a man who
has committed various acts such as betraying the Dharma Initiative and causing their complete genocide by the
Others, the manipulation and deceit towards various people on the show and caused much strife to Sayid
personally including recruiting him to become an assassin during his wife's funeral. When Sayid meets Ben's
younger version he believes that it is his destiny to kill him and prevent all of the bad things he does from ever
happening. However when he does this by shooting him, Ben is taken to the Others where they state that they
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Predestination paradox
could heal him in a mysterious temple but, "his innocence would be lost" and he would "always be one of them."
By trying to prevent Ben from doing the things he did, Sayid actually caused him to become the evil manipulator
that he is and caused all of the evil acts he committed.
• In Artemis Fowl: The Time Paradox, Artemis's mother contracts the deadly magical disease, Spelltropy. To save
his mother, he travels into the past to save the Silky Sifaka lemur, which he kills at age 10 by handing to the
Extinctionists. In the past, Artemis the elder meets Opal Koboi, who follows Artemis into the future. In the
present, Opal gives Artemis's mother Spelltropy-like symptoms, which causes Artemis to time-travel in the first
place.
• In the 2008 episode of Doctor Who: "The Doctor's Daughter", the TARDIS takes the Doctor, Donna, and Martha
to find the source of the Doctor's Daughter's signal. However, the TARDIS arrives early, which leads the Doctor
to the accidental creation of his daughter, thus activating the signal. In the 2010 episode "The Big Bang", the
Doctor is released from the Pandorica by Rory Williams, using the sonic screwdriver supplied by the Doctor after
his release. In the 2011 episode "Let's Kill Hitler", Mels, a friend who Amy and Rory name their daughter Melody
after, turns out to be a pre-regeneration version of River Song, who in the prior episode, "A Good Man Goes To
War", was established to be an alias used by their daughter as an adult, an alias she adopts shortly after
regenerating and hearing the Doctor, Amy, and Rory refer to her as such, having known her only by her alias until
recently.
• In Red vs Blue, when the character Church is thrown back in time in Episode 50, he tries to prevent certain things
from happening, in the process leading to everything becoming the way it was: kicking dirt on a switch hoping it
to be replaced, instead it was kept and later got stuck; giving his captain painkillers to prevent a heart attack, but
killing him because the captain is allergic to aspirin; trying to make the tank not kill him by disabling the
friendly-fire protocol, which later proves his death; telling the tank and robot that they should not leave and build
a robot army, thereby giving them the idea to do it; trying to shoot O'Malley with the rocket launcher only to
shoot Tucker because of the launcher's highly defective targeting system and his inability to aim.
• In the PlayStation 2 video game Shadow Hearts: Covenant Karin Koenig, one of the main protagonists, falls in
love with Yuri Hyuga. She is gently rejected because Yuri still has feelings for the exorcist Alice Elliot, who died
in the previous game. Unrequited love does not stop her from fighting alongside Yuri, though, until at the end of
the game when she is flung into the past and meets Yuri's father. There you finally see a picture she is given
earlier in the game by Yuri's aunt that shows his father, mother and Yuri as a child. It's obvious the woman in the
picture is Karin, thus making her Yuri's mother. She ends up being the only one staying in the past because she
knows she is to become Yuri's mother and assumes the alias "Anne". She also takes back a cross Yuri gave to her,
which is the same cross that belongs to his mother. The cross becomes an Ontological paradox.
• The Black Sabbath song "Iron Man" tells the story of a man who time travels into the future of the world, and sees
the apocalypse. In the process of returning to the present, he is turned into steel by a magnetic field. He is
rendered mute, unable verbally to warn people of his time of the impending destruction. His attempts to
communicate are ignored and mocked. This causes Iron Man to become angry, and have his revenge on mankind,
causing the destruction seen in his vision.
• In The Penguins of Madagascar, the episode "It's About Time" sees Kowalski constructing a time machine called
the "Chronotron". A future Kowalski tells Private to convince his present self not to complete it. After he decides
to destroy the Chronotron, another Kowalski from the future tells Skipper to convince him to save the
Chronotron. When the present Kowalski spots his future selves, a vortex appears. The present Kowalski activates
the Chronotron and goes back in time to talk to Private. When Private points out that if Kowalski had not invented
the Chronotron then he would not have gone back in the first place to tell himself not to make it, the future
Kowalski then goes back in time to talk to Skipper. Rico then throws the Chronotron into the vortex, sealing it.
While a baffled Kowalski tries rationalizing that such a simple thing defies all laws of the universe, Skipper
simply states that Rico is a maverick who makes his own rules, and tells Kowalski to invent something that will
not destroy the world.
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Predestination paradox
• In the Red Dwarf episode "Timeslides", Dave Lister travels back in time using a mutated photograph of a pub in
Liverpool where his band once played a gig to give his teenage self the idea of inventing the Tension Sheet (a
stress relief tool invented by Fred 'Thickie' Holden, a former classmate of Arnold Rimmer, which earned him
millions). This causes him to become rich and famous in the past and never get stuck on Red Dwarf. Arnold
Rimmer, in an attempt to experience fame and fortune for himself, travels back even further in time to his school
days, to give his own younger self the idea of inventing the Tension Sheet instead. Unfortunately for Rimmer,
while he is giving young Rimmer the idea, the conversation is overheard by Thickie Holden (who sleeps in the
next bed) and he is able to patent the idea before young Rimmer can, therefore putting everything back to how it
was at the start of the episode.
• In the PC game Fallout 2, there is a chance that the player may encounter the Guardian of Forever of Star Trek
fame in an Easter egg. Should they use the device, they will be taken back in time to Vault 13, the home of the
Vault Dweller, the player character's ancestor. Using a certain computer in the Vault will result in the water
purification chip being irreparably damaged, thus setting in motion the events of the previous game, that
eventually result in the Vault Dweller being exiled and establishing the player character's tribe. The game
humorously notes that "this comforts [the player] for some reason".
• In the British Television show Misfits, The group repeatedly encounters a man they call Superhoodie, who seems
to know a lot about them. He's almost always seen whenever they have to deal with a dangerous situation and
helps them out of it. The character Alisha Bailey eventually discovers he's really a future version of their friend
Simon Bellamy and begins a relationship with him. He's shot and killed while trying to protect Alisha from a man
who believes he's in a violent video game. He requests she doesn't tell his past self who he is and she agrees.
Alisha, because of future Simon, begins respecting present day Simon and the two become closer. Alisha is killed
by Rachel, a woman who the Misfits killed, but was returned in ghost form by a medium named Jonas to exact
revenge on them. As Alisha dies in Simon's arms, she tells him Superhoodie's true identity. Later, Simon acquires
a one-way time travel ability from a power dealer named Seth, traveling back and becoming Superhoodie. Rudy
Wade, a member of the group, mentions that the paradox will continue and they'll probably just keep going in
circles.
• In The Transformers episode "The Key to Vector Sigma", Optimus Prime assists in the creation of the Aerialbots
who, in the later episode "War Dawn", are sent back through time, thus activating their vital role, and ensuring
Optimus Prime's existence into the future.
• In Sam & Max Season Two there exist 2 examples of this paradox:
• In "Ice Station Santa", Sam and Max must save their future selves from being killed. In "What's New,
Beelzebub?" they are saved by their past selves; this creates an infinite loop of "save and later be saved; the
savers are later saved".
• In "Chariots of the Dogs", Sam and Max are given an egg by their future selves from "What's New,
Beelzebub?" who they also give a remote control too. Later, in What's new Beelzebub S&M give their past
selves the egg and get the remote from them which, once again, creates an infinite loop. However this creates
an inconsistency(paradox) in which the egg has no origin.
• In the Chinese novel Bu Bu Jing Xin, centering the rivalry of Kangxi Emperor's sons for the throne during the
18th century Qing Dynasty, which will results the monarch's fourth son Yinzhen as Yongzheng Emperor. Its main
character, Ma'ertai Ruoxi (Zhang Xiao), a time traveler from the 21st century, aware the princes' feud would leads
to a tragic outcome. However, she is romantically entangled with the three of them, unawares that her relationship
with them would inadvertently leading history to be unfold as written in the future instead of changing it.
• In the Rampage of Haruhi Suzumiya, the fifth in a series of light novels by Nagaru Tanigawa, is about Haruhi
Suzumiya, a girl who can unconsciously change the universe to her tastes (like a deity) and wishes that the
summer holidays would never end. As a result, the summer loops over 15,527 times before her friend Kyon
realises what is happening. He tries to break the time loop many times by trying to stop her leaving a restaurant to
go home at the end of the day. He eventually succeeds by convincing Haruhi to come to a study session, as
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Predestination paradox
neither of them have done their homework yet. Although this was only a 30 page chapter in the book, the
much-anticipated second season aired what was essentially the same episode eight times, with minor differences
in camera angles, the characters' clothes etc. This caused a lot of tension for fans and caused many of them to
drop the series.
• In the 20th episode of the second season of My Little Pony: Friendship Is Magic, "It's About Time," the character
Twilight Sparkle receives a warning from her future self about an impending disaster, but she is pulled back into
the future before she can explain what the disaster is. Twilight drives herself mad with worry in an attempt to
prevent the disaster, which ends up not happening. However, in her efforts, Twilight discovers a spell that will
allow her to travel back in time, and attempts to warn her past self not to worry. She is pulled back into the
present before she can warn her past self not to worry about the future, closing the loop.
• In the web-comic Homestuck, there is a point where one of the characters who time travels often, Dave Strider, is
told he must trust an alien named Terezi and proof of their trust will be a thumbs up from a future Dave nearby.
He would not have trusted Terezi had he not returned to the past and he wouldn't have returned to reassure his
past self if he didn't trust Terezi.
Prophecies
Prior to the use of time travel as a plot device, the self-fulfilling prophecy variant was more common.
In Revenge of the Sith, Anakin Skywalker has visions of his wife dying in childbirth. In his attempt to gain enough
power to save her, he falls to the dark side of the force and becomes Darth Vader. His wife is heartbroken upon
learning this and argues with him. In his anger, he uses his power to hurt her, which eventually leads her to die in
childbirth.
Shakespeare's Macbeth is a classic example of this. The three Witches give Macbeth a prophecy that he will
eventually become king, but the offspring of his best friend will rule after him. Macbeth kills his king and his friend
Banquo. In addition to these prophecies, other prophecies foretelling his downfall are given, such as that he will not
be attacked until a forest moves to his castle, and that no man ever born of a woman can kill him. In the end, fate is
what drives the House of Macbeth mad and ultimately kills them, as Macbeth is killed by a man who was never
'born' as the man was torn from his mother's womb by caesarean section.
In the movie Minority Report, murders are prevented through the efforts of three psychic mutants who can see
crimes before they are committed. When police chief John Anderton is implicated in a murder-to-be, he sets out on a
crusade to figure out why he would kill a man he has yet to meet. Many of the signposts on his journey to meet fate
were predicted exactly as they occur, and his search leads him inexorably to the scene of the crime, where he cannot
stop himself from killing the other man. In the end, the prediction itself is what had set the chain of events in motion.
In Lost, Desmond Hume's future flashes regarding Charlie's deaths eventually lead to his death. Desmond has a
vision in which Charlie pushes a button below a flashing light which allows the other castaways to be rescued just
before he drowns. However when the event occurs, events happen slightly differently than in Desmond's vision and
it is suggested that Charlie may have been able to save himself without jeopardizing the hopes of rescue, if he had
not believed his death was crucial in the rescue of the other castaways.
Yet there are examples of prophecies that happen slowly, if at all. In Red Dwarf: "Stasis Leak", when Lister travels
back in time to meet with Kochanski to marry her, he finds out from his future self from 5 years later that he is going
to pass through a wormhole and end up in a parallel universe version of Earth in 1985 but after 8 whole series, this
has never happened (although similar events happen in "Backwards").
In the Harry Potter Universe by J. K. Rowling a prophecy by Sybill Trelawney is overhe