Paradoxes Situations that seems to defy intuition PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 08 Jul 2014 07:26:17 UTC Contents Articles Introduction 1 Paradox 1 List of paradoxes 4 Paradoxical laughter 16 Decision theory 17 Abilene paradox 17 Chainstore paradox 19 Exchange paradox 22 Kavka's toxin puzzle 34 Necktie paradox 36 Economy 38 Allais paradox 38 Arrow's impossibility theorem 41 Bertrand paradox 52 Demographic-economic paradox 53 Dollar auction 56 Downs–Thomson paradox 57 Easterlin paradox 58 Ellsberg paradox 59 Green paradox 62 Icarus paradox 65 Jevons paradox 65 Leontief paradox 70 Lucas paradox 71 Metzler paradox 72 Paradox of thrift 73 Paradox of value 77 Productivity paradox 80 St. Petersburg paradox 85 Logic 92 All horses are the same color 92 Barbershop paradox 93 Carroll's paradox 96 Crocodile Dilemma 97 Drinker paradox 98 Infinite regress 101 Lottery paradox 102 Paradoxes of material implication 104 Raven paradox 107 Unexpected hanging paradox 119 What the Tortoise Said to Achilles 123 Mathematics 127 Accuracy paradox 127 Apportionment paradox 129 Banach–Tarski paradox 131 Berkson's paradox 139 Bertrand's box paradox 141 Bertrand paradox 146 Birthday problem 149 Borel–Kolmogorov paradox 163 Boy or Girl paradox 166 Burali-Forti paradox 172 Cantor's paradox 173 Coastline paradox 174 Cramer's paradox 178 Elevator paradox 179 False positive paradox 181 Gabriel's Horn 184 Galileo's paradox 187 Gambler's fallacy 188 Gödel's incompleteness theorems 195 Interesting number paradox 213 Kleene–Rosser paradox 214 Lindley's paradox 215 Low birth weight paradox 217 Missing square puzzle 219 Paradoxes of set theory 221 Parrondo's paradox 226 Russell's paradox 231 Simpson's paradox 237 Skolem's paradox 245 Smale's paradox 249 Thomson's lamp 251 Two envelopes problem 253 Von Neumann paradox 265 Miscellaneous 268 Bracketing paradox 268 Buridan's ass 269 Buttered cat paradox 272 Lombard's Paradox 273 Mere addition paradox 274 Navigation paradox 276 Paradox of the plankton 278 Temporal paradox 279 Tritone paradox 280 Voting paradox 282 Philosophy 283 Fitch's paradox of knowability 283 Grandfather paradox 286 Liberal paradox 291 Moore's paradox 295 Moravec's paradox 297 Newcomb's paradox 300 Omnipotence paradox 304 Paradox of hedonism 315 Paradox of nihilism 318 Paradox of tolerance 319 Predestination paradox 320 Zeno's paradoxes 322 Physics 329 Algol paradox 329 Archimedes paradox 329 Aristotle's wheel paradox 331 Bell's spaceship paradox 332 Bentley's paradox 338 Black hole information paradox 338 Braess's paradox 342 Cool tropics paradox 346 D'Alembert's paradox 348 Denny's paradox 357 Ehrenfest paradox 357 Elevator paradox 362 EPR paradox 363 Faint young Sun paradox 374 Fermi paradox 376 Feynman sprinkler 396 Gibbs paradox 399 Hardy's paradox 406 Heat death paradox 409 Irresistible force paradox 410 Ladder paradox 411 Loschmidt's paradox 420 Mpemba effect 422 Olbers' paradox 426 Ontological paradox 431 Painlevé paradox 433 Physical paradox 434 Quantum pseudo-telepathy 439 Schrödinger's cat 442 Supplee's paradox 448 Tea leaf paradox 450 Twin paradox 452 Self-reference 462 Barber paradox 462 Berry paradox 465 Epimenides paradox 467 Grelling–Nelson paradox 470 Intentionally blank page 472 Liar paradox 475 Opposite Day 481 Paradox of the Court 482 Petronius 484 Quine's paradox 488 Richard's paradox 490 Self-reference 492 Socratic paradox 495 Yablo's paradox 497 Vagueness 498 Absence paradox 498 Bonini's paradox 498 Code-talker paradox 499 Ship of Theseus 500 References Article Sources and Contributors 505 Image Sources, Licenses and Contributors 518 Article Licenses License 522 1 Introduction Paradox For other uses, see Paradox (disambiguation). A paradox is a statement that apparently contradicts itself and yet might be true. Most logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself, and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy (questioning whether a ship repaired over time by replacing each of its wooden parts would remain the same ship). Paradoxes can also take the form of images or other media. For example, M.C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word "paradox" often refers to statements that are ironic or unexpected, such as "the paradox that standing is more tiring than walking". Logical paradox See also: List of paradoxes 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: 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'?" Contradiction "This statement is false"; the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldn't come true. This contradicts itself because if the genie grants his wish he did not grant his wish, and if he refuses to grant his wish then he did indeed grant his wish, therefore making it impossible to either grant or not grant his wish because his wish contradicts itself. 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'?" Paradox 2 Other paradoxes involve false statements ("impossible is not a word in my vocabulary", a simple paradox) 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. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, and require extending the context or language in order 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 well-known liar paradox: it is a sentence which cannot be consistently interpreted as either 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. 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 traveller were to kill his own grandfather before his mother or father had been conceived, thereby preventing his own birth. This is a specific example of the more general observation of the butterfly effect, or that a time-traveller's interaction with the past — however slight — would entail making changes that would, in turn, change the future in which the time-travel was yet to occur, and would thus change the circumstances of the time-travel itself. Often a seemingly paradoxical conclusion arises from an inconsistent or inherently contradictory definition of the initial premise. In the case of that apparent paradox of a time traveler killing his own grandfather it is the inconsistency of defining the past to which he returns as being somehow different from the one which leads up to the future from which he begins his trip but also insisting that he must have come to that past from the same future as the one that it leads up to. Quine's classification of paradoxes 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 had been 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.