Paradoxes Situations that seems to defy intuition

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Articles

Introduction 1 1 List of 4 Paradoxical laughter 16

Decision theory 17 17 19 Exchange paradox 22 Kavka's toxin puzzle 34 Necktie paradox 36

Economy 38 38 Arrow's impossibility theorem 41 Bertrand paradox 52 Demographic-economic paradox 53 Dollar auction 56 Downs–Thomson paradox 57 58 59 62 Icarus paradox 65 65 70 Lucas paradox 71 72 73 77 80 St. Petersburg paradox 85

Logic 92 All horses are the same color 92 93 Carroll's paradox 96 Crocodile Dilemma 97 98 Infinite regress 101 102 Paradoxes of material implication 104 107 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 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 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 273 274 Navigation paradox 276 Paradox of the plankton 278 Temporal paradox 279 Tritone paradox 280 Voting paradox 282

Philosophy 283 Fitch's paradox of knowability 283 286 291 Moore's paradox 295 Moravec's paradox 297 Newcomb's paradox 300 paradox 304 Paradox of 315 Paradox of nihilism 318 319 Predestination paradox 320 Zeno's paradoxes 322

Physics 329 Algol paradox 329 Archimedes paradox 329 '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- 462 462 465 paradox 467 Grelling–Nelson paradox 470 Intentionally blank page 472 475 481 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 499 500 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 that apparently contradicts itself and yet might be true. Most logical paradoxes are known to be invalid arguments but are still valuable in promoting . Some paradoxes have revealed errors in assumed to be rigorous, and have caused axioms of mathematics and 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 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 (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: 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'?" "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- 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 , 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 of the initial . 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. In 20th century science, Hilbert's paradox of the Grand Hotel and Schrödinger's cat are famously vivid examples of a theory being taken to a logical but paradoxical end. • 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, which falsely generalizes from true specific statements. • A paradox that is in neither class may be an , 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 and . A fourth kind has sometimes been described since Quine's work. • A paradox that is both true and false at the same time and in the same sense is called a dialetheia. In Western it is often assumed, following Aristotle, that no dialetheia exist, but they are sometimes accepted in Eastern Paradox 3

traditionsWikipedia:Avoid weasel words and in paraconsistent logics. It would be mere equivocation or a matter of degree, for example, to both affirm and deny that "John is here" when John is halfway through the door but it is self-contradictory to simultaneously affirm and deny the event in some sense.

Paradox in philosophy A taste for paradox is central to the of Laozi, , Meister Eckhart, Hegel, Kierkegaard, Nietzsche, and G.K. Chesterton, among many others. Søren Kierkegaard, for example, writes, in the Philosophical Fragments, that But 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.

Paradox in medicine A paradoxical reaction to a drug is the opposite of what one would expect, such as becoming agitated by a sedative or sedated by a stimulant. Some are common and are used regularly in medicine, such as the use of stimulants such as Adderall and Ritalin in the treatment of attention deficit disorder, while others are rare and can be dangerous as they are not expected, such as severe agitation from a benzodiazepine.

References

External links

• Cantini, Andrea (Winter 2012). "Paradoxes and Contemporary Logic" (http:/ / . stanford. edu/ entries/

paradoxes-contemporary-logic/ ). In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

• Spade, Paul Vincent (Fall 2013). "Insolubles" (http:/ / plato. stanford. edu/ entries/ insolubles). In Zalta, Edward N. Stanford Encyclopedia of Philosophy.

• Paradoxes (http:/ / www. dmoz. org/ / Philosophy/ Philosophy_of_Logic/ Paradoxes/ ) at DMOZ

• "Zeno and the Paradox of Motion" (http:/ / www. mathpages. com/ rr/ s3-07/ 3-07. htm) at MathPages.com.

• "Logical Paradoxes" (http:/ / www. iep. utm. edu/ par-log) entry in the Internet Encyclopedia of Philosophy List of paradoxes 4 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. Informally, the term is often used to describe a counter-intuitive result. This list is incomplete; you can help by expanding it [1].

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. Not to be confused with the Barber paradox. • 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 . • Catch-22: 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 of whom it is true to say: if that customer drinks, everybody in the pub drinks. • Paradox of entailment: Inconsistent 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 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. • Ross's paradox: Disjunction introduction poses a problem for imperative by seemingly permitting arbitrary imperatives to be inferred. • 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. • Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess exactly 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? • 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." • : 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." "In a world with no rules, there