Paradox
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/* Logical paradox */
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A '''paradox''' can be an apparently [[Truth|true]] [[Proposition|statement]] or group of statements that leads to a [[contradiction]] or a situation which defies [[intuition (knowledge)|intuition]]; or it can be, seemingly opposite, an apparent [[contradiction]] that actually expresses a [[non-dual]] truth (cf. [[Koan]]). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the [[premise]]s themselves are not all really true or cannot all be true together. The word ''paradox'' is often used interchangeably with ''[[contradiction]]''. Often, mistakenly, it is used to describe situations that are [[irony|ironic]].
The recognition of [[ambiguity|ambiguities]], [[equivocation]]s, and unstated assumptions underlying known paradoxes has led to significant advances in [[science]], [[philosophy]] and [[mathematics]]. But many paradoxes, such as [[Curry's paradox]], do not yet have universally accepted resolutions.
Sometimes the term ''paradox'' is used for situations that are merely surprising. The [[birthday paradox]], for instance, is unexpected but perfectly logical. The logician [[Willard V. O. Quine]] distinguishes ''falsidical paradoxes'', which are seemingly valid, logical demonstrations of absurdities, from ''veridical paradoxes'', such as the birthday paradox, which are seeming absurdities that are nevertheless true.<ref>{{cite book | title = The Ways of Paradox and Other Essays| chapter = The Ways of Paradox | first = Willard | last = Van Orman Quine | authorlink = W. V. O. Quine | year = 1966 | publisher = Random House | page = 5 }}</ref>
Paradoxes in [[economics]] tend to be the veridical type, typically counterintuitive outcomes of economic theory. In [[literature]] a paradox can be any [[contradiction|contradictory]] or obviously untrue statement, which resolves itself upon later [[inspection]].
==Logical paradox==
{{seealso|List of paradoxes}}
Common themes in paradoxes include [[self-reference]], the [[Infinite set|infinite]], [[Begging the question|circular definitions]], and [[confusion of levels]] of [[reason]]ing. Other paradoxes involve [[false statement]]s or [[half-truths]] and the resulting [[cognitive bias|biased]] assumptions.
''For example, consider a situation in which a father and son are driving down the road. The car collides with 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]]. The reader, upon seeing the word [[surgeon]], applies a poll of their knowledge of surgeons (regardless of its depth) and reasons that since the majority of surgeons are male, the surgeon is a man, hence the contradiction: the father of the child, a man, was killed in the crash. The paradox is resolved if it is revealed that the surgeon is a woman, the boy's mother. Other assumptions whose resolution would also resolve the paradox are based on [[cognitive bias]]; the reader, reading terms like "father" and "son" and thinking of a familiar relationship, may assume a traditional family (biological father, biological mother, and son) because other combinations are unknown or disregarded out of prejudicial views. The paradox would resolve itself if it were revealed that the child was adopted and therefore had a biological and adopted father, or if a [[divorce]] resulted in the boy having a father and stepfather, or if a homosexual male couple had adopted a son or entered a committed relationship after one had already fathered a son. Another solution is that the father and son in the car are indeed not related at all - the father being parent to another individual distinct to the one in the car with him. This is because most people read the words "father and son" and immediately conclude that they are referring to two people in the same family, which is not necessarily true.
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 [[logic]]ians and [[philosopher]]s. ''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 false it must be true, and if it is true it must be false. Therefore, it can be concluded the sentence is neither true nor 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 experiment]]s can also yield interesting paradoxes. The [[grandfather paradox]], for example, would arise if a [[time travel]]er were to kill his own grandfather before his father was conceived, thereby preventing his own birth. Under the 'traditional' definition of a paradox, the Grandfather Paradox (and other similar situations) are typically thought to cause spacetime to rip itself apart under the strain of attempting to resolve an 'unresolvable' conclusion (ie, the time traveller killed his grandfather, therefore the time traveller wouldn't be born, therefore his grandfather could not have been killed, therefore he (and the time traveller) are still alive - and so on).
However, if the [[many worlds theory]] is correct, the death of the man does not cause the father of the time traveller and the time traveller to never be born because he is an alternate version of the grandfather.
[[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 person's fifth birthday is the day he turns twenty, if born on a [[leap day]]. Likewise, [[Arrow's impossibility theorem]] involves behaviour of voting systems that is surprising but true.
* A ''falsidical paradox'' establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various [[invalid proof]]s (e.g. that 1 = 2) are classic examples, generally relying on a hidden [[division by zero]]. Another example would be the inductive form of the [[Horse paradox]].
* 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 asserted since Quine's work.
* A paradox which is both true and false at the same time in the same sense is called a [[dialetheism|dialetheia]]. In Western logics it is often assumed, following [[Aristotle]], that no dialetheia exist, but they are sometimes accepted in Eastern traditions and in [[paraconsistent logic]]s. 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 literature==
{{main|Paradox (literature)}}
The paradox as a literary device has been defined as an anomalous juxtaposition of incongruous ideas for the sake of striking exposition or unexpected insight. It functions as a method of literary analysis which involves examining apparently contradictory statements and drawing conclusions either to reconcile them or to explain their presence.<ref name=Rescher>{{cite book | last = Rescher |first = Nicholas | title = Paradoxes: Their Roots, Range, and Resolution | publisher = Open Court | place = Chicago | year = 2001 | isbn = 0812694368}}.</ref>
Literary or rhetorical paradoxes abound in the works of [[Oscar Wilde]] and [[G. K. Chesterton]]; other literature deals with paradox of situation. [[Rabelais]], [[Cervantes]], [[Sterne]], [[Borges]], and Chesterton are all concerned with episodes and narratives designed around paradoxes. Statements such as Wilde’s “I can resist anything except temptation” and Chesterton’s “spies do not look like spies” are examples of rhetorical paradox. Further back, Polonius’ observation in ''Hamlet'' that “though this be madness, yet there is method in’t” is a memorable third.<ref name=Rescher/>
==Moral paradox==
In [[moral philosophy]], paradox in a loose sense plays a role in [[ethics]] debates. For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to armed neighbours actively trying to kill you: if they succeed, you will not be able to love them. But to preemptively attack them or restrain them is not usually understood as loving. This might be better termed an [[ethical dilemma]] rather than a paradox in the strict sense.
Another example is the conflict between an injunction not to [[steal]] and one to care for a family that you cannot afford to feed without stolen money (like the character of [[Robin Hood]]). Such a conflict between two maxims is normally resolved through weakening one or the other of them, e.g. the need for survival is greater than the need to avoid harm to your neighbor). However, as maxims are added for consideration, the questions of which to weaken in the general case and by how much pose issues related to Arrow's theorem (see above); it may be impossible to formulate a single system of ethics rules with a definite order of preference in the general case, a so-called "ethical calculus."
Paradoxes in a more strict sense have been relatively neglected in philosophical discussion within ethics, as compared to their role in other philosophical fields such as logic, epistemology, metaphysics or even the philosophy of science. Important book-length discussions appear in Derek Parfit's ''Reasons and Persons'' and in Saul Smilansky's ''10 Moral Paradoxes''.
==Genetic Paradox==
In zoology, a paradox albino is an animal that exhibits trait of an albino, with little melanin and albino eyes, but may have black or pigmented blotches or half albino, half normal eyes. This trait is most commonly seen with reptiles, though it is still a rare occurrence. It has been seen with boa constrictors, leopard geckos, iguanas and corn snakes as a few examples. These specimens are prized pets for collectors and prices are much higher than the average albino.
==See also==
* [[List of paradoxes]]
* [[Temporal paradox]]
* [[Anachronism]]
* [[Impossible object]]
* [[Formal fallacy]]
* [[Dilemma]]
* [[Puzzle]]
* [[Zeno's paradoxes]]
* [[Self refuting ideas]]
* [[Paradoxes of set theory]]
==References==
* [[R. M. Sainsbury]] (1988). ''Paradoxes''. Cambridge.
* [[W. V. Quine]] (1962). "Paradox". ''[[Scientific American]]'', April 1962, pp. 84–96.
* [[Michael Clarke]] (2002). ''Paradoxes from A to Z''. London: Routledge.
* [[Derek Parfit]] (1984). ''Reasons and Persons''. Oxford: Oxford University Press.
* Saul Smilansky (2007). ''10 Moral Paradoxes''. Malden, MA: Blackwell Publishing.
==Footnotes==
<references/>
==External links==
{{portalpar|Logic}}
*[http://www.paradoxes.co.uk/ Some paradoxes - an anthology]
*[[Stanford Encyclopedia of Philosophy]]:
**"[http://plato.stanford.edu/entries/paradoxes-contemporary-logic/ Paradoxes and Contemporary Logic]" -- by Andrea Cantini.
**"[http://plato.stanford.edu/entries/insolubles Insolubles]" -- by Paul Vincent Spade.
{{Spoken Wikipedia|Paradox.ogg|2005-07-07|SubCat=}}
*{{dmoz|Society/Philosophy/Philosophy_of_Logic/Paradoxes/|Paradoxes}}
*[http://www.mathpages.com/rr/s3-07/3-07.htm "MathPages - Zeno and the Paradox of Motion"]
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