Paradox

For other uses, see Paradox (disambiguation).

A paradox is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition. Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably and wrongly with contradiction; but whereas a contradiction asserts its own opposite, many paradoxes do allow for resolution of some kind.

The recognition of ambiguities, equivocations, 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. This is also the usage in economics, where a paradox is a counterintuitive outcome of economic theory. In literature it can be any contradictory or obviously untrue statement, which resolves itself upon later inspection.

Logical paradox

See also: List of paradoxes

Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning. Other paradoxes involve false statements or half-truths and the resulting 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 prepped 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 the bandwagon fallacy. 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 familial 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 step-father, or if a homosexual male couple had adopted a son or entered a committed relationship after one had already fathered a son.

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 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 experiments can also yield interesting paradoxes. The grandfather paradox, for example, would arise if a time traveler were to kill his own grandfather, 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 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 proofs (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 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 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.

Moral paradox

In moral philosophy, paradox plays a central 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 an armed neighbour actively trying to kill you: if he or she succeeds, you will not be able to love him or her. But to preemptively attack them or restrain them is not usually understood as loving. This might be termed an ethical dilemma. 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. Such a paradox between two maxims is normally resolved through weakening one or the other of the maxims (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".

See also

• List of paradoxes
• Paradox (database)
• Temporal paradox
• 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.

External links

• Some paradoxes - an anthology
• Template:Dmoz
• Insolubles (at the Stanford Encyclopedia of Philosophy)
• "Zeno and the Paradox of Motion"