Paradox

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A paradox (Gk: παράδοξος, "aside belief") 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 resolutions which are accepted by everybody.

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.

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[edit] Etymology

The etymology of paradox can be traced back to at least Plato's Parmenides, where Zeno of Elea, who lived from 490-430 BCE, used the word "paradoxon" to describe some of his seminal philosophic ideas. Zeno sought to illustrate that equal absurdities followed logically from the denial of Parmenides' views. There were apparently 40 'paradoxes of plurality' and other paradoxes that Zeno used to attack the Greek understanding of the physical world. In fact, Zeno's paradoxes of multiplicity and motion, which revealed problems in the Greek idea of space and time, were resolvable only using mathematics discovered in the 19th century.

Later and more frequent usage of the word has been traced to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek παράδοξος paradoxos, which means "contrary to expectation", or "incredible". The word is a fusion of the preposition para, meaning "against" or "beyond", and the noun stem doxa, meaning "belief" or "opinion".

[edit] Logical paradox

Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning. 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 both true and 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.

[edit] Typology

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

[edit] 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.

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