A paradox is an apparently true statement or group of statements that seems to lead to a contradiction or to a situation that 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 recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has led to significant advances in science, philosophy and mathematics.
The word paradox is often used interchangeably with contradiction; but where a contradiction by definition cannot be true, many paradoxes do allow for resolution, though many remain unresolved or only contentiously resolved (such as Curry's paradox). Still more casually, the term is sometimes used for situations that are merely surprising (albeit in a distinctly "logical" manner) such as the Birthday Paradox. This is also the usage in economics, where a paradox is an unintuitive outcome of economic theory.
The etymology of paradox can be traced back to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek παράδοξος paradoxos meaning 'contrary to expectation', 'incredible'. The word is composed of the preposition para which means "against" conjoined to the noun stem doxa, meaning "belief". Compare orthodox (literally, "straight teaching") and heterodox (literally, "different teaching"). The liar paradox and other paradoxes were studied in medieval times under the heading insolubilia.
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 paradox quality.
In moral philosophy, paradox plays a central role in ethics debates. For instance, 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.
Types 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 person may be more than Nine years old on his Ninth birthday. Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but all too 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 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.
- Quine, W. V. (1962). "Paradox". Scientific American, April 1962, pp. 84–96.
- Clarke, Michael (2002). Paradoxes from A to Z. London: Routledge.