For other meanings of "Paradox", see Paradox (disambiguation).
Robert Boyle's self-flowing flask fills itself in this diagram, but perpetual motion machines do not exist.

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; or 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 often 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 of 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 paradoxon. The word is composed of the preposition para which means "by way of", or "according to" conjoined to the noun stem doxa, meaning "what is received". 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 particularly central role in debates on ethics. For instance, an ethical admonition to "love thy neighbor" is not just in contrast with, but in contradiction to an armed neighbor 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 very 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.

Contents

Types of paradoxes[edit]

W. V. Quine (1962) distinguished three classes of paradox:

List of paradoxes[edit]

Quine's classification, of course, is useful only once a paradox has a clear resolution. That question is treated on the page for each individual paradox; the following are grouped thematically.

Logical (except mathematical)[edit]

Semantic paradoxes[edit]

These form a well-known (and well-studied) class having in common that any permissible assignment of semantic value (truth, reference) to an expression immediately implies the assignment of a different value.

Vagueness[edit]

Mathematical and statistical[edit]

File:Monty-hall.png
The Monty Hall paradox: which door do you choose?

Infinity[edit]

Geometry and topology[edit]

Psychological, action, and practical reason[edit]

Physical[edit]

Philosophy[edit]

Economics[edit]

References[edit]

See also[edit]

External links[edit]