A paradox is a true statement or group of statements that leads to a contradiction or a situation which defies intuition; or, inversely, it can be 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 premises 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 ironic. An example of this is hating cucumbers, hating yogurt, but loving tzatziki sauce.
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. 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. Paradoxes in economics tend to be the veridical type, typically counterintuitive outcomes of economic theory. In literature a paradox can be any contradictory or obviously untrue statement, which resolves itself upon later inspection.
Patrick Hughes outlines three laws of the paradox:
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 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 stepfather. 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 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 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 fourth kind has sometimes been asserted since Quine's work.
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.
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.
Another example is the conflict between a moral injunction and a duty that cannot be fulfilled without violating that injunction. For example, take the situation of a person who is obligated to feed his children (the duty) but cannot afford to do so without stealing, which would be wrong (the injunction). 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.