A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition; or, it can be an apparent contradiction that actually expresses a non-dual truth (cf. Koan, Catuskoti). 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.

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.

Logical paradox

Common themes in paradoxes include self-reference, the infinite, circular definitions, and confusion of levels of reasoning.

Patrick Hughes outlines three laws of the paradox:

• Self reference - so all Cretans are liars, said the Cretan, is self referential, because the Cretan describes all Cretans;
• Contradiction - so all Cretans are liars, said the Cretan, is a contradictory because the Cretan is saying that Cretans are liars
• Vicious circularity or infinite regress - so if all Cretans are liars, and the Cretan told us so, then it cannot be true, but if it is not true that ALL Cretans are liars, then SOME Cretan must be a liar, and if there is only one Cretan, the statement stands, but then it is true that all Cretans are liars, so it must be a lie... and so on ad infinitum This is not necessarily a paradox: If there is exists a Cretan other than the Cretan making the claim, and this Cretan is not a liar, then the Cretan making the claim is a liar: He is saying that ALL Cretans are liars when there is in fact some Cretan who is not. A better example of vicious circularity is the statement: "This statement is false".

The Cretan could be telling the truth about all Cretans being liars. They could all be liars, but that doesn't mean that they are lying about everything all of the time. Another example is "I always lie" because if it is true it must also be false.