A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion. A paradox usually involves contradictory-yet-interrelated elements that exist simultaneously and persist over time. They result in "persistent contradiction between interdependent elements" leading to a lasting "unity of opposites".

In logic, many paradoxes exist that are invalid arguments, yet are nevertheless valuable in promoting critical thinking, while other paradoxes have revealed errors in definitions that were assumed to be rigorous, and have caused axioms of mathematics and logic to be re-examined. One example is Russell's paradox, which questions whether a "list of all lists that do not contain themselves" would include itself and showed that attempts to found set theory on the identification of sets with properties or predicates were flawed. Others, such as Curry's paradox, cannot be easily resolved by making foundational changes in a logical system.

Examples outside logic include the ship of Theseus from philosophy, a paradox that questions whether a ship repaired over time by replacing each and all of its wooden parts one at a time would remain the same ship. Paradoxes can also take the form of images or other media. For example, M. C. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly.

Informally, the term paradox is often used to describe a counterintuitive result.

Self-reference, contradiction and infinite regress are core elements of many paradoxes. Other common elements include circular definitions, and confusion or equivocation between different levels of abstraction.

Self-reference occurs when a sentence, idea or formula refers to itself. Although statements can be self referential without being paradoxical ("This statement is written in English" is a true and non-paradoxical self-referential statement), self-reference is a common element of paradoxes. One example occurs in the liar paradox, which is commonly formulated as the self-referential statement "This statement is false". Another example occurs in the barber paradox, which poses the question of whether a barber who shaves all and only those who do not shave themselves will shave himself. In this paradox, the barber is a self-referential concept.

Contradiction, along with self-reference, is a core feature of many paradoxes. The liar paradox, "This statement is false," exhibits contradiction because the statement cannot be false and true at the same time. The barber paradox is contradictory because it implies that the barber shaves himself if and only if the barber does not shave himself.

As with self-reference, a statement can contain a contradiction without being a paradox. "This statement is written in French" is an example of a contradictory self-referential statement that is not a paradox and is instead false.