contradiction

Contradiction is a relationship between two statements or propositions that assert mutually exclusive conditions, such that they cannot both be true in the same sense at the same time. In classical logic, the law of non-contradiction states that a statement and its negation cannot both be true. A contradiction is often formalized as P and not P.

If a system allows a contradiction, the principle of explosion typically follows, meaning any statement becomes derivable from the contradiction, undermining rational inference. By contrast, some non-classical logics, known as paraconsistent logics, are designed to tolerate certain contradictions without explosion.

Contradictions can be categorized as logical, empirical, or conceptual. Logical contradictions arise from the structure of a theory; empirical contradictions arise when observations conflict with conclusions. Conceptual contradictions may reflect ambiguities in language, definitions, or assumptions. Paradoxes, such as the Liar Paradox, involve self-reference and apparent contradictions that resist straightforward resolution.

In practice, contradictions prompt scrutiny: terms may be clarified, premises re-evaluated, or theories revised. In science, conflicting data lead to new experiments or model adjustments. In philosophy and logic, researchers distinguish between dialetheism (accepting some true contradictions) and methodical approaches that avoid inconsistency.

See also non-contradiction, inconsistency, paradox, logic.