Contradiction - Contradiction in Formal Logic

Contradiction in Formal Logic

Note: The symbol (falsum) represents an arbitrary contradiction, with the dual tee symbol used to denote an arbitrary tautology. Contradiction is sometimes symbolized by "Opq", and tautology by "Vpq". The turnstile symbol, is often read as "yields" or "proves".

In classical logic, particularly in propositional and first-order logic, a proposition is a contradiction if and only if . Since for contradictory it is true that for all (because ), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion" or "ex falso quodlibet" ("from falsity, whatever you like").

In a complete logic, a formula is contradictory if and only if it is unsatisfiable.