Tautology (Propositional Logic)

Definition

Tautology (Propositional Logic)

A propositional formula is a tautology (or valid) if it is true under every interpretation.

For an interpretation $I$, a formula $P$ is a tautology if

$$I(P)=I(\top )\text{for every }I.$$

Validity is often written $\models P$. It is the semantic counterpart of the verum $\top$: by validity as equivalence to verum, $P$ is a tautology exactly when $P\equiv \top$. Dually, a tautology is never a contradiction, and its negation $\neg P$ is unsatisfiable by the validity–unsatisfiability duality.