Validity–Unsatisfiability Duality (Propositional Logic)

Definition

Validity–Unsatisfiability Duality (Propositional Logic)

A formula $P$ is valid if and only if its negation $\neg P$ is unsatisfiable:

$$P\text{ valid}⟺\neg P\text{ unsatisfiable}.$$

In the language of models: $P$ is valid when every interpretation is a model of $P$, and $\neg P$ is unsatisfiable when no interpretation is a model of $\neg P$.

Reason

One fact, two spellings

An interpretation $I$ satisfies $\neg P$ exactly when it falsifies $P$:

$$I(\neg P)=I(\top )⟺I(P)=I(\perp ).$$

Validity says the left-hand side holds for every $I$; unsatisfiability of $\neg P$ says the right-hand side holds for every $I$. Since the two sides are equivalent for each single $I$, the two universal claims coincide.