Unsatisfiability–Validity Duality (Propositional Logic)

Definition

Unsatisfiability–Validity Duality (Propositional Logic)

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

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

In other words, $P$ is a contradiction exactly when $\neg P$ is a tautology.

Reason

Mirror image of the validity–unsatisfiability duality

The claim is the validity–unsatisfiability duality applied to $\neg P$ in place of $P$: since [[Knowledge/Double Negation Elimination|$\neg \neg P$ is equivalent to $P$]],

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