Validity as Equivalence to Verum (Propositional Logic)

Definition

Validity as Equivalence to Verum (Propositional Logic)

A formula $P$ is valid if and only if it is equivalent to verum $\top$:

$$P\text{ valid}⟺P\equiv \top .$$

Here $\equiv$ is a global claim: it compares $P$ with the nullary constant formula $\top$ across all interpretations, not under a single one.

Reason

Same value everywhere

Validity means $I(P)=I(\top )$ for every interpretation $I$. That is exactly what $P\equiv \top$ asserts: two equivalent formulas agree in value under every interpretation.