Biconditional Characterization of Equivalence (Propositional Logic)

Definition

Biconditional Characterization of Equivalence (Propositional Logic)

Two formulas $P$ and $Q$ are equivalent if and only if their biconditional $P\leftrightarrow Q$ is valid:

$$P\equiv Q⟺(P\leftrightarrow Q)\text{ valid}.$$

Reason

The truth table of $\leftrightarrow$

For a fixed interpretation $I$, the biconditional is true exactly when both sides agree in value:

$$I(P\leftrightarrow Q)=I(\top )⟺I(P)=I(Q).$$

Therefore ”$P\leftrightarrow Q$ is true under every $I$” and ”$P$ and $Q$ agree under every $I$” are the same universal claim. The result is closely related to the model characterisation of equivalence: both reformulate agreement under all interpretations, once through the models of $P$ and $Q$, once through the biconditional.