Equivalent Formula (Propositional Logic)

Definition

Equivalent Formula (Propositional Logic)

Two formulas $P,Q$ are called equivalent if they evaluate to the same truth value under every interpretation.

$$\forall \text{interpretation}v\left(v(P)=v(Q)\right)$$

Equivalently, two formulas $P,Q$ are called equivalent if they have the same models.

The Model Characterization

Definition

Model Characterization of Equivalence (Propositional Logic)

Two propositional formulas $P,Q$ are equivalent if and only if they have the same models:

$$P\equiv Q⟺\mathrm{Mod}(P)=\mathrm{Mod}(Q),$$

where $\mathrm{Mod}(F)=\{I\mid I(F)=I(\top )\}$ is the set of models of $F$ over all interpretations $I$.

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