Definition:Semantic Equivalence/Predicate Logic

Definition

Let $\mathbf A, \mathbf B$ be WFFs of predicate logic.

Definition 1

Then $\mathbf A$ and $\mathbf B$ are equivalent if and only if:

$\mathbf A \models_{\mathrm{PL_A}} \mathbf B$ and $\mathbf B \models_{\mathrm{PL_A}} \mathbf A$

that is, each is a semantic consequence of the other.

That is to say, $\mathbf A$ and $\mathbf B$ are equivalent if and only if, for all structures $\mathcal A$ and assignments $\sigma$:

$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$ iff $\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf B$

where $\models_{\mathrm{PL_A}}$ denotes the models relation.

Definition 2

Then $\mathbf A$ and $\mathbf B$ are equivalent if and only if:

$\mathbf A \iff \mathbf B$ is a tautology

where $\iff$ is the biconditional connective.

Equivalence of Definitions

The definitions above are all equivalent, as shown on Equivalence of Definitions of Semantic Equivalence for Predicate Logic.