Definition:Semantic Equivalence/Predicate Logic/Definition 1

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

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


 * $\mathbf A \models_{\mathrm{PL_A}} \mathbf B$ 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, for all structures $\AA$ and assignments $\sigma$:


 * $\AA, \sigma \models_{\mathrm{PL_A}} \mathbf A$ $\AA, \sigma \models_{\mathrm{PL_A}} \mathbf B$

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

Also see

 * Definition:Semantic Consequence (Predicate Logic)
 * Definition:Logical Equivalence