Definition:Relative Semantic Equivalence/WFF

Definition

Let $\FF$ be a theory in the language of predicate logic.

Let $\mathbf A, \mathbf B$ be WFFs.

Let $\mathbf C$ be the universal closure of $\mathbf A \iff \mathbf B$.

Then $\mathbf A$ and $\mathbf B$ are semantically equivalent with respect to $\FF$ if and only if:

$\FF \models_{\mathrm{PL}} \mathbf C$

That is, if and only if $\mathbf C$ is a semantic consequence of $\FF$.

Also see

• Definition:Relative Semantic Equivalence of Terms

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions: Definition $\mathrm{II}.8.4$