Definition:Semantic Equivalence/Predicate Logic/Definition 1
Jump to navigation
Jump to search
Definition
Let $\mathbf A, \mathbf B$ be WFFs of predicate logic.
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 $\AA$ and assignments $\sigma$:
- $\AA, \sigma \models_{\mathrm{PL_A}} \mathbf A$ if and only if $\AA, \sigma \models_{\mathrm{PL_A}} \mathbf B$
where $\models_{\mathrm{PL_A}}$ denotes the models relation.
Also see
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\mathrm{II}.8$ Further Semantic Notions