Definition:Semantic Equivalence/Predicate Logic
< Definition:Semantic Equivalence(Redirected from Definition:Semantic Equivalence (Predicate Logic))
Jump to navigation
Jump to search
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 $\AA$ 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.
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.