Definition:Semantic Equivalence
Definition
Let $\mathscr M$ be a formal semantics for a formal language $\LL$.
Let $\phi, \psi$ be $\LL$-WFFs.
Then $\phi$ and $\psi$ are $\mathscr M$-semantically equivalent if and only if:
- $\phi \models_{\mathscr M} \psi$ and $\psi \models_{\mathscr M} \phi$
that is, if and only if they are $\mathscr M$-semantic consequences of one another.
Equivalently, $\phi$ and $\psi$ are $\mathscr M$-semantically equivalent if and only if, for each $\mathscr M$-structure $\MM$:
- $\MM \models_{\mathscr M} \phi$ if and only if $\MM \models_{\mathscr M} \psi$
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.
Definition 1
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\mathbf A \models_{\mathrm{BI}} \mathbf B$ and $\mathbf B \models_{\mathrm{BI}} \mathbf A$
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:
- $\map v {\mathbf A} = T$ if and only if $\map v {\mathbf B} = T$
for all boolean interpretations $v$.
Definition 2
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\map v {\mathbf A} = \map v {\mathbf B}$
for all boolean interpretations $v$.
Definition 3
Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations if and only if:
- $\mathbf A \iff \mathbf B$ is a tautology
where $\iff$ is the biconditional connective.
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.