# 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 $\MM \models_{\mathscr M} \psi$

### Semantic Equivalence for Boolean Interpretations

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.

### Semantic Equivalence for Predicate Logic

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.