# Definition:Semantic Equivalence/Boolean Interpretations

## Definition

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.

## Equivalence of Definitions

The definitions above are all equivalent, as shown on Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations.