# Equivalence of Definitions of Semantic Equivalence for Boolean Interpretations

## Theorem

The following definitions of the concept of Semantic Equivalence for Boolean Interpretations are equivalent:

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.

## Proof

### Definition 1 implies Definition 2

Let $\mathbf A, \mathbf B$ be equivalent according to definition 1.

Let $v$ be a boolean interpretation.

By definition. either $v \left({\mathbf A}\right) = T$ or $v \left({\mathbf A}\right) = F$.

In the first case, it follows by hypothesis that $v \left({\mathbf B}\right) = T$.

In particular, then:

$v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

In the second case, it must be that $v \left({\mathbf B}\right) \ne T$.

That is, $v \left({\mathbf B}\right) = F$, so that:

$v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

Hence $\mathbf A$ and $\mathbf B$ are also equivalent in the sense of definition 2.

$\Box$

### Definition 2 implies Definition 3

Let $\mathbf A, \mathbf B$ be equivalent according to definition 2.

By definition of the boolean interpretation of $\iff$:

$v \left({\mathbf A \iff \mathbf B}\right)= T$ if and only if $v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

Therefore, by hypothesis and definition of tautology:

$\mathbf A \iff \mathbf B$

is a tautology.

$\Box$

### Definition 3 implies Definition 1

Let $\mathbf A, \mathbf B$ be equivalent according to definition 3.

That is, let $\mathbf A \iff \mathbf B$ be a tautology.

From the boolean interpretation of $\iff$, we have:

$v \left({\mathbf A}\right) = v \left({\mathbf B}\right)$

for every boolean interpretation $v$.

Therefore it immediately follows that:

$v \left({\mathbf A}\right) = T$ if and only if $v \left({\mathbf B}\right) = T$

i.e. $\mathbf A$ and $\mathbf B$ are equivalent in the sense of definition 1.

$\blacksquare$