# Definition:Semantic Consequence/Boolean Interpretations

## Definition

Let $\FF$ be a collection of WFFs of propositional logic.

Then a WFF $\mathbf A$ is a semantic consequence of $\FF$ if and only if:

$v \models_{\mathrm{BI}} \FF$ implies $v \models_{\mathrm{BI}} \mathbf A$

where $\models_{\mathrm{BI}}$ is the models relation.

### Semantic Consequence of Single Formula

Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.

#### Definition 1

Then $\mathbf A$ is a semantic consequence of $\mathbf B$ if and only if:

$v \models_{\mathrm{BI}} \mathbf B$ implies $v \models_{\mathrm{BI}} \mathbf A$

for all boolean interpretations $v$.

Here, $\models_{\mathrm{BI}}$ is the models relation.

#### Definition 2

Then $\mathbf A$ is a semantic consequence of $\mathbf B$ if and only if:

$\mathbf A \implies \mathbf B$ is a tautology

where $\implies$ is the conditional connective.

### Notation

That $\mathbf A$ is a semantic consequence of $\FF$ is denoted as:

$\FF \models_{\mathbf{BI}} \mathbf A$