# Definition:Semantic Consequence/Boolean Interpretations/Single Formula

## Definition

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 consequence of $\mathbf B$ for boolean interpretations can be denoted as:

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