Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 1

Definition

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

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.

Notation

That $\mathbf A$ is a semantic consequence of $\mathbf B$ can be denoted as:

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

Also see

• Definition:Semantic Equivalence (Boolean Interpretations)