Definition:Semantic Consequence/Boolean Interpretations

Definition
Let $\mathcal F$ be a collection of WFFs of propositional logic.

Then a WFF $\mathbf A$ is a semantic consequence of $\mathcal F$ iff:


 * $v \models_{\mathrm{BI}} \mathcal F$ implies $v \models_{\mathrm{BI}} \mathbf A$

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

Definition 1
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.

Then $\mathbf A$ is a semantic consequence of $\mathbf B$ iff:


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

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

Definition 2
Let $\mathbf A, \mathbf B$ be WFFs of propositional logic.

Then $\mathbf A$ is a semantic consequence of $\mathbf B$ iff:


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

that is, $\mathbf B \implies \mathbf A$ is a tautology.