Definition:Semantic Consequence/Boolean Interpretations/Single Formula/Definition 2

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:

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

where $\implies$ is the conditional connective.

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)

Sources

• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Definition $1.7$