Definition:Semantic Equivalence/Boolean Interpretations/Definition 1

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

Then $\mathbf A$ and $\mathbf B$ are equivalent for boolean interpretations iff:


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

that is, each is a semantic consequence of the other.

That is to say, $\mathbf A$ and $\mathbf B$ are equivalent iff:


 * $v \left({\mathbf A}\right) = T$ iff $v \left({\mathbf B}\right) = T$

for all boolean interpretations $v$.

Also see

 * Definition:Semantic Consequence (Boolean Interpretations)
 * Definition:Logical Equivalence