Definition:Conditional/Boolean Interpretation

Definition
Let $\mathbf A$ and $\mathbf B$ be propositional formulas.

Let $\mathbf A \implies \mathbf B$ denote the implication operator.

The boolean interpretations for $\mathbf A \implies \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \implies \mathbf B}\right)_\mathcal M = \begin{cases}

T & : \mathbf A_\mathcal M = F \text{ or } \mathbf B_\mathcal M = T \\ F & : \text {otherwise} \end{cases}$

and the boolean interpretations for $\mathbf A \impliedby \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \impliedby \mathbf B}\right)_\mathcal M = \begin{cases}

T & : \mathbf A_\mathcal M = T \text{ or } \mathbf B_\mathcal M = F \\ F & : \text {otherwise} \end{cases}$