Definition:Conditional/Boolean Interpretation

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

Let $\implies$ denote the implication operator.

The truth value of $\mathbf A \implies \mathbf B$ under a boolean interpretation $v$ is given by:


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

T & : v \left({\mathbf A}\right) = F \text{ or } v \left({\mathbf B}\right) = T \\ F & : \text{otherwise} \end{cases}$

and the truth value of $\mathbf A \impliedby \mathbf B$ under a boolean interpretation $v$ is given by:


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

T & : v \left({\mathbf A}\right) = T \text{ or } v \left({\mathbf B}\right) = F \\ F & : \text{otherwise} \end{cases}$