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:


 * $\map v {\mathbf A \implies \mathbf B} = \begin{cases}

\text T & : \map v {\mathbf A} = \text F \text{ or } \map v {\mathbf B} = \text T \\ \text F & : \text{otherwise} \end{cases}$

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


 * $\map v {\mathbf A \impliedby \mathbf B} = \begin{cases}

\text T & : \map v {\mathbf A} = \text T \text{ or } \map v {\mathbf B} = \text F \\ \text F & : \text{otherwise} \end{cases}$