# 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:

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

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

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