\begin{definition}[Definition:Conditional/Boolean Interpretation]
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:
:$\map v {\mathbf A \impliedby \mathbf B} = \begin{cases}
\T & : \map v {\mathbf A} = \T \text{ or } \map v {\mathbf B} = \F \\
\F & : \text{otherwise}
\end{cases}$
\end{definition}