\begin{definition}[Definition:Boolean Interpretation/Formal Semantics]
Let $\LL_0$ be the language of propositional logic.
The boolean interpretations for $\LL_0$ can be interpreted as a formal semantics for $\LL_0$, which we denote by $\mathrm{BI}$.
The structures of $\mathrm{BI}$ are the boolean interpretations.
A WFF $\phi$ is declared ($\mathrm{BI}$-)valid in a boolean interpretation $v$ {{iff}}:
:$\map v \phi = \T$
Symbolically, this can be expressed as:
:$v \models_{\mathrm{BI}} \phi$
\end{definition}