Definition:Model (Boolean Interpretations)

Definition

Let $\LL_0$ be the language of propositional logic.

Let $v: \LL_0 \to \set {T, F}$ be a boolean interpretation of $\LL_0$.

Then $v$ models a WFF $\phi$ if and only if:

$\map v \phi = T$

and this relationship is denoted as:

$v \models_{\mathrm {BI} } \phi$

When pertaining to a collection of WFFs $\FF$, one says $v$ models $\FF$ if and only if:

$\forall \phi \in \FF: v \models_{\mathrm {BI} } \phi$

that is, if and only if it models all elements of $\FF$.

This can be expressed symbolically as:

$v \models_{\mathrm {BI}} \FF$

Also denoted as

Often, when the formal semantics is clear to be $\mathrm {BI}$, the formal semantics of boolean interpretations, the subscript is omitted, yielding:

$v \models \phi$