# Definition:Model (Boolean Interpretations)

Jump to navigation Jump to search

## Definition

Let $\mathcal L_0$ be the language of propositional logic.

Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation of $\mathcal L_0$.

Then $v$ models a WFF $\phi$ iff:

$v \left({\phi}\right) = T$

and this relationship is denoted as:

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

When pertaining to a collection of WFFs $\mathcal F$, one says $v$ models $\mathcal F$ iff:

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

that is, iff it models all elements of $\mathcal F$.

This can be expressed symbolically as:

$v \models_{\mathrm {BI}} \mathcal F$

## 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:

$v \models \phi$