# Definition:Model (Boolean Interpretations)

## 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:

- $v \models \phi$

## Also see

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.7$: Tableaus - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.4.1$: The meaning of logical connectives