# Definition:Model (Boolean Interpretations)

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## 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:

- $v \models \phi$

## Also see

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 1.7$: Tableaus - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**model**:**1.** - 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 - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**model**:**1.**