# Definition:Model (Predicate Logic)

## Definition

Let $\mathcal L_1$ be the language of predicate logic.

Let $\mathcal A$ be a structure for predicate logic.

Then $\mathcal A$ models a sentence $\mathbf A$ if and only if:

$\operatorname{val}_{\mathcal A} \left({\mathbf A}\right) = T$

where $\operatorname{val}_{\mathcal A} \left({\mathbf A}\right)$ denotes the value of $\mathbf A$ in $\mathcal A$.

This relationship is denoted:

$\mathcal A \models_{\mathrm{PL}} \mathbf A$

When pertaining to a collection of sentences $\mathcal F$, one says $\mathcal A$ models $\mathcal F$ if and only if:

$\forall \mathbf A \in \mathcal F: \mathcal A \models_{\mathrm{PL}} \mathbf A$

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

This can be expressed symbolically as:

$\mathcal A \models_{\mathrm {PL}} \mathcal F$

## Also denoted as

Often, when the formal semantics is clear to be $\mathrm{PL}$, the formal semantics for structures of predicate logic, the subscript is omitted, yielding:

$\mathcal A \models \mathbf A$