# Definition:Model (Predicate Logic)

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## 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:

- $\mathcal A \models \mathbf A$

## Also see

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\mathrm{II}.7$ First-Order Logic Semantics: Definition $\mathrm{II.7.11}$

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.4$