Definition:Model (Predicate Logic)

 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 $\mathbf A$ iff:


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

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

This relationship is denoted as:


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

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


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

that is, iff 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

 * Definition:Value of Sentence in Structure for Predicate Logic
 * Definition:Model (Logic)