Definition:Model (Predicate Logic)

Definition

Let $\LL_1$ be the language of predicate logic.

Let $\AA$ be a structure for predicate logic.

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

$\map {\operatorname{val}_\AA} {\mathbf A} = \T$

where $\map {\operatorname{val}_\AA} {\mathbf A}$ denotes the value of $\mathbf A$ in $\AA$.

This relationship is denoted:

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

When pertaining to a collection of sentences $\FF$, one says $\AA$ models $\FF$ if and only if:

$\forall \mathbf A \in \FF: \AA \models_{\mathrm{PL} } \mathbf A$

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

This can be expressed symbolically as:

$\AA \models_{\mathrm {PL} } \FF$

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:

$\AA \models \mathbf A$

Also see

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

Sources

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

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