# Definition:Model (Logic)

## Definition

Let $\mathscr M$ be a formal semantics for a logical language $\LL$.

Let $\MM$ be a structure of $\mathscr M$.

### Model of Logical Formula

Let $\phi$ be a logical formula of $\LL$.

Then $\MM$ is a model of $\phi$ if and only if:

$\MM \models_{\mathscr M} \phi$

that is, if and only if $\phi$ is valid in $\MM$.

### Model of Set of Logical Formulas

Let $\FF$ be a set of logical formulas of $\LL$.

Then $\MM$ is a model of $\FF$ if and only if:

$\MM \models_{\mathscr M} \phi$ for every $\phi \in \FF$

that is, if and only if it is a model of every logical formula $\phi \in \FF$.

## Specific Examples

### Boolean Interpretations

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:

$v \models_{\mathrm {BI}} \FF$

### Predicate Logic

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 known as

If $\MM$ is a model of $\phi$, respectively $\FF$, one sometimes says that $\MM$ models $\phi$, respectively $\FF$.