# Definition:Model (Logic)

## Definition

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

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

### Model of Logical Formula

Let $\phi$ be a logical formula of $\mathcal L$.

Then $\mathcal M$ is a model of $\phi$ iff:

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

that is, if $\phi$ is valid in $\mathcal M$.

### Model of Set of Logical Formulas

Let $\mathcal F$ be a set of logical formulas of $\mathcal L$.

Then $\mathcal M$ is a model of $\mathcal F$ iff:

$\mathcal M \models_{\mathscr M} \phi$ for every $\phi \in \mathcal F$

that is, if it is a model of every logical formula $\phi \in \mathcal F$.

## Specific Examples

### Boolean Interpretations

Let $\mathcal L_0$ be the language of propositional logic.

Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation of $\mathcal L_0$.

Then $v$ models a WFF $\phi$ iff:

$v \left({\phi}\right) = T$

and this relationship is denoted as:

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

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

$\forall \phi \in \mathcal F: v \models_{\mathrm{BI}} \phi$

that is, iff it models all elements of $\mathcal F$.

This can be expressed symbolically as:

$v \models_{\mathrm {BI}} \mathcal F$

### 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 $\mathbf A$ 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 known as

If $\mathcal M$ is a model of $\phi$, respectively $\mathcal F$, one sometimes says that $\mathcal M$ models $\phi$, respectively $\mathcal F$.