# Definition:Structure for Predicate Logic/Formal Semantics

## Definition

Let $\mathcal L_1$ be the language of predicate logic.

### Formal Semantics for Sentences

The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL}$.

For the purpose of this formal semantics, we consider only sentences instead of all WFFs.

The structures of $\mathrm{PL}$ are said structures for $\mathcal L_1$.

A sentence $\mathbf A$ is declared ($\mathrm{PL}$-)valid in a structure $\mathcal A$ if and only if:

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

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

Symbolically, this can be expressed as:

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

### Formal Semantics for WFFs

The structures for $\mathcal L_1$ can be interpreted as a formal semantics for $\mathcal L_1$, which we denote by $\mathrm{PL_A}$.

The structures of $\mathrm{PL_A}$ are pairs $\left({\mathcal A, \sigma}\right)$, where:

$\mathcal A$ is a structure for $\mathcal L_1$
$\sigma$ is an assignment for $\mathcal A$

A WFF $\mathbf A$ is declared ($\mathrm{PL_A}$-)valid in a structure $\mathcal A$ if and only if:

$\sigma$ is an assignment for $\mathbf A$
$\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right] = T$

where $\mathop{ \operatorname{val}_{\mathcal A} \left({\mathbf A}\right) } \left[{\sigma}\right]$ is the value of $\mathbf A$ under $\sigma$.

Symbolically, this can be expressed as one of the following:

$\mathcal A, \sigma \models_{\mathrm{PL_A}} \mathbf A$
$\mathcal A \models_{\mathrm{PL_A}} \mathbf A \left[{\sigma}\right]$