# Definition:Structure for Predicate Logic/Formal Semantics

## Definition

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

### Formal Semantics for Sentences

The structures for $\LL_1$ can be interpreted as a formal semantics for $\LL_1$, 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 $\LL_1$.

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

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

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

Symbolically, this can be expressed as:

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

### Formal Semantics for WFFs

The structures for $\LL_1$ can be interpreted as a formal semantics for $\LL_1$, which we denote by $\mathrm{PL_A}$.

The structures of $\mathrm{PL_A}$ are pairs $\struct {\AA, \sigma}$, where:

$\AA$ is a structure for $\LL_1$
$\sigma$ is an assignment for $\AA$

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

$\sigma$ is an assignment for $\mathbf A$
$\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma = \T$

where $\map {\operatorname{val}_\AA} {\mathbf A} \sqbrk \sigma$ is the value of $\mathbf A$ under $\sigma$.

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

$\AA, \sigma \models_{\mathrm{PL_A} } \mathbf A$
$\AA \models_{\mathrm{PL_A} } \mathbf A \sqbrk \sigma$