Definition:Structure for Predicate Logic

Definition

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

A structure $\AA$ for $\LL_1$ comprises:

$(1): \quad$ A non-empty set $A$;
$(2): \quad$ For each function symbol $f$ of arity $n$, a mapping $f_\AA: A^n \to A$;
$(3): \quad$ For each predicate symbol $p$ of arity $n$, a mapping $p_\AA: A^n \to \Bbb B$

where $\Bbb B$ denotes the set of truth values.

$A$ is called the underlying set of $\AA$.

$f_\AA$ and $p_\AA$ are called the interpretations of $f$ and $p$ in $\AA$, respectively.

We remark that function symbols of arity $0$ are interpreted as constants in $A$.

To avoid pathological situations with the interpretation of arity-$0$ function symbols, it is essential that $A$ be non-empty.

Also, the predicate symbols may be interpreted as relations via their characteristic functions.

Formal Semantics

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:

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

Also known as

A structure for $\LL_1$ is also often called a structure for predicate logic or first-order structure.

The latter formulation is particularly used when the precise vocabulary used for $\LL_1$ is not important.