Definition:Structure for Predicate Logic

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This page is about structures in the context of predicate logic. For other uses, see Definition:Structure.

Definition

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


A structure $\mathcal A$ for $\mathcal L_1$ comprises:

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

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

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

$f_{\mathcal A}$ and $p_{\mathcal A}$ are called the interpretations of $f$ and $p$ in $\mathcal A$, respectively.


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

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


Formal Semantics

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:

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


Also known as

A structure for $\mathcal L_1$ 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 $\mathcal L_1$ is not important.


Also see


Sources