# Definition:Structure for Predicate Logic

*This page is about Structure in the context of Predicate Logic. For other uses, see Structure.*

## 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.

## Also see

## Sources

- 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\text {II}.7$ First-Order Logic Semantics: Definition $\text {II}.7.1$