# Definition:Formal Semantics/Structure

*This page is about structures in the context of formal systems. For other uses, see Definition:Structure.*

## Definition

Let $\mathcal L$ be a formal language.

Part of specifying a formal semantics $\mathscr M$ for $\mathcal L$ is to specify **structures** $\mathcal M$ for $\mathscr M$.

A **structure** can in principle be any object one can think of.

However, to get a useful formal semantics, the **structures** should support a meaningful definition of validity for the WFFs of $\mathcal L$.

It is common that **structures** are sets, often endowed with a number of relations or functions.

### Structure for Predicate Logic

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.