# Definition:Formal Semantics/Structure

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

## Definition

Let $\LL$ be a formal language.

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

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

### Structure for Predicate Logic

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.