# Definition:Formal Semantics/Structure

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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.