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