Definition:Structure for Predicate Logic

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.

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

 * Definition:Language of Predicate Logic
 * Definition:Structure (Formal Systems)