Function (First-Order Logic)

first-order-logic

Definition

Function (First-Order Logic)

Let $I=(\mathcal{U},{\cdot }^I)$ be an interpretation. For a function symbol $f/n\in \mathcal{F}$, the interpretation assigns a function

$$f^I:{\mathcal{U}}^n\to \mathcal{U}$$

Properties

Total

$f^I(t_1,\dots ,t_n)$ is defined for every tuple $(t_1,\dots ,t_n)\in {\mathcal{U}}^n$.

Deterministic

$f^I(t_1,\dots ,t_n)$ is unique for each tuple.

Membership in universe

Every term is interpreted as an object in $\mathcal{U}$. The converse need not hold: not every object need be named by a term.