Term

logic

Definition

Term

A term is a syntactic expression in first-order logic that represents an object in the universe. Terms are defined inductively over a signature $(\mathcal{F},\mathcal{P})$ and a set of variables $\mathcal{V}$:

1. Every variable $x\in \mathcal{V}$ is a term.
2. Every constant $c/0\in \mathcal{F}$ is a term.
3. If $f/n\in \mathcal{F}$ is an $n$-ary function symbol and $t_1,\dots ,t_n$ are terms, then $f(t_1,\dots ,t_n)$ is a term.

Semantics

The meaning of a term is determined by the valuation function ${\mathrm{val}}_{I,\sigma }$, which maps the term to an element of the universe $\mathcal{U}$.

Terms vs. Formulas

Terms represent objects (e.g., $x+1$, $f(a)$), whereas formulas represent statements that can be true or false (e.g., $x+1=y$, $P(x)$).