Mathematical logic and foundations Definition

Term

Definition

Let $\sigma$ be a signature and let $V$ be a set of variables, disjoint from the symbols of $\sigma$ . The set of terms over $\sigma$ and $V$ is the smallest set of finite strings of symbols such that:

every variable $x \in V$ is a term;
every constant symbol $c$ of $\sigma$ is a term;
if $f$ is a function symbol of $\sigma$ of arity $n \ge 1$ and $t_1, \dots, t_n$ are terms, then $f(t_1, \dots, t_n)$ is a term.

A term with no variables is a closed term, or a ground term.

Remarks

In the signature of rings, $\cdot(x, +(y, 1))$ is a term, usually written $x \cdot (y + 1)$ . It is a term by clause 1 for $x$ and $y$ , clause 2 for $1$ , and clause 3 twice. In the signature of ordered sets, $x < y$ is not a term: $<$ is a relation symbol, not a function symbol, so clause 3 does not apply. Every term has a unique reading, so one can define functions on terms by recursion on this construction. In Mathlib terms are FirstOrder.Language.Term.

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Signature concept
Term definition

Tag	`BQQ0FPFZ67`
URI	`m2://BQQ0FPFZ67`
Kind	definition
Writing	W0 stub
MSC	03B10, 03C07
Defines	Term
Uses	Signature
Changed	2 November 2026

Term

Definition

Remarks

Reading path

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