# Ground expression

In mathematical logic, a **ground term** of a formal system is a term that does not contain any free variables.

Similarly, a **ground formula** is a formula that does not contain any free variables. In first-order logic with identity, the sentence
*x* (*x*=*x*) is a ground formula.

A **ground expression** is a ground term or ground formula.

## Examples

Consider the following expressions from first order logic over a signature containing a constant symbol 0 for the number 0, a unary function symbol *s* for the successor function and a binary function symbol + for addition.

*s*(0),*s*(*s*(0)),*s*(*s*(*s*(0))) ... are ground terms;- 0+1, 0+1+1, ... are ground terms.
**x**+*s*(1) and*s*(**x**) are terms, but not ground terms;*s*(0)=1 and 0+0=0 are ground formulae;*s*(1) and ∀**x**: (*s*(**x**)+1=*s*(*s*(**x**))) are ground expressions.

## Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with the set of constant symbols, the set of (individual) variables, the set of functional operators, and the set of predicate symbols.

### Ground terms

Ground terms are terms that contain no variables. They may be defined by logical recursion (formula-recursion):

- elements of C are ground terms;
- If
*f*∈*F*is an*n*-ary function symbol and α_{1}, α_{2}, ..., α_{n}are ground terms, then*f*(α_{1}, α_{2}, ..., α_{n}) is a ground term. - Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).

Roughly speaking, the Herbrand universe is the set of all ground terms.

### Ground atom

A **ground predicate** or **ground atom** or **ground literal** is an atomic formula all of whose argument terms are ground terms.

If *p*∈*P* is an *n*-ary predicate symbol and α_{1}, α_{2}, ..., α_{n} are ground terms, then *p*(α_{1}, α_{2}, ..., α_{n}) is a ground predicate or ground atom.

Roughly speaking, the Herbrand base is the set of all ground atoms, while a Herbrand interpretation assigns a truth value to each ground atom in the base.

### Ground formula

A ground formula or ground clause is a formula without free variables.

Formulas with free variables may be defined by syntactic recursion as follows:

- The free variables of an unground atom are all variables occurring in it.
- The free variables of ¬
*p*are the same as those of*p*. The free variables of*p*∨*q*,*p*∧*q*,*p*→*q*are those free variables of*p*or free variables of*q*. - The free variables of
*x**p*and*x**p*are the free variables of*p*except*x*.

## References

- Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.),
*Handbook of discrete and combinatorial mathematics*, p. 68 - Hodges, Wilfrid (1997),
*A shorter model theory*, Cambridge University Press, ISBN 978-0-521-58713-6 - First-Order Logic: Syntax and Semantics