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

In first-order logic with identity, the sentence is a ground formula, with and being constant symbols. A **ground expression** is a ground term or ground formula.

## Examples

Consider the following expressions in first order logic over a signature containing a constant symbol for the number a unary function symbol for the successor function and a binary function symbol for addition.

- are ground terms,
- are ground terms,
- and are terms, but not ground terms,
- and are ground formulae,

## 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

A **ground term** is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):

- Elements of are ground terms;
- If is an -ary function symbol and are ground terms, then 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**, **ground atom** or **ground literal** is an atomic formula all of whose argument terms are ground terms.

If is an -ary predicate symbol and are ground terms, then 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 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 are the same as those of The free variables of are those free variables of or free variables of
- The free variables of and are the free variables of except

## See also

## 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