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

Examples

Consider the following expressions in first order logic over a signature for addition.

• are ground terms;

• are ground terms;

• are ground terms;

• are terms, but not ground terms;

• are ground formulae.

Formal definitions

What follows is a formal definition for first-order languages the set of predicate symbols.

Ground term

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

1. are ground terms;

2. is a ground term.

3. 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 , or is an atomic formula all of whose argument terms are ground terms.

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 or is a formula without variables.

Ground formulas may be defined by syntactic recursion as follows:

1. A ground atom is a ground formula.

2. are ground formulas.

Ground formulas are a particular kind of closed formulas.

See also

Notes

References