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 $\forall$ 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(z)=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 $C$ the set of constant symbols, $V$ the set of (individual) variables, $F$ the set of functional operators, and $P$ 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 α₁, α₂, ..., α_n are ground terms, then f(α₁, α₂, ..., 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 is an atomic formula all of whose argument terms are ground terms.

If p∈P is an n-ary predicate symbol and α₁, α₂, ..., α_n are ground terms, then p(α₁, α₂, ..., 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 a ground 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 $\forall$ x p and $\exists$ 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