Ground expression

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 with constant symbols $$a$$ and $$b$$, the sentence $$Q(a) \lor P(b)$$ 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 containing the constant symbols $$0$$ and $$1$$ for the numbers 0 and 1, respectively, a unary function symbol $$s$$ for the successor function and a binary function symbol $$+$$ for addition.
 * $$s(0), s(s(0)), s(s(s(0))), \ldots$$ are ground terms;
 * $$0 + 1, \; 0 + 1 + 1, \ldots$$ are ground terms;
 * $$0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0$$ 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.

Formal definitions
What follows is a formal definition for first-order languages. Let a first-order language be given, with $$C$$ the set of constant symbols, $$F$$ the set of functional operators, and $$P$$ 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) Elements of $$C$$ are ground terms;
 * 2) If $$f \in F$$ is an $$n$$-ary function symbol and $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are ground terms, then $$f\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)$$ 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.

If $$p \in P$$ is an $$n$$-ary predicate symbol and $$\alpha_1, \alpha_2, \ldots, \alpha_n$$ are ground terms, then $$p\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)$$ 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) If $$\varphi$$ and $$\psi$$ are ground formulas, then $$\lnot \varphi$$, $$\varphi \lor \psi$$, and $$\varphi \land \psi$$ are ground formulas.

Ground formulas are a particular kind of closed formulas.