In mathematical logic, a ground term of a formal system is a term that does not contain any free variables.
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Similarly, a ground formula is a formula that does not contain any free variables. In first-order logic with identity, the sentence
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.
Formal definition
What follows is a formal definition for first-order languages. Let a first-order language be given, with
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, ..., αn are ground terms, then f(α1, α2, ..., αn) 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 α1, α2, ..., αn are ground terms, then p(α1, α2, ..., αn) 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.