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, the sentence Q ∨ P is a ground formula, with a and b 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 0 for the number 0, a unary function symbol s for the successor function and a binary function symbol + for addition.
What follows is a formal definition for first-order languages. Let a first-order language be given, with C the set of constant symbols, V the set of 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 :
  1. Elements of C are ground terms;
  2. If f ∈ F is an n-ary function symbol and α1, α2,..., αn are ground terms, then f is a ground term.
  3. Every ground term can be given by a finite application of the above two rules.
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 p ∈ P is an n-ary predicate symbol and α1, α2,..., αn are ground terms, then p 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:
  1. The free variables of an unground atom are all variables occurring in it.
  2. 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.
  3. The free variables of ∀x p and ∃x p are the free variables of p except x.