Ground expression

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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 and , the sentence is a ground formula. A ground expression is a ground term or ground formula.


Consider the following expressions in first order logic over a signature containing the constant symbols and for the numbers 0 and 1, respectively, a unary function symbol for the successor function and a binary function symbol for addition.

  • are ground terms;
  • are ground terms;
  • are ground terms;
  • and are terms, but not ground terms;
  • and are ground formulae.

Formal definitions[edit]

What follows is a formal definition for first-order languages. Let a first-order language be given, with the set of constant symbols, the set of functional operators, and the set of predicate symbols.

Ground term[edit]

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

  1. Elements of are ground terms;
  2. If is an -ary function symbol and are ground terms, then 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[edit]

A ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.

If is an -ary predicate symbol and are ground terms, then is a ground predicate or ground atom.

Roughly speaking, the Herbrand base is the set of all ground atoms,[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula[edit]

A ground formula or ground clause 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 and are ground formulas, then , , and are ground formulas.

Ground formulas are a particular kind of closed formulas.

See also[edit]

  • Open formula – formula that contains at least one free variable
  • Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variables


  1. ^ Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.