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.

Examples

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

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

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

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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): \varphi and are ground formulas, then , , and are ground formulas.

Ground formulas are a particular kind of closed formulas.

See also

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

References

  1. Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.