Term that does not contain any variables
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):
- Elements of
are ground terms; - If
is an
-ary function symbol and
are ground terms, then
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, 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:
- A ground atom is a ground formula.
- If
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 variablePages displaying wikidata descriptions as a fallback
- Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variablesPages displaying wikidata descriptions as a fallback
References
- ^ Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.
- Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), Handbook of discrete and combinatorial mathematics, p. 68
- Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
- First-Order Logic: Syntax and Semantics