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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.
Examples
[edit]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):
- 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
[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:
- 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
[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
References
[edit]- ^ 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