Ground expression
Part of Formal languages |
In
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
- 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
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
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
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 variable
- Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variables
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
- ISBN 978-0-521-58713-6
- First-Order Logic: Syntax and Semantics