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Concept
Ground expression
Summary
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.
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.
A 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.
A , or 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, while a Herbrand interpretation assigns a truth value to each ground atom in the base.
A or 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.
Official source
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