Ground expression

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.