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 a and b, 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 0 and 1 for the numbers 0 and 1, respectively, a unary function symbol s for the successor function and a binary function symbol + for addition.

Formal definitions

What follows is a formal definition for first-order languages. Let a first-order language be given, with C the set of constant symbols, F the set of functional operators, and P the set of predicate symbols.

Ground term

A **** is**** a term**** that**** contains**** no**** variables.**** Ground terms may be defined by logical recursion (formula-recursion): Roughly speaking, the Herbrand universe is the set of all ground terms.

Ground atom

A ', ' or **** is an at**om**ic**** formula all of**** whose argument**** terms are ground**** terms.**** If p \in P is an n-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.

Ground formula

A **' or **' is a formula without variables. Ground formulas may be defined by syntactic recursion as follows: Ground formulas are a particular kind of closed formulas.