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 ${\displaystyle a}$and ${\displaystyle b}$, the sentence ${\displaystyle Q(a)\lor P(b)}$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 ${\displaystyle 0}$and ${\displaystyle 1}$for the numbers 0 and 1, respectively, a unary function symbol ${\displaystyle s}$for the successor function and a binary function symbol ${\displaystyle +}$for addition.

• ${\displaystyle s(0),s(s(0)),s(s(s(0))),\ldots }$are ground terms;
• ${\displaystyle 0+1,\;0+1+1,\ldots }$are ground terms;
• ${\displaystyle 0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0}$are ground terms;
• ${\displaystyle x+s(1)}$and ${\displaystyle s(x)}$are terms, but not ground terms;
• ${\displaystyle s(0)=1}$and ${\displaystyle 0+0=0}$are ground formulae.

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

A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):

1. Elements of ${\displaystyle C}$are ground terms;
2. If ${\displaystyle f\in F}$is an ${\displaystyle n}$-ary function symbol and ${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}$are ground terms, then ${\displaystyle f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}$is a ground term.
3. 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 ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.

If ${\displaystyle p\in P}$is an ${\displaystyle n}$-ary predicate symbol and ${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}$are ground terms, then ${\displaystyle p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}$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 ground formula or ground clause is a formula without variables.

Ground formulas may be defined by syntactic recursion as follows:

1. A ground atom is a ground formula.
2. If ${\displaystyle \varphi }$and ${\displaystyle \psi }$are ground formulas, then ${\displaystyle \lnot \varphi }$, ${\displaystyle \varphi \lor \psi }$, and ${\displaystyle \varphi \land \psi }$are ground formulas.

Ground formulas are a particular kind of closed formulas.

• Open formula – Formula that contains at least one free variable
• Sentence (mathematical logic) – In mathematical logic, a well-formed formula with no free variables

• Dalal, M. (2000). "Logic-based computer programming paradigms". In Rosen, K.H.; Michaels, J.G. (eds.). Handbook of discrete and combinatorial mathematics. p. 68.
• Fern, Alan (8 January 2010). "Lecture Notes | First-Order Logic: Syntax and Semantics" (PDF).
• Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 978-0-521-58713-6.
• Alex Sakharov. "Ground Atom". MathWorld. Retrieved 4 May 2025.