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

• ${\displaystyle s(0),s(s(0)),s(s(s(0))),\ldots }${\displaystyle s(0),s(s(0)),s(s(s(0))),\ldots } are ground terms;
• ${\displaystyle 0+1,\;0+1+1,\ldots }${\displaystyle 0+1,\;0+1+1,\ldots } are ground terms;
• ${\displaystyle 0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0}${\displaystyle 0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0} are ground terms;
• ${\displaystyle x+s(1)}${\displaystyle x+s(1)} and ${\displaystyle s(x)}${\displaystyle s(x)} are terms, but not ground terms;
• ${\displaystyle s(0)=1}${\displaystyle s(0)=1} and ${\displaystyle 0+0=0}${\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 ${\displaystyle C}${\displaystyle C} the set of constant symbols, ${\displaystyle F}${\displaystyle F} the set of functional operators, and ${\displaystyle P}${\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 ${\displaystyle C}${\displaystyle C} are ground terms;
2. If ${\displaystyle f\in F}${\displaystyle f\in F} is an ${\displaystyle n}${\displaystyle n}-ary function symbol and ${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are ground terms, then ${\displaystyle f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}${\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 ${\displaystyle p\in P}${\displaystyle p\in P} is an ${\displaystyle n}${\displaystyle n}-ary predicate symbol and ${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}}${\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are ground terms, then ${\displaystyle p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)}${\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,^[1] 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 ${\displaystyle \varphi }${\displaystyle \varphi } and ${\displaystyle \psi }${\displaystyle \psi } are ground formulas, then ${\displaystyle \lnot \varphi }${\displaystyle \lnot \varphi }, ${\displaystyle \varphi \lor \psi }${\displaystyle \varphi \lor \psi }, and ${\displaystyle \varphi \land \psi }${\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.

• Logical expressions
• Mathematical logic

• Articles with short description
• Short description is different from Wikidata
• Use dmy dates from May 2025
• Pages that use a deprecated format of the math tags