Ground expression

Formal definitions

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.

Ground term

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 ${\displaystyle {\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}$ are ground terms, then ${\displaystyle f\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)}$ 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.

Ground atom

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 ${\displaystyle {\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n}$ are ground terms, then ${\displaystyle p\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right)}$ 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.

Ground formula

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 ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are ground formulas, then ${\displaystyle \mathrm{\neg }\phi }$, ${\displaystyle \phi \vee \psi }$, and ${\displaystyle \phi \wedge \psi }$ are ground formulas.

Ground formulas are a particular kind of closed formulas.