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.

Ground term edit

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.

Ground atom edit

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,^[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula edit

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 variablePages displaying wikidata descriptions as a fallback
• Sentence (mathematical logic) – in mathematical logic, a well-formed formula with no free variablesPages displaying wikidata descriptions as a fallback

• Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.), Handbook of discrete and combinatorial mathematics, p. 68
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6
• First-Order Logic: Syntax and Semantics