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Ground expression

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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 $Q(a)\lor P(b)$ 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.

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

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

Elements of $C$ are ground terms;
If $f\in F$ is an $n$ -ary function symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ is a ground term.
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 $p\in P$ is an $n$ -ary predicate symbol and $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ 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

A ground formula or ground clause is a formula without variables.

Ground formulas may be defined by syntactic recursion as follows:

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

Ground formulas are a particular kind of closed formulas.

References

^ Alex Sakharov. "Ground Atom". MathWorld. Retrieved October 20, 2022.

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

Mathematical logic

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Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
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Example axiomatic
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