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, the sentence $Q(a) \lor P(b)$ is a ground formula, with $a$ and $b$ being constant symbols. 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 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 , or 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, while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula

A or 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.

See also

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References

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First-Order Logic: Syntax and Semantics

{{Mathematical logic

Logical expressions
Mathematical logic