Ground expression

Template:Short description Template:Formal languages 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 $Q(a)\vee P(b)$ 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.

• $s(0),s(s(0)),s(s(s(0))),\dots$ are ground terms;
• $0+1,0+1+1,\dots$ 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 Template:Visible anchor is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):

1. Elements of $C$ are ground terms;
2. If $f\in F$ is an $n$-ary function symbol and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ are ground terms, then $f({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ 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 Template:Visible anchor, Template:Visible anchor or Template:Visible anchor is an atomic formula all of whose argument terms are ground terms.

If $p\in P$ is an $n$-ary predicate symbol and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ are ground terms, then $p({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ 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 Template:Visible anchor or Template:Visible anchor 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 $\phi$ and $\psi$ are ground formulas, then $\mathrm{\neg }\phi$, $\phi \vee \psi$, and $\phi \wedge \psi$ 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