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 a constant symbol $0$ for the number $0,$ 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,
• $x+s(1)$ and $s(x)$ are terms, but not ground terms,
• $s(0)=1$ and $0+0=0$ are ground formulae,

Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with $C$ the set of constant symbols, $V$ the set of (individual) variables, $F$ the set of functional operators, and $P$ the set of predicate symbols.

Ground terms

A ground term is a terms that contain no variables. They 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 $\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}$ are ground terms, then $f\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)$ 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

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, 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.

Formulas with free variables may be defined by syntactic recursion as follows:

1. The free variables of an unground atom are all variables occurring in it.
2. The free variables of $\lnot p$ are the same as those of $p.$ The free variables of $p\lor q,p\land q,p\to q$ are those free variables of $p$ or free variables of $q.$
3. The free variables of $\forall x\;p$ and $\exists x\;p$ are the free variables of $p$ except $x.$

References

• 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