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In mathematical logic, a ground term of a formal system is a term that does not contain any free variables.

Similarly, a ground formula is a formula that does not contain any free variables. In first-order logic with identity, the sentence $\forall$ x (x=x) is a ground formula.

A ground expression is a ground term or ground formula.

Examples

Consider the following expressions from 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))) ... are ground terms;
0+1, 0+1+1, ... 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;
s(z)=1 and ∀x: (s(x)+1=s(s(x))) are ground expressions.

Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with the $C$ 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

Ground terms are terms that contain no variables. They may be defined by logical recursion (formula-recursion):

elements of C are ground terms;
If f∈F is an n-ary function symbol and α₁, α₂, ..., α_n are ground terms, then f(α₁, α₂, ..., α_n) 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 or ground atom is an atomic formula all of whose argument terms are ground terms.

If p∈P is an n-ary predicate symbol and α₁, α₂, ..., α_n are ground terms, then p(α₁, α₂, ..., α_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 ground formula or ground clause is a formula without free variables.

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

The free variables of a ground atom are all variables occurring in it;
The free variables of ¬(p) are the same as those of (p). The free variables of (p)∨(q), (p)∧(q), (p)→(q) are those which are free variables of (p) or free variables of (q).
The free variables of ∀ {\displaystyle \forall } x (p) and

0 x (p) are the free variables of p without 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

@@ Line 1: / Line 1: @@
-In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[term (logic)|term]] that does not contain any variables at all, and a '''closed term''' is a term that has no free variables. In [[first-order logic]] all ground terms are closed terms, but in [[lambda calculus]] the closed term {{nowrap|λ ''x''. ''x'' (λ ''y''. ''y'')}} is not a ground term.
+In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[term (logic)|term]] that does not contain any free variables.
-Similarly, a '''ground formula''' is a [[well formed formula|formula]] that does not contain any variables, and a '''closed formula''' or '''sentence''' is a formula that has no free variables. In first-order logic with identity, the sentence {{all}}&nbsp;''x''&nbsp;(''x''=''x'') is not a ground formula.
+Similarly, a '''ground formula''' is a [[well formed formula|formula]] that does not contain any free variables. In first-order logic with identity, the sentence {{all}}&nbsp;''x''&nbsp;(''x''=''x'') is a ground formula.
 A '''ground expression''' is a ground term or ground formula.
@@ Line 12: / Line 12: @@
 * '''x'''+''s''(1) and ''s''('''x''') are terms, but not ground terms;
 * ''s''(0)=1 and 0+0=0 are ground formulae;
-* ''s''('''z''')=1 and ∀'''x''':&nbsp;(''s''('''x''')+1=''s''(''s''('''x'''))) are expressions, but are not ground expressions.
+* ''s''('''z''')=1 and ∀'''x''':&nbsp;(''s''('''x''')+1=''s''(''s''('''x'''))) are ground expressions.
-Ground expressions are necessarily closed.  The last example, ∀'''x''': (''s''('''x''')+1=''s''(''s''('''x'''))), shows that a closed expression is not necessarily a ground expression. So, this formula is a closed formula, but not a ground formula, because it ''contains'' a logical variable, even though that variable is not free.
 == Formal definition ==
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 === Ground atom ===
-A '''ground predicate''' or '''ground atom''' is an [[atomic formula]] all of whose terms are ground terms.  That is,
+A '''ground predicate''' or '''ground atom''' is an [[atomic formula]] all of whose argument terms are ground terms.
 If ''p''∈''P'' is an ''n''-ary predicate symbol and α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>n</sub> are ground terms, then ''p''(α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>n</sub>) is a ground predicate or ground atom.
@@ Line 36: / Line 34: @@
 === Ground formula ===
-A ground formula or ground clause is a formula all of whose arguments are ground atoms.
+A ground formula or ground clause is a formula without free variables.
-Ground formulae may be defined by syntactic recursion as follows:
+Formulas with free variables may be defined by syntactic recursion as follows:
+# The free variables of a ground atom are all variables occurring in it;
-# A ground atom is a ground formula; that is, if ''p''∈''P'' is an ''n''-ary predicate symbol and α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>n</sub> are ground terms, then ''p''(α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>n</sub>) is a ground formula (and is a ground atom);
-# If ''p'' and ''q'' are ground formulae, then ¬(''p''), (''p'')∨(''q''), (''p'')∧(''q''), (''p'')→(''q''), formulas composed with [[logical connective]]s, are ground formulae, too.
+# The free variables of ¬(''p'') are the same as those of (''p''). The free variables of (''p'')∨(''q''), (''p'')∧(''q''), (''p'')→(''q'') are those which are free variables of (''p'') or free variables of (''q'').
+# The free variables of {{all}}&nbsp;''x''&nbsp;(''p'') and {{exists}}&nbsp;''x''&nbsp;(''p'') are the free variables of ''p'' without ''x''.
-# If ''p'' is a ground formula and we can get q from it that way some ( or ) we delete or insert in the p formula, and then the result, q is well-formed and equivalent with p, then q is a ground formula.{{Clarify|date=February 2009}}
-# We can get all ground formulae applying these three rules.
 == References ==