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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, 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):

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, 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:

The free variables of an unground atom are all variables occurring in it.
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 $q.$
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

@@ Line 1: / Line 1: @@
 {{Short description|Term that does not contain any variables}}
-In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[term (logic)|term]] that does not contain any [[Variable (mathematics)|variables]]. Similarly, a '''ground formula''' is a [[well formed formula|formula]] that does not contain any variables.
+In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[Term (logic)|term]] that does not contain any [[Variable (mathematics)|variables]]. Similarly, a '''ground formula''' is a [[Well formed formula|formula]] that does not contain any variables.
-In [[First-order logic#Equality and its axioms|first-order logic with identity]], the sentence ''Q''(''a'') ∨ ''P''(''b'') is a ground formula, with ''a'' and ''b'' being constant symbols. A '''ground expression''' is a ground term or ground formula.
+In [[First-order logic#Equality and its axioms|first-order logic with identity]], the sentence <math>Q(a) \lor P(b)</math> is a ground formula, with <math>a</math> and <math>b</math> being constant symbols. A '''ground expression''' is a ground term or ground formula.
-== Examples ==
+==Examples==
-Consider the following expressions in [[first order logic]] over a [[signature (mathematical logic)|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.
+Consider the following expressions in [[first order logic]] over a [[signature (mathematical logic)|signature]] containing a constant symbol <math>0</math> for the number <math>0,</math> a unary function symbol <math>s</math> for the successor function and a binary function symbol <math>+</math> for addition.
-* ''s''(0), ''s''(''s''(0)), ''s''(''s''(''s''(0))), ... are ground terms,
+* <math>s(0), s(s(0)), s(s(s(0))), \ldots</math> are ground terms,
-* 0{{Hair space}}+{{Hair space}}1, 0{{Hair space}}+{{Hair space}}1{{Hair space}}+{{Hair space}}1, ... are ground terms,
+* <math>0 + 1, \; 0 + 1 + 1, \ldots</math> are ground terms,
-* ''x''{{Hair space}}+{{Hair space}}''s''(1) and ''s''(''x'') are terms, but not ground terms,
+* <math>x + s(1)</math> and <math>s(x)</math> are terms, but not ground terms,
-* ''s''(0) = 1 and 0{{Hair space}}+{{Hair space}}0 = 0 are ground formulae,
+* <math>s(0) = 1</math> and <math>0 + 0 = 0</math> are ground formulae,
-== Formal definition ==
+==Formal definition==
-What follows is a formal definition for [[first-order language]]s. 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 symbol]]s.
+What follows is a formal definition for [[first-order language]]s. Let a first-order language be given, with <math>C</math> the set of constant symbols, <math>V</math> the set of (individual) variables, <math>F</math> the set of functional operators, and <math>P</math> the set of [[predicate symbol]]s.
-=== Ground terms ===
+===Ground terms===
-Ground terms are [[term (logic)|terms]] that contain no variables. They may be defined by logical recursion (formula-recursion):
+A '''{{visible anchor|ground term}}''' is a [[Term (logic)|terms]] that contain no variables. They may be defined by logical recursion (formula-recursion):
-# Elements of ''C'' are ground terms;
+# Elements of <math>C</math> are ground terms;
-# If ''f'' ∈ ''F'' is an ''n''-ary function symbol and α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>''n''</sub> are ground terms, then ''f''(α<sub>1</sub>, α<sub>2</sub>, ..., α<sub>''n''</sub>) is a ground term.
+# If <math>f \in F</math> is an <math>n</math>-ary function symbol and <math>\alpha_1, \alpha_2, \ldots, \alpha_n</math> are ground terms, then <math>f\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)</math> 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 ===
+===Ground atom===
-A '''ground predicate''', '''ground atom''' or '''ground literal''' is an [[atomic formula]] all of whose argument terms are ground terms.
+A '''{{visible anchor|ground predicate}}''', '''{{visible anchor|ground atom}}''' or '''{{visible anchor|ground literal}}''' 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.
+If <math>p \in P</math> is an <math>n</math>-ary predicate symbol and <math>\alpha_1, \alpha_2, \ldots, \alpha_n</math> are ground terms, then <math>p\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)</math> 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 ===
+===Ground formula===
-A ground formula or ground clause is a formula without variables.
+A '''{{visible anchor|ground formula}}''' or '''{{visible anchor|ground clause}}''' is a formula without variables.
 Formulas with free variables may be defined by syntactic recursion as follows:
 # The free variables of an unground 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 free variables of ''p'' or free variables of ''q''.
+# The free variables of <math>\lnot p</math> are the same as those of <math>p.</math> The free variables of <math>p \lor q, p \land q, p \to q</math> are those free variables of <math>p</math> or free variables of <math>q.</math>
-# The free variables of ∀''x''&nbsp;''p'' and ∃''x''&nbsp;''p'' are the free variables of ''p'' except ''x''.
+# The free variables of <math>\forall x \; p</math> and <math>\exists x \; p</math> are the free variables of <math>p</math> except <math>x.</math>
+==References==
+{{reflist}}
-== References ==
 * {{Citation | title=Handbook of discrete and combinatorial mathematics | contribution = Logic-based computer programming paradigms | year=2000 | editor1-last = Rosen | editor1-first = K.H. | editor2-last = Michaels | editor2-first = J.G. | last = Dalal | first = M. | page=68}}
 * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-58713-6 | year=1997}}
@@ Line 47: / Line 52: @@
 {{Mathematical logic}}
-[[Category:Mathematical logic]]
 [[Category:Logical expressions]]
+[[Category:Mathematical logic]]