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 {\displaystyle a} and b {\displaystyle b} , the sentence Q ( a ) ∨ P ( b ) {\displaystyle Q(a)\lor 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 {\displaystyle 0} and 1 {\displaystyle 1} for the numbers 0 and 1, respectively, a unary function symbol s {\displaystyle s} for the successor function and a binary function symbol + {\displaystyle +} for addition.
s ( 0 ) , s ( s ( 0 ) ) , s ( s ( s ( 0 ) ) ) , … {\displaystyle s(0),s(s(0)),s(s(s(0))),\ldots } are ground terms;
0 + 1 , 0 + 1 + 1 , … {\displaystyle 0+1,\;0+1+1,\ldots } are ground terms;
0 + s ( 0 ) , s ( 0 ) + s ( 0 ) , s ( 0 ) + s ( s ( 0 ) ) + 0 {\displaystyle 0+s(0),\;s(0)+s(0),\;s(0)+s(s(0))+0} are ground terms;
x + s ( 1 ) {\displaystyle x+s(1)} and s ( x ) {\displaystyle s(x)} are terms, but not ground terms;
s ( 0 ) = 1 {\displaystyle s(0)=1} and 0 + 0 = 0 {\displaystyle 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 {\displaystyle C} the set of constant symbols, F {\displaystyle F} the set of functional operators, and P {\displaystyle P} the set of predicate symbols.
Ground term A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
Elements of C {\displaystyle C} are ground terms; If f ∈ F {\displaystyle f\in F} is an n {\displaystyle n} -ary function symbol and α 1 , α 2 , … , α n {\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are ground terms, then f ( α 1 , α 2 , … , α n ) {\displaystyle 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 ∈ P {\displaystyle p\in P} is an n {\displaystyle n} -ary predicate symbol and α 1 , α 2 , … , α n {\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are ground terms, then p ( α 1 , α 2 , … , α n ) {\displaystyle 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. Ground formulas may be defined by syntactic recursion as follows:
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