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 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.

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.

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.

A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):

1. Elements of C {\displaystyle C} are ground terms;
2. If f ∈<!-- ∈ --> F {\displaystyle f\in F} is an n {\displaystyle n}-ary function symbol and α<!-- α --> 1 , α<!-- α --> 2 , …<!-- … --> , α<!-- α --> n {\displaystyle \alpha _1,\alpha _2,\ldots ,\alpha _n} are ground terms, then f ( α<!-- α --> 1 , α<!-- α --> 2 , …<!-- … --> , α<!-- α --> n ) {\displaystyle 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.

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<!-- α --> 1 , α<!-- α --> 2 , …<!-- … --> , α<!-- α --> n {\displaystyle \alpha _1,\alpha _2,\ldots ,\alpha _n} are ground terms, then p ( α<!-- α --> 1 , α<!-- α --> 2 , …<!-- … --> , α<!-- 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,^[1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.

A ground formula or ground clause 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 φ<!-- φ --> {\displaystyle \varphi } and ψ<!-- ψ --> {\displaystyle \psi } are ground formulas, then ¬<!-- ¬ --> φ<!-- φ --> {\displaystyle \lnot \varphi }, φ<!-- φ --> ∨<!-- ∨ --> ψ<!-- ψ --> {\displaystyle \varphi \lor \psi }, and φ<!-- φ --> ∧<!-- ∧ --> ψ<!-- ψ --> {\displaystyle \varphi \land \psi } are ground formulas.

Ground formulas are a particular kind of closed formulas.

• Open formula – Formula that contains at least one free variable
• Sentence (mathematical logic) – In mathematical logic, a well-formed formula with no free variables

• Dalal, M. (2000). "Logic-based computer programming paradigms". In Rosen, K.H.; Michaels, J.G. (eds.). Handbook of discrete and combinatorial mathematics. p. 68.
• Fern, Alan (8 January 2010). "Lecture Notes | First-Order Logic: Syntax and Semantics" (PDF).
• Hodges, Wilfrid (1997). A shorter model theory. Cambridge University Press. ISBN 978-0-521-58713-6.