Ground expression explained

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 and b , the sentence 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 and 1 for the numbers 0 and 1, respectively, 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;0+ s( 0 ), s( 0 ) + s( 0 ), s( 0 ) + s(s( 0 )) +0 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 definitions

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

C the set of constant symbols, F the set of functional operators, and P the set of predicate symbols.

Ground term

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

1. Elements of

C are ground terms;

1. 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.

1. 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 , or 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 or 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

\varphi and \psi are ground formulas, then ln ot \varphi , \varphi \lor \psi , and \varphi \land \psi are ground formulas.

Ground formulas are a particular kind of closed formulas.

References

• • • First-Order Logic: Syntax and Semantics