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In
mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, a ground term of a
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
is a
term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...
that does not contain any variables. Similarly, a ground formula is a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
over a
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
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 language First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
s. 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 symbol In mathematical logic, a predicate variable is a predicate letter which functions as a "placeholder" for a relation (between terms), but which has not been specifically assigned any particular relation (or meaning). Common symbols for denoting predi ...
s.


Ground term

A is a
term Term may refer to: * Terminology, or term, a noun or compound word used in a specific context, in particular: **Technical term, part of the specialized vocabulary of a particular field, specifically: ***Scientific terminology, terms used by scient ...
that contains no variables. Ground terms 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 In first-order logic, a Herbrand structure ''S'' is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol '' ...
is the set of all ground terms.


Ground atom

A , or is an
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
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 In first-order logic, a Herbrand structure ''S'' is a structure over a vocabulary σ that is defined solely by the syntactical properties of σ. The idea is to take the symbols of terms as their values, e.g. the denotation of a constant symbol ' ...
is the set of all ground atoms, while a
Herbrand interpretation In mathematical logic, a Herbrand interpretation is an interpretation in which all constants and function symbols are assigned very simple meanings. Specifically, every constant is interpreted as itself, and every function symbol is interpreted a ...
assigns a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progr ...
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: # A ground atom is a ground formula. # If \varphi and \psi are ground formulas, then \lnot \varphi, \varphi \lor \psi, and \varphi \land \psi are ground formulas. Ground formulas are a particular kind of closed formulas.


See also

* *


References

* *
First-Order Logic: Syntax and Semantics
{{Mathematical logic Logical expressions Mathematical logic