![Latex domain of discourse](https://upload.wikimedia.org/wikipedia/commons/4/49/Latex_domain_of_discourse.svg)
In the
formal science
Formal science is a branch of science studying disciplines concerned with abstract structures described by formal systems, such as logic, mathematics, statistics, theoretical computer science, artificial intelligence, information theory, ga ...
s, the domain of discourse, also called the universe of discourse,
universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
, or simply
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
, is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of entities over which certain
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
s of interest in some
formal treatment may range.
Overview
![Giuseppe Peano](https://upload.wikimedia.org/wikipedia/commons/3/3a/Giuseppe_Peano.jpg)
The domain of discourse is usually identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables. Many logicians distinguish, sometimes only tacitly, between the ''domain of a science'' and the ''universe of discourse of a formalization of the science''.
[José Miguel Sagüillo, Domains of sciences, universe of discourse, and omega arguments, History and philosophy of logic, vol. 20 (1999), pp. 267–280.]
Examples
For example, in an
interpretation of
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 ...
, the domain of discourse is the set of individuals over which the
quantifiers range. A proposition such as is ambiguous, if no domain of discourse has been identified. In one interpretation, the domain of discourse could be the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s; in another interpretation, it could be the set of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s. If the domain of discourse is the set of real numbers, the proposition is false, with as counterexample; if the domain is the set of natural numbers, the proposition is true, since 2 is not the square of any natural number.
Universe of discourse
The term "universe of discourse" generally refers to the collection of objects being discussed in a specific
discourse
Discourse is a generalization of the notion of a conversation to any form of communication. Discourse is a major topic in social theory, with work spanning fields such as sociology, anthropology, continental philosophy, and discourse analysis. ...
. In
model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. The concept ''universe of discourse'' is generally attributed to
Augustus De Morgan
Augustus De Morgan (27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.
Biography
Childhood
Augustus De Morgan was b ...
(1846) but the name was used for the first time by
George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ire ...
(1854) on page 42 of his ''
Laws of Thought
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally th ...
''. Boole's definition is quoted below. The concept, probably discovered independently by Boole in 1847, played a crucial role in his philosophy of logic especially in his principle of
wholistic reference
Wholistic reference is reference to the whole—with respect to the context. In its strongest, unqualified form, the principle of wholistic reference is the proposition that each and every proposition, regardless how limited the referents of its n ...
.
Boole’s 1854 definition
See also
*
Domain of a function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function.
More precisely, given a function f\colon X\to Y, the domain of is . ...
*
Domain theory
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...
*
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everythin ...
*
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning unti ...
*
Term algebra
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set ''X'' of variables is exa ...
*
Universe (mathematics)
In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.
In set theory, universes are ...
References
{{DEFAULTSORT:Domain Of Discourse
Semantics
Predicate logic