In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''

universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...

'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also defined as 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. F ...

. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on. 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''.

Etymology

The concept ''universe of discourse'' 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 Ireland. H ...

(1854) on page 42 of his '' Laws of Thought'': 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.

Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...

cited Augustus De Morgan as identifying "that limited class of things which is the special subject of discourse on any particular occasion. Such a class was called by De Morgan, the Universe of Discourse."

Lewis Carroll Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice ...

expressed the need for a universe of discourse as follows:

It sometimes happens that, in one or both of the Terms of a Proposition, the Name consists of Adjectives only, the Substantive being understood. In order to express such a Proposition ''fully'', we must supply the Name of some Class which may be regarded as a Genus of which each Term is a Species...The Genus referred to is called the Universe of Discourse...
Lewis Carroll Charles Lutwidge Dodgson (27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet, mathematician, photographer and reluctant Anglicanism, Anglican deacon. His most notable works are ''Alice ...
(1896)
Symbolic Logic, Part I: Elementary
' page 10

Examples

For example, in an interpretation of first-order logic, 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s; in another interpretation, it could be the set of

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

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. The

binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...

called set membership, expressed as

x \in A

, and meaning that ''x'' belongs to set ''A'', is clear enough. Every binary relation has a converse relation, and the converse of

\in \ \ \text\ \  \ni

. Also, a binary relation must have a domain. The domain of the converse of set membership is the universe of discourse. Any subset of this universe may, or may not, contain ''x''. ''A'' is a subset of this universe, not necessarily restricted to ''A''.

References

{{DEFAULTSORT:Domain Of Discourse Semantics Predicate logic

Etymology

Examples

See also

References