Domain of discourse

In 1847 Augustus De Morgan used U to indicate "everything in the universe spoken of". The phrase universe of discourse was used by George Boole in Laws of Thought (1854): 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 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 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... ==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 sentence 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 numbers; in another interpretation, it could be the set of natural numbers. If the domain of discourse is the set of real numbers, the sentence is false, with as counterexample; if the domain is the set of natural numbers, the sentence is true, since 2 is not the square of any natural number. The binary relation 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 is written \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. ==See also==