Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
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.Giuseppe Peano formalized number theory (arithmetic of positive integers) taking its domain to be the positive integers and the universe of discourse to include all numbers, not just integers.
For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals over which the quantifiers range. 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 no domain of discourse has been identified, a proposition such as ∀x (x2 ≠ 2) is ambiguous. If the domain of discourse is the set of real numbers, the proposition is false, with x = √2 as counterexample; if the domain is the set of naturals, the proposition is true, since 2 is not the square of any natural number.
The term universe of discourse generally refers to the collection of objects being discussed in a specific discourse. 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 (1846) but the name was used for the first time in history by George Boole (1854) on page 42 of his Laws of Thought in a long and incisive passage well worth study. 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 stunning principle of wholistic reference.
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