Domain of discourse
Encyclopedia
In the formal sciences, the domain of discourse, also called the universe of discourse (or simply universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...

), is the set of entities over which certain variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...

s 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.

For example, in an interpretation
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 until they are given some interpretation...

 of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

, the domain of discourse is the set of individuals that the quantifiers range over. In one interpretation, the domain of discourse could be the set of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s; in another interpretation, it could be the set of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

s. If no domain of discourse has been identified, a proposition such as is ambiguous. 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 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 term universe of discourse is generally attributed to Augustus De Morgan (1846) and was also used by George Boole (1854) in his Laws of Thought
The Laws of Thought
The Laws of Thought, more precisely, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, is a very influential 19th century book on logic by George Boole, the second of his two monographs on algebraic logic...

.

A database is a model of some aspect of the reality of an organisation. It is conventional to call this reality the "universe of discourse" or "domain of discourse".

See also

  • Universe (mathematics)
    Universe (mathematics)
    In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...

  • Term algebra
    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 exactly the free magma generated by X...

  • Domain (mathematics)
    Domain (mathematics)
    In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

  • Domain theory
    Domain theory
    Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...

  • Interpretation (logic)
    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 until they are given some interpretation...

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