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Concept
Extensionality
Summary
In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal definitions of objects are the same.
Consider the two functions f and g mapping from and to natural numbers, defined as follows:
To find f(n), first add 5 to n, then multiply by 2.
To find g(n), first multiply n by 2, then add 10.
These functions are extensionally equal; given the same input, both functions always produce the same value. But the definitions of the functions are not equal, and in that intensional sense the functions are not the same.
Similarly, in natural language there are many predicates (relations) that are intensionally different but are extensionally identical. For example, suppose that a town has one person named Joe, who is also the oldest person in the town. Then, the two predicates "being called Joe", and "being the oldest person in this town" are intensionally distinct, but extensionally equal for the (current) population of this town.
The extensional definition of function equality, discussed above, is commonly used in mathematics. Sometimes additional information is attached to a function, such as an explicit codomain, in which case two functions must not only agree on all values, but must also have the same codomain, in order to be equal (in contrast, the usual definition of a function in mathematics means that equal functions must have the same domain).
A similar extensional definition is usually employed for relations: two relations are said to be equal if they have the same extensions.
In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions—with their extension as stated above, so that it is impossible for two relations or functions with the same extension to be distinguished.
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Related concepts (6)
Intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics. Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician and philosopher, who first published it in 1972. There are multiple versions of the type theory: Martin-Löf proposed both intensional and extensional variants of the theory and early impredicative versions, shown to be inconsistent by Girard's paradox, gave way to predicative versions.
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
Extensional and intensional definitions
In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term. An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of nouns, this is equivalent to specifying the properties that an object needs to have in order to be counted as a referent of the term. For example, an intensional definition of the word "bachelor" is "unmarried man".