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Inlogic,extensionality, orextensional equality, refers to principles that judge objects to beequal if they have the same external properties. It stands in contrast to the concept ofintensionality, which is concerned with whether the internal definitions of objects are the same.
The extensional definition of function equality, discussed above, is commonly used in mathematics.A similar extensional definition is usually employed forrelations: two relations are said to be equal if they have the sameextensions.
Inset theory, theaxiom of extensionality states that twosets 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.
Other mathematical objects are also constructed in such a way that the intuitive notion of "equality" agrees with set-level extensional equality; thus, equalordered pairs have equal elements, and elements of a set which are related by anequivalence relation belong to the sameequivalence class.
Type-theoretical foundations of mathematics are generallynot extensional in this sense, andsetoids are commonly used to maintain a difference between intensional equality and a more general equivalence relation (which generally has poorconstructibility ordecidability properties).
There are various extensionality principles in mathematics.
Depending on the chosen foundation, some extensionality principles may imply another. For example it is well known that inunivalent foundations, the univalence axiom implies both propositional and functional extensionality. Extensionality principles are usually assumed as axioms, especially in type theories where computational content must be preserved. However, in set theory and other extensional foundations, functional extensionality can be proven to hold by default.
Consider the twofunctionsf andg mapping from and tonatural numbers, defined as follows:
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 village has just one person named Joe, who is also the oldest person in the village. Then, the two predicates "being called Joe", and "being the oldest person" are intensionally distinct, but extensionally equal for the (current) population of this village.
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