Extensionality thesis

Thesis beloved of logical atomists, logical positivists, and various kinds of nominalists and reductionists. It says that apparent exceptions to Leibniz’s law can be dispensed with; that is, intensions can be reduced to extensions, or roughly, what holds true of objects does not depend on how they are described.

For logical atomism in particular the thesis says that all propositions are truth-functions of certain basic ones (that is, their truth or falsity follows, given the truth or falsity of the basic ones).

An alternative version of the thesis (by Willard Van Orman Quine (1908-2000)) says that only if the above is true can a coherent system of logic be constructed; that is, there is no intensional logic.

Source:
W V O Quine, ‘Reference and Modality’, From a Logical Point of View, 2nd revised edn (1961); reprinted with discussions in L Linsky, ed., Reference and Modality (1971)

Example

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 argument predicates “has one person named”, “is the oldest person in” are intensionally distinct, but extensionally equal for “Joe” in that “town” now.

In mathematics

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.

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.

Other mathematical objects are also constructed in such a way that the intuitive notion of “equality” agrees with set-level extensional equality; thus, equal ordered pairs have equal elements, and elements of a set which are related by an equivalence relation belong to the same equivalence class.

Type-theoretical foundations of mathematics are generally not extensional in this sense, and setoids are commonly used to maintain a difference between intensional equality and a more general equivalence relation (which generally has poor constructibility or decidability properties).