Extensionality

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In logic, extensionality refers to principles that judge objects to be equal if they have the same external properties. It is the opposite concept of intensionality, which is concerned with whether two descriptions are intended to be the same or not.

[edit] Example

Consider the functions f and g from the natural numbers to the 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 "Joe" and "oldest person in the town" are extensionally equal, but intensionally distinct.

[edit] 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 and mathematics formalized in set theory, it is common to identify a relation with its extension, so that it is impossible for two relations with the same extension to be distinguished.

In lambda calculus, a formal system for manipulating number-theoretic functions, extensionality is expressed by the eta-conversion rule, which allows conversion between any two expressions that denote the same function.

In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements.

[edit] References

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