Extensionality


Extensionality

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.

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.

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.

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • extensionality — extensionality, axiom of …   Philosophy dictionary

  • extensionality — noun see extensional …   New Collegiate Dictionary

  • extensionality — noun a) The principle that objects are equal if and only if their observed properties are the same, regardless of internal processes that lead to those properties. b) The principle that functions are equal if and only if they operate on the same… …   Wiktionary

  • extensionality — ex·ten·sion·al·i·ty …   English syllables

  • extensionality — ikˌstenchəˈnaləd.ē, (ˌ)ekˌ noun ( es) : the quality or state of being extensional …   Useful english dictionary

  • extensionality, axiom of — Basic axiom of set theory . It asserts that sets are identical if and only if they have the same members …   Philosophy dictionary

  • Axiom of extensionality — In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo Fraenkel set theory. Formal statement In the formal language of… …   Wikipedia

  • axiom of extensionality — noun One of the axioms in axiomatic set theory, equivalent to the statement that two sets are equal if and only if they contain the same elements …   Wiktionary

  • extension — extensionality, extensionalism, n. extensional, adj. extensionally, adv. extensionless, adj. /ik sten sheuhn/, n. 1. an act or instance of extending. 2. the state of being extended. 3. that by which something is extended; an addition: a four room …   Universalium

  • Mereology — In philosophy and mathematical logic, mereology (from the Greek μέρος, root: μερε(σ) , part and the suffix logy study, discussion, science ) treats parts and the wholes they form. Whereas set theory is founded on the membership relation between a …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.