Extensionality

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.

Contents

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  1 Example
  


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  2 In mathematics
  


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  3 See also
  


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  4 References
  

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).

See also

• Duck typing


• Identity of indiscernibles


• Structural typing


• Univalence axiom

References

• Intensional Logic (Stanford Encyclopedia of Philosophy)


• 
  equality in nLab
  

v t e Mathematical logic
General	Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory
Systems	Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus
Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram
Propositional calculus and Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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