In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, 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.
Example
Consider the two
functions ''f'' and ''g'' mapping from and to
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, 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 predicates "being called Joe", and "being the oldest person in this town" are intensionally distinct, but extensionally equal for the (current) population of this town.
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 mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
, in which case two functions must not only agree on all values, but must also have the same codomain, in order to be equal (in contrast, the usual definition of a function in mathematics means that equal functions must have the same
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
).
A similar extensional definition is usually employed for
relations: two relations are said to be equal if they have the same
extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ex ...
.
In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, 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 pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s have equal elements, and elements of a set which are related by an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
belong to the same
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
.
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)equalityin
nLab
{{Mathematical logic
Set theory
Concepts in logic
Equivalence (mathematics)