Extensional and intensional definitions: Difference between revisions

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{{Short description|Classification of definitions in mathematics, philosophy, and logic}}

'''Extensional and intensional definitions''' are two key ways in which the [[wikt:entity|object(s)]] or [[concept]](s) a [[terminology|term]] refers to can be [[definition|defined]].

In [[logic]], '''Extensionalextensional and intensional definitions''' are two key ways in which the [[wikt:entity|objectObject (sphilosophy)|objects]], [[concept]]s, or [[conceptreferent]](s) a [[terminology|term]] refers to can be [[definition|defined]]. They give [[Meaning (linguistic)|meaning]] or denotation to a term.

An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used.

An '''extensional definition''' ofgives ameaning conceptto ora term formulates its meaning by specifying its '''extension''', that is, every [[object (philosophy)|object]] that falls under the definition of the concept or term in question.

For example, in set theory one would extensionally define the set of [[Square number|square numbers]] as {0, 1, 4, 9, 16, <math>\dots</math>}, while an intensional definition of the set of the square numbers could be {<math>x \mid x</math> is the square of an integer}.

==Intensional definition==

{{also|Intension}}

InAn [[logic]] and [[mathematics]], an '''intensional definition''' gives the [[Meaning (linguistic)|meaning]] ofto a term by specifying necessary and sufficient conditions for when the term should be used. In the case of [[nouns]], this is equivalent to specifying the [[Property (philosophy)|properties]] that an [[Object (philosophy)|object]] needs to have in order to be counted as a [[referent]] of the term.

For example, an intensional definition of the word "bachelor" is "unmarried man". This definition is valid because being an unmarried man is both a necessary condition and a sufficient condition for being a bachelor: it is necessary because one cannot be a bachelor without being an unmarried man, and it is sufficient because any unmarried man is a bachelor.<ref name="Cook">Cook, Roy T. "Intensional Definition". In ''A Dictionary of Philosophical Logic''. Edinburgh: Edinburgh University Press, 2009. 155.</ref>

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[[Genus–differentia definition|Definition by genus and difference]], in which something is defined by first stating the broad category it belongs to and then distinguished by specific properties, is a type of intensional definition. As the name might suggest, this is the type of definition used in [[Linnaean taxonomy]] to categorize living things, but is by no means restricted to [[biology]]. Suppose one defines a miniskirt as "a skirt with a hemline above the knee". It has been assigned to a ''genus'', or larger class of items: it is a type of skirt. Then, we've described the ''differentia'', the specific properties that make it its own sub-type: it has a hemline above the knee.

IntensionalAn intensional definition may also appliesconsist toof rules or sets of [[axiom]]s that define a [[set (mathematics)|set]] by describing a procedure for generating all of its members. For example, an intensional definition of ''[[square number]]'' can be "any number that can be expressed as some integer multiplied by itself". The rule—"take an integer and multiply it by itself"—always generates members of the set of square numbers, no matter which integer one chooses, and for any square number, there is an integer that was multiplied by itself to get it.

Similarly, an intensional definition of a game, such as [[chess]], would be the rules of the game; any game played by those rules must be a game of chess, and any game properly called a game of chess must have been played by those rules.

==Extensional definition==

{{also|Extension (semantics)}}

An '''extensional definition''' of a concept or term formulates its meaning by specifying its '''extension''', that is, every [[object (philosophy)|object]] that falls under the definition of the concept or term in question.

An extensional definition gives meaning to a term by specifying its [[Extension (semantics)|extension]], that is, every [[object (philosophy)|object]] that falls under the definition of the term in question.

For example, an extensional definition of the term "nation of the world" might be given by [[List of sovereign states|listing all of the nations of the world]], or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for [[finite sets]] and only practical for relatively [[Small set (category theory)|small sets]], is a type of ''[[enumerative definition]]''.

Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a [[set (mathematics)|set]] tells the questioner enough about the nature of that set.

For example, an extensional definition of the term "nation of the world" might be given by listing all of the nations of the world, or by giving some other means of recognizing the members of the corresponding class. An explicit listing of the extension, which is only possible for finite sets and only practical for relatively small sets, is a type of ''[[enumerative definition]]''.

ThisAn isextensional similardefinition possesses similarity to an [[ostensive definition]], in which one or more members of a set (but not necessarily all) are pointed outto as examples., Thebut oppositecontrasts approachclearly iswith thean [[intensional definition]], which defines by listing properties that a thing must have in order to be part of the set captured by the definition.

Extensional definitions are used when listing examples would give more applicable information than other types of definition, and where listing the members of a [[set (mathematics)|set]] tells the questioner enough about the nature of that set.

==Etymology==

This is similar to an [[ostensive definition]], in which one or more members of a set (but not necessarily all) are pointed out as examples. The opposite approach is the [[intensional definition]], which defines by listing properties that a thing must have in order to be part of the set captured by the definition.

The terms "[[intension]]" and "[[Extension (semantics)|extension]]" were introduced before 1911 by [[Constance Jones]]<ref>{{cite web

| title =Emily Elizabeth Constance Jones: Observations on Intension and Extension

| url =https://plato.stanford.edu/entries/emily-elizabeth-constance-jones/#ObseInteExte

| website =Stanford Encyclopedia of Philosophy

| date = 7 August 2020

| access-date = 19 November 2020}}</ref> and formalized by [[Rudolf Carnap]].<ref>{{cite SEP |url-id=logic-intensional |title=Intensional logic |last=Fitting |first=Melvin}}</ref>

== See also ==

* [[{{annotated link|Comprehension (logic)]]}}

* [[{{annotated link|Extension (predicate logic)]]}}

* [[{{annotated link|Extension (semantics)]]}}

* [[{{annotated link|Extensional context]]}}

* {{annotated link|Extensionalism}}

* [[{{annotated link|Extensionality]]}}

* [[Intension]]

* {{annotated link|Intension}}

* [[Ostensive definition]]

* {{annotated link|Intensional logic}}

* [[{{annotated link|Ostensive definition]]}}

== References ==

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{{Defining}}

[[Category:Necessity and sufficiency]]

[[Category:Definition]]

[[Category:Logic]]