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In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
, extensional and intensional definitions are two key ways in which the objects,
concept Concepts are defined as abstract ideas A mental representation (or cognitive representation), in philosophy of mind Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the bo ...

concept
s, or
referent A referent () is a person or thing to which a name – a linguistics, linguistic Phrase, expression or other symbol – reference, refers. For example, in the sentence ''Mary saw me'', the referent of the word ''Mary'' is the particular person calle ...
s a
term
term
refers to can be
defined A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions ...

defined
. They give
meaning Meaning most commonly refers to: * Meaning (linguistics), meaning which is communicated through the use of language * Meaning (philosophy), definition, elements, and types of meaning discussed in philosophy * Meaning (non-linguistic), a general ter ...
or denotation to a term.


Intensional definition

An intensional definition gives meaning to a term by specifying necessary and sufficient conditions for when the term should be used. In the case of
nouns A noun () is a word In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) ...
, this is equivalent to specifying the
properties Property (''latin: Res Privata'') in the abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether phys ...
that an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
needs to have in order to be counted as a
referent A referent () is a person or thing to which a name – a linguistics, linguistic Phrase, expression or other symbol – reference, refers. For example, in the sentence ''Mary saw me'', the referent of the word ''Mary'' is the particular person calle ...
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.Cook, Roy T. "Intensional Definition". In ''A Dictionary of Philosophical Logic''. Edinburgh: Edinburgh University Press, 2009. 155. This is the opposite approach to the
extensional definition In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...
, which defines by listing everything that falls under that definition – an extensional definition of ''bachelor'' would be a listing of all the unmarried men in the world. As becomes clear, intensional definitions are best used when something has a clearly defined set of properties, and they work well for terms that have too many referents to list in an extensional definition. It is impossible to give an extensional definition for a term with an
infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infinite'' (EP), debut EP of American musi ...

infinite
set of referents, but an intensional one can often be stated concisely – there are infinitely many even numbers, impossible to list, but the term "even numbers" can be defined easily by saying that even numbers are
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
multiples of two. 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 Linnaean taxonomy can mean either of two related concepts: # the particular form of biological classification In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical stru ...
to categorize living things, but is by no means restricted to
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

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. An intensional definition may also consist of rules or sets of
axiom An axiom, postulate or assumption is a statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s that define a set by describing a procedure for generating all of its members. For example, an intensional definition of ''
square number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
'' 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 Chess is a board game Board games are tabletop game Tabletop games are game with separate sliding drawer, from 1390–1353 BC, made of glazed faience, dimensions: 5.5 × 7.7 × 21 cm, in the Brooklyn Museum (New Yor ...

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

An extensional definition gives meaning to a term by specifying its extension, that is, every
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
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 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 An enumerative definition of a concept or term is a special type of extensional definition that gives an explicit and exhaustive listing of all the objects that fall under the concept or term in question. Enumerative definitions are only possible ...
''. 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 tells the questioner enough about the nature of that set. An extensional definition possesses similarity to an
ostensive definitionAn ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers ...
, in which one or more members of a set (but not necessarily all) are pointed to as examples, but contrasts clearly with an
intensional definition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label ...
, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.


History

The terms "
intension In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ...

intension
" and " extension" were introduced by Constance Jones and formalized by
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle The Vienna Circle (german: Wiener Krei ...
.


See also

*
Comprehension (logic) In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ac ...
*
Extension (predicate logic) The extension of a predicate Predicate or predication may refer to: Computer science *Syntactic predicate (in parser technology) guidelines the parser process Linguistics *Predicate (grammar), a grammatical component of a sentence Philosophy a ...
*
Extension (semantics) In any of several fields of study that treat the use of signs — for example, in linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studyi ...
* Extensional context *
Extensionalism Extensionalism, in the philosophy of language, in logic and semantics, is the view that all languages or at least all scientific languages should be extensional. It has been described as the default option for the scientism in the nineteenth century ...
*
Extensionality In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents stateme ...
*
Intension In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include ...

Intension
*
Intensional logic Intensional logic is an approach to predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and ...
*
Ostensive definitionAn ostensive definition conveys the meaning of a term by pointing out examples. This type of definition is often used where the term is difficult to define verbally, either because the words will not be understood (as with children and new speakers ...


References

{{Defining Necessity Definition Logic