intensional definition

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logic Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...
, extensional and intensional definitions are two key ways in which the objects,
concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by sev ...
s, or referents a term refers to can be defined. They give meaning 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, this is equivalent to specifying the properties that an 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.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, 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 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 is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
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 Taxonomy (biology), biological classification (taxonomy) set up by Carl Linnaeus, as set forth in his ''Systema Naturae'' (1735) and subsequent works. In the ta ...
to categorize living things, but is by no means restricted to
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of Cell (biology), cells that proce ...
. 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 axioms that define a set by describing a procedure for generating all of its members. For example, an intensional definition of ''
square number In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
'' 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 between two Player (game), players. It is sometimes called international chess or Western chess to distinguish it from chess variant, related games, such as xiangqi (Chinese chess) and shogi (Japanese chess). The current ...
, 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 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''. 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 definition, 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, 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" and " extension" were introduced before 1911 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 and an advocate of logical positivism. He ...
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