In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, extensional and intensional definitions are two key ways in which the
objects,
concept
A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs.
Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...
s, or
referent
A referent ( ) is a person or thing to which a name – a linguistic expression or other symbol – refers. For example, in the sentence ''Mary saw me'', the referent of the word ''Mary'' is the particular person called Mary who is being spoken o ...
s a
term refers to can be
defined. They give
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 gives meaning to a term by specifying every
object that falls under the definition of the term in question.
For example, in set theory one would extensionally define the set of
square numbers as , while an intensional definition of the set of the square numbers could be .
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
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Philosophy and science
* Property (philosophy), in philosophy and logic, an abstraction characterizing an ...
that an
object needs to have in order to be counted as a
referent
A referent ( ) is a person or thing to which a name – a linguistic expression or other symbol – refers. For example, in the sentence ''Mary saw me'', the referent of the word ''Mary'' is the particular person called Mary who is being spoken o ...
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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 (taxonomy) set up by Carl Linnaeus, as set forth in his ''Systema Naturae'' (1735) and subsequent works. In the taxonomy of Linnaeus th ...
to categorize living things, but is by no means restricted to
biology
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
. 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 true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s that define a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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 for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...
, 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
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 possib ...
''.
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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
In logic, extensional and intensional definitions are two key ways in which the objects, concepts, or referents a term refers to can be defined. They give meaning or denotation to a term.
An intensional definition gives meaning to a term by sp ...
, which defines by listing properties that a thing must have in order to be part of the set captured by the definition.
Etymology
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.
...
.
See also
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References
{{Defining
Necessity and sufficiency
Definition
Logic