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In logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
 and mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition
Definition

A definition is a statement of the Meaning of a word or phrase. The term to be defined is known as the definiendum . The words which define it are known as the definiens ....
, that is, the necessary and sufficient conditions
Necessary and sufficient conditions

In logic, the words necessity and sufficiency refer to the implicational relationships between Statement . The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true....
 for belonging to the set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 being defined.

For example, an intensional definition of "bachelor" is "unmarried man." Being an unmarried man is an essential property of something referred to as a bachelor. It is a necessary condition: one cannot be a bachelor without being an unmarried man.






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In logic
Logic

Logic is the study of the principles of valid demonstration and inference. Logic is a branch of philosophy, a part of the classical Trivium . The word derives from Greek language ?????? , fem....
 and mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition
Definition

A definition is a statement of the Meaning of a word or phrase. The term to be defined is known as the definiendum . The words which define it are known as the definiens ....
, that is, the necessary and sufficient conditions
Necessary and sufficient conditions

In logic, the words necessity and sufficiency refer to the implicational relationships between Statement . The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true....
 for belonging to the set
Set

A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics....
 being defined.

For example, an intensional definition of "bachelor" is "unmarried man." Being an unmarried man is an essential property of something referred to as a bachelor. It is a necessary condition: one cannot be a bachelor without being an unmarried man. It is also a sufficient condition: any unmarried man is a bachelor.

This is the opposite approach to the extensional definition
Extensional definition

An extensional definition of a concept or term formulates its meaning by specifying its extension , that is, every object that falls under the definition of the concept or term in question....
, which defines by listing everything that falls under that definition — an extensional definition
Extensional definition

An extensional definition of a concept or term formulates its meaning by specifying its extension , that is, every object that falls under the definition of the concept or term in question....
 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 it works well for sets that are too large to list in an extensional definition. It is impossible to give an extensional definition for an infinite
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
 set, but an intensional one can often be stated concisely — there is an infinite number of even numbers
Even and odd numbers

In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2....
, impossible to list, but they can be defined by saying that even numbers are integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 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

Linnaean taxonomy is a method of classifying living things, originally devised by Carolus Linnaeus , although it has changed considerably since his time....
 to categorize living things, but is by no means restricted to biology
Biology

Biology is a branch of the natural sciences concerned with the study of living organisms and their interaction with each other and their environment ....
. Suppose we define a miniskirt as "a skirt with a hemline above the knee." We've assigned it 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.

Intensional definition also applies to rules or sets of axiom
Axiom

In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evidence, or subject to necessary decision....
s that generate all members of the set being defined. For example, an intensional definition of "square number
Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself....
" can be "any number that can be expressed as some integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 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

Chess is a recreational and competitive game played between two Player . Sometimes called Western chess or international chess to distinguish it from History of chess and other chess variants, the current form of the game emerged in Southern Europe during the second half of the 15th century after evolving from similar, much older...
, 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.

See also


  • Intension
    Intension

    Intension refers to the possible things a word or phrase could describe. It stands in contradistinction to extension , which refers to the actual things the word or phrase does describe....
  • Comprehension (logic)
    Comprehension (logic)

    In logic, the comprehension of an object is the totality of intensions, that is, attributes, characters, marks, properties, or qualities, that the object possesses, or else the totality of intensions that are pertinent to the context of a given discussion....
  • Extensional definition
    Extensional definition

    An extensional definition of a concept or term formulates its meaning by specifying its extension , that is, every object that falls under the definition of the concept or term in question....
  • Ostensive definition
    Ostensive definition

    An 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 or because of the nature of the term ....