Intensional definition
Encyclopedia
In logic
and mathematics
, an intensional definition gives the meaning of a term by specifying all the properties required to come to that definition
, that is, the necessary and sufficient conditions
for belonging to the set 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
, 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 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
set, but an intensional one can often be stated concisely — there is an infinite number of even numbers
, impossible to list, but they can be defined by saying that even numbers are integer
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
to categorize living things, but is by no means restricted to biology
. 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
s that generate all members of the set being defined. 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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...
, that is, the necessary and sufficient conditions
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. 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.-Definitions:A necessary condition...
for belonging to the set 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 is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
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 formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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 can mean either of two related concepts:# the particular form of biological classification set up by Carl Linnaeus, as set forth in his Systema Naturæ and subsequent works...
to categorize living things, but is by no means restricted to biology
Biology
Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...
. 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 proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
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 is the square of an 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 formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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 two-player board game played on a chessboard, a square-checkered board with 64 squares arranged in an eight-by-eight grid. It is one of the world's most popular games, played by millions of people worldwide at home, in clubs, online, by correspondence, and in tournaments.Each player...
, 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
- 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 definitionExtensional definitionAn 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....
- IntensionIntensionIn linguistics, logic, philosophy, and other fields, an intension is any property or quality connoted by a word, phrase or other symbol. In the case of a word, it is often implied by the word's definition...
- Ostensive definitionOstensive 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 or because of the nature of the term...