Intensional definition

In logic and mathematics, an intensional definition gives the meaning of a term by specifying all the properties required to come 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 axioms 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.