Extensional and intensional definitions

Extensional and intensional definitions are two key ways in which the object(s) or concept(s) a term refers to can be defined.

In logic and mathematics, an intensional definition gives the meaning of 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.

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 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 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.

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