Extensional and intensional definitions

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 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 {0, 1, 4, 9, 16,

…

{\displaystyle \dots }

}, while an intensional definition of the set of the square numbers could be {

x

∣

x

{\displaystyle x\mid x}

is the square of an integer}.

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 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.