Title: Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.
Author: Whitehead, Alfred North, 18611947.

76
INTRODUCTICN
[CHAP.
To explain the theory of classes, it is necessary first to explain the distinction between extensional and intensional functions. This is effected 'by
the following definitions:
The truthvalue of a proposition is truth if it is true, and falsehood if it is
false. (This expression is due to Frege.)
Two propositions are said to be equivalent when they have the same truthvalue, i.e. when they are both true or both false.
Two propositional functions are said to beformally equivalent when they
are equivalent with every possible argument, i.e. when any argument which
satisfies the one satisfies the other, and vice versa. Thus " is a man" is
formally equivalent to "2 is a featherless biped"; "' is an even prime" is
formally equivalent to " is identical with 2."
A function of a function is called extensional when its truthvalue with any
argument is the same as with any formally equivalent argument. That is to
say, f( )z) is an extensional function of fz if, provided ^ is formally equivalent to ^z,f(4z)) is equivalent to f(2z), Here the apparent variables o and
* are necessarily of the type from which arguments can significantly be
supplied tof. We find no need to use as apparent variables any functions
of nonpredicative types; accordingly in the sequel all extensional functions
considered are in fact finctions of predicative functions*.
A function of a function is called intensional when it is not extensional.
The nature and importance of the distinction between intensional and
extensional functions will be made clearer by some illustrations. The proposition "'x is a man' always implies 'x is a mortal'" is an extensional function
of the function " S is a man," because we may substitute, for "x is a man,"
' x is a featherless biped," or any other statement which applies to the same
objects to which " x is a man " applies, and to no others. But the proposition
"A believes that 'x is a man' always implies 'x is a mortal"' is an intensional
function of " is a man," because A may never have considered the question
whether featherless bipeds are mortal, or may believe wrongly that there are
featherless bipeds which are not mortal. Thus even if "x is a featherless
biped" is formally equivalent to " x is a man," it by no means follows that a
person who believes that all men are mortal must believe that all featherless
bipeds are mortal, since he may have never thought about featherless bipeds,
or have supposed that featherless bipeds were not always men. Again the
proposition " the number of arguments that satisfy the function 4)! z is n" is
an extensional function of 4! z, because its truth or falsehood is unchanged if
we substitute for 40!z any other function which is true whenever 4! z is true,
and false whenever 4! z is false. But the proposition "A asserts that the
number of arguments satisfying ^! z is n" is an intensional function of )! 2,
* Cf. p. 56.
