Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/98

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 truth-value 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 truth-value, i.e. when they are both true or both false.

Two propositional functions are said to be formally 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 "$\scriptstyle {\hat {x}}$ is a man" is formally equivalent to "$\scriptstyle {\hat {x}}$ is a featherless biped"; "$\scriptstyle {\hat {x}}$ is an even prime" is formally equivalent to "$\scriptstyle {\hat {x}}$ is identical with 2."

A function of a function is called extensional when its truth-value with any argument is the same as with any formally equivalent argument. That is to say, $\scriptstyle {f(\phi {\hat {z}})}$ is an extensional function of $\scriptstyle {\phi {\hat {z}}}$ if, provided $\scriptstyle {\psi {\hat {z}}}$ is formally equivalent to $\scriptstyle {\phi {\hat {z}}}$, $\scriptstyle {f(\phi {\hat {z}})}$ is equivalent to $\scriptstyle {f(\psi {\hat {z}})}$. Here the apparent variables $\scriptstyle {\phi }$ and $\scriptstyle {\psi }$ are necessarily of the type from which arguments can significantly be supplied to $\scriptstyle {f}$. We find no need to use as apparent variables any functions of non-predicative types; accordingly in the sequel all extensional functions considered are in fact functions of predicative functions^[1].

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 "'$\scriptstyle {x}$ is a man' always implies '$\scriptstyle {x}$ is a mortal'" is an extensional function of the function "$\scriptstyle {\hat {x}}$ is a man," because we may substitute, for "$\scriptstyle {x}$ is a man," "$\scriptstyle {x}$ is a featherless biped," or any other statement which applies to the same objects to which "$\scriptstyle {x}$ is a man" applies, and to no others. But the proposition "$\scriptstyle {A}$ believes that '$\scriptstyle {x}$ is a man' always implies '$\scriptstyle {x}$ is a mortal'" is an intensional function of "$\scriptstyle {\hat {x}}$ is a man," because $\scriptstyle {A}$ 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 "$\scriptstyle {x}$ is a featherless biped" is formally equivalent to "$\scriptstyle {x}$ 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 $\scriptstyle {\phi !{\hat {z}}}$ is $\scriptstyle {n}$" is an extensional function of $\scriptstyle {\phi !{\hat {z}}}$, because its truth or falsehood is unchanged if we substitute for $\scriptstyle {\phi !{\hat {z}}}$ any other function which is true whenever $\scriptstyle {\phi !{\hat {z}}}$ is true, and false whenever $\scriptstyle {\phi !{\hat {z}}}$ is false. But the proposition "$\scriptstyle {A}$ asserts that the number of arguments satisfying $\scriptstyle {\phi !{\hat {z}}}$ is $\scriptstyle {n}$" is an intensional function of $\scriptstyle {\phi !{\hat {z}}}$,

1. Cf. p. 56.