Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/101

 such a case what is symbolically, though not really, the same function $$\scriptstyle{f}$$ can receive as arguments functions of various different types. If, with a given argument $\scriptstyle{\phi\hat z}$, the function $\scriptstyle{f(\phi\hat z)}$, so interpreted, is equivalent to $$\scriptstyle{f(\psi!\hat z)}$$ whenever $$\scriptstyle{\psi!\hat z}$$ is formally equivalent to $\scriptstyle{\phi\hat z}$, then $$\scriptstyle{f\{\hat z(\phi z)\}}$$ is equivalent to $$\scriptstyle{f(\phi\hat z)}$$ provided there is any predicative function formally equivalent to $\scriptstyle{\phi\hat z}$. At this point, we make use of the axiom of reducibility, according to which there always is a predicative function formally equivalent to $\scriptstyle{\phi\hat z}$.

As was explained above, it is convenient to regard an extensional function of a function as having for its argument not the function, but the class determined by the function. Now we have seen that our derived function is always extensional. Hence if our original function was $\scriptstyle{f(\psi!\hat z)}$, we write the derived function $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined where "$\scriptstyle{\hat z(\phi z)}$" may be read "the class of arguments which satisfy $\scriptstyle{\phi\hat z}$," or more simply "the class determined by $\scriptstyle{\phi\hat z}$." Thus "$\scriptstyle{f\{\hat z(\phi z)\}}$"|undefined will mean: "There is a predicative function $$\scriptstyle{\psi!\hat z}$$ which is formally equivalent to $$\scriptstyle{\phi\hat z}$$ and is such that $$\scriptstyle{f(\psi!\hat z)}$$ is true." This is in reality a function of $\scriptstyle{\phi\hat z}$, but we treat it symbolically as if it had an argument $\scriptstyle{\hat z(\phi z)}$. By the help of the axiom of reducibility, we find that the usual properties of classes result. For example, two formally equivalent functions determine the same class, and conversely, two functions which determine the same class are formally equivalent. Also to say that $$\scriptstyle{x}$$ is a member of $\scriptstyle{\hat z(\phi z)}$, i.e. of the class determined by $\scriptstyle{\phi\hat z}$, is true when $$\scriptstyle{\phi x}$$ is true, and false when $$\scriptstyle{\phi x}$$ is false. Thus all the mathematical purposes for which classes might seem to be required are fulfilled by the purely symbolic objects $\scriptstyle{\hat z(\phi z)}$, provided we assume the axiom of reducibility.

In virtue of the axiom of reducibility, if $$\scriptstyle{\phi\hat z}$$ is any function, there is a formally equivalent predicative function $\scriptstyle{\psi!\hat z}$; then the class $$\scriptstyle{\hat z(\phi z)}$$ is identical with the class $\scriptstyle{\hat z(\psi!z)}$, so that every class can be defined by a predicative function. Hence the totality of the classes to which a given term can be significantly said to belong or not to belong is a legitimate totality, although the totality of functions which a given term can be significantly said to satisfy or not to satisfy is not a legitimate totality. The classes to which a given term $$\scriptstyle{a}$$ belongs or does not belong are the classes defined by $\scriptstyle{a}$-functions; they are also the classes defined by predicative $\scriptstyle{a}$-functions. Let us call them $\scriptstyle{a}$-classes. Then "$\scriptstyle{a}$-classes" form a legitimate totality, derived from that of predicative $\scriptstyle{a}$-functions. Hence many kinds of general statements become possible which would otherwise involve vicious-circle paradoxes. These general statements are none of them such as lead to contradictions, and many of them such as it is very hard to suppose illegitimate. The fact that they are rendered possible by the axiom of reducibility, and that they would otherwise be excluded by the vicious-circle principle, is to be regarded as an argument in favour of the axiom of reducibility.