Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/104

 which, again, is equivalent to which, in virtue of the axiom of reducibility, is equivalent to  Thus our definition of the use of $$\scriptstyle{\hat z(\phi z)}$$ is such as to satisfy the conditions (2) and (3) which we laid down for classes, i.e. we have

Before considering classes of classes, it will be well to define membership of a class, i.e. to define the symbol "$\scriptstyle{x\in\hat z(\phi z)}$," which may be read "$\scriptstyle{x}$ is a member of the class determined by $\scriptstyle{\phi\hat z}$." Since this is a function of the form $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined it must be derived, by means of our general definition of such functions, from the corresponding function $\scriptstyle{f\{\phi!\hat z\}}$.|undefined We therefore put This definition is only needed in order to give a meaning to "$\scriptstyle{x\in\hat z(\phi z)}$"; the meaning it gives is, in virtue of the definition of $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined  It thus appears that "$\scriptstyle{x\in\hat z(\phi z)}$" implies $\scriptstyle{\phi x}$, since it implies $\scriptstyle{\psi!x}$, and $$\scriptstyle{\psi!x}$$ is equivalent to $\scriptstyle{\phi x}$; also, in virtue of the axiom of reducibility, $$\scriptstyle{\phi x}$$ implies "$\scriptstyle{x\in\hat z(\phi z)}$," since there is a predicative function $$\scriptstyle{\psi}$$ formally equivalent to $\scriptstyle{\phi}$, and $$\scriptstyle{x}$$ must satisfy $\scriptstyle{\psi}$, since $$\scriptstyle{x}$$ (ex hypothesi) satisfies $\scriptstyle{\phi}$. Thus in virtue of the axiom of reducibility we have i.e. $$\scriptstyle{x}$$ is a member of the class $$\scriptstyle{\hat z(\phi z)}$$ when, and only when, $$\scriptstyle{x}$$ satisfies the function $$\scriptstyle{\phi}$$ which defines the class.

We have next to consider how to interpret a class of classes. As we have defined $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined we shall naturally regard a class of classes as consisting of those values of $$\scriptstyle{\hat z(\phi z)}$$ which satisfy $\scriptstyle{f\{\hat z(\phi z)\}}$.|undefined Let us write $$\scriptstyle{\alpha}$$ for $\scriptstyle{\hat z(\phi z)}$; then we may write $$\scriptstyle{\hat\alpha(f\alpha)}$$ for the class of values of $$\scriptstyle{\alpha}$$ which satisfy $\scriptstyle{f\alpha}$. We shall apply the same definition, and put where "$\scriptstyle{\beta}$" stands for any expression of the form $\scriptstyle{\hat z(\psi!z)}$.

Let us take "$\scriptstyle{\gamma\in\hat\alpha(f\alpha)}$" as an instance of $\scriptstyle{F\{\hat\alpha(f\alpha)\}}$.|undefined Then

Thus we find