Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/103

 But this is a contradiction. Hence "$\scriptstyle{\alpha\in\alpha}$" and "$\scriptstyle{\alpha\sim\in\alpha}$" must always be meaningless. In general, there is nothing surprising about this conclusion, but it has two consequences which deserve special notice. In the first place, a class consisting of only one member must not be identical with that one member, i.e. we must not have $\scriptstyle{\iota'x=x}$. For we have $\scriptstyle{x\in\iota'x}$, and therefore, if $\scriptstyle{x=\iota'x}$, we have $\scriptstyle{\iota'x\in\iota'x}$, which, we saw, must be meaningless. It follows that "$\scriptstyle{x=\iota'x}$" must be absolutely meaningless, not simply false. In the second place, it might appear as if the class of all classes were a class, i.e. as if (writing "$\scriptstyle{\text{Cls}}$"|undefined for "class") "$\scriptstyle{\text{Cls}\in\text{Cls}}$"|undefined were a true proposition. But this combination of symbols must be meaningless; unless, indeed, an ambiguity exists in the meaning of "$\scriptstyle{\text{Cls}}$,"|undefined so that, in "$\scriptstyle{\text{Cls}\in\text{Cls}}$,"|undefined the first "$\scriptstyle{\text{Cls}}$"|undefined can be supposed to have a different meaning from the second.

As regards the above requisites, it is plain, to begin with, that, in accordance with our definition, every propositional function $$\scriptstyle{\phi\hat z}$$ determines a class $\scriptstyle{\hat z(\phi z)}$. Assuming the axiom of reducibility, there must always be true propositions about $\scriptstyle{\hat z(\phi z)}$, i.e. true propositions of the form $\scriptstyle{f\{\hat z(\phi z)\}}$.|undefined For suppose $$\scriptstyle{\phi\hat z}$$ is formally equivalent to $\scriptstyle{\psi!\hat z}$, and suppose $$\scriptstyle{\psi!\hat z}$$ satisfies some function $\scriptstyle{f}$. Then $$\scriptstyle{\hat z(\phi z)}$$ also satisfies $\scriptstyle{f}$. Hence, given any function $\scriptstyle{\phi\hat z}$, there are true propositions of the form $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined i.e. true propositions in which "the class determined by $\scriptstyle{\phi\hat z}$" is grammatically the subject. This shows that our definition fulfils the first of our five requisites.

The second and third requisites together demand that the classes $$\scriptstyle{\hat z(\phi z)}$$ and $$\scriptstyle{\hat z(\psi z)}$$ should be identical when, and only when, their defining functions are formally equivalent, i.e. that we should have Here the meaning of "$\scriptstyle{\hat z(\phi z)=\hat z(\psi z)}$" is to be derived, by means of a two-fold application of the definition of $\scriptstyle{f\{\hat z(\phi z)\}}$,|undefined from the definition of  by the general definition of identity.

In interpreting "$\scriptstyle{\hat z(\phi z)=\hat z(\psi z)}$," we will adopt the convention which we adopted in regard to $\scriptstyle{(}$$\scriptstyle{x)(\phi x)}$ and $\scriptstyle{(}$$\scriptstyle{x)(\psi x)}$, namely that the incomplete symbol which occurs first is to have the larger scope. Thus $$\scriptstyle{\hat z(\phi z)=\hat z(\psi z)}$$ becomes, by our definition, which, by eliminating $\scriptstyle{\hat z(\psi z)}$, becomes  which is equivalent to