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

But this is a contradiction^[1]. 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 "$\scriptstyle {\text{Cls}}$" for "class") "$\scriptstyle {{\text{Cls}}\in {\text{Cls}}}$" were a true proposition. But this combination of symbols must be meaningless; unless, indeed, an ambiguity exists in the meaning of "$\scriptstyle {\text{Cls}}$," so that, in "$\scriptstyle {{\text{Cls}}\in {\text{Cls}}}$," the first "$\scriptstyle {\text{Cls}}$" 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 $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$. 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 $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, 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

$\scriptstyle {{\hat {z}}(\phi z)={\hat {z}}(\psi z).\equiv :\phi x.\equiv _{x}.\psi x}$.

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 $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, from the definition of

"$\scriptstyle {\chi !{\hat {z}}=\theta !{\hat {z}}}$,"

which is

$\scriptstyle {\chi !{\hat {z}}=\theta !{\hat {z}}.=:(f):f!\chi !{\hat {z}}.\supset .f!\theta !{\hat {z}}\quad {\text{Df}}}$

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,

$\scriptstyle {(\exists \chi ):\phi x.\equiv _{x}.\chi !x:\chi !{\hat {z}}={\hat {z}}(\psi z)}$,

which, by eliminating $\scriptstyle {{\hat {z}}(\psi z)}$, becomes

$\scriptstyle {(\exists \chi ):.\phi x.\equiv _{x}.\chi !x:.(\exists \theta ):\psi x.\equiv _{x}.\theta !x:\chi !{\hat {z}}=\theta !{\hat {z}}}$,

which is equivalent to

$\scriptstyle {(\exists \chi ,\theta ):\phi x.\equiv _{x}.\chi !x:\psi x.\equiv _{x}.\theta !x:\chi !{\hat {z}}=\theta !{\hat {z}}}$,

1. This is the second of the contradictions discussed at the end of Chapter II.