Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/105

If we now extend the axiom of reducibility so as to apply to functions of functions, i.e. if we assume

$\scriptstyle {(\exists g):f(\psi !{\hat {z}}).\equiv _{\psi }.g!(\psi !{\hat {z}})}$,

we easily deduce

$\scriptstyle {\vdash :(\exists g):f\{{\hat {z}}(\psi !z)\}.\equiv _{\psi }.g!\{{\hat {z}}(\psi !z)\}}$,

i.e.

$\scriptstyle {\vdash :(\exists g):f\beta .\equiv _{\beta }.g!\beta }$.

Thus

$\scriptstyle {\vdash :\gamma \in {\hat {\alpha }}(f\alpha ).\equiv .f\gamma }$.

Thus every function which can take classes as arguments, i.e. every function of functions, determines a class of classes, whose members are those classes which satisfy the determining function. Thus the theory of classes of classes offers no difficulty.

We have next to consider our fifth requisite, namely that "$\scriptstyle {{\hat {z}}(\phi z)\in {\hat {z}}(\phi z)}$" is to be meaningless. Applying our definition of $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, we find that if this collection of symbols had a meaning, it would mean

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

i.e. in virtue of the definition

$\scriptstyle {x\in \psi !{\hat {z}}.=.\psi !x\quad {\text{Df}}}$,

it would mean

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

But here the symbol "$\scriptstyle {\psi !(\psi !{\hat {z}})}$" occurs, which assigns a function as argument to itself. Such a symbol is always meaningless, for the reasons explained at the beginning of Chapter II (pp. 41—3). Hence "$\scriptstyle {{\hat {z}}(\phi z)\in {\hat {z}}(\phi z)}$" is meaningless, and our fifth and last requisite is fulfilled. As in the case of $\scriptstyle {f(}$℩$\scriptstyle {x)(\phi x)}$, so in that of $\scriptstyle {f\{{\hat {z}}(\phi z)\}}$, there is an ambiguity as to the scope of $\scriptstyle {{\hat {z}}(\phi z)}$ if it occurs in a proposition which itself is part of a larger proposition. But in the case of classes, since we always have the axiom of reducibility, namely

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

which takes the place of $\scriptstyle {\mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$, it follows that the truth-value of any proposition in which $\scriptstyle {{\hat {z}}(\phi z)}$ occurs is the same whatever scope we may give to z$\scriptstyle {{\hat {z}}(\phi z)}$, provided the proposition is an extensional function of whatever functions it may contain. Hence we may adopt the convention that the scope is to be always the smallest proposition enclosed in dots or brackets in which $\scriptstyle {{\hat {z}}(\phi z)}$ occurs. If at any time a larger scope is required, we may indicate it by "$\scriptstyle {[{\hat {z}}(\phi z)]}$" followed by dots, in the same way as we did for $\scriptstyle {[(}$℩$\scriptstyle {x)(\phi x)]}$.

Similarly when two class symbols occur, e.g. in a proposition of the form $\scriptstyle {f\{{\hat {z}}(\phi z),{\hat {z}}(\psi z)\}}$, we need not remember rules for the scopes of the two symbols, since all choices give equivalent results, as it is easy to prove. For the preliminary propositions a rule is desirable, so we can decide that the class symbol which occurs first in the order of writing is to have the larger scope.