Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/106

The representation of a class by a single letter $\scriptstyle {\alpha }$ can now be understood. For the denotation of $\scriptstyle {\alpha }$ is ambiguous, in so far as it is undecided as to which of the symbols $\scriptstyle {{\hat {z}}(\phi z)}$, $\scriptstyle {{\hat {z}}(\psi z)}$, $\scriptstyle {{\hat {z}}(\chi z)}$, etc. it is to stand for, where $\scriptstyle {\phi z}$, $\scriptstyle {\psi z}$, $\scriptstyle {\chi z}$, etc. are the various determining functions of the class. According to the choice made, different propositions result. But all the resulting propositions are equivalent by virtue of the easily proved proposition:

$\scriptstyle {\vdash :\phi x\equiv _{x}\psi x.\supset .f\{{\hat {z}}(\phi z)\}\equiv f\{{\hat {z}}(\psi z)\}}$."

Hence unless we wish to discuss the determining function itself, so that the notion of a class is really not properly present, the ambiguity in the denotation of $\scriptstyle {\alpha }$ is entirely immaterial, though, as we shall see immediately, we are led to limit ourselves to predicative determining functions. Thus "$\scriptstyle {f(\alpha )}$," where $\scriptstyle {\alpha }$ is a variable class, is really "$\scriptstyle {f\{{\hat {z}}(\phi z)\}}$," where $\scriptstyle {\phi }$ is a variable function, that is, it is

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

where $\scriptstyle {\phi }$ is a variable function. But here a difficulty arises which is removed by a limitation to our practice and by the axiom of reducibility. For the determining functions $\scriptstyle {\phi {\hat {z}}}$, $\scriptstyle {\psi {\hat {z}}}$, etc. will be of different types, though the axiom of reducibility secures that some are predicative functions. Then, in interpreting $\scriptstyle {\alpha }$ as a variable in terms of the variation of any determining function, we shall be led into errors unless we confine ourselves to predicative determining functions. These errors especially arise in the transition to total variation (cf. pp. 15, 16). Accordingly

$\scriptstyle {fa=.(\exists \psi ).\phi !x\equiv _{x}\psi !x.f\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$

It is the peculiarity of a definition of the use of a single letter [viz. $\scriptstyle {\alpha }$] for a variable incomplete symbol that it, though in a sense a real variable, occurs only in the definiendum, while "$\scriptstyle {\phi }$," though a real variable, occurs only in the definiens. Thus "$\scriptstyle {f{\hat {\alpha }}}$" stands for

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

and "$\scriptstyle {(\alpha ).f\alpha }$" stands for

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

Accordingly, in mathematical reasoning, we can dismiss the whole apparatus of functions and think only of classes as "quasi-things," capable of immediate representation by a single name. The advantages are two-fold: (1) classes are determined by their membership, so that to one set of members there is one class, (2) the "type" of a class is entirely defined by the type of its members. Also a predicative function of a class can be defined thus

$\scriptstyle {f!\alpha =.(\exists \psi ).\phi !x\equiv _{x}\psi !x.f!\{\psi !{\hat {z}}\}\quad {\text{Df.}}}$

Thus a predicative function of a class is always a predicative function of any predicative determining function of the c1ass, though the converse does not hold.