Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/73

here a new objection, namely the following: A proposition is not a single entity, but a relation of several; hence a statement in which a proposition appears as subject will only be significant if it can be reduced to a statement about the terms which appear in the proposition. A proposition, like such phrases as "the so-and-so," where grammatically it appears as subject, must be broken up into its constituents if we are to find the true subject or subjects^[1]. But in such a statement as "$\scriptstyle {p}$ is a man," where $\scriptstyle {p}$ is a proposition, this is not possible. Hence "$\scriptstyle {\{(x).\phi x\}}$ is a man" is meaningless.

V. The Hierarchy of Functions and Propositions.

We are thus led to the conclusion, both from the vicious-circle principle and from direct inspection, that the functions to which a given object $\scriptstyle {a}$ can be an argument are incapable of being arguments to each other, and that they have no term in common with the functions to which they can be arguments. We are thus led to construct a hierarchy. Beginning with $\scriptstyle {a}$ and the other terms which can be arguments to the same functions to which $\scriptstyle {a}$ can be argument, we come next to functions to which $\scriptstyle {a}$ is a possible argument, and then to functions to which such functions are possible arguments, and so on. But the hierarchy which has to be constructed is not so simple as might at first appear. The functions which can take $\scriptstyle {a}$ as argument form an illegitimate totality, and themselves require division into a hierarchy of functions. This is easily seen as follows. Let $\scriptstyle {f(\phi {\hat {z}},x)}$ be a function of the two variables $\scriptstyle {\phi {\hat {z}}}$ and $\scriptstyle {x}$. Then if, keeping $\scriptstyle {x}$ fixed for the moment, we assert this with all possible values of $\scriptstyle {\phi }$, we obtain a proposition:

$\scriptstyle {(\phi ).f(\phi {\hat {z}},x)}$.

Here, if $\scriptstyle {x}$ is variable, we have a function of $\scriptstyle {x}$; but as this function involves a totality of values of $\scriptstyle {\phi {\hat {z}}}$^[2], it cannot itself be one of the values included in the totality, by the vicious-circle principle. It follows that the totality of values of $\scriptstyle {\phi {\hat {z}}}$ concerned in $\scriptstyle {(\phi ).f(\phi {\hat {z}},x)}$ is not the totality of all functions in which $\scriptstyle {x}$ can occur as argument, and that there is no such totality as that of all functions in which $\scriptstyle {x}$ can occur as argument. It follows from the above that a function in which $\scriptstyle {\pi {\hat {z}}}$ appears as argument requires that "$\scriptstyle {\phi {\hat {z}}}$ should not stand for any function which is capable of a given argument, but must be restricted in such a way that none of the functions which are possible values of "$\scriptstyle {\phi {\hat {z}}}$ should involve any reference to the totality of such functions. Let us take as an illustration the definition of identity. We might attempt to define "$\scriptstyle {x}$ is identical with $\scriptstyle {y}$" as meaning "whatever is true of $\scriptstyle {x}$ is true of $\scriptstyle {y}$," i.e. "$\scriptstyle {\phi x}$ always implies $\scriptstyle {\phi y}$." But here,

1. Cf. Chapter III.
2. When we speak of "values of $\scriptstyle {\phi {\hat {z}}}$" it is $\scriptstyle {\phi }$, not $\scriptstyle {z}$, that is to be assigned. This follows from the explanation in the note on p. 42. When the function itself is the variable, it is possible and simpler to write $\scriptstyle {\phi }$ rather than $\scriptstyle {\phi {\hat {z}}}$, except in positions where it is necessary to emphasize that an argument must be supplied to secure significance.