Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/83

classes, the axiom of reducibility becomes unnecessary. The assumption of the axiom of reducibility is therefore a smaller assumption than the assumption that there are classes. This latter assumption has hitherto been made unhesitatingly. However, both on the ground of the contradictions, which require a more complicated treatment if classes are assumed, and on the ground that it is always well to make the smallest assumption required for proving our theorems, we prefer to assume the axiom of reducibility rather than the existence of classes. But in order to explain the use of the axiom in dealing with classes, it is necessary first to explain the theory of classes, which is a topic belonging to Chapter III. We therefore postpone to that Chapter the explanation of the use of our axiom in dealing with classes.

It is worth while to note that all the purposes served by the axiom of reducibility are equally well served if we assume that there is always a function of the $\scriptstyle {n}$th order (where $\scriptstyle {n}$ is fixed) which is formally equivalent to $\scriptstyle {\phi {\hat {x}}}$, whatever may be the order of $\scriptstyle {\phi {\hat {x}}}$. Here we shall mean by "a function of the $\scriptstyle {n}$th order" a function of the $\scriptstyle {n}$th order relative to the arguments to $\scriptstyle {\phi {\hat {x}}}$; thus if these arguments are absolutely of the $\scriptstyle {m}$th order, we assume the existence of a function formally equivalent to $\scriptstyle {\phi {\hat {x}}}$ whose absolute order is the $\scriptstyle {m+n}$th. The axiom of reducibility in the form assumed above takes $\scriptstyle {n=1}$, but this is not necessary to the use of the axiom. It is also unnecessary that $\scriptstyle {n}$ should be the same for different values of $\scriptstyle {m}$; what is necessary is that $\scriptstyle {n}$ should be constant so long as $\scriptstyle {m}$ is constant. What is needed is that, where extensional functions of functions are concerned, we should be able to deal with any $\scriptstyle {a}$-function by means of some formally equivalent function of a given type, so as to be able to obtain results which would otherwise require the illegitimate notion of "all $\scriptstyle {a}$-functions"; but it does not matter what the given type is. It does not appear, however, that the axiom of reducibility is rendered appreciably more plausible by being put in the above more general but more complicated form.

The axiom of reducibility is' equivalent to the assumption that "any combination or disjunction of predicates^[1] is equivalent to a single predicate," i.e. to the assumption that, if we assert that $\scriptstyle {x}$ has all the predicates that satisfy a function $\scriptstyle {f(\phi !{\hat {z}})}$, there is some one predicate which $\scriptstyle {x}$ will have whenever our assertion is true, and will not have whenever it is false, and similarly if we assert that $\scriptstyle {x}$ has some one of the predicates that satisfy a function $\scriptstyle {f(\phi !{\hat {z}})}$. For by means of this assumption, the order of a non-predicative function can be lowered by one; hence, after some finite number of steps, we shall be able to get from any non-predicative function to a formally equivalent predicative function. It does not seem probable that

1. Here the combination or disjunction is supposed to be given intensionally. If given extensionally (i.e. by enumeration), no assumption is required; but in this case the number of predicates concerned must be finite.