V.
The Axiom of Reducibility.


CS.A propositional function of x may, as we have seen, be of any order; hence any statement about Уall properties of xФ is meaningless. (A Уproperty of xФ is the same thing as a Уpropositional function which holds of x.Ф But it is absolutely necessary, if mathematics is to be possible, that we should have some method of making statements which will usually be equivalent to what we have in mind when we (inaccurately) speak of Уall properties of x.Ф This necessity appears in many cases, but especially in connection with mathematical induction. We can say, by the use of any instead of all, УAny property possessed by 0, and by the successors of all numbers possessing it, is possessed by all finite numbers.Ф But we can not go on to: УA finite number is one which possesses all properties possessed by 0 and by the successors of all numbers possessing them.Ф If we confine this statement to all first-order properties of numbers, we can not infer that it holds of second-order properties. For example, we shall be unable to prove that if m, n are finite numbers, then m + n is a finite number. For, with the above definition, Уm is a finite numberФ is a second-order property of m; hence the fact that m + 0 is a finite number, and that, if m + n is a finite number, so is m + n + 1, does not allow us to conclude by induction that m + n is a finite number. It is obvious that such a state of things renders much of elementary mathematics impossible.
CT.The other definition of finitude, by the non-similarity of whole and part, fares no better. For this definition is: УA class is said to be finite when every one-one relation whose domain is the class and whose converse domain is contained in the class has the whole class for its converse domain.Ф Here a variable relation appears, i.e., a variable function of two variables; we have to take all values of this function, which requires that it should be of some assigned order; but any assigned order will not enable us to deduce many of the propositions of elementary mathematics.
CU.Hence we must find, if possible, some method of reducing the order of a propositional function without affecting the truth or falsehood of its values. This seems to be what common-sense effects by the admission of classes. Given any propositional function φx, of whatever order, this is assumed to be equivalent, for all values of x, to a statement of the form Уx belongs to the class α.Ф Now this statement is of the first order, since it makes no allusion to Уall functions of such-and-such a type.Ф Indeed its only practical advantage over the original statement φx is that it is of the first order. There is no advantage in assuming that there really are such things as classes, and the contradiction about the classes which are not members of themselves shows that, if there are classes, they must be something radically different from individuals. I believe the chief purpose which classes serve, and the chief reason which makes them linguistically convenient, is that they provide a method of reducing the order of a propositional function. I shall, therefore, not assume anything of what may seem to be involved in the common-sense admission of classes, except this: that every propositional function is equivalent, for all its values, to some predicative function.
CV.This assumption with regard to functions is to be made whatever may be the type of their arguments. Let φx be a function, of any order, of an argument x, which may itself be either an individual or a function of any order. If φ is of the order next above x, we write the function in the form φ!x; in such a case we will call φ a predicative function. Thus a predicative function of an individual is a first-order function; and for higher types of arguments, predicative functions take the place that first-order functions take in respect of individuals. We assume, then, that every function is equivalent, for all its values, to some predicative function of the same argument. This assumption seems to be the essence of the usual assumption of classes; at any rate, it retains as much of classes as we have any use for, and little enough to avoid the contradictions which a less grudging admission of classes is apt to entail. We will call this assumption the axiom of classes, or the axiom of reducibility.
CW.We shall assume similarly that every function of two variables is equivalent, for all its values, to a predicative function of those variables, where a predicative function of two variables is one such that there is one of the variables in respect of which the function becomes predicative (in our previous sense) when a value is assigned to the other variable. This assumption is what seems to be meant by saying that any statement about two variables defines a relation between them. We will call this assumption the axiom of relations or the axiom of reducibility.
CX.In dealing with relations between more than two terms, similar assumptions would be needed for three, four, ... variables. But these assumptions are not indispensable for our purpose, and are therefore not made in this paper.
CY.By the help of the axiom of reducibility, statements about Уall first-order functions of xФ or Уall predicative functions of αФ yield most of the results which otherwise would require Уall functions.Ф The essential point is that such results are obtained in all cases where only the truth or falsehood of values of the functions concerned are relevant, as is invariably the case in mathematics. Thus mathematical induction, for example, need now only be stated for all predicative functions of numbers; it then follows from the axiom of classes that it holds of any function of whatever order. It might be thought that the paradoxes for the sake of which we invented the hierarchy of types would now reappear. But this is not the case, because, in such paradoxes, either something beyond the truth or falsehood of values of functions is relevant, or expressions occur which are unmeaning even after the introduction of the axiom of reducibility. For example, such a statement as УEpimenides asserts ψxФ is not equivalent to УEpimenides asserts φ!xФ even though ψx and φ!x are equivalent. Thus УI am lyingФ remains unmeaning if we attempt to include all propositions among those which I may be falsely affirming, and is unaffected by the axiom of classes if we confine it to propositions of order n. The hierarchy of propositions and functions, therefore, remains relevant in just those cases in which there is a paradox to be avoided.