Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

II]

THE AXIOM OF REDUCIBILITY

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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 nth order (where n is fixed) which is formally equivalent
to Ox', whatever may be the order of Ox'. Here we shall mean by "a function
of the nth order" a function of the nth order relative to the arguments
to O5~; thus if these arguments are absolutely of the mth order, we assume
the existence of a function formally equivalent to Obx whose absolute order is
the m+nth. The axiom of reducibility in the form assumed above takes
n= 1, but this is not necessary to the use of the axiom. It is also unnecessary that n should be the same for different values of m; what is
necessary is that n should be constant so long as m is constant. What is
needed is that, where extensional functions of functions are concerned, we
should be able to deal with any 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 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* is equivalent to a single predicate,"
i.e. to the assumption that, if we assert that x has all the predicates that
satisfy a function f(o! z), there is some one predicate which x will have
whenever our assertion is true, and will not have whenever it is false, and
similarly if we assert that x has some one of the predicates that satisfy
a functionf (b! z). For by means of this assumption, the order of a nonpredicative 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
* 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.