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

SECTION B]

THE AXIOM OF REDUCIBILITY

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and taking arguments of a given order. For example, second-order functions
of an individual x are always derived by generalization from a matrix
f! (?! z,!,. Y!,z......),
where the functions f, <, a,... are predicative. It is possible, therefore,
without loss of generality, to use no apparent variables except such as are
predicative.
We require, however, a means of symbolising a function whose order is
not assigned. We shall use " cx " or "f(! z)" or etc. to express a function
(b orf) whose order, relatively to its argument, is not given. Such a function
cannot be made into an apparent variable, unless we suppose its order previously fixed. As the only purpose of the notation is to avoid the necessity
of fixing the order, such a function will not be used as an apparent variable;
the only functions which will be so used will be predicative functions, because,
as we have just seen, this restriction involves no loss of generality.
We have now to state and explain the axiom of reducibility.
It is important to observe that, since there are various types of propositions and functions, and since generalization can only be applied within
some one type (or, by means of systematic ambiguity, within some welldefined and completed set of types), all phrases referring to " all propositions "
or " all functions," or to "some (undetermined) proposition" or "some (undetermined) function," are prima facie meaningless, though in certain cases
they are capable of an unobjectionable interpretation. Contradictions arise
from the use of such phrases in cases where no innocent meaning can be
found.
If mathematics is to be possible, it is absolutely necessary (as explained
in the Introduction, Chapter II) 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 " all properties of x." (A " property of x" may be
defined as a propositional function satisfied by x.) 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 frx, of whatever order, this is assumed to be equivalent, for all values
of x, to a statement of the form "x belongs to the class a." Now assuming
that there is such an entity as the class a, this statement is of the first order,
since it involves no allusion to a variable function. Indeed its only practical
advantage over the original statement frx 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. It would seem that the sole purpose which classes
serve, and one main reason which makes them linguistically convenient, is