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

68

INTRODUCTION

[CHAP. II

In spite of the contradictions which result from unnoticed typical
ambiguity, it is not desirable to avoid words and symbols which have
typical ambiguity. Such words and symbols embrace practically all the
ideas with which mathematics and mathematical logic are concerned: the
systematic ambiguity is the result of a systematic analogy. That is to say, in
almost all the reasonings which constitute mathematics and mathematical
logic, we are using ideas which may receive any one of an infinite number
of different typical determinations, any one of which leaves the reasoning
valid. Thus by employing typically ambiguous words and symbols, we are
able to make one chain of reasoning applicable to any one of an infinite
number of different cases, which would not be possible if we were to forego
the use of typically ambiguous words and symbols.
Among propositions wholly expressed in terms of typically ambiguous
notions practically the only ones which may differ, in respect of truth or
falsehood, according to the typical determination which they receive, are
existence-theorems. If we assume that the total number of individuals
is n, then the total number of classes of individuals is 2", the total number
of classes of classes of individuals is 22", and so on. Here n may be
either finite or infinite, and in either case 2" >n. Thus cardinals greater
than n but not greater than 2" exist as applied to classes,but not as applied
to classes of individuals, so that whatever may be supposed to be the
number of individuals, there will be existence-theorems which hold for
higher types but not for lower types. Even here, however, so long as
the number of individuals is not asserted, but is merely assumed hypothetically, we may replace the type of individuals by any other type, provided
we make a corresponding change in all the other types occurring in the
same context. That' is, we may give the name "relative individuals" to
the members of an arbitrarily chosen type r, and the name "relative classes
of individuals" to classes of "relative individuals," and so on. Thus so long
as only hypotheticals are concerned, in which existence-theorems for one
type are shown to be implied by existence-theorems for another, only
relative types are relevant even in existence-theorems. This applies also
to cases where the hypothesis (and therefore the conclusion) is asserted, provided the assertion holds for any type, however chosen. For example, any
type has at least one member; hence any type which consists of classes, of
whatever order, has at least two members. But the further pursuit of these
topics must be left to the body of the work.