errata p. 68. line 20, for Ie !,leA" :"’.7U.ti .. classes of cla.sses."
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 systenlatic analogy. That is to say, in almost all the reasonings which constitute mathematics and mathematical logic, we are using ideas which may receive anyone of an infinite number of different typical determinations, anyone 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 anyone 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 n , the total number of classes of classes of individuals is 2 2 ?l, and so on. Here n may be either finite or infinite, and in either case 2 n > n. Thus cardinals greater . . 0’1 t r.. Co , . than n but not greater than 2 n eXIst as applIed 1jO 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-theorelns which hold for higher types but not for lower types. Even here, however, so long as the nUlnber of individuals is not asserted, but is merely assumed hypo- thetically, 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 7’, and the name "relative classes of individuals" to classes of "relative individuals," and so on. Thus so long as only hypotheticals are concerned, ill 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, pro- vided tbe 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. ’ .a
