The Principles of Mathematics (1903)

§ 71

Class may be defined either extensionally or intensionally. That is to say, we may define the kind of object which is a class, or the kind of concept which denotes a class: tihs is the precise meaning of the opposition of extension and intension in this connection. But although the general notion can be defined in this two-fold manner, particular classes, except when they happen to be finite, can only be defined intensionally, i.e. as the objects denoted by such and such concepts. I believe this distinction to be purely psychological: logically, the extensional definition appears to be equally applicable to infinite classes, but practically, if we were to attempt it, Death would cut short our laudable endeavour before it had attained its goal. Logically, therefore, extension and intension seem to be on a par. I will begin with the extensional view.(§ 71 ¶ 1)

When a class is regarded as defined by the enumeration of its terms, it is more naturally called a collection. I shall for the moment adopt this name, as it will not prejudge the question whether the objects denoted by it are truly classes or not. By a collection I mean what is conveyed by A and B or A and B and C, or any other enumeration of definite terms. The collection is defined by the actual mention of the terms, and the terms are connected by and. It would seem that and represents a fundamental way of combining terms, and that just this way of combination is essential if anything is to result of wihch a number other than 1 can be asserted. Collections do not presuppose numbers, since they result simply from the terms together with and: they could only presuppose numbers in the particular case where the terms of the collection themselves presuppose numbers. There is a grammatical difficulty which, since no method exists of avoiding it, must be pointed out and allowed for. A collection, grammatically, is singular, whereas A and B, A and B and C, etc. are essentially plural. This grammatical difficulty arises from the logical fact (to be discussed presently) that whatever is many in general forms a whole which is one; it is, therefore, not removable by a better choice of technical terms.(§ 71 ¶ 2)

The notion of and was brought into prominence by Bolzano[57]. In order to understand what infinity is, he says, we must go back to one of the simplest conceptions of the understanding, in order to reach an agreement concerning the word that we are to use to denote it. This is the conception which underlies the conjunction and, which, however, if it is to stand out as clearly as is required, in many cases, both by the purposes of mathematics and by those of philosophy, I believe to be best expressed by the words A system (Inbegriff) of certain things, or a whole consisting of certain parts. But we must add that every arbitrary object A can be combined in a system with any others B, C, D, ..., or (speaking still more correctly) already forms a system by itself[58], of which some more or less important truth can be enunciated, provided only that each of the presentations A, B, C, D, … in fact represents a different object, or in so far as none of the propositions A is the same as B, A is the same as C, A is the same as D, etc., is true. For if e.g. A is the same as B, then it is certainly unreasonable to speak of a system of the things A and B.(§ 71 ¶ 3)

The above passage, good as it is, neglects several distinctions which we have found here necessary. First and foremost, it does not distinguish the many from the whole which they form. Secondly, it does not appear to observe that this method of enumeration is not practically applicable to infinite systems. Thirdly, and this is connected with the second point, it does not make any mention of intensional definition nor of the notion of a class. What we have to consider is the difference, if any, of a class from a collection on the one hand, and from the whole formed of the collection on the other. But let us first examine further the notion of and.(§ 71 ¶ 4)

Anything of which a finite number other than 0 or 1 can be asserted would be commonly said to be many, and many, it might be said, are always of the form A and B and C and .... Here A, B, C, ... are each one and are all different. To say that A is one seems to amount to much the same as to say that A is not of the form A1 and A2 and A3 and .... To say that A, B, C, ... are all different seems to amount only to a condition as regards the symbols: it should be held that A and A is meaningless, so that diversity is implied by and, and need not be specially stated.(§ 71 ¶ 5)

A term A which is one may be regarded as a particular case of a collection, namely as a collection of one term. Thus every collection which is many presupposes many collections which are each one: A and B presupposes A and presupposes B. Conversely some collections of one term presuppose many, namely those which are complex: thus A differs from B is one, but presupposes A and difference and B. But there is not symmetry in this respect, for the ultimate presuppositions of anything are always simple terms.(§ 71 ¶ 6)

Every pair of terms, without exception, can be combined in the manner indicated by A and B, and if neither A nor B be many, then A and B are two. A and B may be any conceivable entities, any possible objects of thought, they may be points or numbers or true or false propositions or events or people, in short anything that can be counted. A teaspoon and the number 3, or a chimaera and a four-dimensional space, are certainly two. Thus no restriction whatever is to be placed on A and B, except that neither is to be many. It should be observed that A and B need not exist, but must, like anything that can be mentioned, have Being. The distinction of Being and existence is important, and is well illustrated by the process of counting. What can be counted must be something, and must certainly be, though it need by no means be possessed of the further privilege of existence. Thus what we demand of the terms of our collection is merely that each should be an entity.(§ 71 ¶ 7)

The question may now be asked: What is meant by A and B? Does this mean anything more than the juxtaposition of A with B? That is, does it contain any element over and above that of A and that of B? Is and a separate concept, which occurs besides A, B? To either answer there are objections. In the first place, and, we might suppose, cannot be a new concept, for if it were, it would have to be some kind of relation between A and B; A and B would then be a proposition, or at least a propositional concept, and would be one, not two. Moreover, if there are two concepts, there are two, and no third mediating concept seems necessary to make them two. Thus and would seem meaningless. But it is difficult to maintain this theory. To begin with, it seems rash to hold that any word is meaningless. When we use the word and, we do not seem to be uttering mere idle breath, but some idea seems to correspond to the word. Again some kind of combination seems to be implied by the fact that A and B are two, which is not true of either separately. When we say A and B are yellow, we can replace the proposition by A is yellow and B is yellow; but this cannot be done for A and B are two; on the contrary, A is one and B is one. Thus it seems best to regard and as expressing a definite unique kind of combination, not a relation, and not combining A and B into a whole, which would be one. This unique kind of combination will in future be called addition of individuals. It is important to observe that it applies to terms, and only applies to numbers in consequence of their being terms. Thus for the present, 1 and 2 are two, and 1 and 1 is meaningless.(§ 71 ¶ 8)

As regards what is meant by the combination indicated by and, it is indistinguishable from what we before called a numerical conjunction. That is, A and B is what is denoted by the concept of a class of which A and B are the only members. If u be a class-concept of which the propositions A is a u, B is a u are true, but of which all other propositions of the same form are false, then all u's is the concept of a class whose only terms are A and B; this concept denotes the terms A, B combined in a certain way, and A and B are those terms combined in just that way. Thus A and B are the class, but are distinct from the class-concept and from the concept of the class.(§ 71 ¶ 9)

The notion of and, however, does not enter into the meaning of a class, for a single term is a class, although it is not a numerical conjunction. If u be a class-concept, and only noe proposition of the form x is a u be true, then all u's is a concept denoting a single term, and this term is the class of which all u's is a concept. Thus what seems essential to a class is not the notion of and, but the being denoted by some concept of a class. This brings us to the intensional view of classes.(§ 71 ¶ 10)

§ 71 n. 1. Paradoxien des Unendlichen, Leipzig, 1854 (2nd ed., Berlin, 1889), § 3.

§ 71 n. 2. i.e. the combination of A with B, C, D, ... already forms a system.