The Principles of Mathematics (1903)

§ 73

Great difficulties are associated with the null-class, and generally with the idea of nothing. It is plain that there is such a concept as nothing, and that in some sense nothing is something. In fact, the proposition nothing is not nothing is undoubtedly capable of an interpretation which makes it true--a point which gives rise to the contradictions discussed in Plato's Sophist. In Symbolic Logic the null-class is the class which has no terms at all; and symbolically it is quite necessary to introduce some such notion. We have to consider whether the contradictions which naturally arise can be avoided.(§ 73 ¶ 1)

It is necessary to realize, in the first place, that a concept may denote although it does not denote anything. This occurs when there are propositions in which the said concept occurs, and which are not about the said concept, but all such propositions are false. Or rather, the above is a first step towards the explanation of a denoting concept which denotes nothing. It is not, however, an adequate explanation. Considre, for example, the proposition chimaeras are animals or even primes other than 2 are numbers. these propositions appear to be true, and it would seem that they are not concerned with the denoting concepts, but with what these concepts denote; yet that is impossible, for the concepts in question do not denote anything. Symbolic Logic says that these concepts denote the null-class, and that the propositions in question assert that the null-class is contained in certain other classes. But with the strictly extensional view of classes propounded above, a class which has no terms fails to be anything at all: what is merely and solely a collection of terms cannot subsist when all the terms are removed. Thus we must either find a different interpretation of classes, or else find a method of dispensing with the null-class.(§ 73 ¶ 2)

The above imperfect definition of a concept which denotes, but does not denote anything, may be amended as follows. All denoting concepts, as we saw, are derived from class-concepts; and a is a class-concept when x is an a is a propositional function. The denoting concepts associated with a will not denote anything when and only when x is an a is false for all values of x. This is a complete definition of a denoting concept which does not denote anything; and in this case we shall say that a is a null class-concept, and that all a's is a null concept of a class. Thus for a system such as Peano's, in which what are called classes are really class-concepts, technical difficulties need not arise; but for us a genuine logical problem remains.(§ 73 ¶ 3)

The proposition chimaeras are animals may be easily interpreted by means of formal implication, as meaning x is a chimaera implies x is an animal for all values of x. But in dealing with classes we have been assuming that propositions containing all or any or every, though equivalent to formal implications, were yet distinct from them, and involved ideas requiring independent treatment. Now in the case of chimaeras, it is easy to substitute the pure intensional view, according to which what is really stated is a relation of predicates: in the case in question the adjective animal is part of the definition of the adjective chimerical (if we allow ourselves to use this word, contrary to usage, to denote the defining predicate of chimaeras). But here again it is fairly plain that we are dealing with a proposition which implies that chimaearas are animals, but is not the same proposition--indeed, in the present case, the implication is not even reciprocal. By a negation we can give a kind of extensional interpretation: nothing is denoted by a chimaera which is not denoted by an animal. But this is a very roundabout interpretation. On the whole, it seems most correct to reject the proposition altogether, while retaining the various other propositions that would be equivalent to it if there were chimaeras. By symbolic logicians, who have experienced the utility of the null-class, this will be felt as a reactionary view. But I am not at present discussing what should be done in the logical calculus, where the established practice appears to me the best, but what is the philosophical truth concerning the null-class. We shall say, then, that, of the bundle of normally equivalent interpretations of logical symbolic formulae, the class of interpretations considered in the present chapter, which are dependent upon actual classes, fail where we are concerned with null class-concepts, on the ground that there is no actual null-class.(§ 73 ¶ 4)

We may now reconsider the proposition nothing is not nothing--a proposition plainly true, and yet, unless carefully handled, a source of apparently hopeless antinomies. Nothing is a denoting concept, which denotes nothing. The concept which denotes is of course not nothing, i.e. it is not denoted by itself. The proposition which looks so paradoxical means no more than this: Nothing, the denoting concept, is not nothing, i.e. is not what itself denotes. But it by no means follows from this that there is an actual null-class: only the null class-concept and the null concept of a class are to be admitted.(§ 73 ¶ 5)

But now a new difficulty has to be met. The equality of class-concepts, like all relations which are reflexive, symmetrical, and transitive, indicates an underlying identity, i.e. it indicates that every class-concept has to some term a relation which all equal class-concepts also have to that term--the term in question being different for different sets of equal class-concepts, but the same for the various members of a single set of equal class-concepts. Now for all class-concepts which are not null, this term is found in the corresponding class; but where are we to find it for null class-concepts? To this question several answers may be given, any of which may be adopted. For we now know what a class is, and we may therefore adopt as our term the class of all null class-concepts or of all null propositional functions. These are not null-classes, but genuine classes, and to either of them all null class-concepts have the same relation. If we then wish to have an entity analogous to what is elsewhere to be called a class, but corresponding to null class-concepts, we shall be forced, wherever it is necessary (as in counting classes) to introduce a term which is identical for equal class-concepts, to sbustitute everywhere the class of class-concepts equal to a given class-concept for the class corresponding to that class-concept. The class corresponding to the class-concept remains logically fundamental, but need not be actually employed in our symbolism. The null-class, in fact, is in some ways analogous to an irrational in Arithmetic: it cannot be interpreted on the same principles as other classes, and if we wish to give an analogous interpretation elsewhere, we must substitute for classes other more complicated entities--in the present case, certain correlated classes. The object of such a procedure will be mainly technical; but failure to understand the procedure will lead to inextricable difficulties in the interpretation of the symbolism. A very closely analogous procedure occurs constantly in Mathematics, for example with every generalization of number; and so far as I know, no single case in which it occurs has been rightly interpreted either by philosophers or by mathematicians. So many instances will meet us in the course of the present work that it is unnecessary to linger longer over the point at present. Only one possible misunderstanding must be guarded against. No vicious circle is involved in the above account of the null-class; for the general notion of class is first laid down, is found to involve what is called existence, is then symbolically, not philosophically, replaced by the notion of a class of equal class-concepts, and is found, in this new form, to be applicable to what corresponds to null class-concepts, since what corresponds is now a class which is not null. Between classes simpliciter and classes of equal class-concepts, there is a one-one correlation, which breaks down in the sole case of the class of null class-concepts, to which no null-class corresponds; and this fact is the reason for the whole complication.(§ 73 ¶ 6)