The Principles of Mathematics (1903)

§ 66

To bring clearly before the mind what is meant by class, and to distinguish this notion from all the notions to which it is allied, is one of the most difficult and important problems of mathematical philosophy. Apart from the fact that class is a very fundamental concept, the utmost care and nicety is required in this subject on account of the contradiction to be discussed in Chapter X. I must ask the reader, therefore, not to regard as idle pedantry the apparatus of somewhat subtle discriminations to be found in what follows.(§ 66 ¶ 1)

It has been customary, in works on logic, to distinguish two standpoints, that of extension and that of intension. Philosophers have usually regarded the latter as more fundamental, while Mathematics has been held to deal specially with the former. M. Couturat, in his admirable work on Leibniz, states roundly that Symbolic Logic can only be built up from the standpoint of extension[54]; and if there really were only these two points of view, his statement would be justified. But as a matter of fact, there are positions intermediate between pure intension and pure extension, and it is in these intermediate regions that Symbolic Logic has its lair. It is essential that the classes with which we are concerned should be composed of terms, and should not be predicates or concepts, for a class must be definite when its terms are given, but in general there will be many predicates which attach to the given terms and no others. We cannot of course attempt an intensional definition of class as the class of predicates attaching to the terms in question and to no others, for this would involve a vicious circle; hence the point of view of extension is to some extent unavoidable. On the other hand, if we take extension pure, our class is defined by the enumeration of its terms, and this method will not allow us to deal, as Symbolic Logic does, with infinite classes. Thus our classes must in general be regarded as objects denoted by concepts, and to this extent, the point of view of intension is essential. It is owing to this consideration that the theory of denoting is of such great importance. In the present chapter we have to specify the precise degree in which extension and intension respectively enter into the definition and employment of classes; and throughout the discussion, I must ask the reader to remember that whatever is said has to be applicable to infinite as well as finite classes.(§ 66 ¶ 2)

§ 66 n. 1. La Logique de Leibniz, Paris, 1901, p. 387.