The Principles of Mathematics (1903)

§ 20

In this calculus there are very much fewer new primitive propositions—in fact, two seem sufficient—but there are much greater difficulties in the way of non-symbolic exposition of the ideas embedded in our symbolism. These difficulties, as far as possible, will be postponed to later chapters. For the present, I shall try to make an exposition which is to be as straightforward and simple as possible.(§ 20 ¶ 1)

The calculus of classes may be developed by regarding as fundamental the notion of class, and also the relation of a member of a class to its class. This method is adopted by Professor Peano, and is perhaps more philosophically correct than a different method which, for formal purposes, I have found more convenient. In this method we still take as fundamental the relation (which, following Peano, I shall denote by ∈) of an individual to a class to which it belongs, i.e. the relation of Socrates to the human race which is expressed by saying that Socrates is a man. In addition to this, we take as indefinables the notion of a propositional function and the notion of such that. It is these three notions that characterize the class-calculus. Something must be said in explanation of each of them.(§ 20 ¶ 2)