Hardly anything more unwelcome can befall a scientific writer   than to have one of the foundations of his edifice shaken after the   work is finished. This is the position into which I was put by a   letter from Mr Bertrand Russell as the printing of this volume was   nearing completion. The matter concerns my Basic Law (V). I have   never concealed from myself that it is not as obvious as the others   nor as obvious as must properly be required of a logical   law. Indeed, I pointed out this very weakness in the foreword to the   first volume, p. VII. I would gladly have dispensed with this   foundation if I had known of some substitute for it. Even now, I do   not see how arithmetic can be founded scientifically, how the   numbers can be apprehended as logical objects and brought under   consideration, if it is not—at least   conditionally—permissible to pass from a concept to its   extension. May I always speak of the extension of a concept, of a   class? And if not, how are the exceptions to be recognised? May one   always infer from the extension of one concept's coinciding with   that of a second that every object falling under the first concept   also falls under the latter? These questions arise from Mr Russell's   communication. …What is at stake here is not my approach to a   foundation in particular, but rather the very possibility of any   logical foundation of   arithmetic. (p. 253)[3]