The second result still more directly concerns a philosophical logic. It is this: Since the few types of relations which this sort of analysis reveals as the fundamental ones in exact science are of such importance, the logic of the present day is especially required to face the questions: _What is the nature of our concept of relations?_ What are the various possible types of relations? Upon what does the variety of these types depend? What unity lies beneath the variety? As a fact, logic, in its modern forms, namely, first that symbolic logic which Boole first formulated, which Mr. Charles S. Peirce and his pupils have in this country already so highly developed, and which Schroeder in Germany, Peano's school in Italy, and a number of recent English writers have so effectively furthered--and secondly, the logic of scientific method, which is now so actively pursued, in France, in Germany, and in the English-speaking countries--this whole movement in modern logic, as I hold, is rapidly approaching _new solutions of the problem of the fundamental nature and the logic of relations_. The problem is one in which we are all equally interested. To De Morgan in England, in an earlier generation, and, in our time, to Charles Peirce in this country, very important stages in the growth of these problems are due. Russell, in his work on the _Principles of Mathematics_ has very lately undertaken to sum up the results of the logic of relations, as thus far developed, and to add his own interpretations. Yet I think that Russell has failed to get as near to the foundations of the theory of relations as the present state of the discussion permits. For Russell has failed to take account of what I hold to be the most fundamentally important generalization yet reached in the general theory of relations. This is the generalization set forth as early as 1890, by Mr. A. B. Kempe, of London, in a pair of wonderful but too much neglected, papers, entitled, respectively, _The Theory of Mathematical Form_, and _The Analogy between the Logical Theory of Classes and the Geometrical Theory of Points_. A mere hint first as to the more precise formulation of the problem at issue, and then later as to Kempe's special contribution to that problem, may be in order here, despite the impossibility of any adequate statement. III