22 Mr Van Horn, An Axiom in Symbolic Logic. 



An Axiom in Symbolic Logic. By C. E. Van Horn, M.A. 

 (Commimicated by Mr Q. H. Hardy.) 



[Received 30 August 1916: read 30 October 1916.] 



Philosophy's task is a search for the primal and fundamental 

 elements of the world. Its face is turned in the opposite direction 

 to that of science and mathematics. Philosophy hands back to 

 them its results, and they as best they can construct systematic 

 bodies of doctrine that purport to show us what the world may bo 

 on the one hand (science) and what the world might be on the 

 other (mathematics). As philosophy advances in the pursuit of its 

 task it is continually vacating old ground to science and mathe- 

 matics. The history of this change of boundary can be traced in 

 the changes in the nomenclature of human knowledge : Natural 

 Philosophy has become Physics ; Mental Philosophy has become 

 Psychology ; Moral Philosophy is becoming the inductive science 

 of Ethics. Thus (paradoxically speaking) philosophy's advance is 

 to be marked by the retreat of her boundaries. 



It is interesting to Avatch this retreat in a field occupied b}' 

 philosophy from its very beginning, and until recently supposed to 

 be its permanent possession. I refer to the field of the foundations 

 of mathematics. Here large areas once occupied by philosophy by 

 sovereign right of long control are slowly passing into the possession 

 of pure mathematics; and by the way both are gainers by the 

 transfer*. 



To facilitate the mathematical treatment of these new areas a 

 new instrument of investigation had to be invented, namely, Mathe- 

 matical, or Symbolic, Logic. This new logic, which is infinitely 

 more powerful than the traditional logic, and which embraces all 

 that is really self-consistent in the old logic, makes possible a 

 precise and easy handling of all the highly abstract and complex 

 ideas occurring in the noAv fields. For example, both philosophy 

 and the old logic found themselves involved in many a tangle on 

 questions concerning classes and relations because neither possessed 

 the requisite instruments of analysis. Again, philosophy had 

 wandered into a veritable labyrinth of difficulties concerning 

 infinity, quantity, continuity, and so on. Here too the secret of 

 the trouble lay in the inadequacy of the instruments of analysis 

 afforded by the traditional logic. 



* Much valuable light is thrown upon the details of this process in the writings 

 of Bertrand Russell, especially in the preface and introductory chapters of the 

 Frincipia Mathematica, Vol. i. 1910; and more recently in his Scientific Method in 

 Philosophy, 1914. 



