Ml' Van Horn, An Axiom in Symbolic Logic 28 



Nuw however the matter is all changed. Philosophy, equipped 

 with the latest instruments of mathematical logic, is able to deal 

 successfidly with the problems of these fields. In fact so fully have 

 these ideas been analysed that at last philosophy as such has 

 relinquished these fields to pure mathematics. Even more, the 

 whole field of deduction has now become the foundation-branch of 

 mathematics and has developed into a precise Calculus of Pro- 

 positions. Out of it grow by easy stages the Calculus of Classes 

 and the Calculus of Relations, and these in turn grow by equally 

 easy stages into all the manifold branches of pure mathematics as 

 more commonly known. It is in these and similar ways that 

 philosophy and pure mathematics are both gainers by the transfer 

 of the fields recently acquired by mathematics from philosophy. 



It is now easy to understand why the axioms of mathematical 

 logic (and so of all pure mathematics) lie in the borderland between 

 philosophy and mathematics, and are thus the concern of the 

 philosopher equally with the mathematician. To depart entirely 

 from our figures and adopt others, the rootage of mathematics is in 

 philosophy. It is here too that we meet the innovations of mathe- 

 matical logic that appear so fantastic to the philosopher trained 

 only in the old logic. Its definitions and treatment of some of the 

 common terms of language seem so at variance with what the 

 traditional logician is familiar with that he often views the new 

 logic as the victim of some delusion. It appears however from the 

 nature of the case itself that many of those peculiarities, which 

 from the view-point of traditional logic would be described as 

 abnormal, do not deserve to be so described ; that in fact it is in 

 the theories of the traditional logician and philosopher that the 

 abnormalities really occur*. 



In order to indicate what seems to me a possible simplification 

 of the axiomatic basis of mathematical logic I wish to introduce in 

 a new form an idea advocated by Shelfer. Its importance lies in 

 the fact that in terms of it Sheffer was able to define the four 

 fundamental operations of logic, namely. Negation, Disjunction, 

 Implication, and Conjunction or Joint Assertion. It is a familiar 

 fact that Kronecker found the use of certain auxiliary quantities 

 (let us call them ' parameters ') of great value in his algebraic 

 investigations, the chief value lying in the fact that their dis- 

 appearance led to desired relations among numbers essential to his 

 investigations. It is a precisely similar use of Sheffer's idea that 

 I desire to make in the field of the philosophy of logic. In terms 

 of it I define, after him, the four fundamental operations of logic. 

 Then, unlike him, I work by means of an axiom to eliminate that 

 idea from the formulae, and in so doing to arrive at the desired 



* Cf. Russell, Scientific Method in Philosophij, chap. i. 



