952 



SCIENCE 



[N. S. Vol. XXX. No. 783 



the outgrowth and prolongation of the 

 former, and that the twain are one as the 

 branches and upper stem of' a tree are con- 

 tinuous with the lower stem and the roots. 

 To any one who knows something of the 

 immensity of modern mathematics, some- 

 thing of the continent of doctrine that the 

 term connotes, something of the countless 

 variety and the infinite complexity of the 

 ideas and propositions that compose the 

 body and constitution of the science, the 

 simple thesis in question is really astound- 

 ing. And one demands that the thesis be 

 explicated in terms in order that one may 

 know precisely and concretely in detail 

 what it constates. What, we wish to be 

 informed, are the logical primitives that, 

 it is alleged, are capable, though so few, of 

 supporting so great a burden? Before 

 attempting to meet this demand, I beg to 

 remind you of the fact that, given a logic- 

 ally coherent or autonomous body of propo- 

 sitions, it is always in some degree a matter 

 of arbitrary choice, though probably never 

 one of complete indifference which of the 

 propositions are taken as fundamental and 

 which as derivative — that is, which are 

 assumed and which proved. In every case 

 the choice is to be guided by considerations 

 of expedience, of interest, or of economy, 

 but seems never to be coerced by necessity 

 or by "the nature of things." Questions 

 of relative interest, however, and of relative 

 expedience and economy are matters of 

 judgment. Accordingly it is not a matter 

 for surprise that several systems of logical 

 primitives have been devised and submitted, 

 differing any two of them in respect of one 

 or more elements but agreeing all of them 

 as to the adequaej' of a small number of 

 elements, and that among investigators in 

 the field it remains a moot question which 

 of the systems, if any one of them enjoys 

 that distinction in comparison with the rest, 

 is to be preferred. 



The system that I shall present here is 

 that which Russell has adopted in his great 

 synthesis of modern logic and modern 

 mathematics, "The Principles of Mathe- 

 matics," and which with slight modifica- 

 tions has been so delightfully expounded by 

 Couturat in his "Les Principes des Mathe- 

 matiques ' ' and his ' ' Traite de Logistique. ' ' 

 I have thought it best to gather together 

 all the primitive elements of the three 

 branches of logic for compact presentation 

 in a single uninterrupted list under their 

 appropriate headings, reserving commen- 

 tary for a subsequent stage. Moreover, 

 despite the somewhat forbidding appear- 

 ance, at first glance, of logical symbolism, 

 I have decided to present primitive propo- 

 sitions in symbolic form, employing for this 

 purpose the symbolism of Peano slightly 

 modified by selection from that of Schroder. 

 Indeed this symbolism is not difficult to 

 master; and if at first it seems a thing of 

 so frightful mean that to be hated needs 

 but to be seen, yet, seen often enough to 

 become familiar with its face, we come first 

 to endure, and then to embrace it as a con- 

 venient and potent means of clarity and 

 economy alike of thought and of expression. 

 It is a moot question which one, if indeed 

 any one, of the three varieties of the logical 

 calculus is primordial to the other two. 

 As, however, discourse of any kind, whether 

 about classes or about relations, would seem 

 to be difficult if not impossible without 

 propositions, I shall follow the leading of 

 common sense and begin with 



The Logic of Propositions.— In addition 

 to the notions, truth and its negative, 

 which, though they are constantly em- 

 ployed, seem neither to admit of effective 

 definition nor to be strictly coordinate with 

 any other indispensable notion, the primi- 

 tive notions in propositional logic are 



(1) Material Implication, 



(2; Formal Implication. 



