September 22, 1916] 



SCIENCE 



413 



which in some respects increased in the 

 learned world at the time of the Renais- 

 sance. Mankind then changed its author- 

 ity, and this fact temporarily acted as an 

 emancipation. But the main fact, and we 

 can find complaints 2 of it at the very com- 

 mencement of the modern movement, was 

 the establishment of a reverential attitude 

 towards any statement made by a classical 

 author. Scholars became commentators on 

 truths too fragile to bear translation. A 

 science which hesitates to forget its found- 

 ers is lost. To this hesitation I ascribe the 

 barrenness of logic. Another reason for 

 distrust of logical theory and of mathe- 

 matics is the belief that deductive reason- 

 ing can give you nothing new. Tour con- 

 clusions are contained in your premises, 

 which by hypothesis are known to you. 



In the first place this last condemnation 

 of logic neglects the fragmentary, discon- 

 nected character of human knowledge. To 

 know one premise on Monday, and another 

 premise on Tuesday, is useless to you on 

 Wednesday. Science is a permanent record 

 of premises, deductions and conclusions, 

 verified all along the line by its correspond- 

 ence with facts. Secondly, it is untrue that 

 when we know the premises we also know 

 the conclusions. In arithmetic, for ex- 

 ample, mankind are not calculating boys. 

 Any theory which proves that they are 

 conversant with the consequences of their 

 assumptions must be wrong. We can ima- 

 gine beings who possess such insight. But 

 we are not such creatures. Both these an- 

 swers are, I think, true and relevant. But 

 they are not satisfactory. They are too 

 much in the nature of bludgeons, too exter- 

 nal. We want something more explanatory 

 of the very real difficulty which the ques- 

 tion suggests. In fact, the true answer is 

 embedded in the discussion of our main 

 2 E. g., in 1551 by Italian schoolmen. 



problem of the relation of logic to natural 

 science. 



It will be necessary to sketch in broad 

 outline some relevant features of modern 

 logic. In doing so I shall try to avoid the 

 profound general discussions and the 

 minute technical classifications which oc- 

 cupy the main part of traditional logic. It 

 is characteristic of a science in its earlier 

 stages — and logic has become fossilized in 

 such a stage — to be both ambitiously pro- 

 found in its aims and trivial in its han- 

 dling of details. We can discern four de- 

 partments of logical theory. By an analogy 

 which is not so very remote I will call these 

 departments or sections the arithmetic sec- 

 tion, the algebraic section, the section of 

 general-function theory, the analytic sec- 

 tion. I do not mean that arithmetic arises 

 in the first section, algebra in the second 

 section, and so on; but the names are sug- 

 gestive of certain qualities of thought in 

 each section which are reminiscent of anal- 

 ogous qualities in arithmetic, in algebra, in 

 the general theory of a mathematical func- 

 tion, and in the analysis of the properties 

 of particular functions. 



The first section — namely, the arithmetic 

 stage — deals with the relations of definite 

 propositions to each other, just as arith- 

 metic deals with definite numbers. Con- 

 sider any definite proposition; call it "p." 

 We conceive that there is always another 

 proposition which is the direct contradic- 

 tory to "p"; call it "not-p." When we 

 have got two propositions, p and q, we can 

 form derivative propositions from them, 

 and from their contradictories. We can 

 say, "At least one of p or q is true, and 

 perhaps both." Let us call this proposi- 

 tion "p or q." I may mention as an aside 

 that one of the greatest living philosophers 

 has stated that this use of the word " or ' ' — 

 namely, "p or q" in the sense that either 

 or both may be true — makes him despair of 



