402 THE POPULAR SCIENCE MOXTHLY 



ing ought, I think, to take account of the fact that there is nowadays 

 a new logic, that this new logic is in considerable part the work of 

 men whose attention has been attracted to the nature of the deductive 

 process by a wide experience of mathematical procedure, and that this 

 new logic shows us with regard to the syllogism, for instance, two things, 

 first that the essence of the syllogism does not consist in the fact that 

 a particular case is brought under a general principle; and secondly, 

 that the syllogism is by no means the only form of deductive reasoning. 

 From the point of view of the new logic, the student has upon his 

 hands the problem that Poincare has so well stated at the outset of his 

 book, " Science and Hypothesis." This is the problem presented by the 

 enormous Fecundity of the Deductive Process. Our own American logi- 

 cian, Charles Peirce, long since called attention to this fecundity. It is 

 a fact of much philosophical importance. What I mean by the fecundity 

 of deduction as a logical fact can be suggested by what Poincare men- 

 tioned, and also by a summary of the matters to which Peirce has 

 frequently called attention. Poincare states the case thus: From the 

 point of view of the older interpretation of the nature of the syllogism 

 it would seem impossible that a deductive science such as mathematics 

 could do anything but draw out of premises what it had already more 

 or less overtly or secretly put into them. Xothing novel could result. 

 And in fact if the reasoning of mathematics were all of tlie kind that 

 Professor Pillsbury supposes to be the typical deductive reasoning, 

 mathematical science would consist in a process as stupid and monoto- 

 nous as the process of taking the major premise. All men are mortal, 

 and then looking up all the names in a directory and solemnly con- 

 cluding to write opposite to each name the fact that since this person 

 is a man he is mortal. But now as a fact mathematical science consists 

 of nothing of the kind. The situation is actually this: you can write 

 upon a few sheets of paper an accurate statement of a set of principles 

 from which the whole science of the quantities of ordinary algebra can 

 be deduced. That is to say, the principles of the ordinary mathematical 

 analysis are capable of such a brief statement as this. But the conse- 

 quences of these principles are such that novel results in vast numbers 

 are annually discovered. These results are not stowed away in the 

 premises in any such way as that in which the mortality of this man 

 is stowed away in the assertion of the mortality of all men. 



Poincare, in the passage to which I have referred, suggests his own 

 theory to account for the fecundity of mathematical analysis. His 

 theory may as a logical theory be questioned. But tlie fecundity to 

 which he attracts attention ought to be a commonplace to any one who 

 has looked into any branch of mathematics with care. Peirce has called 

 attention to this fact, and speaking as a logician has gone further. 

 Following a lead of De Morgan's, Peirce has shown that any proposition 



