PREFACE TO THE SECOND EDITION. xxvii 



the nature of an axiom ? Where would the science of 

 arithmetic be now if an analysis of the nature of number 

 itself were a necessary preliminary to a development of 

 the results of its laws ? In recent times there have been 

 enormous additions to the mathematical sciences, but very 

 few attempts at psychological analysis. In the Alex- 

 andrian and early medieval schools of philosophy, much 

 attention was given to the nature of unity and plurality 

 chiefly called forth by the question of the Trinity. In 

 the last two centuries whole sciences have been created 

 out of the notion of plurality, and yet speculation on the 

 nature of plurality has dwindled away. This present 

 treatise contains, in the eighth chapter, one of the few 

 recent attempts to analyse the notion of number itself. 



If further illustration is needed, I may refer to the 

 differential calculus. Nobody calls in question the formal 

 truth of the results of that calculus. All the more exact 

 and successful parts of physical science depend upon its 

 use, and yet the mathematicians who have created so 

 great a body of exact truths have never decided upon 

 the basis of the calculus. "What is the nature of a limit 

 or the nature of an infinitesimal ? Start the question 

 among a knot of mathematicians, and it will be found 

 that hardly two agree, unless it is in regarding the question 

 itself as a trifling one. Some hold that there are no such 

 things as infinitesimals, and that it is all a question of 

 limits. Others would argue that the infinitesimal is the 

 necessary outcome of the limit, but various shades of 

 intermediate opinion spring up. 



Now it is just the same with logic. If the forms of 

 deductive and inductive reasoning given in the earlier 

 part of this treatise are correct, they constitute a definite 

 addition to logical science, and it would have been absurd 

 to decline to publish such results because I could not at 

 the same time decide in my own mind about the psy- 

 chology and philosophy of the subject. It comes in short 

 to this, that my book is a book on Formal Logic and 



