The student who has completed some elementary study of symbolic logic and wishes to pursue the subject further finds himself in a discouraging situation. He has, perhaps, mastered the contents of Venn's 'Symbolic Logic' or Couturat's admirable little book, 'The Algebra of Logic', or the chapters concerning this subject in Whitehead's 'Universal Algebra'. If he read German with sufficient ease, he may have made some excursions into Schröder's 'Vorlesungen über die Algebra der Logik'. These all concern the classic, or Boole-Schröder algebra, and his knowledge of symbolic logic is probably confined to that system. His further interest leads him almost inevitably to Peano's 'Formulaire de Mathematiques', 'Principia Mathematica' of Whitehead and Russell, and the increasingly numerous shorter studies of the same sort. And with only elementary knowledge of a single kind of development of a small branch of the subject, he must attack these most difficult and technical of treatises, in a new notation, developed by methods which are entirely novel to him, and bristling with logico-metaphysical difficulties. If he is bewildered and searches for some means of further preparation, he finds nothing to bridge the gap. Schroder's work would be of most assistance here, but this was written some twenty-five years ago; the most valuable studies are of later date, and radically new methods have been introduced.
What such a student most needs is a comprehensive survey of the subject - one which will familiarize him with more than the single system which he knows, and will indicate not only the content of other branches and the alternative methods of procedure, but also the relation of these to the Boole-Schröder algebra and to one another. The present book is an attempt to meet this need, by bringing within the compass of a single volume, and reducing to a common notation (so far as possible), the most important developments of symbolic logic. If, in addition to this, some of the requirements of a "handbook" are here fulfilled, so much the better. But this survey does not pretend to be encyclopaedic. A gossipy recital of results achieved, or a superficial account of methods, is of no more use in symbolic logic than in any other mathematical discipline. What is presented must be treated in sufficient detail to afford the possibility of real insight and grasp. This aim has required careful selection of material.
The historical summary in Chapter I attempts to follow the main thread of development, and no reference, or only passing mention, is given to those studies which seem not to have affected materially the methods of later researches. In the remainder of the book, the selection has been governed by the same purpose. Those topics comprehension of which seems most essential, have been treated at some length, while matters less fundamental have been set forth in outline only, or omitted altogether. My own contribution to symbolic logic, presented in Chapter V, has not earned the right to inclusion here; in this, I plead guilty to partiality. The discussion of controversial topics has been avoided whenever possible and, for the rest, limited to the simpler issues involved. Consequently, the reader must not suppose that any sufficient consideration of these questions is here given, though such statements as are made will be, I hope, accurate. Particularly in the last chapter, on "Symbolic Logic, Logistic, and Mathematical Method", it is not possible to give anything like an adequate account of the facts. That would require a volume at least the size of this one. Rather, I have tried to set forth the most important and critical considerations "somewhat arbitrarily and dogmatically, since there is not space for argument" and to provide such a map of this difficult territory as will aid the student in his further explorations.
Proofs and solutions in Chapters II, III, and IV have been given very fully. Proof is of the essence of logistic, and it is my observation that students "even those with a fair knowledge of mathematics" seldom command the technique of rigorous demonstration. In any case, this explicitness can do no harm, since no one need read a proof which he already understands. I am indebted to many friends and colleagues for valuable assistance in preparing this book for publication: to Professor W A Merrill for emendations of my translation of Leibniz, to Professor J H McDonald and Dr B A Bernstein for important suggestions and the correction of certain errors in Chapter II, to Mr J C Rowell, University Librarian, for assistance in securing a number of rare volumes, and to the officers of the University Press for their patient helpfulness in meeting the technical difficulties of printing such a book. Mr Shirley Quimby has read the whole book in manuscript, eliminated many mistakes, and verified most of the proofs. But most of all, I am indebted to my friend and teacher, Josiah Royce, who first aroused my interest in this subject, and who never failed to give me encouragement and wise counsel. Much that is best in this book is due to him.
C I Lewis.
Berkeley, 10 July 1917.