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SYMBOLIC LOGIC BY CLARENCE IRVING LEWIS AND COOPER HAROLD LANGFORD Second Edition DOVER PUBLICATIONS, INC. Copyright © 1932, 1959 by Clarence I. Lewis and Cooper H. Langford. All rights reserved under Pan American and In ternational Copyright Conventions. Published in Canada by General Publishing Com pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC 2. This Dover edition, first published in 1959, is an unabridged and corrected republication of the first edition published by The Century Company in 1932. This Dover edition also contains a new Appendix by Clarence Irving Lewis. International Standard Book Number: 0-486-60170-6 Library of Congress Catalog Card Number: 59-4164 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014 PREFACE Symbolic logic does not presuppose either logic or mathe matics. Since it is itself logic in an exact form, and since it constitutes the theoretical foundation of mathematics in general, it is capable of being understood without previous training in either of these branches. The purpose of this book is that of providing an introduction to the subject in a form which is consonant with this priority of symbolic logic and which makes minimum demands on the reader in the way of previous technical training. However, discussion is not confined to the elements. The first five chapters cover those topics which lend themselves most easily to beginning study, and certain later portions are so written that a general view of the topic in question may be gained without technical difficulties. Chapters I-V follow the same general plan of development as Lewis’s Survey of Symbolic Logic (now out of print); but the subject-matter is here somewhat more condensed, and certain parts which are mathematically but not logically important have been omitted. An attempt has been made to keep the exposition as simple and clear as possible, and the extreme of rigor has sometimes been sacrificed to this consideration. The remainder of the book is independent of Chapters I-V, and these may well be omitted by readers who already have an elementary knowledge of symbolic logic. Chapter VI provides a new foundation for the subject, by the more rigorous ‘logistic’ method, an understanding of which is essential for those who wish to go on to the study of Principia Mathematica and similar con temporary treatises. Chapters VII and VIII presuppose this preceding development, but are not themselves essential to the understanding of anything further in the book. These chapters touch upon some of the most difficult and abstract questions connected with the subject, but they are also of the first im portance for the nature and purport of symbolic logic. Chap ter IX contains a general development of the theory of proposi tions, and it, together with VI, provides a foundation for the PREFACE subsequent discussion. In XI and XII, application is made of the logic of propositions to the theory of sets of postulates, the first of these chapters giving a general and elementary treatment, and the second dealing with a special topic not appropriate to an introductory discussion. Postulational technique is impor tant at many points in mathematics; but the fundamental con ceptions that occur are logical rather than mathematical in a strict sense, and we have confined the discussion very largely to developing and illustrating these conceptions. In Chapter XIII we endeavor to give some account of the paradoxes that have arisen in logic and of the bearing of these paradoxes on logical theory. The choice of characters for symbolic logic continues to be a problem, since no set of universally accepted conventions has as yet emerged. In this book we have adopted the symbols used by Whitehead and Russell in Principia Mathematica wherever the idea to be symbolized coincides precisely with one which occurs in that work. For the rest, we have been guided partly by historical precedents and partly by considerations of conven ience and clarity. Chapters I-VIII and Appendix II have been written by Mr. Lewis; Chapters IX-XIII and Appendix I, by Mr. Langford. Each of us has given the other such assistance as he might; but final responsibility for the contents of the chapters is as indicated. Important assistance in the writing of the book has been received from many sources. Our indebtedness to the publica tions of Professors Lukasiewicz and Tarski, as well as to Ludwig Wittgenstein’s Tractatus Logico-philosophicus, will be evident in Chapters VII and VIII; and we have drawn freely upon the writings of Professor E. V. Huntington for materials used in Chapters XI and XII. We are also indebted, especially in Chap ters IX, XI, and XII, to the published and unpublished views of Professor H. M. Sheffer, with whom both of us have been associated. Dr. William T. Parry of Harvard University has made extensive and important contributions to the development of Chapter VI. Our indebtedness to him for work contained in Appendix II, and also to Dr. M. Wajsberg of the University of Warsaw, to Mr. Paul Henle of Harvard University, and to the published mono PREFACE graph of Professor Oskar Becker, is more specifically acknowl edged in that place. Proof of one critical theorem in Chapter VI was provided by Professor S. L. Quimby, of Columbia University. Dr. David Wilson has read Chapters VII and VIII in manuscript, correcting several mistakes; and Dr. Kurt E. Rosinger has read the whole of Chapters I-VIII, verifying the proofs and eliminat ing numerous errors. C. I. Lewis. C. H. Langford. EDITOR'S INTRODUCTION The present volume is a most unusual-one aDd meets a long- felt and important need. Recent years have witnessed a consid erable growth of interest in symbolic logic in the United States. Americans have made important contributions to the subject, but these contributions have usually been in detached articles in the journals. Professor Lewis's A Survey of Symbolic Logic in 1918 traced the history of the most important developments of symbolic logic from Leibniz to the twentieth century and dis cussed the relation of a “ system of strict implication" to systems of material implication and to the classic algebra of logic. But there has been no authoritative treatment of the field of sym bolic logic in the light of developments of the last fifteen years, a treatment which would present constructively the achievements in symbolic logic by one who had taken part in making the achievements possible and believed heartily in the importance of this branch of philosophic thought. This present volume is just such a treatment. Its two authors have had wide experi ence both as teachers of symbolic logic in the classroom and as writers on important problems in the field. Their experience is adequate guarantee that the book will prove useful and stimu lating to many readers. The book is suited to two classes of readers. On the one hand it offers, especially in Chapters I to V, an introduction to symbolic logic for elementary students. It here presents the material in a form available for classroom use as well as for private study; and it orients the reader by furnishing a discussion of the con nection between symbolic logic and traditional non-symbolic EDITOR'S INTRODUCTION logic. On the other hand, it contains sections that develop original material or bring together important material by other authors which has not previously been assembled and has not appeared except in technical articles. These sections concern the specialist; but as they raise problems of general and often controversial nature, they will also appeal to many others who have philosophical interests. A complete list of the topics of these sections is not needed here and can easily be derived from table of contents and index. But it should be noted that the distinction between the logic of intensions and the logic of extensions, presented earlier by Professor Lewis in his Survey, is carried out with new force and is made basic to the discussion of the entire volume. The conception of consistency between propositions, which, it is contended, is incapable of definition in terms of material implication, is thus brought into harmony with the current mathematical conception. A new operation, con verse substitution, is presented that makes possible the proof of existence propositions. The matrix method is developed further than ever before and is employed to defend the idea of plurality of logical truth; that is, it is so handled as to free the question of mathematical validity from ordinary logical intuitions and thus supports the supposition of a variety of non-Aristotelian logics akin to the variety of non-Euclidean geometries. The issues here raised and argued lead on to problems concerning the nature of logic and truth and other fundamental philosophical ideas. The present volume is therefore fitted to introduce the elementary student to the field of symbolic logic and to interest the specialist. And its bearings on broader problems of logic make it a challenge to serious students of philosophy. Sterling P. Lamprecht, General Editor. Amherst College. CONTENTS CHAPTER PAGE i. Introduction.................................................................... 3 ii. The Boole-Schroder Algebra................................. 27 h i. The Logic of Terms.................................................... 49 IV. The Two-valued Algebra......................................... 78 V. Extension of the Two-valued Algebra to Propositional Functions................................... 90 VI. The Logistic Calculus of Unanaltzed Propo sitions ......................................................................... 122 Section 1. General Properties of the Elementary Functions of Propositions............... 123 Section 2. Material Implication........................... 136 Section 3. Further Theorems................................ 147 Section 4. Consistency and the Modal Functions 153 Section 5. The Consistency Postulate and Its Consequences.................................... 166 Section 6. The Existence Postulate and Exist ence Theorems.................................... 178 VII. Truth-Value Systems and the Matrix M ethod 199 VIII. Implication and Deducibility................................. 235 IX. The General Theory of Propositions............... 263 X. Propositions of Ordinary Discourse.................. 310 XI. POSTULATIONAL TECHNIQUE: DEDUCTION................ 335 XII. Postulational Technique: Deducibility........... 398 XIII. The Logical Paradoxes.............................................. 438 APPENDIX i. The Use of Dots as Brackets............................... 486 ii. The Structure of the System of Strict Impli cation .......................................................................... 492 h i. Final Note on System S2.......................................... 503 Index................................................................................... 515