AUGUSTUS DE MORGAN AND THE LOGIC OF RELATIONS The New Synthese Historical Library Texts and Studies in the History of Philosophy

VOLUME 38

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The titles published in this series are listed at the end of this volume. DANIEL D. MERRILL

Department of Philosophy, Oberlin College, USA

AUGUSTUS DE MORGAN AND THE LOGIC OF RELATIONS

KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON Library of Congress Cataloging. in·Publication Data

Merrill, Daniel D. (Daniel Davy) Augustus De Morgan and the lOglC of relations / Daniel D. Merril. p. cm. --

1. De Morgan, Augustus, 1806-1871--Contributions in logic of relations. 2. Inference. 3. Syllogism. 4. Logic, Symbolic and mathematical--History--19th cenTury. I. Title. II. Series. BC185.M47 1990 160' .92--dc20 90-34935

ISBN·13: 978·94·010·7418·6 e-ISBN -13: 978-94-009-2047-7 DOl: 10.1007/978-94-009-2047-7

Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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All Rights Reserved © 1990 Kluwer Academic Publishers Softcover reprint ofthe hardcover I st edition 1990 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. TABLE OF CONTENTS

PREFACE Vll

CHAPTER 1. The Traditional Syllogism 1 1. Whately and the Revival of Formal Logic 2 2. Euclid and the Syllogism 10 3. Reid, Hamilton and Mansel on Relational Inferences 15

CHAPTER II. First Thoughts on the Copula 26 1. The Two Copulas 26 2. First Notions of Logic 35 3. Relations and Identity 43

CHAPTER III. Generalizing the Copula 48 1. The Abstract Copula 49 2. The Bicopular Syllogism and the Composition of Relations 60 3. Oblique Inferences and De Morgan's Dictum 79

CHAPTER IV The Problem of Form and Matter 89 1. "Sundry Perversions of the Syllogistic Form" 90 2. The Material Copula 95 3. De Morgan's Response 99 4. The Issues 103 5. Heads and Tails 110

CHAPTER V The Logic of Relations 113 1. Philosophical Preliminaries 114 2. General Logic of Relations 117 3. Properties of Relations 124 4. Singular Relational Syllogisms (Unit Syllogisms) 129 5. Quantified Relational Syllogisms 136 6. The Limited Unit Syllogism 143 7. The Ordinary Syllogism and the Relational Syllogism 145

v vi TABLE OF CONTENTS

CHAPTER VI. The Logic of Relations and the Theory of the Syllogism 149 1. The Two Views 150 2. Objective View-The Basic Account 151 3. Objective View-The Relational Form 156 4. The Subjective View 164

CHAPTER VII. Logic and Mathematics 170 1. "A Mathematical Logic" 170 2. Algebraic Techniques and Analogies in Logic 174 3. Logic and Geometrical Proof 176 4. Logic and Algebraic Reasoning 180 5. Form in Algebra and Logic 183 6. Conclusions 193

CHAPTER VIII. A Rigorous Formulation 196 1. Basic Issues 196 2. The System D 200 3. Properties of Inclusion and Identity 202 4. De Morgan's Basic Identities 206 5. Theorem K 208 6. Properties of Relations 212 7. Additional Inclusion Laws 217 8. The Full System of Three-Relation Terms 219 9. De Morgan's Logic with Identity 221 10. More Properties of Relations 223 11. A Surrogate for Quantification Theory 229 12. Postscript-1864 235 13. De Morgan's Conjectures 240

NOTES 245

INDEX 255 PREFACE

The middle years of the nineteenth century saw two crucial develop• ments in the history of modern logic: George Boole's algebraic treat• ment of logic and Augustus De Morgan's formulation of the logic of relations. The former episode has been studied extensively; the latter, hardly at all. This is a pity, for the most central feature of modern logic may well be its ability to handle relational inferences. De Morgan was the first person to work out an extensive logic of relations, and the purpose of this book is to study this attempt in detail. Augustus De Morgan (1806-1871) was a British mathematician and logician who was Professor of Mathematics at the University of London (now, University College) from 1828 to 1866. A prolific but not highly original mathematician, De Morgan devoted much of his energies to the rather different field of logic. In his Formal Logic (1847) and a series of papers "On the Syllogism" (1846-1862), he attempted with great ingenuity to reformulate and extend the tradi• tional syllogism and to systematize modes of reasoning that lie outside its boundaries. Chief among these is the logic of relations. De Mor• gan's interest in relations culminated in his important memoir, "On the Syllogism: IV and on the Logic of Relations," read in 1860. De Morgan made important contributions to the study of the categorical syllogism, and he invented many novel notations and rules for this purpose. The sheer variety of his approaches to the syllogism give his work an almost chaotic quality, despite the systematic motiva• tions behind it. In the course of these investigations, he introduced into logic the concept of the universe of discourse and rediscovered what have come to be called De Morgan's laws-e.g., that the comple• ment of the intersection of two classes is the union of the complements of the two classes. De Morgan's greatest achievement in logic was undoubtedly his discovery of the logic of relations. Some earlier logicians, including Aristotle, had noted the existence of relational arguments, such as, "Every circle is a figure; therefore, anyone who draws a circle draws a figure." It appears, though, that De Morgan was the first logician to sense the pervasiveness of relational arguments and to see their systematic importance for logic. This belief in the importance of relational arguments led De Morgan to create much of the logic of relations as we know it today. He made

vii viii PREFACE the fundamental discovery that relations could be compounded so that, for instance, one can compound the relations brother and parent to get their relative product, brother of a parent-i.e., uncle. Once this and other forms of relational composition are introduced, a vast array of inferences comes into view. Among them is the inference which De Morgan called Theorem K, which states that if the product of a relation L with a relation M is included in a relation N, then the product of the complement of N with the converse of M is included in the complement of L. This means that the statement "Every brother of a parent of a person is an uncle of that person," implies "Every non-uncle of a child of a person is a non-brother of that person." De Morgan was also the first person to study the properties of relations, such as transitivity and symmetry (which he called con• vertibility). Once again, complex logical inferences abound. De Mor• gan notes, for instance, that the transitivity of the ancestor relation implies that "Among non-descendents are contained all ancestors of non-descendents. " Such inferences are hardly trivial, and De Morgan's memoir on the logic of relations is filled with similar examples. We must admire not only his ingenuity in constructing the logic of relations but also his ear for valid inferences. The logic of relations has long since been incorporated into quantifi• cation theory, but it remains of interest in its own right. For De Morgan, it was as if he had discovered a new logical world. After developing this new logic, he proclaimed, "And thus in logic, as in mathematics, the horizon opens with the height gained: generalization suggests detail, which again suggests generalization, and so on ad injinitum"(S4,235). While realizing the great importance of the logic of relations for later developments in logic, it is important to place De Morgan's pioneering work in the context of his own thought. This is one of the main objectives of this book. We will see how De Morgan's logic of relations grew out of two idiosyncratic features of his thought: an analysis of propositions which turned all categorical propositions into relational propositions; and the resulting attempt to work out syllogis• tic logic within the framework of the logic of relations. This framework both motivated and limited De Morgan's logic of relations. Chapter One sets the stage. I first consider some central features of Archbishop Whately'S Elements of Logic (1826), which is often cre- PREFACE ix dited with restoring the study of formal logic in Great Britain. Also included is a discussion of the importance of relational inferences for the logic of geometry and of attempts by Reid, Hamilton and others to deal with such inferences. In Chapters Two through Four, I trace the development of De Morgan's very distinctive relational account of judgment and infer• ence. He moves from what I will call the doctrine of the two copulas to the doctrine of the abstract copula to the bicopular syllogism and the composition of relations. In the process, all judgments, even subject• predicate judgments, are seen as having a relational form. The tradi• tional copula comes to express merely another material relation, no different from, say, "is joined to." As a result, all inferences, even those of the traditional syllogism, are grounded in the logic of rela• tions. De Morgan's radical views on form and matter in logic become clarified only in replying to the criticism that he had gone beyond formal logic. This relational analysis of inference cried out for a logic of relations, which is the subject-matter of De Morgan's 1860 paper. In Chapter Five I outline the main features of this logic, and in Chapter Six I show how he applied it to several analyses of the traditional syllogism. All of this is described in considerable detail, so that the reader can get an idea of how the system is supposed to work. It is also evaluated critically. De Morgan did not develop this logic systematically. In Chapter Eight, I formulate a rigorous version of this logic, providing a full list of assumptions and a logical order for the theorems. Additional theorems are deduced, and the system is augmented in various ways, so that even a surrogate for quantification theory is obtained. It is natural to wonder about the extent to which De Morgan's work as a mathematician was linked to his logic of relations. In Chapter Seven I examine this issue from several points of view. While De Morgan's concern for the nature of mathematical reasoning had only a marginal impact on his logic of relations, his relational approach to judgment and inference could well have been suggested by very general mathematical considerations. But there is no evidence that the 1860 logic of relations was substantially influenced by his study of mathematics. This book grows out of my earlier studies in the history of the logic of relations which were supported by Oberlin College, through sab- x PREFACE batical leaves and a Research Status appointment. I greatly appreciate a fellowship which I received from the American Council of Learned Societies (supported, in turn, by the National Endowment for the Humanities), which made it possible for me to spend the 1981-82 academic year in London examining De Morgan's manuscripts and letters. Most of these papers are housed in the Paleography Room in the Senate House Library of the University of London. I wish to express my deep appreciation to the staff members of the Paleography Room, who were unfailingly courteous and helpful in responding to my innumerable requests for assistance during the year in London. Many other libraries provided access to other portions of De Mor• gan's correspondence, and I acknowledge their help as well. These include: University College of the University of London; the British Library; the University Library and University Archives of Cambridge University; St. John's College and Trinity College, Cambridge Uni• versity; The Bodleian Library, University of Oxford; Trinity College, Dublin; The Royal Society of London; and The Royal Astronomical Society. I am also in the debt of the Oberlin College Library, both for its fine collection of works on 19th century logic and for the assistance of its interlibrary loan staff. Portions of this book have been presented at departmental seminars at the London School of Economics, Chelsea College of the University of London, and the State University of New York at Buffalo; and, at the conference on "The Birth of Mathematical Logic," held in March, 1983, at the State University of New York College at Fredonia. This book has benefited from the discussion which followed each of these presentations. I am indebted to Peter Heath for his excellent edition of De Morgan's logical writings and for his encouragement in the initial stage of this project. It has been a pleasure to discuss topics from this book with John Corcoran, Randall Dipert, Ivor Grattan-Guinness, Calvin Jongsma, and Joan Richards. In faculty seminars at Oberlin College, I have been helped by comments from my colleagues Norman Care, Robert Grimm, David Love, Alfred MacKay, Peter McInerney, and Ira Steinberg. I am especially indebted to John Corcoran for his detailed comments on an earlier draft of this book. Any errors which remain are, of course, my responsibility. I wish to thank the Open Court Publishing Co. for allowing the quotations from their reprint editions of De Morgan's Formal Logic, PREFACE xi edited by A.E. Taylor, and On the Study and Difficulty of Mathe• matics. The quotations from On the Syllogism and Other Logical Writings, edited by Peter Heath, appear with the permission of Rout• ledge. Many thanks go to Karen Barnes and Linda Robinson, and to Candace Shaw and Marian Zelman, for the gracious and efficient way in which they did the word processing needed for the several drafts of this rather complex manuscript. Finally, I am deeply grateful to my wife, Marly, for the support which she has given me over the years. In a project which took considerably longer than expected, her understanding and encourage• ment have been invaluable.