Table Of ContentAUGUSTUS 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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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·P ublication Data
Merrill, Daniel D. (Daniel Davy)
Augustus De Morgan and the lOglC of relations / Daniel D. Merril.
p. cm. -- <The New synthese historical library; 38)
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
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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,
