Augustus De Morgan and the Logic of Relations
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 ext
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The New Synthese Historical Library Texts and Studies in the History of Philosophy VOLUME 38
Series Editor: NORMAN KRETZMANN, Cornell University
Associate Editors: DANIEL ELLIOT GARBER, University of Chicago SIMO KNUUTTILA, University of Helsinki RICHARD SORABJI, University of London
Editorial Consultants: JAN A. AERTSEN, Free University, Amsterdam ROGER ARIEW, Virginia Polytechnic Institute E. JENNIFER ASHWORTH, University of Waterloo MICHAEL AYERS, Wadham College, Oxford GAIL FINE, Cornell University R. J. HANKINSON, University of Texas JAAKKO HINTIKKA, Boston University, Finnish Academy PAUL HOFFMAN, Massachusetts Institute of Technology DAVID KONSTAN, Brown University RICHARD H. KRAUT, University of Illinois, Chicago ALAIN DE LIBERA, Ecole Pratique des Hautes Etudes, Sorbonne DAVID FATE NORTON, McGill University LUCA OBERTELLO, Universita degli Studi di Genova ELEONORE STUMP, Virginia Polytechnic Institute ALLEN WOOD, Cornell University
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. -- Y w>x y>z w>z.
Rather than thinking of this as the concatenation of three inequalities, we construe it instead as the substitution of "w" for "x" and of "z" for "y" in the first premise. This algebraic analogy provides a natural transition to the main problem with De Morgan's use of the substitutional model. In the algebraic case, no problems arise so long as we have simple inferences such as (3.55). But the full use of substitution in De Morgan's dictum is strictly analogous to the substitution of unequal terms in all kinds of inequalities; and this is fraught with peril, for sometimes it works and sometimes it does not. Take the inference, (3.56)
a - (x - b) > 5 w>x a-(w-b»5
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CHAPTER III
Is this inference valid? It is easy to perform calculations to determine whether this substitution holds (it doesn't), but the result would not be something that can be stated as a simple rule of substitution. When "x" is the full term in an inequality, there is no problem in framing a rule to govern substitutions from further inequalities in which it enters as a full term; but this does not hold when we allow the "x" to occur in complex terms containing other letters. An analogous problem arises with De Morgan's dictum. When it is applied to traditional syllogistic arguments, no problem arises. But when it is applied to parts of complex terms, serious difficulties ensue. There is a certain irony in this, for it is precisely the case of complex terms which provides the real justification for framing the new dictum in terms of substitution. The predicational language of the old dictum would have sufficed for the traditional syllogism; and it is only the need to operate on sub-parts
