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De Morgan's Logic DE MORGAN'S LOGIC Michael E. Hobart and Joan L. Richards 1 INTRODUCTION: THE SYMBOLIC TURN IN LOGIC Looking back on a more than a decade of work in logic, Augustus De Morgan (1806-1871) remembered that from the beginning "in my own mind I was facing Kant's assertion that logic neither has improved since the time of Aristotle, nor of its own nature can improve, except in perspicuity, accuracy of expressions and the like" .1 For Immanuel Kant, writing at the very end of the eighteenth century, the longevity of Aristotle's logic showed that the Greek had essentially gotten it right; for De Morgan it was evidence of stagnation. From his position in the progressive nineteenth century, De Morgan set out in the 1840s to move the study of logic forward by building on the foundation Aristotle had laid. In so doing, De Morgan brought a very specific view of progress to his logical enterprise. "Every science that has thriven has thriven upon its own symbols: logic, the only science which is admitted to have made no improvements in century after century, is the only one which has grown no new symbols. ''2 De Morgan's logical program, then, entailed creating a set of symbols that would show him the way to move logic beyond its Aristotelian base into an ever-expanding future. For many, De Morgan's determination to create an operational set of logical symbols has marked him as a pioneer of the symbolic logic of the late nineteenth and early twentieth centuries, and he is routinely credited with the laws that bear his name and with the logic of relations. Beyond this appreciation, the rest of his more than twenty years of logical effort is generally seen to have little im- portance in the development of logic; most modern readers shrug it off as odd, clumsy, or old fashioned. Thus dismissing the bulk of De Morgan's logic because of its unfamiliarity is myopic at best, however, for it erases the complex dynamic of change that supported the development of modern symbolic logic. And un- derstanding tile roots of twenty-first century symbolic logic requires entering the nineteenth-century world in which they developed. 1Augustus De Morgan, "On the Syllogism, No. III, and on Logic in general", Transactions of the Cambridge Philosophical Society, p. 173, 1858 lOS, p. 74]. (Hereafter, [$3]; see fn. #21 for a complete listing of De Morgan's major logical works and the abbreviations we use in citations.) De Morgan's reference is to Kant's statement: "Logic, by the way, has not gained much in content since Aristotle's times and indeed it cannot, due to its nature. But it may well gain in exactness, definiteness and distinctness." Immanuel Kant, Logic. Translated, with an Introduction by Robert S. Hartman and Wolfgang Schwarz, New York: Dover Publications, Inc., p. 23, 1974. 2[$3, p. 184] [OS, p. 88] (ital. De Morgan's). Handbook of the History of Logic. Volume 4: British Logic in the Nineteenth Century. Dov M. Gabbay and John Woods (Editors) @ 2008 Elsevier BV. All rights reserved. 284 Michael E. Hobart and Joan L. Richards De Morgan was an erudite scholar and prolific writer, with wide-ranging and catholic concerns; his particular combination of intellectual and moral gravity with high-brow and omnipresent wit runs through not only his many published books, but also the more than 700 articles he wrote for the popular Penny Cyclopedia, and the well over a thousand reviews he published in the Athenaeum. He ad- dressed subjects ranging from algebra to gymnastics, from chemical change in the Eucharist to astro-theology, but mathematics, logic, and their histories remained his dominant interests throughout his career. The logic of Augustus De Morgan was primarily rooted in his identity as a Cambridge-educated Englishman of the mid-nineteenth century. Initially, that identity was much more tied up with mathematics than with logic. De Morgan began the serious study of mathematics as a student at Cambridge, where math- ematics was the center of a liberal education designed to educate the Anglican clergy. 3 The pairing of mathematics and theology that supported De Morgan's Cambridge education stemmed from a position developed in John Locke's Essay Concerning Human Understanding. In this magisterial work, Locke had exempted mathematics and theology from the uncertainties of empirical knowledge by plac- ing them in a separate category of demonstrative knowledge. The mathematical focus of De Morgan's Cambridge education is a testament to the enduring power of Locke's coupling of mathematics and theology as exemplars of certain knowledge. In Locke's construction, the certainty of mathematical knowledge was attested to by the extra-ordinarily tight fit between symbols and their positive, spatial meanings. The power of this connection is manifest in geometry, where diagrams play a crucial role in illustrating and clarifying proofs. Yet, by the end of the seventeenth century, a great deal of mathematics did not fit the geometrical model of certainty. In the one hundred and fifty years before Locke published his work, algebra had moved from virtual non-existence to the center of mathematical de- velopment. At the heart of this transformation lay the creation of numerous and recognizably modern conventions of notation -- e.g. such functional symbols as +, --, x, +, =, <, >, V/, oe, ~,dy plus the use of superscripts (2,3...n .... x,y,z...) for powers and of letters for knowns (a, b, c,...) and unknowns (x, y, z,...).4 Algebra's power lay precisely in the way its symbolical exuberances broke through the constraints of meaning that held geometry fast. Throughout the English eighteenth century the problem of algebraic certainty was usually glossed over, but late in the century some of England's political and 3For the nineteenth-century development of the Cambridge mathematical education, see: An- drew Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics, Chicago: University of Chicago Press, 2003. There has been considerable interest in the interaction of mathematics and theology within that education, much of it focused on William Whewell. The articles gathered in Menachem Fisch and Simon Schaffer, William Whewell: A Composite Pot- trait, Oxford: Clarendon Press, 1991, are useful as an initial entr~ into that material. 4For the history of mathematical notation, the magisterial work of Florian Cajori (A History of Mathematical Notations. Two vols. Vol. I: Notations in Elementary Mathematics; vol. II: Notations Mainly in Higher Mathematics, New York: Dover Publications Inc., 1993 [c. 1928- 1929]) remains unsurpassed. De Morgan's Logic 285 religious radicals were determined to bring it to the fore. 5 In 1797, the radical Unitarian William Prend, published A Treatise on Algebra in which he insisted that all mathematical truth -- including algebraic truth -- lay in the close tie between symbol and subject matter. Algebra was just generalized arithmetic, and the subject matter of arithmetic had been known since antiquity to be the numbers with which we count the objects around us. There is no such thing as a negative object, Frend insisted, and with that he eliminated negative numbers from algebra. Prend's was an extreme position -- to follow his lead was essentially to destroy all algebraic developments since the sixteenth century -- but in the Lockean world of Cambridge mathematics his stubborn insistence that the validity of mathematical symbols depended on their interpretations was difficult to refute. Within five years of Prend's negative-denying treatise, the Cambridge tutor Robert Woodhouse tried to answer its extremism with a re-examination of the connection between mathematics and meaning. Woodhouse's writing was too "dif- ficult and perplexed ''6 for his book have a broad impact, but by the 1810s others moved to free English mathematics from the constraints of the meanings-based Lockean approach. In the 1810s the more effective communicators of the short- lived, but highly influential Cambridge Analytical Society (ca. 1812-1817), set out to introduce the symbology of French analysis into England. In 1816 John Herschel, Charles Babbage, and George Peacock translated Sylvestra Francois Lacroix's Elementary treatise on the differential and integral calculus for the ben- efit of their countrymen; by the end of the decade, Cambridge students were abandoning Isaac Newton's conceptually rich fluxional notation in favor of the operational power of Gottfried Leibniz's ~dy notation. This change in symbology carried with it a challenge to English meaning-based mathematics that was deep enough to focus the thinking of a generation of Englishmen. 7 5For a fuller treatment of these developments see Helena M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on New- ton's "Universal Arithmetic", Cambridge: Cambridge University Press, 1997. 6penny Cyclopaedia (1833-43), s.v. [Augustus De Morgan], "Robert Woodhouse". 7For the work of the Analytical Society see Philip C. Enros, "Cambridge University and the adoption of analytics in early Nineteenth-Century England", in H. Mehrtens, et al. (eds.), Social history of Nineteenth-Century mathematics, Boston: Birkhauser, pages 135-48, 1981; "The Ana- lytical Society 1812-1813," Historia Mathematica, 10: 24-47, 1983; Harvey W. Becher, "William Whewell and Cambridge mathematics", Historical Studies in the Physical Sciences, 11: 1-48, 1981; Menachem Fisch, "'The emergency which has arrived': The problematic history of 19th- century British algebra -- a programmatic outline", British Journal for the History of Science, 27: 247-276, 1994. For the influence of French developments in particular on De Morgan's early career, see Maria Panteki, "French 'logique' and British 'logic': on the origns of Augustus De Morgan's early logical inquiries, 1805-1835", Historia Mathematica, 30: 278-340, 2003. Classic studies in which nineteenth-century English algebraists are evaluated for their formal content include: Eric Temple Bell, The Development of Mathematics, New York: McGraw Hill, 1954; L.
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