DOCSLIB.ORG

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

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
AUGUSTUS DE MORGAN and the LOGIC of RELATIONS the New Synthese Historical Library Texts and Studies in the History of Philosophy AUGUSTUS DE MORGAN AND THE LOGIC OF RELATIONS 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. -- <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 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. KIuwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper 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.
Load more

Recommended publications
  • George Boole?
    iCompute For more fun computing lessons and resources visit: Who was George Boole? 8 He was an English mathematician 8 He believed that human thought could be George Boole written down as ‘rules’ 8 His ideas led to boolean logic which is Biography for children used by computers today The story of important figures in the history of computing George Boole (1815 – 1864) © iCompute 2015 www.icompute -uk.com iCompute Why is George Boole important? 8 He invented a set of rules for thinking that are used by computers today 8 The rules were that some statements can only ever be ‘true’ or ‘false’ 8 His idea was first used in computers as switches either being ‘on’ or ‘off’ 8 Today this logic is used in almost every device and almost every line of computer code His early years nd 8 Born 2 November 1815 8 His father was a struggling shoemaker 8 George had had very little education and taught himself maths, French, German and Latin © iCompute 2015 www.icompute -uk.com iCompute 8 He also taught himself Greek and published a translation of a Greek poem in 1828 at the age of 14! 8 Aged 16, the family business collapsed and George began working as a teacher to support the family 8 At 19 he started his own school 8 In 1840 he began having his books about mathematics published 8 In 1844, he was given the first gold medal for Mathematics by the Royal Society 8 Despite never having been to University himself, in 1849 he became professor of Mathematics at Queens College Cork in Ireland 8 He married Mary Everett in 1855 8 They had four daughters between 1956-1864
    [Show full text]
  • Explanations
    © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. CHAPTER 1 Explanations 1.1 SALLIES Language is an instrument of Logic, but not an indispensable instrument. Boole 1847a, 118 We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. De Morgan 1868a,71 That which is provable, ought not to be believed in science without proof. Dedekind 1888a, preface If I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems whilst the root drives into the depthswx . Frege 1893a, xiii Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. Russell 1903a, 451 1.2 SCOPE AND LIMITS OF THE BOOK 1.2.1 An outline history. The story told here from §3 onwards is re- garded as well known. It begins with the emergence of set theory in the 1870s under the inspiration of Georg Cantor, and the contemporary development of mathematical logic by Gottlob Frege andŽ. especially Giuseppe Peano. A cumulation of these and some related movements was achieved in the 1900s with the philosophy of mathematics proposed by Alfred North Whitehead and Bertrand Russell.
    [Show full text]
  • How Peircean Was the “'Fregean' Revolution” in Logic?
    HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0.
    [Show full text]
  • Mathematical Language and Mathematical Progress by Eberhard Knobloch
    Mathematical language and mathematical progress By Eberhard Knobloch In 1972 the space probe Pioneer 10 was launched, carrying a plaque which contains the first message of mankind to leave our solar system1: The space probe is represented by a circular segment and a rectangle designed on the same scale as that of the man and the woman. This is not true of the solar system: sun and planets are represented by small circles and points. We do not know whether there are other intelligent beings living on stars in the universe, or whether they will even understand the message. But the non-verbal, the symbolical language of geometry seemed to be more appropriate than any other language. The grammar of any ordinary language is so complicated that still every computer breaks down with regards to it. 1 Karl Märker, "Sind wir allein im Weltall? Kosmos und Leben", in: Astronomie im Deutschen Museum, Planeten - Sterne - Welteninseln, hrsg. von Gerhard Hartl, Karl Märker, Jürgen Teichmann, Gudrun Wolfschmidt (München: Deutsches Museum, 1993), p. 217. [We reproduce the image of the plaque from: http://en.wikipedia.org/wiki/Pioneer_plaque ; Karine Chemla] 1 The famous Nicholas Bourbaki wrote in 19482: "It is the external form which the mathematician gives to his thought, the vehicle which makes it accessible to others, in short, the language suited to mathematics; this is all, no further significance should be attached to it". Bourbaki added: "To lay down the rules of this language, to set up its vocabulary and to clarify its syntax, all that is indeed extremely useful." But it was only the least interesting aspect of the axiomatic method for him.
    [Show full text]
  • George Boole. Selected Manuscripts on Logic and Its Philosophy (Science Networks
    George Boole. Selected Manuscripts on Logic and its Philosophy (Science Networks. Historical Studies, vol. 20). Ivor Grattan-Guinness/G´erard Bor- net, eds. Birkh¨auser Verlag: Basel/Boston/Berlin, 1997, e 37.30, ISBN 0- 8176-5456-9, lxiv + 236 pp. Reviewed by Volker Peckhaus Universit¨at Paderborn Kulturwissenschaftliche Fakult¨at Warburger Str. 100, D – 33098 Paderborn E-mail: [email protected] George Boole, the founder of the algebra of logic, did not stop working on logic after having published his seminal An Investigation of the Laws of Thought (1854 ). Among his aims was to write a philosophical comple- ment to his writings in the algebra of logic, but he died before he was able to finish it. All this was known already to his contemporaries and although leading authorities like Augustus De Morgan, Bertrand Russell, Philip E. B. Jourdain and Louis Couturat were involved in attempts to publish Boole’s posthumous logical papers, only bits and pieces from his logical manuscripts have been published up to now. This insufficient situation comes now to an end with this scholarly edition by Ivor Grattan-Guinness and G´erard Bornet. The editors have compiled a selection of about 40 % of Boole’s unpublished manuscripts on logic “from a scattered and chronologically unclear textual universe” (p. xxv). The bad condition of the manuscripts is due to Boole’s working style. He usually started each time from scratch producing not really substantial texts but notes and further work-plans. This is explained by Boole himself when he reported his plans to Augustus De Morgan in March 1859 (quoted p.
    [Show full text]
  • An Introduction to Symbolic Logic
    An Introduction to Symbolic Logic Guram Bezhanishvili and Wesley Fussner∗ 1 Introduction This project is dedicated to the study of the basics of propositional and pred- icate logic. We will study it based on Russell and Whitehead's epoch making treatise Principia Mathematica [9]. Published in three volumes between 1910 and 1913, Principia was a culmination of work that had been done in the preceding century on the foundations of mathematics. Since the middle of the nineteenth century, such thinkers as George Boole (1815{1864), Augustus De Morgan (1806{1871), Charles Sanders Peirce (1839{1914), Ernst Schr¨oder (1841-1902), Gottlob Frege (1848{1925), and Giuseppe Peano (1858{1932) had been developing axiomatic bases for logic and the foundations of math- ematics. This research program found its culmination in Principia, which had a tremendous influence on the development of logic and the foundations of mathematics in the twentieth century. Logic is a branch of science that studies correct forms of reasoning. It plays a fundamental role in such disciplines as philosophy, mathematics, and computer science. Like philosophy and mathematics, logic has ancient roots. The earliest treatises on the nature of correct reasoning were written over 2000 years ago. Some of the most prominent philosophers of ancient Greece wrote of the nature of deduction more than 2300 years ago, and thinkers in ancient China wrote of logical paradoxes around the same time. However, though its roots may be in the distant past, logic continues to be a vibrant field of study to this day. Modern logic originated in the work of the great Greek philosopher Aris- totle (384{322 bce), the most famous student of Plato (c.427{c.347 bce) and ∗Department of Mathematical Sciences; New Mexico State University; Las Cruces, NM 88003, USA; [email protected]; [email protected].
    [Show full text]
  • Download Article (PDF)
    Nonlinear Engineering 2015; 4(1): 39–41 Majid Khan*, Tariq Shah, and Ali Shahab A brief description about the fathers of computer and information sciences Abstract: In this article, we are mainly presenting a trib- 1985) [1]. We will utilize both this book [1] and the life story ute to the fathers of computer and information sciences, composed by O’connor and Robertson for the Mactutor George Boole and Claude Elwood Shannon with their hard- History of Mathematics document [2]. Also, we have taken ships and achievements. This piece of writing also elabo- some of historical memories from the book of Thomas [3]. rates the applications of George Boole’s and Claude Shan- George Boole was born in November 1815 in Lincoln, non’s works in dierent disciplines. England. His father was an ordinary tradesman. He gave Boole his rst mathematics lessons and planted in him the Keywords: George Boole; Computer Science; Claude Shan- passion of learning. A family friend ,who was a local book- non; Information Science seller, helped him learn basic Latin. By the age of 12, Boole was beginning to translate Latin poetry. By 14, the adoles- DOI 10.1515/nleng-2014-0029 cent Boole was uent in French, German, and Italian as Received December 31, 2014; accepted February 27, 2015. well. His love for poetry and novels was remarkable. His capabilities in higher maths did not indicate un- til he was 17 years of age . He read his maiden progressed arithmetic book, in particular Lacroix’s Dierential and 1 The Contributions of George Integral Calculus. Since his father’s business zzled, he Boole was compelled to earn his bread to look after his family.
    [Show full text]
  • 2016 Clifton Henry.Pdf
    The Foundations of Mathematics History, Concepts, and Applications in Higher Level Mathematics by Clifton Henry A CAPSTONE PROJECT submitted in partial fulfillment of the requirements for the acknowledgement Honors Distinction Mathematics School of Engineering, Mathematics, and Science TYLER JUNIOR COLLEGE Tyler, Texas 2016 When people hear the word mathematics, some think numbers and others think solving for x and y. In all truth, the field of mathematics is more than numbers and equations. Mathematics was founded based off the principle of counting in 50,000 B.C.E. Mathematical contributions and ideas increased from the Babylonian Empire in 2000 BC to the 18th century, where mathematicians and philosophers developed ideas in algebra, geometry and calculus. It can be argued that mathematical contributions in the 19th century shaped the field of mathematics and set in place the true foundations of mathematics. In the 19th century, mathematical logic was founded, followed by the development of set theory, and then came the concept of mathematical function. Logic, set theory, and function are some of the foundations of mathematics, and the three revolutionized the field of mathematics as a whole. Mathematical fields, such as topology, abstract algebra, and probability rely heavily on the three foundations. Instead of being seen as numbers and equations, mathematics is the field of logic and proof that is based off the mathematical foundations of logic, set theory, and function. It is important to know how the ideas were developed and put to use in order for mathematics to have its meaning and significance. After understanding how the foundations were developed, it is important to understand how they apply in advanced mathematical fields such as topology, abstract algebra, and probability.
    [Show full text]
  • On the Interplay Between Logic and Philosophy : a Historical Perspective Séminaire De Philosophie Et Mathématiques, 1992, Fascicule 7 « Logique Et Philosophie », , P
    Séminaire de philosophie et mathématiques YEHUDA RAV On the Interplay between Logic and Philosophy : a Historical Perspective Séminaire de Philosophie et Mathématiques, 1992, fascicule 7 « Logique et philosophie », , p. 1-21 <http://www.numdam.org/item?id=SPHM_1992___7_A1_0> © École normale supérieure – IREM Paris Nord – École centrale des arts et manufactures, 1992, tous droits réservés. L’accès aux archives de la série « Séminaire de philosophie et mathématiques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON THE INTERPLAY BETWEEN LOGIC AND PHILOSOPHY: A HISTORICAL PERSPECTIVE Yehuda RAV* ABSTRACT In this historical essay, we examine the reciprocal influences of philosophical doctrines and logic, their interrelations with language, and the place of mathematics in these developments. Our concern in this essay is the interplay between logic and philosophy. The spectrum of philosophical traditions and topics is wide, ranging from inspirational, aphoristic, and poetic wisdom-searching philosophies to stark anti- metaphysical logical positivism. However, logic has flourished only within those philosophical traditions in which critical discussion and debates played a major role. "Formal logic,
    [Show full text]
  • John Venn on the Foundations of Symbolic Logic: a Non-Conceptualist Boole
    UvA-DARE (Digital Academic Repository) "A terrible piece of bad metaphysics"? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic Verburgt, L.M. Publication date 2015 Document Version Final published version Link to publication Citation for published version (APA): Verburgt, L. M. (2015). "A terrible piece of bad metaphysics"? Towards a history of abstraction in nineteenth- and early twentieth-century probability theory, mathematics and logic. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:25 Sep 2021 chapter 7 John Venn on the foundations of symbolic logic: a non-conceptualist Boole 0. Introduction: Venn,
    [Show full text]
  • Augustus De Morgan (1806–1871)
    AUGUSTUS DE MORGAN (1806–1871) AUGUSTUS DE MORGAN was born in the month of June at Madura in the presid- ency of Madras, India; and the year of his birth may be found by solving a conundrum ¢ ¡ proposed by himself, “I was ¡ years of age in the year .” The problem is indeterm- inate, but it is made strictly determinate by the century of its utterance and the limit to a man’s life. His father was Col. De Morgan, who held various appointments in the service of the East India Company. His mother was descended from James Dodson, who computed a table of anti-logarithms, that is, the numbers corresponding to exact logarithms. It was the time of the Sepoy rebellion in India, and Col. De Morgan re- moved his family to England when Augustus was seven months old. As his father and grandfather had both been born in India, De Morgan used to say that he was neither English, nor Scottish, nor Irish, but a Briton “unattached,” using the technical term ap- plied to an undergraduate of Oxford or Cambridge who is not a member of any one of the Colleges. When De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England, and her son received his elementary edu- cation at various schools of no great account. His mathematical talents were unnoticed till he had reached the age of fourteen. A friend of the family accidentally discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses, and explained to him the aim of Euclid, and gave him an initiation into demonstration.
    [Show full text]
  • George Boole, John Venn and CS Peirce
    Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce Janet Heine Barnett∗ 9 March 2013 1 Introduction On virtually the same day in 1847, two major new works on logic were published by prominent British mathematicians. Augustus De Morgan (1806{1871) opened his Formal Logic with the following description of what is known today as `logical validity' [6, p. 1]: The first notion which a reader can form of Logic is by viewing it as the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments, so that the conclusion may contain no inaccuracy which was not previously asserted in the premises. It has so far nothing to do with the truth of the facts, opinions, or presump- tions, from which an inference is derived; but simply takes care that the inference shall certainly be true, if the premises be true. Whether the premises be true or false, is not a question of logic, but of morals, philosophy, history, or any other knowledge to which their subject-matter belongs: the question of logic is, does the conclusion certainly follow if the premises be true? The second of these new nineteenth century works on logic was The Mathematical Analysis of Logic by George Boole (1815{1864). Like De Morgan, Boole sought to stretch the boundaries of traditional syllogistic1 logic by developing a general method for representing and manipulating all logically valid inferences or, as DeMorgan described it to Boole in a letter dated 28 November 1847, to develop `mechanical modes of making transitions, with a notation which represents our head work' [22, p.
    [Show full text]