Publications des Archives Henri-Poincaré Publications of the Henri Poincaré Archives

Textes et Travaux, Approches Philosophiques en Logique, Mathématiques et Physique autour de 1900 Texts, Studies and Philosophical Insights in , and Physics around 1900 Éditeur/Editor: Gerhard Heinzmann, Nancy, France Louis Couturat−Traite´ de Logique algorithmique

Edited by Oliver Schlaudt and Mohsen Sakhri, with an introduction and annotations by Oliver Schlaudt

Birkhäuser Editors: Dr. Oliver Schlaudt Dr. Mohsen Sakhri Universitat¨ Heidelberg Laboratoire d’ Histoire des Sciences et de Philosophisches Seminar Philosophie - Archives Henri Poincare´ Schulgasse 6 UMR 7117 CNRS - Nancy-Universite´ 69117 Heidelberg Universite´ Nancy 2 Germany 91 avenue de la Libe´ ration - BP 454 54001 Nancy Cedex France

ISBN 978-3-0346-0410-9 e-ISBN 978-3-0346-0411-6 DOI 10.1007/978-3-0346-0411-6

Library of Congress Control Number: 2010927519

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Springer Basel AG is part of Springer Science+Business Media www.birkhauser.ch Preface

Louis Couturat (1868–1914) was an outstanding intellectual of the turn of the nineteenth to the twentieth century. He is known for his work in the of mathematics, for his critical and editorial work on Leibniz, for his attempt to popularise modern logic in France, for his commitment to an international auxiliary language, as well as for his extended correspondence with scholars and mathematicians from Great Britain, the United States, Italy, and Germany. From his correspondence we know of four unpublished manuscripts on logic and its history, which were largely complete and some of which must have been of considerable size. We publish here for the first time in a critical edition the only one of these manuscripts that has been rediscovered: the Traité de Logique algorithmique, presumably written in the years 1899–1901. It is a highly interesting document of the academic reception and popularisation of symbolic logic in France. It provides evidence of the discussions and controversies which accompanied the creation of logic as a new branch of science. At the same time it completes the picture of Couturat’s work, which has been opened up to systematic study by the publication of important parts of his correspondence during the last decade. We append the article on Symbolic Logic of 1902 which Couturat wrote in collaboration with Christine Ladd- Franklin for Baldwin’s Dictionary of Philosophy and Psychology. This article, as now becomes evident, is a sort of résumé of the Traité; at the same time it points the way to Couturat’s Algèbre de la Logique of 1905. It thus helps to situate the Traité in Couturat’s œuvre. The same purpose is served by the second document appended, a short part of Couturat’s report of the first International Congress of Philosophy, which took place in Paris in 1900. This report documents Couturat’s reception of Platon Poretsky, whose work was of considerable importance for the outline of L’Algèbre de la Logique and marks the main difference between this later work and the Traité. – Since history of modern logic already attracts a lot of attention, the introduction focusses on Couturat and his perspective on modern logic in order to provide information the reader may lack. Finally a critical apparatus should help the reader to find his way through the Traité and to understand its genesis.

Acknowledgements

We are much obliged to the Laboratoire d’Histoire des Sciences et de Philosophie – Archives Henri Poincaré (UMR 7117 CNRS / Nancy-Université), especially to Gerhard Heinz- mann, for the generous intellectual and material support which rendered possible the realisation of this project, as well as to the CDELI (Centre de documentation et d’étude sur la langue internationale) at the municipal library of La Chaux-de-Fonds (Switzerland) for granting the printing licence for the manuscript. We are furthermore indebted to vi Preface a number of colleagues for their help and advice which were essential for our project. We would like to thank Paolo Mancuso (Berkeley), Peter McLaughlin (Heidelberg), Philippe de Rouilhan (Paris), Fabien Schang (Nancy/Dresden), Anne-Françoise Schmid (Lyon/Paris), Christian Thiel (Erlangen), and Paul Ziche (Utrecht). Contents

I Introduction 1

I. Presentation of the Manuscript 3

II. Biographical Note 3

III. Dating of the Manuscript 5

IV. Origin and Meaning of the Term “Logique algorithmique” 7

V. Characterisation of the Manuscript 9

VI. Couturat’s Interest in 11

VII. Editorial Policy 32

VIII. Editorial Symbols in the Presentation of the Text 33

II Transcription of the Manuscript 35

Tome I 37

I. Définitions et notations : A. Logique des concepts 37

II. Définitions et notations : B. Logique des propositions 49

III. Principes 63

IV. Lois de la multiplication et de l’addition 75

V. Lois de la négation 93

VI. Développement des fonctions 107

VII. Théorie des équations 121

Appendice II. Sur les opérations inverses : Soustraction et division 145

Tome II 157

VIII. Théorie des inégalités 157

IX. Calcul des propositions constantes 185 viii Contents

X. Calcul des jugements variables (ou des probabilités) 205

XI. Comparaison avec la Logique classique 225

XII. Conclusions 241

Editor’s Appendix: 251

A: Louis Couturat and Christine Ladd-Franklin: Symbolic Logic, 1902 251

B: Couturat on Schröder and Poretsky, on the Ist International congress 258

of Philosophy, Paris 1900

III Critical Apparatus 261

Variants and Annotations 263

Table of Correspondence with L’Algèbre de la Logique 294

List of Signs and Abbreviations 295

Table of Figures 296

Bibliography 297

Index Nominum 311

Index Rerum 312 Part I

Introduction I. Presentation of the Manuscript

The present text is a transcription of Louis Couturat’s manuscript entitled “Traité de Logique algorithmique”. Of this manuscript, only the tenth chapter has been published – posthumously in 1917. The present edition of the entire text is based on the only known copy, an undated handwritten version preserved at the CDELI (Centre de documentation et d’étude sur la langue internationale) at the library of La Chaux-de-Fonds in Switzerland. It was found there by Mohsen Sakhri in 2003 in the course of his research on Couturat’s work on international auxiliary languages. With dismay we must note that according to the CDELI the remaining papers of Couturat, including the manuscript of the Traité, have in the meantime been seriously damaged and partly destroyed by water during construction work. The only version available for study thus is a photocopy of the manuscript kept at the Archives Henri Poincaré at Nancy University (France). The manuscript consisted of two volumes of about 200 sheets each. The sheets are mostly used on one side; if used, the versos contain additional notes or corrections. Three handwritings can be distinguished: The rather uniform main text is in Couturat’s clear and easily legible handwriting; additional notes in different ink were presumably added by Couturat himself; and some notes that correspond to the 1917 edition of chapter X of the manuscript are in a different hand. On the last two pages of the second volume are placed a table of contents and a table of figures. The table of contents additionally indicates three appendices: I. Sur les signes adoptés, II. Soustraction et division (15 p.), and III. Solutions générales symétriques. However only one of them – on the logical operations of division and substraction – is given at the end of the first volume.

II. Biographical Note

Alexandre-Louis Couturat was an outstanding intellectual of the French Third Republic. He was born on January 17, 1868 in Ris-Orangis near Paris and died on August 3, 1914, on the eve of the First World War, when on the way from Paris to his country house in Bois le Roi he was involved in a fatal traffic accident with an army transporter during the mobilisation of the French army.1 Couturat is today still known as a Leibniz scholar and as an important figure in the propagation of modern logic in France. Couturat’s most creative period as a philosophical writer as well as a tireless propagator of the international auxiliary language in the name of peace and internationalism coincided with the impact of the Dreyfus Affair on the French society. He expressed his pacifist convictions – coupled with profound knowledge in the history of philosophy – in a

1The main sources of his biography are André Lalande’s synopsis L’Œuvre de Louis Couturat, published in 1914 in the Revue de métapysique et de morale, and the obituary Couturat’s friend Louis Benaerts published in 1915 in L’annuaire de l’Association amicale de secours des Anciens Elèves de l’Ecole Normale Supérieure. These articles have been reprinted together with Arnold Reymond’s short obituary from 1915, but this booklet is difficult to find today. There are two monographs on Couturat, one from the Argentinean historian of mathematics Claro C. Dassen (1873–1941) from 1939, and a more recent one from Ubaldo Sanzo, published in 1991. Dassen’s book, reviewed by Quine in 1940, benefits from the author’s correspondence with Couturat from January 1902 until at least 1911 on the subject of Ido.

O. Schlaudt, M. Sakhri (eds.), Louis Couturat – Traité de Logique algorithmique, Publications des Archives Henri Poincaré, DOI 10.1007/978-3-0346-0411-6_1, © Springer Basel AG 2010 4 Introduction polemic over Kant’s notion of war with the conservative writer Ferdinand Brunetière, a key figure of the Dreyfus Affair. After a very successful academic education in philosophy at the Ecole Normale Supérieure (ENS) 1887–1890 and then in mathematics at the ENS as well as at the Faculté des sciences where he attended lectures given by Jules Tannery, Picard, Jordan and Poincaré in 1890–1892, he was appointed as a lecturer in 1894 at the University of Toulouse where he stayed for one year. 1896 saw the defence of his two theses prepared during an academic leave of absence of two years: the principle thesis De l’infini mathématique which marks an important – though not uncriticised1 – step in the philosophical reception of Cantor in France, and the secondary thesis in Latin De Platonis mythicis.2 After a further year of studies in physics at the Sorbonne, Couturat accepted a position as a lecturer at the University of Caen. In Toulouse, Couturat had taught Plato, and still in 1900 at the First International Congress of Philosophy, he presented a paper, Le système de Platon exposé dans son développement historique. But when he went to Caen in October 1897 he devoted his lectures and research activities almost exclusively to philosophy of mathematics and to logic. About this time he also started his important exchange of letters with (1896) and (1897).3 After two years in Caen he successfully applied for a leave of absence and went back to Paris, where he organised the logic section of the first International Congress of Philosophy, which took place in August 1900. Aside from his unexpected replacement of Bergson at the Collège de France in 1905/06, this leave marked the end of his teaching activity.4 In 1900 and 1901 he spent some time in Hanover where he sifted through Leibniz’s unpublished manuscripts on logic, the existence of which had been indicated to him by the Italian mathematician Giovanni Vacca. This work yielded the book La Logique de Leibniz d’après des documents inédits (1901) and the Opuscules et fragments inédits de Leibniz (1903). The following years saw the publications of his popularisation of Russell’s Principles of Mathematics, Les principes des mathématiques (1904), and of his short textbook L’Algèbre de la Logique (1905) which earned him a reputation as an important representative of modern logic in France. During the years 1905/06 he had a famous quarrel with his former teacher Henri Poincaré on the rôle of intuition in logic, which underlines Couturat’s turning away from the then dominant Kantian philosophy. Already at the first International Congress of Philosophy there occured for the first time in Couturat’s writings the idea of an international auxiliary language. His commitment to Ido increasingly absorbed Couturat. It gave rise to dozens of articles and works. The most important of them is probably the Histoire de la langue

1Cf. Tannery 1897 and Dugac 1983. 2A comprehensive report of Couturat’s defence before a jury composed of his director Boutroux, Jules Tannery, Evellin, Séailles and Egger for De l’infini, and of Brochard, Croiset, Bouché-Leclercq and Decharme for De Platonis mythicis was published in RMM, Supplément du numéro de septembre 1896, pp. 13-20. 3These important correspondences have meanwhile been edited, cf. Schmid 2001 as well as Luciano and Roero 2005; in the following we refer to the former by indicating the date of the letter prefixed with a C for Couturat and a R for Russell, the volume and page number, and to the latter by indicating the date and page number. For a survey of the published part of Couturat’s letters see part B of the bibliography. 4In 1902 he applied unsuccessfully for the chair of general history of science at the Collège de France, cf. Paul 1976, p. 393. Introduction 5 universelle (1903) written together with Léopold Leau. They emphasised the necessity of a universal language as a linkage of nations and of the scientific community. From 1900 to 1908, Couturat held the office of treasurer of the Délégation pour l’adoption d’une langue auxiliaire internationale; in 1908 he was secretary of the delegation’s committee. In the same year he founded the journal Progreso, and from 1910 on he was secretary of the Akademio di la linguo internacino Ido. This engagement led from 1908 on to a more serious interest in and also resulted in some articles on the relation of language and logic which are of distinctive interest.

III. Dating of the Manuscript

The Traité de Logique algorithmique is, besides Logique mathématique, the Manuel de logistique and the Histoire de la Logistique, one of four unpublished manuscripts Couturat mentioned in his correspondence.1 It is the only one that has been rediscovered. It must however have been accessible during the first years after Couturat’s unexpected death in , since its tenth chapter was published posthumously in 1917 in the Revue de Métaphysique et de Morale. The anonymous editor – presumably André Lalande, with whom Couturat collaborated for the Vocabulaire technique et critique de la philosophie and who in his 1915 study L’Œuvre de Couturat had already proved to be acquainted with Couturat’s remaining papers, including his correspondence with Russell – adjoined the following annotation:

The present article is extracted from an unfinished treatise on algorithmic logic written by Couturat perhaps a long time ago, definitely before 1902, perhaps a long time before that. [. . . ] After this period he abandoned publication; he completely revised this first version in order to transform it into a Manual of which, as we hope, will be published soon. (1917b, p. 291)

1For references to the Traité see the information given in this section. In C 30.08.04 Couturat reports the completion of his Logique mathématique for the publisher Naud, with whom he had agreed upon this project in January 1904 (cf. letter to Peano, p. 62, and C 11.02.1904; Couturat sent a résumé of this book to Peano in July 1904, p. 69. In October 1904 he still waited for page proofs, which shows that he indeed had delivered the completed manuscript, cf. p. 77). It purported to present Peano’s system completed by Russell’s logic of relations and some methodological considerations. For reasons not mentioned in the correspondence the book was never published. In December 1904, after having finished L’Algébre de la Logique, Couturat announced his project of the Manuel de Logistique for the publisher Alcan (C 18.12.1904, II/453). In January he reported his ongoing work to Russell (C 22.01.1905) and to Peano (05.01.1905, p. 85). In July 1905 he announced to Peano (p. 89) the upcoming completion of the book, containing his two articles already published in the journal Enseignement mathématique (1900f and 1900g). One chapter of the Manuel has been published posthumously in 1917 (1917a), which gives altogether a quite precise idea of the book. Couturat, having made good progress with this project, stopped it because of his unexpected call to the Collège de France as a substitute for (cf. his letters to Russell C 10.11.1905, II/546, and to Peano, p. 93). At the Collège de France he lectured on the history of modern logic and took up again a book on this subject, the Histoire de la Logistique, the fourth manuscript, which he mentioned for the last time in his correspondence with Russell in C 22.07.1906 II/614, and in his correspondence with Peano in October 1906 (p. 120). 6 Introduction

The available information, in particular Couturat’s correspondence with Bertrand Rus- sell and Giuseppe Peano published in 2001 and in 2005 respectively, confirms this dating. Indeed Couturat was charged with giving lectures at the university of Caen (Normandy) from November 1897 on.1 For his second academic year in Caen he announced a course “Studies of the diverse systems of algorithmic logic: Boole, de Morgan, Stanley Jevons, Delbœuf, Peirce, MacColl, Schröder, Peano, etc. On the relations of mathematics and logic; on the scope of the mathematical method. The Idea of universal algebra (White- head).”2 This course, mainly inspired by a preliminary reading of Whitehead’s Treatise on Universal Algebra3, lies at the origin of the present manuscript. In September 1899, one year later, Couturat wrote to Russell: “I concentrate on writing my course on algorithmic logic which I intend to publish next year” (C 05.09.1899, I/135). A little more than one year later he confirms: “I still intend to write an Algorithmic Logic in two volumes” (C 03.01.1901, I/219). The consequences to be drawn from this last utterance are not unambiguous: On the one hand, the specification of the two volumes seems to refer to our manuscript; on the other hand the expressed intention to write the treatise in the future contradicts the letter of September 1899 reporting the already initiated project of writing up the lecture notes. Most probably the production of the Traité consisted of two major parts, the relatively homogeneous collection of his lecture notes and, as the annotations in the manuscript also suggest, a later rough proof-reading. The first part may even just predate the more careful reading of Whitehead, since one finds his name subsequently added four times, while the original version does not mention it. Of course it is possible, too, that the additional annotations originate in the preparatory work for L’Algèbre de la Logique in 1904. There is also another curious detail which confirms the dating of the manuscript: In the Traité Couturat still used the spelling “Leibnitz”4, abandoned in his La Logique de Leibniz (1901a, p. vii, note 1). In his cor- respondence with Russell, one can localise this change in the spelling quite accurately between November 1900 and January 1901 (cf. C 05.11.1900 and C 03.01.1901). For reasons that are not evident from (the published parts of) his correspondence, Couturat abandoned the plan to publish the Traité de logique algorithmique sometime after January 1901. He in any case interrupted his work on modern logic in order to complete his study La Logique de Leibniz, published in 1901, as well as his Opuscules et fragments inédits de Leibniz, finally published in 1903.5 It seems that the only work on logic he published in the meantime is the article Symbolic Logic for Baldwin’s Dictionary of Philosophy and Psychology, written in collaboration with Christine Ladd-Franklin (cf. Appendix A, p. 251). This article, as can be seen now, is in fact a précis of our Traité,

1Letter to Peano, p. 7; letter to Russell, C 09.11.1897, I/69. 2RMM, Supplément Septembre 1898, p. 2; cf the letters to Peano and to Bettazzi, p. 9 and p. 190. 3In C 08.07.1898 he reports having read the introduction only; the complete reading for his review in RMM followed in 1899 (C 03.02.1899 and C 05.09.1899). 4In a letter to Peano, Couturat explains in June 1899, p. 24: «Pour Leibnitz, j’ai adopté l’orthographe que préconise M. Boutroux, pour des raisons d’analogie, tout en sachant que l’auteur et les éditeurs écrivaient Leibniz. Mais je n’y tiens pas autrement.» 5In a letter to Peano he later confirmed this break during 1901, 1902 and 1903 (p. 85). – It should be remarked that this work on Leibniz again was partly inspired by the reading of Whitehead (and thereafter of Grassmann); cf. C 13.05.1900. Introduction 7 differing from it only in some minor details, like the propositional interpretation of “o” and “1” and the explicit discussion of the Ladd-Franklin-formula. In 1902 Couturat was looking forward to returning to his studies on logic after having finished the Opuscules (C 08.07.1902, I/282). In the middle of 1903 we find Couturat again working on logic, however no longer on Boole and Schröder, whose works were crucial to the Traité, but on Peano and Russell – a “complete change” in his own words (C 10.06.1903). In 1904 he announces his popularisation of Russell’s Principles of Mathematics of 1903, and at the same time also a booklet on mathematical logic, i. e. in particular on Peano’s symbolism (C 11.02.1904). Nevertheless he returned in 1904 to algorithmic logic and wrote his book on the Algebra of Logic. At this time algebraic logic was of course no longer an end in itself for Couturat, but a didactical means: the book was supposed first of all to facilitate the book on mathematical logic (C 15.11.1904, C 22.01.1905). Mathematical logic, i. e. Peano’s symbolism, was then considered by Couturat as more fundamental than algebraic logic, though admittedly less practicable (C 07.05.1905). Couturat reported the completion of L’Algèbre de la logique in December 1904. It contains a considerable part of the Traité in a compressed form (cf. the table of correspondence, p. 294). There are only a few points which cannot be found in the Traité. These mainly concern issues with which Couturat may have become acquainted at the International Congress of Philosophy in Paris in 1900, where Russell, Schröder, MacColl, Peano, Johnson, and in particular Platon Poretsky were present. Primarily the writings of the latter had a considerable influence on L’Algèbre de la logique (cf. Appendix B, p. 258). Couturat however suppressed the exposition of the calculus of variable propositions, i. e. of probabilities (ch. X), as well as the detailed comparison to classical logic (ch. XI). In the conclusion of L’Algèbre de la logique he declared: The foregoing exposition is far from being exhaustive; it does not pretend to be a complete treatise on the algebra of logic, but only undertakes to make known the elementary principles and theories of that science. (1905a, § 60, p. 94) Since he seems to allude to the present manuscript in speaking of a “traité complet d’Algèbre de la Logique”, we feel all the more justified in publishing it, though the completion of L’Algèbre de la Logique might finally have contributed to Couturat’s decision to abandon the Traité.

IV. Origin and Meaning of the Term “Logique algorith- mique”

Algebraic or algorithmic logic resulted from the application of the mathematical method to logic, achieved particularly in the work of Boole and Schröder.1 By the term “logic” must be understood first and foremost classical logic, i. e. the theory of the syllogism

1For a short account of the history of algebraic logic see Jourdain’s introduction to the English translation of L’Algèbre de la Logique, published in 1914; for a detailed study cf. Grattan-Guinness 2000.