around 1900) (English and French Edition)

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 Logic, Mathematics and Physics around 1900 Éditeur/Editor: Gerhard Heinzmann, Nancy, France

Louis Couturat − Trait e´ de Logique algorithmique Edited by Oliver Schlaudt and Mohsen Sakhri, with an introduction and annotations by Oliver Schlaudt

Birkhäuser

Editors:

Dr. Oliver Schlaudt Universit¨at Heidelberg Philosophisches Seminar Schulgasse 6 69117 Heidelberg Germany

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

Dr. Mohsen Sakhri Laboratoire d’Histoire des Sciences et de Philosophie - Archives Henri Poincare´ UMR 7117 CNRS - Nancy-Universite´ Universite´ Nancy 2 91 avenue de la Libe´ ration - BP 454 54001 Nancy Cedex France

e-ISBN 978-3-0346-0411-6

Library of Congress Control Number: 2010927519

© Springer Basel AG 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Printed on acid-free paper 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 philosophy 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 From his correspondence we know of four unpublished manuscripts on logic and its history, which were largely complete and some of which must have been of considerable size.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 LaddFranklin 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 Heinzmann, 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 Algebraic Logic

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 Ido 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 1 The 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 Giuseppe Peano (1896) and Bertrand Russell (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 1 Cf.

Tannery 1897 and Dugac 1983. 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. 3 These 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. 4 In 1902 he applied unsuccessfully for the chair of general history of science at the Collège de France, cf. Paul 1976, p. 393. 2A

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 linguistics 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 August 1914, 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 mathematical logic which, as we hope, will be published soon. (1917b, p. 291) 1 For 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 Henri Bergson (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 Russell 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 (Whitehead).”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 correspondence 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, 1 Letter

to Peano, p. 7; letter to Russell, C 09.11.1897, I/69. Supplément Septembre 1898, p. 2; cf the letters to Peano and to Bettazzi, p. 9 and p. 190. 3 In 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). 4 In 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.» 5 In 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. 2 RMM,

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 algorithmique” 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 1 For 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.

8

Introduction

or the logic of concepts. The application of mathematical method was made possible by the discovery of structural similarities between classical logic and algebra. Insofar as algebraic logic applies a method borrowed from mathematics, it subsumes logic under mathematics. This becomes notably clear in Alfred North Whitehead’s Treatise on Universal Algebra from 1898, a comparative study of Boole’s symbolic logic, Hamilton’s calculus of quaternions and Hermann Grassmann’s calculus of extension (Ausdehnungslehre), all subsumed under a general notion of calculus. Due to this mathematical treatment, logical propositions figure in algebraic logic mainly in an uncommon form, namely as equations. Its main subject hence is the theory of equations and the development of functions. In the Traité these subjects are completed by a chapter on the theory of inequalities and on probabilities, interpreted – following Boole and MacColl – as truth coefficients. After these first successful steps were taken in algebraic logic authors like Peirce and again Schröder started working on relational logic. For Couturat, though acquainted with these efforts, relational logic became important only in 1901, due to Russell’s work on this topic. Relational logic was essential for the logicist programme of reducing mathematics to logic, i. e. reducing mathematical reasoning to logical inference and introducing the basic concepts of mathematics by means of the logical primitive terms. In so far, the rise of relational logic turned the relation of logic and mathematics upside down. In the years from 1900 onward Couturat became an ardent advocate of the logicist programme. The Traité however predates these developments and originates in an epoch where Couturat was still sceptical with regard to relational logic (see my remarks below). It is an interesting fact that Couturat, notwithstanding his changing attitudes towards logic, in particular towards relational logic, always preferred the notion of “algorithmic logic”, that Church traced back to Castillon (1803) and Delbœuf (1877).1 In two articles (1904e and 1912a), where Couturat discussed the different occurring names for the new logics, he always opts in favour of “logique algorithmique” (or “logistique” as was proposed independently by Couturat, Lalande, and Itelson). He rejected the term “symbolic logic” because it overemphasises the use of symbols, the term “Algebra of logic” because it presents the subject as a branch of mathematics, as a special algebra instead of logic, and finally also the easily misunderstood term “mathematical logic” (1904e, p. 1042, 1905e, p. 706, 1912a, p. 138). In our manuscript, “algorithmic logic” however still stands simply for algebraic logic, i. e. a logic providing rules for solving logical problems by means of standardised calculating procedures (cf. ch. VII.4). The Traité thus documents the completion of a development originating in the nineteenth century, rather than the beginning of the logicist programme that was of fundamental importance in the early twentieth century. The study of algebraic logic was nevertheless of philosophical importance for Couturat, as I will show in the following sections.

1 Church 1956, p.56-7, note 124, and 1936, p. 126-7 and p. 129. Delbœuf’s treatise was indeed part of the course Couturat held at Caen in 1899; Castillon is mentioned by Couturat in connection with his lecture at Collège de France, C 24.11.1905, II/555.

Introduction

9

V. Characterisation of the Manuscript As the information used for dating the manuscript has already revealed, the Traité is essentially based on lecture notes, which suggests the character of a compendium or a survey. A cursory reading indeed confirms the character of a popularisation or synopsis: A popularisation in that it compiles recent developments in algebraic logic (Boole, Peirce, Schröder, MacColl, Venn, Whitehead) and presents them in French; a synopsis in that it clearly shortens long and fatiguing algebraic procedures (e. g. of equation solving) in order to show the reader the central ideas, and in so far as it neglects – knowingly or due to lack of familiarity – more subtle problems connected with concept formations in algebraic logic1 . The Traité is in this sense an interesting historical document on the propagation of symbolic logic in France at the turn of the twentieth century. This characterisation in addition corresponds to how Couturat understood his own work and the rôle he played in the intellectual life of his epoch. Thus, he programmatically declared in a letter to Russell: “In my review I will leave aside all logical difficulties and subtleness in order to bring out the mathematical principles that constitute, from the point of view of general philosophy, the most important part of your work.” (C 19.11.1903, I/338) In a different letter to Russell he willingly admitted: “the debate surpasses my competency and rather concerns the inventive authors (like you, Whitehead, Peano) . . . ” (C 27.07.05). Schmid concludes from that, that Couturat did not conceive of himself as an “inventive author” (Schmid 2001, I/ 37) and comes to the following general characterisation: [. . . ] the intellectual rôle of Couturat: correspondent of scholars and philosophers, intermediary in the circulation of ideas between several countries (particularly England, Italy, Germany, and the United States). His concern was not to invent, but to identify the results of recent research in order to keep up a vivid thinking against the conservatism of the institutions. (2001, I/341) One is tempted to say that Schmid sketches the image of a modern Marin Mersenne, whose merit consists rather in detecting, transmitting, and connecting with a keen sense recent trends in science than in contributing to them in an inventive manner. This image indeed seems to be fair in a number of its characterisations. The last point in particular, concerning Couturat’s undogmatic activity in academia, is confirmed by the Traité itself, which, as we have seen, emerged from a course on symbolic logic which 1 Cf. for example the issues raised by Husserl’s and Frege’s critique of Schröder (analysed in Heinzmann 1992), published several years before the composition of the Traité. The reader will note that Couturat underestimated in particular the importance of the copula ε (ch. II.28, cf. also 1899b, p. 628) as well as the problems that come along with the concept of a “universe of discourse”. Couturat avoided these latter problems simply by abandoning any absolute meaning of the notion of universe. According to him the universe is always to be understood relatively to a given problem (ch. I.10). In ch. VII.2 he even speaks of the “universe of the problem“ instead of the “universe of discourse”. The necessity of the distinction between inclusion < and membership ε was acknowledged by Couturat in 1901 (cf. the letter to Russell C 27.01.1901, I/232), i. e. shortly after the termination of his work on the Traité. Couturat anyhow did not go into the problems connected with the concept of the “universe of discourse”, indicated by Russell, R 20.10.1903.

10

Introduction

must have been quite exceptional at that time1 . In the winter 1905/06, when Couturat taught history of mathematical logic at the Collège de France, he still complained of the lack of approval.2 As regards the first point – concerning Couturat’s lack of originality – one should nevertheless be careful when jumping to the suggested conclusion. Similarly Russell judged Couturat’s Principes des mathématiques to be a too dogmatic synopsis, presenting his theses as apodictic, although they were rather intended as problematic. In a letter to Ottoline Morrell he expressed his rather severe opinion of Couturat: Then when my “Principles of Math’cs” came out, he wrote a short book professing to explain its doctrines simply to the French public. He left out all the doubts and difficulties, all the places where consistency had led me into paradox, and everything which he imagined calculated to shock, and at the same time put the thing forward as a dogmatic doctrine finally solving a host of difficulties. In consequence he made me appear absurd, and took to the international language, which now occupies him wholly.3 But in fact there are good reasons for doubting Russell’s judgement, at least in this rigour. Couturat’s Principes des mathématiques are in fact the result of an autonomous and critical work. This finds its most obvious expression in the theory of extensive magnitudes presented in Chapter V, essentially based on the works of Otto Hölder, Edward V. Huntington and Cesare Burali-Forti. This supplement in effect answers a failing of Russell’s “absolute” theory of magnitude, the inability to grasp the additive property of extensive magnitudes, which however is constitutive for the numerical expression of magnitudes and hence for the essential purpose of measurement.4 Surely Couturat didn’t dare to confront Russell directly. Instead he tried to reconcile the opposing posi1 I infer this judgement from Couturat’s own reports, since there is unfortunately no survey of logic teaching activities in France of the 19th and the early 20th century. In C 22.01.1905, II/467, Couturat reports the abandoning of the idea to teach logic at the Sorbonne because of the low expected effect in view of the traditional program and the traditional requirements for passing examinations. Having begun his lecture on the history of logic at the Collège de France, he confirms: «C’est peut-être la seule fois que depuis cent ans on aura fait un cours de Logique formelle au Collège de France.» (C 10.11.1905). In his inaugural lecture, he emphasises that it was indeed in 1838, when Barthélemy Saint-Hilaire (1805–1895) lectured on Aristotle’s logic, that anyone was concerned with logic at the Collège de France. A remarkable exception to this absence of modern logic is presented by Louis Liard (1846–1917) who published a book on contemporary English logicians in 1878. We have no information about Liard’s teaching activities, but we know at least that he was important for Couturat’s career: It was Liard, then director of higher education, who supported Couturat’s journey to Hanover where he studied the manuscripts of Leibniz at the Royal Library (cf. Couturat 1903a, p. ii), and it was Liard too, who encouraged Couturat to accept the position at the university of Caen in 1897 (cf. Dassen 1939, p. 97). 2 At the beginning, Couturat was quite pleased about the large audience (C 12.12.1905, II/565); but some weeks later he already had cause for complaint: «Je dois vous dire que l’auditoire se fait de plus en plus rare: le ‹public› n’y vient naturellement pas, et les étudiants se trouvent obligés d’assister à des conférences de philosophie qui ont lieu aux mêmes heures.» (C 06.02.1906, II/593) 3 Letter to Ottoline Morrell, March 25 1912, Russell Archive no. 400, quoted with permission of the Bertrand Russell Archives Copyright Permissions Commitee, McMaster University, Hamilton. I am grateful to Kenneth Blackwell and Carl Spadoni for transcribing the quotation and for granting the permission to publish. For a French translation cf. Schmid 2001, I/339. 4 Cf. Schlaudt 2009b, p. 288-293 and Nagel 1931, p. 326-7, who criticised Russell in the same sense but did not notice the ambiguity in Couturat’s text which he also quoted.

Introduction

11

tions (cf. C 06.06.1904, C 26.06.1904) and ended up in his Principes des mathématiques with a presentation rightly characterised by Schmid as a rather awkward compromise (2001, II/416-7, ann. 3). Be that as it may, Couturat’s Principes des mathématiques give good evidence of an awareness of existing problems as well as of a certain originality. If Couturat’s philosophical originality did not result in contributions to the development of logic, it nevertheless found expression in the presentation of the subject matter. These hints should serve as a guideline when looking for the originality of the Traité, as I will do in the following section.

VI. Couturat’s Interest in Algebraic Logic The Traité owes its content to the writings of the founders of symbolic logic, i. e. first and foremost to George Boole, Charles Sanders Peirce, Ernst Schröder, Hugh MacColl, Alfred North Whitehead and others. Since Couturat himself constantly spells out these debts, itemising them here in detail does not promise to yield unforeseen insights. The presentation and composition of the Traité however is of interest, as I pointed out in the last section. It is governed by several particular preferences or decisions, which are held together by a general aspect. The particular preferences are (1.) calculus vs. ordinary language, (2.) analytical vs. synthetical reasoning, (3.) algebra of logic vs. logic of algebra, (4.) equivalence vs. implication, (5.) extension vs. comprehension, and (6.) structure vs. interpretation. The common general motivation behind these preferences can be found in Couturat’s interest in formalisation of thought, more precisely in his discovery (or rather recovery of Leibniz’ discovery) of non-quantitative formalisation as a part of algebra in a more general sense, the characteristica universalis. This framework allowed Couturat, as we will see, to conceive the study of logic as a way to study the mind itself. Quantitative formalisation, or briefly quantification, was already one of the major topics of Couturat’s De l’infini mathématique of 1896. Measurement theory had become an important issue during the nineteenth century, primarily in consequence of two important developments in science: on the one hand the emergence of quantitative psychology due to Herbart and Fechner, and on the other hand the growing interest in the foundations of mathematics that led, among other things, to an axiomatization of quantity. Both developments are reflected in Hermann von Helmholtz’s epoch-making essay Messen und Zählen. Erkenntnistheoretisch betrachtet from 1887 (cf. Schlaudt 2009a, pp. 173 et seqq.). This essay was for Couturat, too, indeed one of the most important references of his De l’infini. In this book Couturat studied the concept of number on the one hand and the concept of quantity or magnitude on the other hand. He developed thereby a highly differentiated and refined conceptual framework, whose effectiveness he proved anew two years later in his reply to Russell’s On the Relations of Number and Quantity (1897). In this still neo-hegelian essay, later judged by its author to be “unmitigated rubbish” (1959, p. 41), Russell claimed to show that each alternative in the conception of quantity leads to contradictions, thus coming finally to the conclusion of an “inadequacy of thought to sense, or, if we prefer it, of the

12

Introduction

fundamental irrationality of sense” (p. 341). Such a conclusion was not acceptable for Couturat who held reality to be for us “what we think is true, and the true is that which reason understands and affirms as existing” (Bowne 1966, p. 30). In his reply (1898b) Couturat showed that the alleged antinomies can in large part be resolved simply by attentively distinguishing between (kinds of) quantities, determinate states of quantity, numerical expressions of quantities, and concrete instantiations of abstract quantities. This is the vocabulary he had developed in his De l’infini mathématique. The basic idea of this book is, that the concepts of number and quantity are independent of one another and that both are independent of experience. This conception led him to a consideration of number and quantity as isomorphic systems and thus to a conception of measurement which anticipates its currently prevailing definition as “assignment of numbers according to rules” (Stevens 1946), which in effect underlies the formal study of numerical representations as it was accomplished in the second half of the twentieth century (Krantz et al. 1971–1990). Couturat’s De l’infini already shows an interest in, and a familiarity with, algebra in the narrow sense, i. e. as a calculus of number and quantity. His main source seems to have been Henri Padé’s Leçons d’algèbre published in 1892. The acquaintance with Whitehead’s Universal Algebra, the first volume of which was published in 1898 and which Couturat read shortly thereafter (see above), seems to be at the origin of Couturat’s wider idea of formalisation. This comparative study offered to Couturat the idea of alternative ways of formalisation which, though they do not proceed in a quantitative manner, permit rigorous reasoning. These alternative ways of formalisation of thought found for Couturat their most striking expression in the two formulae: a + a = 2a

and

a + a = a.

These formulae characterise two different algebras proceeding according to different rules: The first characterises the quantitative calculus, in which taking a two times gives 2a, thus leading to numerical expressions; the second characterises the logical calculus, in which considering a a second time does not add anything to a (ch. IV.5 of the Traité as well as 1901a, p. 365-6, and 1900d, p. 331). Classical logic can therefore rightly be considered as a qualitative algebra. In turn traditional algebra could be considered as the logic of quantity. Both points of view lead to the same result: Logic and algebra are subsumed under a general or universal algebra which conversely is specified with the special sense given to the relation designated by the sign of equality “=”: quantitative equality leads to ordinary algebra as the calculus of number and quantity, especially Grassmann’s extensional calculus; identity and mutual inclusion lead to the calculus of concepts and classes (intensional and extensional respectively); equivalence leads to the calculus of propositions; similitude, congruence, and equipollence finally lead to special geometric calculi.1 The reading of Whitehead, and then of Grassmann, led Couturat to a systematic study of Leibniz, where he found that the project of formalising thought without first quantifying it had been anticipated. In Leibniz this endeavour was coupled 1 1900d, p. 331, 1901a, p. 318 et seqq. and p. 410. – As regards the idea of the universal algebra in the work of Schröder, cf. Peckhaus 1994.

Introduction

13

with a critique of Spinoza and Descartes, who, according to Leibniz, had failed to extend the geometrical method to metaphysics (1901a, p. 94, and p. 280). The subject of his critique was nota bene not the attempt to subject other topics to a rigorous and secure method, but the failure to examine the preconditions of these efforts. This lack of methodology resulted in a procedure more geometrico, which merely imitated the mathematical method without establishing a real calculus in metaphysics. The latter however would have resulted in a revision and broadening of the concept of mathematics itself, transcending the narrow borders of the “science of quantity” (1901a, p. 290-1). This is the path leading from Leibniz’ efforts to establish a metaphysical calculus to modern algebraic logic, as I will discuss in more detail below. It is noteworthy that Couturat not only appropriated Whitehead’s ideas as a generalisation of his own former studies, but conversely also applied a fundamental standpoint of his De l’infini mathématique to universal algebra. One of the noticeable traits of this work is its distinction between a logical and a rational point of view: According to Couturat, concept formation is essentially underdetermined by logic; there are always various ways to form concepts and to extend theories which are equally possible from the point of view of logic. They thus demand a supplementary justification from a rational point of view (cf. Bowne 1966, p. 29 and Schlaudt 2009b, p. 220-2). In arithmetic, the topic of De l’infini, the issue in question is the broadening of the number concept: Although the extension of number from integer number to whole number, fractional number and so on is logically possible, Couturat still demands a rational justification for it. This justification is provided by the criterion of applicability to quantity. Whole numbers, for example, logically introduced as sets of pairs of integers, can be regarded as numbers in so far as there are quantities adequately representable by them. The same reasoning holds, following Couturat, with regard to the logical calculus. In his review of Whitehead (1900d, p. 328), Couturat explains that, from a purely logical point of view, the calculus can be performed without regarding the meaning of the symbols (this is even one of the major features of symbolic logic); from a philosophical point of view however, the raison d’être of the symbols remains to be explored. This is the task Spinoza and Descartes missed. Couturat’s account of this specific value of symbolic representations, allowing the application of a calculus, will be examined below. However, we can already get a first idea of how Couturat adopted the more general point of view provided by Whitehead’s Universal Algebra and how he also kept at once his own original philosophical standpoint. In addition we can observe that the intermixture of the Traité with measurement-theoretic issues (the discussion of the measure of extension and comprehension, of the measure of probability, of the notion of homogeneity, of Fechner’s psychophysical law, of mathematical variables, etc.) is not merely due to the accidental intellectual biography of its author, but is indicative of the fact that algebraic logic and measurement are two sides of one and the same coin, namely, formalisation of thought. I will now examine the particular consequences that Couturat drew from this general idea for the conception of his treatise on symbolic logic. 1. Calculus vs. ordinary language. – The calculus of symbolic logic provides the advantages of absolute rigour together with purely mechanical performance. It even permits the construction of logical machines, “mills into which the premises are fed

14

Introduction

and which turn out the conclusions by the revolution of a crank”, as Peirce put it (1887, p. 165). This mechanical reasoning not only rules out error, but also permits us to save human intellectual power or, as one may put it, permits us to effectuate complicated logical inferences which surpass our limited natural powers (cf. ch. XII). Ordinary languages on the contrary are afflicted with equivocality, inexactness, and ambiguity. This is the opposition as Couturat sees it. It is this last advantage of the calculus, the advantage of saving intellectual power, which permits Couturat to understand the logical calculus as a realisation of Descartes’ idea of method.1 This trait has been emphasised by MacColl, who began his Calculus of Equivalent Statements with the following considerations: Symbolical reasoning may be said to have pretty much the same relation to ordinary reasoning that machine-labour has to manual labour. In the case of machine-labour we see some ingeniously contrived arrangement of wheels, levers, &c., producing with speed and facility results which the hands of man without such aid could only accomplish slowly and with difficulty, or which they would be utterly powerless to accomplish at all. In the case of symbolical reasoning we find in an analogous manner some regular system of rules and formulæ, easy to retain in the memory from their general symmetry and interdependence, economising or superseding the labour of the brain, and enabling any ordinary mind to obtain by simple mechanical processes results which would be beyond the reach of the strongest intellect if left entirely to its own resources. (1880, p. 45) We find the same rhetoric in W. E. Johnson: As a material machine is an instrument for economising the exertion of force, so a symbolic calculus is an instrument for economising the exertion of intelligence. And, employing the same analogy, the more perfect the calculus, the smaller would be the amount of intelligence applied as compared with the results produced. (1892, p. 3) It is also on this intuitive level that Couturat could establish the equivalence of mechan1 Grossmann argued in his sociological study on Descartes’ method from 1946 that Descartes’ ideas arose by a kind of generalisation from experiences with early machines in manufactural production; he thus linked the birth of rationality in the scholastic world with the emergence of automatic machines, (2009, p. 163, p. 188, p. 197 and p. 215). With his algebraic method, Descartes created an “intellectual auxiliary means” (p. 163), “a kind of intellectual machine operating automatically, quickly, and surely” (p. 223-4). This conception is of particular interest, as will become clear, for Couturat’s idea of logic that was also closely linked to the concepts of method and mechanization. According to Grossmann, concepts gained from technique can in turn be applied to nature, as the editors point out (p. 23), by taking the machine as a model of (the relevant part of) nature. Nature itself is transformed by that into a huge machine, the “machina mundi” (p. 10 and p. 190). As regards our case, we find a similar situation: The logical calculus as well as logic machines thus provide a model for the human intellect as regards its faculty to reason logically. We will see the resources of Couturat’s philosophy for such a conception of the mind in what follows. This leads to Leibniz’ famous statement that “omne opus mentis nostrae esse computationem” (cf. Couturat 1901, p. 458, and Marciszewski and Murawski 1995, p. 76). As Couturat pointed out, in Leibnizian methaphysics finally the whole nature appears as an “admirable calculating machine” (1901, p. 256).

Introduction

15

ical and formal reasoning.1 As regards machines, the connection between mechanical and formal reasoning is immediately evident: Once a machine has been built (once the calculus has been established), it can be used without the slightest idea of how it works and what it is good for, and even an untrained worker is able to effectuate work of the highest accuracy. This idea becomes even more evident in regard to early clocks and in regard to machines from antiquity whose inventors did not succeed in mechanising the whole procedure to be effectuated. In this case of partial mechanisation, a repeated intervention of a keeper is necessary, i. e. intervention of someone who understands what purpose the machine should serve and who consequently knows how the gaps in its performance can be filled by hand.2 Couturat’s critique of the early calculus of relations (as well as of Whitehead’s notation) will consist exactly in the existence of such gaps which cannot be filled without considering the sense of the utterances in question (see below, point 3). Here a lack of mechanical performability is interpreted as indicating an imperfect formalisation. Mechanical and formal reasoning are thus equated. As regards the advantages of the calculus compared to ordinary language – mechanical rigour vs. ambiguity –, Couturat at first sight only repeats commonplaces. His distinction between ordinary language and calculus however rests on a remarkable philosophical conception. This conception essentially follows from Couturat’s antipsychologism. This conviction marks a constant element of Couturat’s philosophy. A first outline is already found in the preface of his thesis De l’infini from 1896 (pp. viii-xii). Couturat later spelled out this critique of psychologism, accompanied by “sociologism” and “moralism”, in his inaugural lecture at Collège de France in 1905. His arguments against psychologism are at first sight common too: Logic is the normative science of correct reasoning whereas psychology can at best be the natural history of the soul.3 But Couturat went one decisive step further. He argued that introspective psychology is not even competent to investigate reasoning, the latter being a “non- or sub-conscious process”. Introspection has no privileged access to reasoning, being the action of the “esprit” (in the following translated as mind), not of the “conscience” (consciousness). We find hence in Couturat the very same “extrusion of thoughts from the mind” which 1 This equivalence was later questioned by the specification of the idea of the algorithm, particularly in the works of Alonzo Church and Alan Turing on undecidability of arithmetic and first-order logic in 1936 (cf. Krämer 1988, p. 138 et seqq.). Semidecidability of predicate logic, formulated by Thoralf Skolem and Jacques Herbrand in the 1920s, nevertheless provided a basis for further research on the mechanization of deductive reasoning. The appearance of computer technology in the 1950s additionally stimulated this research and efforts towards automated theorem provers (cf. Marciszewski and Murawski 1995, p. 209 et seqq. and p. 228-30). 2 Grossmann 2009, p. 197. Freudenthal 1982 analysed the momentous ideological differences resulting from taking either a perfect scientist’s watch or an artisan’s clock as a model of the physical universe, as was done by Leibniz and by Newton and Clarke respectively in their famous controversy. As the artisan’s clock demands repeated interventions of a keeper, it suggests, taken as a model, a universe that is not causally closed and leaves space for divine interventions. 3 For the critique of psychologism, in particular the arguments of Husserl and Frege, cf. Kusch 1994. As Kusch points out, Husserl observed that the normative-descriptive-distinction alone does not provide a conclusive argument against psychologism, since thought as it ought is a special case of thought as it in fact occurs (p. 44). In Couturat too this distinction is a mere preliminary to the key argument.

16

Introduction

Dummett pointed out in Frege.1 But unlike Frege, Couturat did not move thoughts to a “third realm” of objective but non-causal entities. Couturat drew the conclusion that reasoning cannot be studied empirically unless it has been externalised, for example in the form of language, i. e. written or spoken utterances: In order to get an accurate picture of reason’s subliminal operations we have to compare it to an invisible machine whose products only can be conceived and whose structure can only be discovered in terms of the traces it produces and the forms it impresses upon them. (1896b, p. xi; cf. also 1906b, p. 321 et seqq.) This point of view suprisingly resembles the approach of analytic philosophy in that the latter, too, rejected Frege’s “third realm” and accounted for the objectivity of thoughts by localising them in language, understood as an externalisation essentially shared by a community of competent users. It is however a difficult question whether Couturat performed a linguistic turn, i. e. attempted to analyse thought by means of the analysis of language, which would make him a kind of analytic philosopher avant la lettre (according to the definition in Dummett 1994, p. 25). The concrete significance of Couturat’s approach essentially depends on (1) what counts as a trace of the invisible mind and (2) how the relation between the mind and its empirical traces is conceived. As regards the first question, in 1896 it was nothing else than science in its most recent developments which Couturat considered as the mind’s proper work and hence as the subject to be studied. This conception is reflected in his thesis De l’infini. Some years later, at the time of the Traité, science is replaced by symbolic logic. This change was justified, as we have seen, by the discovery that logic, in its algebraic form, can be conceived as a “calculus of quality” that complements the quantitative calculus of science. We can thus state a first result concerning the philosophical significance of algorithmic logic for Couturat: Algorithmic logic was indeed a promising candidate for studying the objective structure of the mind. This highlights the philosophical importance that the study of algorithmic logic had for Couturat when he wrote the Traité. However, this approach does not fit the idea of a linguistic turn. Remember that at that time Couturat conceived logic as opposed to language. Couturat considered natural language rather as an obstacle to precise reasoning, and it is the aim of his analysis to show the advantages of the calculus as compared to ordinary language. In his inaugural lecture at Collège de France he still holds that logic does not arise from the study of languages, but on the contrary permits one to judge and criticise the different languages, which are nothing but coarse and imperfect instruments of the mind (1906b, p. 328). It is in 1911/12, i. e. after a considerable gap in Couturat’s philosophical bibliography, that we find him to be much more liberal on this point, probably as a result of his linguistic studies, which came along with his increasing engagement for Ido.2 Couturat now emphasises that language is among the manifestations of thought (pensée) the most universal and 1 Dummett 1994, p. 22 et seqq.; Couturat, by the way, studied Frege’s works not before 1903, cf. his letter to Frege of February 11, 1904, in Gabriel et al., 1976, p. 25. 2 Couturat mentions the lectures the linguist Antoine Meillet (1866–1936) held at the Collège de France in 1911. Cf. Dassen 1939, p. 176.

Introduction

17

– for all that – the most adequate. He expresses the conviction that the mind (esprit) indeed impresses its pattern on language by continuously reshaping it. Linguistic forms therefore mirror patterns of thought. They are part of a “universal grammar” which has been partially detected, and partially is still to be detected, in linguistic (i. e. empirical) research (1912b, p. 1-2).1 Logic is thereby nevertheless still considered as a regulative element: Logic defines the “ideal language”, helps to identify the essential elements of language, and to eliminate interfering factors like the impact of tradition.2 Nonetheless Couturat finally professes that “theoretical logic” and “logic inherent to our languages” are basically the same (1911a, p. 510-6). Here, finally, language takes over the place that science and logic had occupied before. Thus, although Couturat never explicitly expressed the conviction that “philosophical problems are problems which may be solved (or dissolved) either by reforming language, or by understanding more about the language we presently use” (Rorty 1997, p. 3), we indeed find in his late writings the idea of an analysis of thought by means of an analysis of language, i. e. a kind of linguistic turn.3 This idea is in addition based on anti-psychologism in so far as the latter demands that thought be studied not by means of introspection, but by means of an analysis of its externalisations. This linguistic turn nevertheless does not rest on a principle somehow equivalent to Frege’s Context Principle, but rather on the (empirical) assumption of a “universal grammar”. But this is the later Couturat, where logic has been reduced to a kind of regulative element; around 1900, on the contrary, algorithmic logic was still considered as the sole way to grasp the mind, whereas language was associated with ambiguity and error. Before turning to a more detailed analysis of the relation of calculus and ordinary language, I will shortly discuss the second question, i. e. the question how to conceive the relation between the mind and its traces that are the proper subject of philosophical investigation. In the introduction of L’Algèbre de la Logique, Couturat wrote: It belongs to the realm of philosophy to decide whether, and in what measure, this calculus corresponds to the actual operations of the mind, and is adapted to translate or even to replace argument; we cannot discuss this point here. (1905a, § 1) Couturat himself is hence not explicit on this point. There is certainly a lot that 1 Giuculescu 1983, p. 122, pointed out the resemblance to the the idea of a general grammar in Noam Chomsky. But since Chomsky himself (1968, p. 12 et seqq.) dates back the origin of this idea to the PortRoyal Grammar the fact that it appears in Couturat should not be overemphasised. As Savatovsky 1992, p. 99, suggests, the idea of universal grammar was at that time even rather outdated. 2 As regards the relation of psychology and logic in Couturat as well as the rôle of logic in the creation of an international auxiliary language cf. Ziche 2009. As Lalande reported (1913, p. 373), Couturat’s 1912 article provoked an animated discussion in the Société de philosophie in which notably Meillet, Vendryes and Lévy-Bruhl took part. 3 This story does not lack irony: It is known that Russell was disappointed by the fact that Couturat devoted himself from 1900 on increasingly to the propagation of the international auxiliary language Ido. It was however this engagement for Ido that led Couturat via his linguistic studies to a point of view quite close to what Russell himself had written in his Principles of Mathematics: “The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers . . . and in what follows, grammar, though not our master, will yet be taken as our guide.” (1903, p. 42).

18

Introduction

can be inferred from his works. Instead of tackling a historical analysis I will restrict myself to sketching the possible answers in order to appreciate the systematic interest of Couturat’s approach. A plausible model for the study of the mind-logic-connection would be mechanics in so far as it, too, studies causes that are known solely by their effects (like the moving force of a body and gravitation). But one notes that this approach does not guarantee the scientific character of the investigation (as is shown e. g. by the “entelechia” introduced by Couturat’s contemporary Driesch, explicitly in reference to Newton’s gravitation). In this approach, the mind risks being nothing more than a qualitas occulta incorporating human beings’ faculty of reason. In addition, this approach links the mind more closely to its traces than Couturat’s anti-psychologism probably permits. To remove the mind too far from its traces risks on the other hand turning Couturat’s whole project into a kind of physico-theology, i.e. the attempt to demonstrate the existence and properties of God from his works, simply replacing “God” by “mind”. Couturat’s intention unquestionably has to be located somewhere in between these two approaches. In order to avoid the problems that accompany both of them I propose a third reading that surely does not fit Couturat’s own intentions, but instead points out the promising resources in his philosophy. The point is: If algorithmic logic is to be understood as the work of the mind, and if in addition it is impossible to compare the material manifestations of the mind to the mind itself, and if thirdly it is an essential feature of the logical algorithm, as we will see, to replace the mind, it is quite plausible to say that the mind can only be grasped by taking the logical algorithm as its model. When Leibniz declared “omne opus mentis nostrae esse computationem”, we now can interpret this as taking computation as a model of the reasoning process and thus conceiving the latter as computation rather than, e. g., as “inner speech”, that is, reasoning modeled by language. This approach seems at first sight to come along with a loss, i. e. the loss of an explanation of language and logic in terms of reasoning. But as we have already remarked, the ability to reason looks quite similar to a qualitas occulta. What counts is thus what we gain within this framework. And this gain is that our approach suggests a genetic theory of the mind that accounts for the gnoseological primacy of algorithmic logic: It is viewed from the perspective of the primacy of the effect vis-à-vis the cause (the working mind), of the material representation vis-à-vis the entity represented (the laws of the mind), and of the means vis-à-vis the end (sound reasoning). This approach resembles Sellars’ idea that “the semantical characterisation of overt verbal episodes is the primary use of semantical terms, and that overt linguistic events as semantically characterised are the model for the inner episodes [i. e. for thoughts] introduced by the theory” (1956, § 58, p. 319). In our case we are concerned not with single thoughts, but with the process of reasoning, and the model is not provided by language, but by the logical algorithm. The real difficulty now consists in this, that we need a model that really does what the mind is intended to do, i. e. reasoning. This brings us back to Couturat’s main point, the analysis of the logical algorithm as compared to language. Couturat’s analysis indeed yields what we are looking for and hence is even reevaluated by putting it into a framework that admittedly does not correspond to Couturat’s original intentions, given his rejection of “sociologism”.

Introduction

19

I thus complete the preceding considerations and turn now to a closer investigation of how Couturat conceived the supremacy of the calculus as compared to natural language in the epoch of the Traité. He does so in two steps: Firstly, his anti-psychologist conception has the remarkable consequence of equalising the different forms of reasoning, – spoken, diagrammatical, algebraical, and even mechanical – as different externalisations of reasoning which are situated within a continuum.1 The peculiarity of Couturat’s view becomes evident as opposed to Husserl’s, who distinguished categorically between language as a “method of symbolico-systematical utterance of mental phenomena”, and calculus as a “method of symbolico-systematical reasoning” (1891b, p. 258). Accordingly language can be a calculus, but a mere calculus could never serve as a language. In Couturat on the contrary, language is no longer the proper medium of thought, replaced by symbolic logic as an artificial representation. Both are proper or artificial to the same extent. This permits Couturat on the one hand to legitimise the calculus with respect to language, and on the other hand to discuss its advantages compared to language. This is what the second step consists in. Both language and the logical calculus are symbolic representations. The answer to the question in what the value of the latter consists, is taken from, or at least inspired by, Leibniz’ characteristica universalis, i. e. the attempt to extend mathematics to metaphysical subjects.2 As regards numbers and quantities, the work is already done by ordinary algebra. As regards geometry, in so far as it treats of spatial quantities with both direction and magnitude, the application of algebra had been achieved only by the detour of analytic geometry, where the algebraic symbols stand for numbers which themselves encode quantities. Leibniz sought a way to represent geometric entities by symbols without the intermediary of number. But this was not accomplished until the work of Grassmann and Hamilton. The last and most ambitious extension of the mathematical method consists in establishing a calculus of metaphysical subjects. Here too Leibniz sought to establish a calculus based on symbols representing ideas. Thereby these symbols should not only provide a “stenography” or a “tachygraphy”, i. e. a abridged notation which facilitates reasoning, but a “logical pasigraphy” or “ideography” which permits one to establish a calculus which replaces reasoning (1901a, p. 89 and p. 101). He experimented, for example, with methods inspired by integer factorisation. Denoting the primitive ideas by prime numbers should lead to a unique numerical representation of complex ideas: human (6) = rational (2) × animal (3). Although none of his efforts were successful, they are among the few constructive ones in an epoch which has seen mainly precipitate ad-hoc-applications of mathematical methods to philosophical subjects and surveys of those approaches which, while sometimes of considerable measurement-theoretic interest, are purely critical in purpose.3 1 Logic machines, as Couturat himself mentions them in L’Algèbre de la Logique, § 49, and in La Logique de Leibniz, p. 116 note 2, are well situated within this continuum too: On the one hand early logic machines were nothing else than logical diagrams manipulable in a reversible way (cf. Gardner 1958, Leibniz p. 113 note 2); on the other hand there is, as regards the purely formal nature of the operations, no real difference between setting into operation a machine’s clockwork by pulling a lever and applying mechanical rules e. g. of adding or multiplying numbers in decimal notation, as we will explain in detail. 2 Compare also the description of the characteristica given by Peckhaus 1996, p. 31. 3 As an example we mention Thomas Reid’s Essay on Quantity (1748), a critique of Hutcheson’s quanti-

20

Introduction

Couturat’s project was of course much less ambitious than Leibniz’. He did not seek to establish a metaphysical calculus, he was simply looking for a conceptual framework that would permit him to understand algorithmic logic. The analysis of successfully working algorithms should yield important heuristics. The basic arithmetic operations of addition, substraction, multiplication, and division provide a simple but instructive example. Couturat knew it from Cournot’s careful study De l’origine et des limites de la correspondance entre l’algèbre et la géométrie (1847).1 As Cournot pointed out, these operations depend on the decimal notation for numbers. It is a feature of this notational system that it permits the application of simple transformation rules which correspond to the basic arithmetic operations of addition, substraction, multiplication, and division. These rules are applied in a purely mechanical way and permit computations with numbers of any size which surpass one’s ability to perform them mentally. This is an advantage of the decimal notation compared to roman numerals.2 In establishing a metaphysical calculus, the notational system thus plays a crucial rôle. The symbols are characterised by the feature of not merely representing objects (as hieroglyphs, chemical symbols or roman numerals do, too) but also providing reasoning (1901a, p. 81). Couturat speaks of “manageable” or “expressive” symbols (1901a, p. 87, and C 11.02.1904, II/351). In short one can say that algebraic symbols are characterised by an operative value, using a notion borrowed from Krämer (1991, p. 88 et seqq). By operative value I mean that the symbols are constructed in such a way that the application of transformation-rules results in signs which symbolise, according to the very same construction rules, the result of the corresponding operations on the symbolised objects. To establish such a system of operative symbols in all fields of intellectual interest – mathematics, geometry, and metaphysics – is the problem of the characteristica universalis. The ideas that Couturat found in Cournot indeed enabled him in his interpretation of Leibniz’ characteristica universalis to go far beyond what, e. g., Trendelenburg had presented in 1856, and to highlight a point that even Cassirer missed two years later in his study on Leibniz (1902, pp. 135-138). The story however does not end with the notational system. Once the desired ideographical symbols are given for the domain in question, their combination and manipulation is governed by a set of rules – as Couturat emphasises repeatedly. These rules form the calculus ratiocinator, which serves to draw in a mechanical way all conclusions from given tative approach to moral philosophy. Cf. also the introduction to the German translation, Schlaudt 2009a, pp. 21-4. As regards practical and moral questions, it can be stated that the theory of probability came closer to a formal theory of rational decision than any other approach (cf. Daston 1995). Understood as the calculus of variable judgments, the theory of probability becomes a part of algorithmic logic and hence can also be found in Couturat’s Traité (ch. X). 1 Couturat 1901a, p. 106, and Cournot 1847 p. 3 et seqq. Couturat added a second example, the Leibnizean symbolism for the infinitesimal calculus (1901a, p. 84-5, 1906a, p. 213) which illustrates Cournot’s point too, but in a less manifest way. 2 Detlefsen et al. 1976 have shown that for roman numerals, too, algorithmic procedures for arithmetical operations can be established. This demands a reevaluation of roman numerals, but does not affect the main point, that algorithmic procedures essentially depend on appropriate notational systems. – A. von Humboldt in his study on systems of numerals offers a quite natural explanation of how the decimal notation was established. According to him (1829, p. 217), the early notational systems arose themselves from the use of material calculation devices. It is hence not surprising that they in turn support algorithms.

Introduction

21

premisses.1 In this regard it is of special interest that Couturat in the Traité used solely the word “rule” whereas he prefered later, in L’Algèbre de Logique (1905), Principes des mathématiques (1905) and Prinzipien der Logik (1912), where the idea of the calculus became less prominent, to speak of “principles”. Obviously in our Traité “rule” is not merely a different name for principles or axioms, but has really to be understood as an instruction about how to manipulate material symbols. All these aspects are subsumed in Couturat’s notion of calculus, which I provide here since it is missing in the more technical Traité: A calculus is the art of manipulating and combining certain substitutional signs according to a set of rules in such a manner that the result of the operations, having been interpreted, expresses a proposition about the objects designated. (1900d, p. 324) I recapitulate the important consequences which follow from this conception of the calculus. First we state that the calculus applies to signs tout court (i. e. not to ideas through the medium of signs, but to the signs themselves). The subject matter of algorithmic logic is hence not concepts, but letters (1901a, p. 284). Couturat furthermore emphasises the physical or material nature of the calculus (1900d, p. 327). From this follows, that the calculus replaces reasoning.2 Alternatively we can say that it is the calculus itself that reasons. This may seem astonishing, but in fact in the mechanical execution of the rules of the arithmetic algorithm, too, arithmetic operations actually are carried out though nobody intentionally calculates. I stressed this point in the model theory of the mind. These points taken altogether justify attributing to Couturat an operative notion of logic, “operative” being understood as the claim that logic consists, basically, in manipulating material signs according to rules (Lorenzen 1955, p. 4). One should however not overemphasise this point, since it is not certain whether the operative aspect really has its roots in Couturat’s philosophical convictions or if it rather resulted from an accentuation of the operative nature of algebra, which can be admitted even by someone who has no operative notion of logic and mathematics at all. The peculiarity of Couturat’s concept of algorithmic logic indeed does not lie in his notion of the calculus which was not uncommon at his time3 , but rather in his analysis of the operative characteristic of the symbols the calculus is based on.4 1 As regards the characteristica as the basis for the calculus ratiocinator, cf. 1901a, p. 96. In so far as the calculus is based on the characteristica, Frege’s opposition of both in his critique of Boole is misconceived (Booles rechnende Logik und die Begriffsschrift, published posthumously in Frege 1969, pp. 9-52; cf. in particular p. 13). Cf. the clarifying discussion in Peckhaus 2004, pp. 7-8. I will discuss van Heijenoort’s conception of logic, which took up Frege’s critique of Boole, below, point 3, p. 26. 2 Couturat 1899b, p. 617, letter to Peano, p. 6, 1901a, p. 55, p. 88 et seqq, and p. 101. The complementary thesis, that language too is a – admittedly imperfect – kind of reasoning machine that replaces the mind, has been explored by Dugas 1896, esp. ch. IV, § 1. 3 Cf. for example Whitehead 1898, p. 4-5, and Husserl 1891b, p. 246 and p. 259, but of course also the quotations from MacColl and Johnson given above. Lorenz 1971, p. 47, refers to the following definitions from Leibniz: “nihil aliud enim est calculus quam operatio per characteres” and “Calculus vel operatio consistit in relationum productione facta per transmutationes formularum, secundum leges quasdam praescriptas facto”. For more historical details cf. also Lorenz 1976. 4 A comparison to Husserl’s critique of Schröder elucidates this point. In his review of Schröder’s

22

Introduction

There is a last point which yet needs to be clarified. I emphasised that, on the one hand, the calculus applies to mere letters independent of the interpretation given to them. On the other hand, in the definition of the calculus quoted above, Couturat refers to the interpretation of the calculus. And elsewhere he explains that in algorithmic logic the “intuitive rules of manipulation of signs” correspond to the abstract laws of logic (1901a, p. 89). One should note that Couturat, though he defines the calculus as a mere play with symbols1 , is far from being a nominalist. Quite the contrary, militant anti-nominalistic attacks are a common characteristic of his work. By “nominalism” Couturat obviously did not understand the denial of the existence of abstract objects. He used the term rather in the sense of conventionalism which tends to degrade rational justification to mere utilitarian considerations.2 Couturat’s attack consequently consists in showing that the calculus is not merely conventional. The main argument is provided by his theory of representation in so far as it enabled him to refer to the correspondence between the manipulations of symbols and manipulations of things represented symbolically (1900d, p. 325, 1901a, p. 89, and still in 1905b, p. 276-7). This is the raison d’être of the calculus. It is thus a consequence of his “rational” point of view that Couturat always refers to an interpreted calculus (see the definition quoted above) though he regards as its main feature the abstraction from all content. This kind of “anti-nominalism” of course has nothing to do with realism with respect to universals. On the contrary, it permits Couturat to maintain a critical attitude towards premature reifications of ideas – an attitude present in the Traité too (ch. XII). – I have elaborated on Couturat’s distinction between calculus and language. The remaining points can now however be explained quite easily. 2. Analytical vs. synthetical reasoning. There is a straightforward consequence of the mechanical aspect, which is of considerable philosophical importance: In so far as the process of reasoning can be replaced or performed by a machine, it cannot rest on intuition, at least not in a somehow constitutive manner (1906a, p. 214). Couturat reserves for this characteristic the notion of analyticity (cf. C 18.12.1903, II/456). Algebra der Logik, Husserl emphasises the difference between symbolic language (lingua characteristica) and the algebraic procedure (calculus rationcinator). He asserts that the latter is not necessarily given with the former (1891b, p. 258). But this is only true if the symbols are considered as mere and arbitrary representations without further analysis of their features. Couturat, who provides such an analysis, thus is obviously not affected by this critique. 1 1901a, p. 96: «jeu d’écriture», and p. 101: «jeu de symboles». 2 This reading is suggested by Loi 1976. Couturat indeed labeled the position of his former teacher Poincaré as nominalism (1898a, p. 361; cf. also Sanzo 1975, p. 405 and p. 414-5). One may be surprised that nonetheless both Couturat and Poincaré rejected Edouard Le Roy’s (1870–1954) position as nominalist (Poincaré 1902, General Conclusions of part III; Couturat 1900b). The difference consists in this: Poincaré rejected the claim that science is entirely conventional, whereas he still argued for the weaker thesis that science contains, or partially rests on, irreducible conventional elements. Couturat did not even accept this weaker claim. What he had in mind obviously was a rational justification of each single term of a scientific theory, whereas the existence of conventional elements allows at best a justification of larger parts of theories, if not only of whole theories. That both Couturat and Poincaré share anti-nominalism has been well remarked in Lalande 1928, II/518. Couturat there added: «Il est vrai que le point de départ du nominalisme scientifique n’est pas l’ancien nominalisme des logiciens; mais il reste un caractère commun entre les deux doctrines, qui justifie la communauté d’appellation: refuser toute valeur objective à nos concepts, et par suite aux lois scientifiques.» (op. cit., p. 517-8).

Introduction

23

In the Traité, Couturat is not unambiguous on that point: In establishing the full distributive law he indeed refers to intuition (ch. IV.2), whereas he completely rejects it from abstract theory on the occasion of discussing geometrical illustrations (ch. II.31). Finally however he indeed draws the conclusion that reasoning is purely analytical (ch. XII). This immediately leads to the question whether mathematics rests on intuition as it was raised in the famous controversy with Henri Poincaré in 1905 and 1906.1 To answer this question demands clarifying the relations between logic and mathematics. In his article La philosophie des mathématiques de Kant, annexed to the Principes des mathématiques de 1905, Couturat, arguing for logicism, i. e. the claim that mathematics rests entirely on logic, and identifying this with the claim that mathematics is purely analytic, rejected the synthetic character of mathematics.2 In the Traité he still argues the contrary: Here Couturat argues for the synthetical character of mathematics, perhaps even more explicitly than in his published works such as his article on the definition of number (1900a). He emphasises that the very method of mathematics, deduction, must be synthetic in character, since it leads to more general theorems, whereas logic always descends from the general to the particular. Here, by the way, Couturat is not far from Poincaré (1902, ch. I). If the very method of deduction is regarded as synthetic, it is in consequence not even sufficient to admit that the mathematical theorems follow in an analytic manner from synthetic principles and axioms. This latter way was chosen later by Cassirer in order to reconcile Kant with logicism.3 Couturat nota bene distinguished in the Traité between logical and mathematical reasoning. Bowne has already remarked in her 1966 study that this distinction still seemed natural to both Russell and Couturat in 1896, whereas Peano already had abandoned it (Bowne 1966, p. 46). All in all one will still find a rather Kantian idea of mathematics in the Traité; but nevertheless Couturat absolutely advocates the analyticity of reasoning as a consequence of its algebraisation. 3. Algebra of logic vs. logic of algebra. Couturat distinguished between two main currents of modern logic, the system Boole-Schröder on the one hand and the system Peano-Russell on the other hand. He later adopted – referring to a bon mot of Gregor Itelson – the labellings “algebra of logic” and “logic of algebra”, the first being a logic in algebraic form and hence an application of mathematical methods, the latter on the contrary being logic as concerned with the foundations of mathematics.4 Initially, i. e. in 1 Cf. Poincaré’s articles, reprinted in Heinzmann 1986, Couturat 1906a, and also Heinzmann 1985, p. 16 et seqq. as well as Sanzo 1975. 2 Cf. C 16.11.1903, II/332. Russell did not agree, neither with this identification nor with Couturat’s use of the notion of analyticity, cf. Schmid 2001, note 4 on p. 278, notes 7 and 8 on p. 334. 3 Cassirer 1907. Omitting the question of historical correctness of their interpretations, one can (in a simplified manner) say that the later Couturat (1901a) made of Kant an anti-Leibnizian and then used modern logic, anticipated by Leibniz, as an argument against Kantianism (1904b, 1904c), whereas Cassirer (1902) ascribed to Leibniz a kind of kantian point of view and then reconciled Kant with modern logic and mathematics (1907). – In 1905b, p. 249, Couturat denied explicitly the intermediate solution discussed in the Traité, that the synthetical character of mathematics could be inferred from the fact that the mathematical definitions are synthetical. 4 Couturat 1904e, p.1042, and letter to Russell, C 15.11.1904, II/450; as regards the relation of algebraic logic to mathematics cf. 1905a, p. 95: «la véritable logique des Mathématiques est la logique des relations [. . . ] l’Algèbre de la Logique est une logique mathématique, par sa forme et par sa méthode; mais il ne faut pas la prendre pour la Logique des mathématiques». For a detailed study cf. Grattan-Guinness 2000.

24

Introduction

particular at the time of the composition of the Traité, this distinction was connected to a critique of the logic of algebra, which Couturat suspected of not permitting mechanical reasoning and hence of not being purely formal. As is well known, Couturat later revised his reluctance vis-à-vis the system Peano-Russell, becoming even one of its most important adherents in France. He nevertheless never dropped the distinction itself. I shall have a more detailed look at this unexpected fact. It seems that the reading of Peano marked the very beginning of Couturat’s study of modern logic. In a letter to Russell, Couturat confessed that it was Peano who “initiated” him to algorithmic logic. As a matter of fact he continues: I do not appreciate his symbolism, but I acknowledge his and his school’s contributions to mathematical logic. (C 01.10.1900, I/199) He criticised severely Peano’s notation for its inability to support a veritable algorithm.1 In spite of the great importance of Whitehead for Couturat’s studies in logic and on Leibniz, he had the same criticism of his notation (1900d, p. 336-7 and p. 339). The works of MacColl, Boole, and Schröder were, besides Whitehead’s Treatise on Universal Algebra, indeed the next to be studied by Couturat (C 08.07.1898, C 29.12.1898, letter to Peano, p. 8), and here he found the proper algorithm I described above. This step in Couturat’s studies found its expression in our Traité. We nevertheless will have a look beyond. Couturat’s main worries concerning logic of algebra were connected to the logic of relations. In the Traité, the irreducibility of relations to ordinary predicates is only shortly mentioned. Also one finds no trace of the vocabulary of relational logic in it.2 Relational logic was yet known to him at that time, as is shown by his review of Schröder’s Algebra der Logik (1900g, p. 83-102). In La Logique de Leibniz (1901) Couturat had already emphasised the logic of relations, which gained in importance due to Russell’s work. In 1902 he confessed in a letter to Russell: “I expect much from the Logic of relations that is the real future logic” (C 27.03.1902). In the Principes des mathématiques finally the “calculus of relations” stands on an equal footing with the calculi of propositions and of classes. Nevertheless Couturat still had doubts concerning the status of mathematical logic, as he wrote to Russell: I somewhat deplore [. . . ] that the real and complete Logistic does not show the character of simplicity and of almost mechanical regularity possessed by the Algebra of logic. (C 07.05,1905, II/494). These doubts concerned primarily relational logic: Anyhow, I make the following remark: Logic of relations does not seem to me to constitute a formal algorithm like Logic of P or of Cls. Here is 1 Schmid 2001, I/107, ann. 2, Couturat 1899b, p. 643; C 03.01.1901, and Russell 1901c, p. 357-8, who accepted Couturat’s critique without attaching much importance to it. – It may be surprising that Couturat nevertheless announced to Peano in 1899 a chapter on his logical system in his “projected book” (letter from 04.06.1899, p. 23). Unfortunately it is not clear if he alludes to the Traité, where Peano’s name only casually appears. – For the rôle of this critique in the discussions of this epoch cf. Peckhaus 2004. 2 Couturat in particular alludes several times to the principle of hypothetic reasoning (a > b.b > c :> .a > c)) without referring to the transitivity of implication expressed by this principle. Later he indeed did so (1905b, p. 11).

Introduction

25

what I want to say: the latter provides formal rules for the quasi mechanical transformation and combination of formulae; there is nothing comparable in the Logic of relations as you conceive it: there, all operations executed are guided and instructed by the sense of the formulae; they only translate common sense reasonings that can be transported by language too, even though with less exactness and precision. Altogether, your Logic of relations appears to me to be less a calculus than a notation, a logical stenography. This, by the way, has from the beginning been my impression of Peano’s notation in so far as it goes beyond the bounds of Schröder’s Logic. (C 16.04.1905, II/485) Obviously, Couturat in 1905, i. e. after having finished the Principes des mathématiques (at least the parts on logic), repeats his early critique without any modification. We see that Couturat kept his early distinction between algebra of logic and logic of algebra as well as the reservations concerning the latter. What did indeed change was this: Couturat attached much less importance to the algorithmic aspect of logic. Around 1900 this aspect was essential for him; in 1905 he treated it as merely desirable. In 1912 Couturat finally prefixed his Prinzipien der Logik1 by the following quotation of Russell: Symbolic logic considered as a calculus has undoubtedly much interest on its own account; but this aspect has hitherto been too much emphasised at the expense of the aspect in which symbolic logic is merely the most elementary part of mathematics, and the logical prerequisite of all the rest.2 Couturat hence never revised his judgements on mathematical logic, but only the importance he attached to the aspects criticised. This also explains why he published his L’Algèbre de la Logique notwithstanding his conversion to the Peano-Russell account.3 Taking into account a new aspect, he also criticised Schröder and the algebra of logic in a way that didn’t play any rôle in the Traité. In 1905 he wrote to Russell: one makes it too easy on oneself if, like Schröder, one implicitly supposes all logical principles and proceeds like in algebra. (C 22.01.1905, II/465) This point merits elaboration, since Couturat here discusses the limitations of algorithmic logic and hence of the Traité. In the conclusion of L’Algèbre de la Logique Couturat specified which principles he thought of and why they have to be tacitly presupposed in algebraic logic (1905a, p. 95). The principles in question are the principle of deduction and the principle of substitution. The first says that, given a true implication p > q, and 1 Actually

written in 1910, cf. C 29.06.1910, II/635. 1912a, p. 137; see also Russell 1906, p. 187 and the letter C 22.07.1906, II/612-3. 3 In a letter to Russell he declares: «Je compose en ce moment un petit manuel Algèbre de la Logique, principalement d’après Schröder; je trouve que cette doctrine, quelle que soit sa portée, doit être distinguée de la Logique mathématique, et mérite d’être exposé à part, comme Algèbre plutôt que comme Logique (comme dans l’Universal Algebra de Whitehead). Cela allégera mon manuel de Logique mathématique, qui est consacré au contraire exclusivement au système Peano-Russell, comme je l’appelle, devançant l’Histoire.» (C 15.11.1904, II/449) Russell recognised the didactical utility of algebraic logic as a step towards mathematical logic, R 01.01.1905, II/462. 2 Couturat

26

Introduction

given too that the hypothesis p is true, the thesis q is also true, so that it can be asserted separately. The latter principle says that, in a general formula, a general or indeterminate term may be replaced by a particular or individual term. These two principles have the peculiarity of not being expressible by means of symbols. They rather provide the basis for any symbolical expression. One could be tempted to express the former principle by the formula: p > q.p :> .q But to assert this proposition only means to express the whole hypothetical judgment that q if p > q and p, whereas the principle of deduction permits one to affirm q separately, as it is the aim of deduction. A similar reasoning holds for the principle of substitution. In the formula ∀xP(x) > P(a) P(a) should express that P(x) holds for the individual a substituted for the variable x; but since the individual can be chosen at pleasure, P(a) must be true for any a; hence “a” again is nothing else than a variable. On the contrary, in order to apply the given formula to real particulars, the principle of substitution must already be used. The proposed formula hence presupposes the principle in question and thus fails to express it symbolically.1 “Here we have,” concluded Couturat in Die Prinzipien der Logik (1912a, p. 145), “a striking proof of the necessary limitations of symbolic logic.” There is a last point in connection with the algebra of logic / logic of algebra distinction I would like to discuss. As is well known, Jean van Heijenoort also distinguished two traditions that he supposed to be fundamentally different: the algebraic tradition Boole-Peirce-Schröder, and the quantification-theoretic tradition Frege-PeanoRussell. Based on Frege’s critique of Boole, he referred to these two traditions as logic as calculus and logic as language respectively (1967). He explained elsewhere: Unlike Boole and De Morgan, Frege did not consider that the task of logic is to investigate, with the use of ordinary language, combinatorial relations between propositions, but rather that it is to set up a reconstructed language, free from ambiguity and vagueness, in which mathematics (and perhaps other sciences) could be translated and developed. This new language becomes the logician’s language. It is not a mere manipulation of signs, because it is, like any language, understood (according to certain rules). But is is self-sufficient: it is not interpreted in another system, such as naive set theory for instance. (1992, p. 244) Van Heijenoort’s distinction and the evaluation coming along with it have been criticised as artificial and unfounded given Schröder’s influence on Löwenheim and Skolem 1 1905b, p. 11-2 and 1912a, p. 144-5; Couturat discussed this point with Russell. In C 22.07.1906, II/613, he insisted on the distinction between the premisses of a syllogism and the principle permitting the syllogistic deduction. He criticised Russell, who had placed the principle “Syll” among the premisses of a syllogism. Russell admitted this point in R 21.08.1906, II/616.

Introduction

27

(Moore 1977, p. 469), on Tarski (Béziau 1998/2000, p. 112), his, and Peirce’s, development of a quantification theory (Anellis 1994, p. 103, Hintikka 1997, p. 143), and finally his conception of algebra (Peckhaus 1993 and 1994). Van Heijenoort’s conception of logic and its history is a controversial issue and one might wonder how it relates to Couturat’s, for, on the one hand, Couturat, too, distinguished between these two traditions of logic, algebra of logic and logic of algebra. In addition, Couturat held to this distinction notwithstanding his changing attitudes towards both strands of logic. He thus maybe even provides further arguments for the adequacy of Heijenoort’s distinction. On the other hand, Couturat’s idea of logic, at least at the time he wrote the Traité, is plainly incompatible with both van Heijenoort’s contempt for the Boole-Schrödertradition and his qualification of the Frege-tradition as “logic as language” (see above, point 1). Where van Heijenoort wrote in regard to Frege’s logic: “We now have a lingua, not simply a calculus” (1967, p. 325), Couturat would have said contrariwise in regard to Boole’s logic: “We now have a calculus, not simply a lingua.” He emphasised this point in a letter to Russell, cutting off thereby inopportune reference to Leibniz: Anyhow, I hold that these two things have to be completely separated: a logical symbolism comprising an algorithm and presuming the analysis of thought, on the one hand, and, on the other hand, a universal language, serving the same services as our written and spoken languages. It has been the error of Leibniz to confound them (at least at the beginning, for from 1678 on, he studies the Universal Language separately from the logical calculus). (C 03.01.1901, I/217) Couturat was however well aware of the limits of both language and algebraic logic. Language, he proceeds in the letter to Russell, is an universal, but coarse instrument, whereas logic is more precise, but, for this reason, more limited in scope. Universality and precision, as van Heijenoort seems to claim at the same time for Frege’s Formelsprache, are hence considered by Couturat to be incompatible, at least in full extent. Admittedly, Couturat’s idea of universality of language, i. e. universal applicability, was much less sophisticated than Hintikka’s reading of universality in the sense of being inescapable (1997, p. ix) which probably was also intended by van Heijenoort, as suggested by the quotation given above. It is anyhow a difficult question whether the later Couturat, i. e. after the reevaluation of Peano-Russell around 1901, is closer to van Heijenoort. For Couturat then considered modern logic rather to be the result of the synthesis of Boole-Schröder-Peirce and Weierstraß-Cantor-Peano (1905b, p. 5). Given his persistent dissatisfaction with the lack of mechanicity or mechanical performability of Peano’s symbolism, Couturat might have considered this symbolism again as a mere lingua. So, even if there is a certain factual convergence of Couturat and van Heijenoort, there still rests a conceptual incompatibility between them. 4. Equivalence vs. implication. The algebraic standpoint has another rather trivial implication which also is reflected by the Traité. Algebra, understood as a general method, consists of rules for transforming equations. The logical counterpart consists of statements of equivalence between two propositions or of identity of classes. Couturat

28

Introduction

thus defined algebraic logic as the theory of functions and equations.1 In logic however, considered as concerned with deduction, the most important relation is implication of propositions and inclusion of classes respectively. This conflict between the equational and the implicational point of view does not yet necessarily lead to contradictions, since the implication p > q is logically equivalent to the equivalence p = pq and conversely the equivalence p = q can be defined as mutual implication p > q. ∧ .q > p.2 It may however pose a problem if one seeks to eliminate all implications in favour of equivalences. Jevons’ claim to having done so provoked MacColl’s critique (1881, p. 41): The syllogism Barbara, e. g., with the aid of implications symbolically expressed as a > b.b > c :> . a > c, would turn into the clumsy equation a = ab.b = bc :=: .a = ab.b = bc.a = ac. Couturat indeed attached great importance to logical equations in the Traité. In the conclusions he even mentioned the principle of subtitution, on which Jevons had based his equational logic. As Schmid rightly observes, Couturat was an algebraist used to equations and interested in the first instance in structural analogies between the different calculi, which requires the equational point of view (I/228, 304 and 310). When he in the Traité introduces the rules for transforming inclusions into equations (ch. III.10), Couturat thus refers to the equations as the “ideal of algebraic logic”. Nevertheless Couturat seems not to have connected any philosophical claim to the technical importance of equations in algebraic logic. In the Traité he first introduces implication as the basic relation and then defines equivalence as mutual implication. The alternative implicational vs. equational logic is thus of minor importance for us here. I hence pass on to the next point: 5. Extension vs. comprehension. The logical calculus is, as was well known at the epoch of the Traité, capable of two distinct interpretations, conceptual and propositional. Couturat discusses this at length in the text. He even reserves for this characteristic the notion “duality of interpretation”, substituting the terms “correlation” or “reciprocity” for what Boole traditionally described as duality (see below). Concepts present again a twofold aspect: comprehension and extension. With the Logique de Port Royal one can understand them as the totality of all attributes comprehended in the concept and the totality of all subjects that the concept applies to respectively (Part I, ch. VI) – a point of view clearly present in the Traité (ch. I.2). The extensional interpretation can 1 Cf.

the letters to Peano, p. 14: «Algèbre de la Logique, c-à-d la théorie des fonctions et des équations». Russell 1906, p. 189: “The above proposition enables us to transform every implication into an equivalence, which is an advantage if we wish to assimilate symbolic logic as far as possible with ordinary algebra. But when symbolic logic is regarded as an instrument of proof, we need implications, and it is merely inconvenient to substitute equivalences.” and Russell 1901c, p. 353: “It has been one of the bad effects of the analogy with ordinary Algebra that most formal logicians (with the exception of Frege and Mr. MacColl) have shown more interest in logical equations than in implications. Two propositions in logic are said to be equal when there is a mutual implication – a quite exceptional and not especially important case. What is desired, in actual reasoning, is deduction, and this does not as a rule require equations.” 2 Cf.

Introduction

29

be extended to propositions by replacing them either by the totality of the cases where they are true or by the totality of the temporal instances where they are true (1899b, p. 623, Traité, ch. II.15). This extensional interpretation of the calculus was of particular interest for Couturat because of its close connection to the algorithmic or mechanical aspect of reasoning. Actually, the view of mechanical reasoning not only accentuates the mechanical aspect of formal reasoning, but in addition involves the idea that reasoning can really be performed by machines. The first logical machines, such as the Stanhope Demonstrator or Venn’s Logical-diagram machine, were indeed nothing but reversibly manipulable geometric diagrams.1 The use of diagrams in logic was introduced by Leibniz, Euler, Lambert and others (cf. Gardner 1958 and Davenport 1956). Couturat mentions them in the Traité, in La Logique de Leibniz, and in L’Algèbre de la Logique. The diagrammatic method in turn is justified by the isomorphism between the logical calculus and topology. This becomes immediately evident exactly in its set-theoretic interpretation, where the sets (consisting of individuals) correspond to manifolds or geometric domains (consisting of points), and where the set-theoretic relations of inclusion and exclusion have an immediate topological counterpart. This justifies Couturat’s talk of “geometrical schemes” of logical relations.2 The more sophisticated constructions of later logical machines were, though no longer of diagrammatic nature, nevertheless still based on isomorphisms to different domains. Besides the machines based on the isomorphism to topology (Stanhope, Venn), there are machines based on the isomorphism to certain mechanical principles (as Annibale Pastore presented in 1906), and finally electrical machines based on the isomorphism to electrical network theory (theoretically explored in Shannon 1936). Couturat’s Traité of course predates the latter machines. The state of development achieved at the end of the nineteenth century was nevertheless sufficient not only to let Couturat prefer the extensional to the comprehensional standpoint, but to make him declare that the former is the standpoint proper to modern logic: Algorithmic logic necessarily is logic of extension. (1900d, p. 340) In his article on Whitehead (1900d, p. 340) he argues positively: Due to the reduction of both concepts and propositions to extensions, logic is capable of mathematical treatment. In La Logique de Leibniz (1901, p. 22 and p. 32) he complements the negative part of the argument and shows that comprehensional relations are not susceptible of geometric representation. This altogether led him to reject completely the comprehensional standpoint and to adopt the extensional point of view without any reservation, though there are arguments which demand a more differentiated discussion.3 His arguments of 1 Cf. Harley 1879, Venn 1880, p. 15-8, and also Couturat’s discussion of Leibniz’s diagrammatical representations as well as their extension to diagrams with movable parts, La Logique de Leibniz, 1901, p. 26 et seqq. and p. 113 note 2. 2 1901a, p. 22 et seqq. – In 1900g, p. 51, Couturat even speaks of an “intuitive application in geometry”; in 1917a, p. 47-48, he omitted the use of geometric illustrations because they might give rise to the mistaken idea that logic is based on intuition. 3 Couturat became acquainted with these doubts by Russell’s critique of La Logique de Leibniz (1903, p. 189): “There is, no doubt, a certain broad truth in this statement: the Logical Calculus undoubtedly requires a point of view more akin to that of extension than to that of intension. But it would seem that the

30

Introduction

course are much less ambitious than e. g. Schröder’s. Obviously they are far from being conclusive and are open to criticism, such as those raised by Husserl (1891a). Nevertheless we acquire an idea of how Couturat came to favour the extensional interpretation and thus to declare that algorithmic logic is essentially set theory with regard to their mutual inclusion and exclusion (1900d, p. 340, 1901a, p. 305). In the Traité he finally even abandons the talk of a threefold interpretation of the logical calculus (concepts, propositions, and sets) and speaks instead of its proper interpretation as set theory and its threefold application on the logic of concepts, of propositions, and on topology (ch. II.31). 6. Structure vs. interpretation. I have already mentioned the twofold interpretation of algebraic logic, the calculus of concepts and the calculus of propositions. Concepts, as to their extension, directly lead to logic as set theory or the calculus of classes. But propositions, too, as we have seen, can be regarded from the extensional point of view (see above, point 5). This assimilates them to classes. On the other hand, classes can be transformed into propositions with the help of the copula “ ε ”. At all instances the class a can be replaced by the proposition “x ε a”, i. e. “x is an a”. Both interpretations of algebraic logic hence are mutually transformable (ch. II.29). It is an interesting question whether Couturat nonetheless had a preference for one of the two possible interpretations. In his 1899 review of Peano, probably written slightly before the Traité, Couturat clearly was in favour of the conceptual interpretation. The reason for this is simple. As we have seen, Couturat considered as the most remarkable characteristic of the logical calculus the validity of the formula a + a = a. There is in fact a deeper reason for that in the structure of the calculus, i. e. the fact that in every formula the implication can be reversed if the signs for logical multiplication and addition are interchanged (and vice versa). This is called, according to Boole, duality. It also leads to the consequence that the distributive laws holds for addition too: a(b + c) = ab + ac

and

a + bc = (a + b)(a + c).

The duality however is, as Couturat observes, not invariant with respect to the different interpretations, propositional and conceptual, of the logical calculus. Peano e. g. began with the propositional calculus. He introduced as primitive terms implication, conjunction and negation of propositions as well as the copula “ε”, permitting one to replace each class a by the proposition “x ε a”, “x is an a”. Multiplication he defined then by the help of addition. This procedure has the disadvantage, as Couturat criticised in his 1899 review (and repeated in the Traité, ch. V.9), of disguising the duality of the calculus of classes (1899b, p. 627 and p. 643). He thus criticised Peano’s approach from an algebraist’s point of view, i. e. the point of view of someone who is primarily interested in truth lies somewhere between the two, in a theory not yet developed. This results from the consideration of infinite classes. [. . . ] There appears to be here a logical problem, as yet unsolved and almost unconsidered; and in any case, the matter is less simple than M. Couturat represents it as being.” Couturat responded to that with curiosity (C 12.05.1903, I/291). – Couturat recognised in his study La Logique de Leibniz that Leibniz tended to a “comprehensional” point of view, which he criticised (p. 23-24). This aspect of Leibniz was later appreciated particularly by Raili Kauppi 1960, but cf. also the discussions in Rescher (1954) and Dummett (1956).

Introduction

31

the structure of the calculus and who therefore wants the structural characteristics to become apparent. In the Traité however, there appear arguments in favour of the propositional interpretation. Above all it is the discussion of “mixed” and “inhomogeneous” formulae that challenged the conceptual interpretation (ch. II.27). Mixed formulae are characterised by the occurence of several copulae of different order as, e. g., in the expression a > b.b > c :> . a > c, providing three first-order copulae > bound together by one second-order copula >. In the propositional interpretation, the sign “>” is translated in all four occurrences as implication. In the conceptual interpretation however, only the first-order occurrences are translated as inclusions, while the second-order occurrence still has to be read as an implication. One hence can observe an asymmetry, which indicates a superiority of the calculus of propositions. As Couturat observes elsewhere (ch. IV.10), theorems from the calculus of propositions in addition are necessary for the proofs of theorems in the calculus of classes. Furthermore, Couturat considered the propositional calculus to be of greater generality, because it applies not only to predicative judgements, but also to relational judgements (ch. IX.1, ann. 1 – this argument, of course, lost its validity with Wiener’s definition of the ordered pair in 1914, which made it possible to reduce the calculus of relations to that of classes, cf. Grattan-Guinness 2000, p. 241). These observations counteract the preference for the conceptual interpretation, though even Russell recognised Couturat’s critique of Peano (Russell 1901c, p. 358). In the Traité it seems that both strands counterbalance one another. Couturat thus comes to a solomonic judgement: While in 1899 he preferred the conceptual interpretation of the calculus because of its advantage as regards the manifestness of its typical structure, he abstained from any interpretational preference in the Traité for the very same reason. In the Traité Couturat argues, that the algebraist is interested only in the structure of the calculus and hence disregards its possible interpretations (ch. II.30). He thus has no preference of any of the possible interpretations. But there is a still deeper point. As Couturat argues in his article on the logical definition of number (1900a), interpretation always involves intuition. In order to apply the logical calculus to sets, they first have to be conceived as sets, i. e. as “multiplicities of unities”, as assemblages of elements whose unity and difference has been recognised. But this presupposes, according to Couturat, the intuition of unity and multiplicity.1 Rigour and analyticity thus exclude interpretation and restrict logic to structure. Later Couturat again changed his mind and preferred the propositional interpretation, essentially for the reasons he already mentioned in the Traité, i. e. the problem of mixed formulae (1917a, written in 1906, p. 41). In addition, he meanwhile adopted from Russell the concept of propositional functions which permits one to define concepts by means of propositions (ibid. p. 15). Couturat provided in this article an additional analysis of the differences between the calculi of classes and of propositions that elucidates some remarks of the Traité, too: 1 1900a,

p. 33; cf. also the discussion with Russell, Schmid 2001, I/330-339.

32

Introduction

All these divergences between the calculi of classes and of propositions result from the fact that propositions can have only two different values, true and false, whereas classes, in general, are susceptible to a lot of different values distinct from Nothing and Everything, so to speak intermediate between those two extrema. (1917a, p. 43) This analysis finally explains Couturat’s account of the calculus of constant judgements as a calculus of sets in a universe containing only one single element (ch. IX.7). We have thus acquired an idea of Couturat’s interest in algebraic logic. We have explored its twofold origin in Couturat’s own earlier research and in the work of Whitehead, and we have seen how the Traité in its concrete form resulted from this. The guiding interest behind these efforts concerns non-quantitative ways of formalising thought. On this conception Couturat based the importance logic obtains as an objective study of the mind itself, whereas the mind cannot be conceived by introspection but has to be modeled according to its external manifestations. As a remarkable trait of Couturat’s early philosophy of logic, i. e. at the time he wrote the Traité, I have highlighted the attempt to found the calculus on a theory of symbolic representation. Inspired by Leibniz’ characeristica universalis, Couturat brought out what could be called a theory of the operationability or expressibility of logical symbols.

VII. Editorial Policy Despite several annotations indicating points to be further elaborated on, the Traité presents a quite readable text that seems on the whole to have attained its final or at least close to final form. We have thus felt justified in aiming at an edition that presents the text in an easily readable form. Hence we have reproduced variants, crossed out words and suppressed passages, together with additional annotations, in a critical apparatus following the text. The greater part of these alterations originate immediately from the first redaction and are to a large extent of stylistic concern, i. e. they seem to have been made in order to avoid linguistic repetitions etc. There are however also a number of alterations which reveal an interesting hesitation between alternative formulations and are thus of importance for a more careful interpretation. Later insertions – of whole passages as well as of notes and single key-words – have nevertheless been marked in the text in order to give a more precise idea of the manuscript and its genesis as well as to remind the reader of its character as an uncompleted text. With the same intention we preserved Couturat’s original spelling and notation, e. g. in the case of abbreviations Couturat habitually used (càd for c’est-à-dire, 1e for premier etc.). The numbering of the footnotes, originally beginning at each sheet of the manuscript with 1, has been adopted to the pages of our edition. Also, a couple of minor scribal errors in formulae have been tacitly corrected. A list of these corrections is attached to the annotations (p. 293). Only the second volume of the manuscript presented a consecutive pagination; in the first volume, only the pages preceding a page used for figures and the last page of every chapter respectively provide a page number. The first 36 pages – rectos as

Introduction

33

well as versos if used, i. e. up to the first page of chapter two – display a consecutive numbering in a hand writing differing from Couturat’s; this numbering however does not fit the one implied by the table of contents. It seems that Couturat only counted the recto of each page, skipping the versos as well as the rectos exclusively used for figures – with the exception of page 61, showing the figures 1–5. On this basis the intended page numbering was reconstructed and is indicated beside the text. Obviously Couturat skipped in his numbering the pages following on § 15 of chapter VII, i. e. § 16 and the appendix at the end of Volume 1. The numbers 184a – 184r were attributed to these pages.

VIII. Editorial Symbols in the Presentation of the Text Editorial symbols within the maintext: text

simply underlined words and passages, reproduced in italics

NAME

names, doubly underlined, reproduced in small capitals

H

I

added passages, written in the margin, at the bottom of the page or on the verso

T

U

items added in a different ink, presumably by Couturat in the course of a later revision

{

}

J

K

items added presumably by the editor of the tenth chapter in RMM, 1917 items added by the editors of this volume

¿– ?

undecipherable word

¿– – –?

undecipherable phrase

1., 2., 3. . . .

numbering of the paragraphs, also added in a different ink, reproduced in bold numerals

(1) , (2) , (3) ,

Couturat’s annotations at the bottom of the page

...

1, 2, 3

...

annotations of the editors, referring to the critical apparatus

a, b, c,

...

variants reproduced in the critical apparatus

34

Introduction

[187]

pagebreaks of the manuscript (for the reconstruction of the pagination see above)

[187bis]

verso of the page indicated

[187∗ ]

page not accounted for in the original pagination, following the indicated page

||187bis|

pagebreaks in annotations

Editorial symbols within the critical apparatus differing from those used in the maintext: text h

cancelled words and passages struck out linewise i

paragraphs struck out diagonally in a different ink

text

simply underlined words and passages

Name

names, doubly underlined

*

+

||187|

words added between the lines pagebreaks

Abbreviations used by Couturat in the text, presumbably intended to be replaced by the typesetter càd

c’est-à-dire (that is to say, i. e.)

1er ,1e , 2e etc.

premier, première, deuxième etc.

Names of authors accompanied by a year, as e. g. “Couturat 1905a”, refer to the bibliography in the critical apparatus.

Part II

Transcription of the Manuscript

Traité de Logique algorithmique (I)1

[0]

CHAPITRE I

[1]

I. Définitions et notations : A. Logique des concepts – 1. L’acte élémentaire de la pensée est le jugement. Un jugement est l’affirmationa(1) d’un rapport entre plusieures idées, ces idées pouvant être générales (concepts) ou singulières (idée de tel objet ou être individuel). Une proposition est l’énoncé d’un jugement. La Logique de termes absolus (la seule que nous étudierons ici) ne s’occupe que d’une seule espèce de jugementsc : Les jugements de prédication (ou d’attribution) qui consistent à affirmerd un rapport entre un sujet et un attribut (ou prédicat). Cee rapport s’exprime dans le langage par le coplule «est». Exemples : la neige est blanche, la mer est salée. La Logique étudie aussi les jugement d’existence, où le verbe est encore le verbe être, mais cette fois avec un sens de prédication : le mot «est» signifie alors «est existant», il implique l’attribut d’existence. Ilf est souvent remplacé par le verbe «exister» ou par la locution (française) : «Il y a». Exemples : Dieu est ; Les antipodes existent ; il y a des cygnes noirs. | Il va sans dire qu’il y a bien d’autres relations possibles et concevables que celles [2] qu’exprime le verbe être : telles sont toutes les relations exprimées par les autres verbesg : Pierre aimeh Paul ; Jean pense à Jacques ; etc. A ces relations il faut ajouter certaines relations qui, pour s’exprimer par des locutions où figure le verbe êtrei , n’en sont pas moins distinctes de la relation de prédication ; Exemples : Pierre est frèrej de Paul ; le cercle O est égal au cercle O0 , etc. Il est clair, en effet, que la copule véritable, cellek qui exprime une relation entre les deux termes du jugement, n’est pas la copule ordinaire «est», mais bien : «est frére de», «est égal à». Sans doute la Loqique grammaticale considère ici comme attributs les locutionsl : «frère de Paul», «égal au cercle O0 » ; elle enseigne même quem tous les verbes équivalent au verbe être joint à un attribut, et décompose les jugements précités comme suit : «Pierre est aimant-Paul ;» «Jean est pensant-à-Jacques ;» mais ce sont évidemment là des détoursn et des | fictions qui ne correspondent nullement à la pensé véritablement exprimé et [3] qui la dénaturent. De même, quand on exprime les mêmes jugements sous une autre (1) Nousb entendons ici «affirmation» au sens le plus général, comprenant aussi la négation. La négation d’un rapport est est l’affirmation du rapport contraire.

O. Schlaudt, M. Sakhri (eds.), Louis Couturat – Traité de Logique algorithmique, Publications des Archives Henri Poincaré, DOI 10.1007/978-3-0346-0411-6_2, © Springer Basel AG 2010

38

Traité de Logique algorithmique

forme : «Pierre et Paul sont frères ;» «Les cercles O et O0 sont égaux,» il saute aux yeux que les attributs grammaticauxa «frères» et «égaux» exprimentb la relation affirmée entre les deux termes qui figurent comme sujets, et ne sont pas de véritables prédicatsc de ces sujets : car s’il en était ainsi, on devrait pouvoir les attribuer à chacun d’eux séparément, et dire : «Pierre est frère,» ou : «Le cercle O est égal,» ce qui n’a pas de sens.(1) Toutes les relations autres que la relation de prédication, qu’elles s’expriment ou non au moyern du verbe être, restente en dehors de la Logique des termes absolus, et forment l’objet d’une autre Logique, beaucoup plus compliquée, et qu’il convient par suite de n’aborder qu’après celle-ci : la Logique des termes relatifs ou simplement des relations.(2) [4] | 2. Définissons maintenant la relation de prédication. Cette relation peut se concevoir en deux sens différents, à deux points de vue opposé : celui de l’extension et celui de la compréhension. On sait que la compréhension d’une idée (d’un concept) est son contenu intelligible, càd l’ensemble des caractères, des qualités ou des propriétés «connotés» par cette idéeg . L’extension d’une idée, d’autre part, est l’ensemble des objets «dénotés» ou désignés par cette idée.h Si l’idée est singulière, càd représente un seul individu, son extension est réduite à cet individu unique ; sa compréhension embrasse en revanche toutes les propriétés de cet individu. Si l’idée est un concept abstrait et général, sa compréhension se compose de tous les caractères qu’on a retenus dans l’opération de l’abstraction ; son extension est constitutée par la collection des objets auxquels ce concept s’applique en vertu de la généralisation. On dit souvent que l’extension d’un concept est en raison inverse de sa compré[5] hension ; cela est vrai | en gros et d’une manière métaphorique. Il est certain, en général, que plus un concept est abstrait, càd moins il comprendi de caractères distinctifs, plus il est général, càd plus il dénote d’individus. Mais cela n’est pas toujours ni nécessairement vrai : par exemple, si l’on ajoute à l’idée de polygone régulier l’attribut «inscriptible (au cercle)», l’idée de polygone régulier inscriptible aura la même extension que la première ; et si l’on ajoute l’attribut «circonscriptible (au cercle)», l’idée plus complexe encore de polygone régulier inscriptible et circonscriptible aura toujours la même extension.(3) D’autre part, la formule précitée prétend à une précision mathématique qui est illusoire. On peut sans doute évaluer mathématiquement l’extension d’un concept par [6] le nombre des individus qu’il dénote (que ce nombre soit d’ailleurs fini ou infini) ; | mais il est impossible d’évaluerl mathématiquement sa compréhension, et l’on peut tout au plus dire vaguement qu’elle augmente ou diminue ; elle est susceptible de la distinction du plus et du moins, mais nullement d’une mesure précise. On ne peut évidemment pas la définir comme le nombre des caractères connotés par le concept ; car (1) C’est

là un exemple des ambiguïtés et des confusions qu’engendre l’emploid du langage usuel. par Charles Santiago PEIRCE (1870.) (3) Cela n’a rien d’étonnant quandj on sait que tout polygone régulier est à la fois inscriptible et circonscriptible ; mais, que ce soit là un jugement analytique ou synthétique (question délicate et épineuse), il est certain que les attributs inscriptible et circonscriptible ajoutentk quelque chose à la notion de polygone régulier, au moins pour ceux qui ignorent la Géométrie et qui apprennent pour la première fois ce théorème. (2) Crééf

Chapitre I – Définitions et notations : A. Logique des concepts

39

à quoi reconnaîtra-t-on l’unité et l’individualité d’un caractère, càd d’une abstraction, quand il est déjà difficile et scabreux de déterminer l’unité et l’individualité d’un objet concret ?(1) Or si l’on ne peut pas mesurer ni évaluer en nombres un des deux termes comparés, il est vain de formuler entre eux une loi mathématique rigoureuseb telle que celle de la proportionnalité inversec ; à proprement parler, elle n’a pas de sens, et ne doit être entendue qu’au figuré . 3. Cela posé, la relation de prédication consiste, d’une partd , en ce que l’attribut est contenue dans la compréhension duf sujet, et, d’autre partg , en ce que le sujet est contenu dans l’extension de l’attribut. En particulier, un concept peut avoir | pour [7] attributs tous les éléments de sa compréhension, et pour sujets tous les individus qui forme son extension. On a beaucoup discuté pour savoir lequel de ces deux points de vue opposés est primitif et fondamental dans le jugement, et lequel prédomine généralement dans l’esprit. Nous n’entrerons pas dans cette discussion, attendu que c’est là une question psychologique qui n’a rien à voir avec la Logique. Nous reconnaissons volontiers que c’est le point de vue de la compréhension qui domine le plus souvent, que la plupart des jugements sont conçus et pensés en compréhension. Mais cette constatation psychologique ne nous oblige nullement à préférer ce point de vue en Logique et ne peut nous empêcher d’adopter le point de vue contraire, s’il nous paraît préférable pour des raisons purement logiques. Or c’est bien le point de vue de l’extension qui doit être préféré, dans l’intérêt de la clarté et de la rigueur que réclame la Logique et surtout la Logique algorithmique. C’est ce que prouve déjà l’histoire de la Logique, qui esth restée stationnaire | tant [8] qu’elle est restée fidèle au point de vue de la compréhension, adopté par Aristote, et qui n’a commencé à faire des progrès que lorsquei le point de vue de l’extension fut en faveur, grâce aux mathématiciens comme Euler. Ce fait historique a d’ailleurs une raison théorique : c’est que les rapports de compréhension sont vagues, flottants et variables, donc difficiles à définir et à préciser ; que l’on songe, par exemple, à la difficulté qu’il y a à déterminerj si tel jugement est analytique ou synthétique, càd si son attribut est ou n’est pas contenu dans la compréhension du sujet. Au contraire, les rapports d’extension sont relativement clairk et précis : les ensembles d’individusl ou d’objets qui constituent les extensions respectives des concepts comportent des relations nettes et bien définies, que nous exposerons bientôt. De plus, les relations d’extension sont analogues à des relations numériques, puisque les ensembles en question peuvent se réduirem par abstraction à des collections d’unités, càd à des nombres. Elles sont donc susceptibles d’un traitement mathématique, d’unen notation et d’un calcul analogues à ceux de l’Algèbre. | Pour toutes ces raisons, c’est le point de vue de l’extension qu’a définitivement [9] adopté la Logique moderne, à l’exclusion de celui de la compréhension. Il serait déjà préférable si les deux points de vue étaient équivalents ; il l’est à plus forte raison s’ilo est plus général que le point de vue opposé. Or, si tous les rapports de compréhension peuvent se traduire par des rapports d’extension, il ne semble pas qu’inversement, tous les rapports d’extension puissent se traduire en rapports de compréhension. (1) V.a

De l’Infini mathématique.1

40

Traité de Logique algorithmique

Pour le prouver, sans aller chercher des exemplesa singuliers et des cas exceptionnels, il suffit de rappeler que la Logique aristotélicienne admetb , outre les jugements universels, les jugements particuliers, de la forme : Quelques A sont B. Or il est évident que de tels jugements ne peuvent être pensés qu’en extension, et non en compréhension.c On conçoit en effet qu’une partie (ou peut être même la totalité) de la classe des Ad soite contenue dans la classe des B ; mais que signifierait de dire : L’attribut B fait quelquefois [10] partie de la compréhension du sujet A ? Les concepts A et B étant | essentiellement constants, leur rapportf de compréhension ne saurait être variable et contingent : ou s’il est contingent, c’est qu’il n’existe pas, càd que B n’est jamais compris dans A. Ainsi, au point de vue de la compréhension, les jugements particuliers n’ont pas de sens ni de raison d’être : un concept est ou n’est pas contenu dans la compréhension d’un autre, et il n’y a pas de milieu ; il n’existe donc à ce point de vue que des jugements universels. Si Aristote ag cru pouvoir considérer des jugements particuliers, c’est par une inconséquence qui consiste, au fond, à mêler les deux ordres de considérations, comme cela a lieu sans cesse d’ailleursh dans la Logique classique. 4. Désormais, chaque concept sera donc représenté par son extension, càd par l’ensemble des objets individuels auxquels il s’applique. Cet ensemble s’appellera la classe correspondant à ce concept, ou brièvement sa classe. Toutes les relations entre les concepts devront se traduirei par leurs rapports d’extension, càd par des relations [11] entre leurs classes.(1) Chacune de ces classes, avons-nous dit, | est une collection d’unités, càd d’objets ou d’êtres individuels et bien distincts. On peut se demander à quoi l’on reconnaîtra l’unité de chacun de ces individus. Nous répondrons : à ce fait qu’il est représenté tout entier et tout seul par le concept correspondant. Cette unité peut être toute fictive et idéale. Dans une armée, il y a un nombre déterminé de soldats, de compagnies, de régiments. Qu’est-ce qui permet de déterminerk ce nombre ? C’est que l’on sait exactement reconnaître un soldat, une compagnie, un régiment, et cela au moyen du concept général de soldat, de compagnie, de régiment, que l’on applique tour à tour à chaque soldat, à chaque compagnie, à chaque régiment. Les individus qui composent une classe sont appelés les éléments de cette classe. Il est impossible de concevoir une classe sans la concevoir comme composée d’élémentsl , puisqu’une classe est par définition un ensemble, une collection d’individus. Il nous paraît donc au moins inutilem de chercher, comme M. SCHRÖDER, à faire abstraction des éléments pour concevoir les classes, et à définir leurs relations sans considérer les éléments qui les constituent. [11bis] |Les concepts étant remplacésn par leurs classes, la lettre qui représente un concept représentera proprement sa classe dans toutes les formules. Nous représenterons les concepts et les classes par des minuscules italiques a, b, c . . . x, y, z. Les individus de la classe a (dénotés par le concept a) s’appelleront par abréviation «les a» commeo on dit : les hommesp , pour désigner l’ensemble des êtres qui constituent l’extension du concept «homme». Chacunq de ces éléments, pris à part, s’appellera «un a.» (1) Le Calcul des classes quej la Logique a conduit à inventer, a par suite la plus grande analogie avec la Théorie des ensembles, créée par les mathématiciens modernes pour servir de fondement logique à leur science.

Chapitre I – Définitions et notations : A. Logique des concepts

41

| 5. La première relation qu’on ait à formulera entre deux concepts est naturelle- [12] ment la relation de prédication, qu’exprime le jugement universel : «[Tout] a est b.»b Cec jugement, avons-nous dit, signifie, au point de vue de l’extension, que la classe des ad est contenue (tout entière) dans la classe des be . Nous avons donc besoin d’un signe pour représenter l’inclusion d’une classef dans une autre. Nous adopterons à cet effet le signe < (signe d’inégalité en Arithmétique et en Algèbre), et nous écrirons le jugement en question comme suit : a < b(1) Exactementg , cette formule signifie que chacun des éléments de la classe a est un élément de la classe b ; que chacun des a est un b ; càdh que tout a est b. Comme on le voiti , c’est là une relation claire et distincte entre les deux classes a et b : il suffirait qu’un seul a ne fût pas contenu dans la classe b pour que la relation n’eût pas lieu, et partout pour que la proposition précédente fût fausse. Il importe de remarquer que cette proposition ne nous apprendj rien de plus touchant l’extension de la classe b : elle n’implique pas que la | classe b contienne [13] d’autres éléments que les a ; elle n’implique pas non plus qu’elle ne contienne que les a. 6. Dans le cas où la classe b ne contient aucun élément de plusk que la classe a , càd aucun élément autre que les a, il est vrai de dire que tous les b sont des a, ce qui s’écrit : b < a. Mais d’autre part, dans ce cas, les deux classes a et bl sont identiques, càd ont le même contenu, puisque tout élément de l’une est élément de l’autre. Il convient d’avoir un signe pour représenter cette relation d’identité ; nous adopterons pour cela le signe = (signe d’égalité en Arithmétique et en Algèbre), et nous traduirons cette relation par la formule suivante : a = b(2) Il importe de remarquer, pour prévenir une confusion entre le sens mathématique et le sens logique de ce signe, qu’il ne représentem pas une simple égalité ou équivalence numérique, mais une identité absolue : il ne signifie pas par exemple(3) que les classes a et b ont le même nombre d’éléments, mais qu’elles ont exactement les mêmes éléments, qu’elles | ont un contenu, une composition identiques. [14] La définition précédente de l’égalité implique une proposition qu’il nous reste à énoncer explicitementn : (1) Qui se lit «a est contenu dans b» ou simplement «a dans b». – La formule : b > a sera considérée comme identique à la formule a < b. ||12bis| La relation b < a sera dite l’inverse de la relation a < b. L’inversion d’une relation consiste, en général, dans la permutation de ses deux éléments. TIntroduire et jusitifier le mot inclusion (subsumption, inégalité) antécédant, conséquent ; inférieur, supérieur.U (2) Qui se lit : «a égale b.» (3) Comme dans la théorie des ensembles.

42

Traité de Logique algorithmique

Toutesa les fois qu’on a en même temps b < a;(1)

a < b, on a aussi nécessairement : a = b,

et réciproquement. En effet, 1◦ quand on a la fois : «Tout a est b»b , et «Tout b est a» il en résulte que les deux classes a et b sont identiques. 2◦ Réciproquement, affirmerc que deux classes a et b sont identiques, càd que chaque élément de l’une fait partie de l’autre, c’est dire que «Tout a est b» et que «Tout b est a». Il y a donc équivalence complète entre la relation d’égalité (d’identité) d’une part et l’ensemble des deux relations (inverses) d’inclusion entre les mêmes termes. Cette équivalence peut servir à formuler la définition de l’égalité. Elle établit une connexion fondamentale entre la relation d’identité et la relation d’inclusion, ou (en style grammatical) entre la copule égale et la copule est. TRelation symétrique.U [15] | 7. Nous venons de définir les deux relations principales qu’on peut concevoir entre deux classes, les deux copules (< et =) sur lesquelles repose toute la Logique. Il convient d’observer que les termes de ces relations peuvent être aussi bien des individus que des classes. Par exemple, si dans l’inclusion : a b, ce qui montre bien que les deux relations sont inverses. même que, tout en définissant le nombree : une collection d’unités, on peut admettre le nombre 1, correspondant au cas particulier où la collection ne comprend qu’une seule unité. (3) Verso.||15bis| En règle générale, nous désignons les individus ou éléments des classes par des minuscules grecques. (2) De

Chapitre I – Définitions et notations : A. Logique des concepts

43

l’addition, la multiplication et la négation. Mais tandis que cette dernière porte sur un terme unique, les deux premières portent sur deux termes (au moins), et sont à proprement parler des combinaisons de classes. C’est elles que nous allons définir d’abord. La somme logique de deux classes a et b est l’ensemble de tous les élémentsa contenus dans ces deux classes (soit dans l’une soit dans l’autre, soit dans les deux). Elle s’écrit : a+b comme une somme arithmétique. Seulement, il importe de remarquer que s’il y a des éléments communs aux deux classes a et b, ils ne figurent pas deux fois dans la somme, mais une seule, de sorte que, dans ce cas, la somme logique n’a pas pour nombre la somme arithmétique des nombres de a et de b, mais un nombre plus petit. | Dans le cas contraire, càd lorsque les classes additionnées n’ont aucun élément [17] commun, elles sont dites disjointes, et la somme elle-même est qualifiée de disjointeb .(1) Le produit logique de deux classes a et b est l’ensemble de tous les éléments communs à ces deux classes (càd contenus à la fois dans l’une et dans l’autre). Il s’écrit : a×b

ou

ab

comme un produit arithmétique ; mais la multiplication logique n’a qu’une analogie toute formelle avec la multiplication arithmétique : car tandis que le produit de deux nombres entiers est toujours plus grand que chacun d’eux, le produit de deux classes est au plus égal à l’une d’elles, et généralement plus petit que toutes deux. Il n’ya donc entre le produit logique de deux classes et le produit arithmétique de leurs nombres aucune correspondance, même partielle, comme celle qui existe entre la somme de deux classes et la somme de leurs nombres.(2) (Verso) H|Il y a un cas particulier à prévoir ; c’est celui où les deux classes a et b n’ont [17bis] aucun élément commun. Leur produit existe-t-il encore, càd est-il encore une classe ? Nous pourrons bientôt répondre à cette question : Oui, leur produit existe, seulement il est nul, càd identique à la classe 0. [12.]e I |10.1 On peut maintenant définir deux classes particulières qui jouent un rôle [19] cardinal en logique : la classe 0 et la classe1. On les définit formellement comme suit : 0<x

x