Hugh Maccoll After One Hundred Years

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Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences

15-1 | 2011 Hugh MacColl after One Hundred Years

Amirouche Moktefi and Stephen Read (dir.)

Electronic version URL: http://journals.openedition.org/philosophiascientiae/351 DOI: 10.4000/philosophiascientiae.351 ISSN: 1775-4283

Publisher Éditions Kimé

Printed version Date of publication: 1 April 2011 ISBN: 978-2-84174-551-7 ISSN: 1281-2463

Electronic reference Amirouche Moktef and Stephen Read (dir.), Philosophia Scientiæ, 15-1 | 2011, “Hugh MacColl after One Hundred Years” [Online], Online since 01 April 2011, connection on 15 January 2021. URL: http:// journals.openedition.org/philosophiascientiae/351; DOI: https://doi.org/10.4000/philosophiascientiae. 351

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Tous droits réservés Preface Amirouche Mokteﬁ LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université & IRIST (EA 3424), Université de Strasbourg, France Stephen Read Department of Logic and Metaphysics, University of St. Andrews, Scotland

“It all turns upon that little word if, Mr. Burton.” [MacColl 1891, 955]

This volume is devoted to the life and work of Hugh MacColl (1837- 1909), the Scottish mathematician, philosopher and novelist. Although MacColl is nowadays remembered mainly for his contributions to the science of logic, his work covers many other domains and witnesses the richness and variety of his intellectual interests. MacColl’s early writings (from 1861 onwards) were devoted to mathematics, notably geometry, algebra and the theory of probabilities. In the late 1870s and early 1880s, he published several papers on logic that brought him to the attention of his contemporaries. In these writings, he expressed his preference for propositional calculus instead of the term/class approach, widespread in his day. From around 1884, and for about thirteen years, he devoted much of his time to literature and published two novels. In the late 1890s however, he returned to his logical investigations and published several papers that led him into controversy with some of the leading logicians of his time, notably on the issues of modalities, implication and existence. Eventually, in 1906 he published a comprehensive exposition of his logical theory [MacColl 1906]. In his late years, he continued to write on science, philosophy and religion and ultimately published a collection of essays on these issues [MacColl 1909].

Philosophia Scientiæ, 15 (1), 2011, 1–6. 2 Amirouche Mokteﬁ & Stephen Read Greetings from Boulogne

The richness and originality of MacColl’s intellectual production made him well known in his lifetime, but his name almost vanished in secondary literature after his death. Except for some occasional references, one has to wait until the 1980s to see MacColl’s work rediscovered thanks to the research carried out at Erlangen by Christian Thiel, Volker Peckhaus and Anthony Christie. This work was pursued in the 1990s, mainly by Michael Astroh and Shahid Rahman and culminated in the organisation of a colloquium on ‘Hugh MacColl and the Tradition of Logic’, that took place in Greifswald (29 March - 1 April 1998), the proceedings of which appeared as [Astroh & Read 1998]. Those proceedings together with a biographical paper published shortly afterwards [Astroh, Grattan-Guinness & Read 2001; reprinted in this volume] became the starting point of all the research carried out since, notably by Rahman and his colleagues in Lille [Rahman & Redmond 2007]. In 2009, on the occasion of the centenary of MacColl’s death, a commemorative meeting was held in Boulogne-sur-Mer (France), the city where he lived from 1865 to his death in 1909. All but one of the papers in the present volume were first presented at that centenary meeting. Although MacColl might have gained some local recognition in his lifetime, it appears that he left little impression in Boulogne as his name and work sank into oblivion. In this respect, the centenary meeting was an opportunity to rediscover MacColl in his own town, as his grave was restored on this occasion and a reception was organised at the city hall. It is obvious that MacColl’s life in Boulogne influenced his scientific career, achievements and acquaintances, as it kept him outside of the Victorian intellectual establishment, although maybe not so very far (see Stein Olsen’s paper). Apparently, this position as an outsider both stimulated MacColl’s original thinking and prevented him from gaining more academic recognition. A simple look at MacColl’s biography shows that one remarkable and invariable feature of his life was his search for recognition (see the paper by Astroh, Ivor Grattan-Guinness and Stephen Read). In his papers, MacColl often claimed the superiority of his logical method over those of his contemporaries, and frequently insisted, in an unsubtle manner, that he had discovered his methods alone, rarely recognising any influence from authors who preceded him or with whom he interacted (see for instance Irving Anellis’ paper on MacColl’s relationship with Peirce and Schröder). The Boulogne meeting was an opportunity to review the situation on MacColl’s legacy one hundred years after his death. It was also an occa- Preface 3 sion to make a survey of the progress in scholarship since the Greifswald meeting, more than ten years ago, as many participants attended both meetings. One feature that is confirmed, as one can easily observe in this volume, is the plurality of MacColl’s interests, and hence, the interdisciplinary approaches of his scholars. Although logic held a prominent place in both meetings, other intellectual domains were also discussed, namely mathematics and literature. They shed new lights into little-known but important aspects of MacColl’s personality and thinking, as are his activities as a contributor to the mathematical columns of the Educational Times (see James Tattersall’s paper) and his duties as a mathematical reviewer for the journal The Athenaeum (see Francine Abeles and Amirouche Moktefi’s joint paper).

On the way to pluralism

Even within logical studies, the focus is no longer on the MacColl- Russell relationship. MacColl is now seen as a logician working within a large intellectual community in which he was certainly a respected member, although not necessarily much followed. In this respect, MacColl’s position as a “logical” outsider should be moderated, for most symbolic logicians were in any case outsiders within the academic scene which was dominated by philosophically rather than mathematically trained logicians. In his first logical phase, MacColl was just one among many logicians who invented a symbolic notation and claimed it to be the most effective for solving logical problems. We reproduce at the end of this volume a rarely cited paper by MacColl which is very useful in coming to understand how he worked in those years, as it explains the principles on which MacColl based his symbolical notation, and provides an account of its development. Whatever his position within the intellectual scene of his time, MacColl proved to be an ingenious and innovative logician. His contributions cover such areas as propositional logic (see Jean-Marie Chevalier’s paper), the logic of fictions (see Rahman’s paper) and modalities (see Fabien Schang’s paper). One particular issue that raised some debate among historians of logic concerns MacColl’s place as a precursor of logical pluralism. The contributions of Grattan-Guinness and John Woods to this volume tend to show that MacColl was not a pluralist in the modern sense of that word, although he certainty raised questions and made several steps in that direction. As often happens with priority disputes, a close look shows indeed that historical developments are too complex to 4 Amirouche Moktefi & Stephen Read credit a single man and most innovations turn out to be the culmination of slow and gradual improvements. MacColl might not have judged such a reading as “either just or generous” [Astroh, Grattan-Guinness & Read 2001, 92], but that is the way it is. However, thanks to his explorations beyond classical logic, MacColl certainly did contribute to opening the way to pluralism. MacColl lived and worked within a crucial period in the history of logic. His logical writings help to understand both the mature phase of the algebra of logic and the early years in the development of logistics, as Louis Couturat used to call those two mathematical logic trends. It is the wish of the editors that the present volume will convince readers of the historical and philosophical interest and importance of MacColl’s work as a logician, and that it will also stimulate their curiosity to discover the other writings of this multi-facetted author, notably in mathematics and literature. An edition of MacColl’s scientific correspondence (both private and public) and a comprehensive bibliography of his writings are currently in preparation. It is hoped that these achievements will promote in the near future the project of editing MacColl’s Collected Writings, as already alluded to in [Astroh & Read 1998, iv].

Acknowledgments

This volume would not exist without the help of all those who contributed to the organisation of the MacColl centenary meeting (Boulogne-sur-Mer, 9–10October 2009). We are especially grateful to Bruno Béthouart, Jacques Dubucs, Gerhard Heinzmann, and Shahid Rahman. We would also like to thank Michael Astroh, Sandrine Avril, Anny Bégard, Christian Berner, Pierre-Édouard Bour, Peggy Cardon, Emmanuelle Jablonsky, Christian Mac Coll, Tony Mann, Gildas Nzokou, Max Papyle, Bernard Quéhen, Monique Randon, Manuel Rebuschi, Juan Redmond, Isabelle Roblin, Fabien Schang, Oliver Schlaudt, Sandrine Souraya, Florence Thill, Tero Tulenheimo, Catherine Wadoux, and Scott Walter for their help and support at different stages of the preparation of the centenary meeting or of the present volume. Finally we would like to express our gratitude to all the authors, referees and proofreaders of the papers included in this volume. The achievement of this project benefited from the material support of different institutions to which we express here our debt: IHPST – UMR 8590 (Université Paris,1 – CNRS), STL – UMR 8163 (Université Lille 3 – CNRS), LHSP AHP – UMR 7117 (Nancy-Université – CNRS), the British Society for the History of Mathematics, IRIST – EA 3424 Preface 5

(Université de Strasbourg), HLLI – EA 4030 (Université du Littoral – Boulogne sur Mer), Université Nancy2, Université du Littoral Côte d’Opale, MSH Lorraine (EManMath project), Région Lorraine, and Ville de Boulogne-sur-Mer.

References

Astroh, Michael, Grattan-Guinness, Ivor, & Read, Stephen 2001 A survey of the life and work of Hugh MacColl (1837-1909), History and Philosophy of Logic, 22, 81–98. Astroh, Michael, & Read, Stephen (eds.) 1998 Hugh MacColl and the Tradition of Logic, special issue of the Nordic Journal of Philosophical Logic, 3 (1). MacColl, Hugh 1891 Mrs. Higgins’s strange lodger, Murray’s Magazine, 10, 937-976. 1906 Symbolic Logic and its Applications, London: Longmans, Green, & Co. 1909 Man’s Origin, Destiny, and Duty, London: Williams & Norgate. Rahman, Shahid, & Redmond, Juan 2007 Hugh MacColl: an Overview of his Logical Work with Anthology, London: College Publications.

A survey of the life of Hugh MacColl (1837-1909) Michael Astroh Institut für Philosophie, Universität Greifswald, Germany Ivor Grattan-Guinness Middlesex University, England Stephen Read Department of Logic and Metaphysics, University of St. Andrews, Scotland

[Editors’ note: This article appeared ﬁrst in the journal History and Philosophy of Logic, 22, 2001, 81–98. It is reprinted here with the kind permis- sion of the authors and the publisher. The original article can be consulted on the journal’s web site at: www.informaworld.com. The French abstract below has been provided by the editors.]

[Résumé : Le logicien écossais Hugh MacColl est bien connu pour ses contributions innovantes aux logiques modales et non-classiques. Cependant, jusque- là, nous disposions de peu d’informations biographiques sur son cheminement académique et culturel, sa situation personnelle et professionnelle, et sa position au sein de la communauté scientiﬁque de la période victorienne. Le présent article présente un certain nombre de découvertes récentes à ce sujet.]

Abstract: The Scottish logician Hugh MacColl is well known for his innovative contributions to modal and non-classical logics. However, until now little biographical information has been available about his academic and cultural background, his personal and professional situation, and his position in the scientiﬁc community of the Victorian era. The present article reports on a number of recent ﬁndings.

Philosophia Scientiæ, 15 (1), 2011, 7–30. 8 Michael Astroh et al. Introduction

Contrary to a widespread assumption the modern history of modal logic did not start with C. I. Lewis’ Survey of Symbolic Logic [Lewis 1918]. His eminent work was preceded by some 20 years by H. MacColl’s fifth article on ‘The Calculus of Equivalent Statements’. This article was read at the London Mathematical Society on 12 November 1896. Some months later it was published in the Society’s Proceedings [MacColl 1896-1897]. During the following years MacColl presented his logic primarily in a series of articles on ‘Symbolical Reasoning’. All of them were published between 1897 and 1906 in Mind, the journal in which C. I. Lewis first wrote on strict implication (in 1912). In Symbolic Logic and its Applications [MacColl 1906] MacColl set forth his final version of a logic for propositions qualified as certain, impossible, contingent, true or false. A main fragment of his logic has turned out to be equivalent to the logic T introduced by Feys (1937, 1938) and von Wright (1951) many decades later, i.e. to the smallest normal epistemic modal logic. 1 In recent years a number of investigations into MacColl’s logic and its philosophical background have been carried out, and an edition of his Collected Papers on Logic, Mathematics and Philosophy is being prepared. 2 Until now, scarcely anything was known about MacColl’s life and person alike. The present article sums up a number of recent findings.

Scottish origins

Hugh MacColl was born in Strontian, Argyllshire, on 11 or 12 January 1837. 3 Other sources say that he was born in the Scottish Highlands, probably in Glenfinnan on the borders of Invernessshire and Argyllshire. 4 Most likely, MacColl’s second marriage contract, naturally signed by himself, is the more reliable source. This document confirms that he was the son of John MacColl and Martha, née Macrae. Hugh, named after his grandfather, was their youngest child. He had three brothers and two sisters. John MacColl was a shepherd and tenant-farmer from Glencamgarry in Kilmalie (between Glenfinnan and Fort William), and he married Martha Macrae in the parish of Kilmalie on 6 February 1823. Their first child, Alexander, was born on 6 April and baptised in the Church of Scotland on 22 April 1824. The marriage and baptism are recorded in the Parish Registers of the Church. However, John MacColl and his family belonged to the Episcopal Church of Scotland (as did many MacColls—and also Sir Walter

1. Cf. [Read 1999]. 2. For a survey cf. Astroh and Read [Astroh & Read 1999, Introduction, iv]. 3. Cf. [Ville de Boulogne 1887]. 4. Cf. Poggendorﬀ et al. [Poggendorﬀ 1863-1938, vol. 3, pt. 2, 850]. A survey of the life of Hugh MacColl (1837-1909) 9

Scott, for example), so that none of the later births is registered in the records of the established Church. The second son, John, was born in 1826 or 1827 at Glenfinnan. Apparently, he suffered from asthma. As late as 1875 his bad health made him and his family emigrate to Australia. He died in 1892 at Clifton Hill, Victoria. Around 1829 a first daughter, named Christine, was born. There is no further information about her or about her younger sister who, seemingly, was also older than Hugh.

Hugh MacColl in 1886 5

The third son, Malcolm (known in the family as ‘Callum’), born at Glenﬁnnan on 27 March 1831, was the most famous of the MacColls’ children. A memoir and edition of his voluminous correspondence was published by G. W. E. Russell in 1914 [Russell 1914, esp. 1–16]. It contains a useful chapter on Malcolm’s early life, and provides most of what little evidence we have concerning his younger brother Hugh’s family background. It is also a delightful book to read; Malcolm MacColl was clearly quite a character. Martha Macrae, who came from Letterfearn in Kintail, was herself a Presbyterian who joined the Episcopal Church on her marriage. She spoke only Gaelic. According to Russell’s memoir, John MacColl was suﬃciently educated to teach his boys elementary Latin, Greek and mathematics. However, he died at the age 45 when his youngest son was only three years old. This early loss had a far-reaching impact at least on Hugh MacColl’s life. In particular, his scholarly education did not respond adequately to the needs of his talents. He was almost 40 years old and living on private teaching in

5. By courtesy of Mrs. Rosemarie Maconchy. 10 Michael Astroh et al. northern France when he took his BA in mathematics as an external student of the University of London. Likewise, Hugh’s elder brother John was not content with his early formation as a schoolmaster. When he was nearly 50 years old he managed to receive an Oxonian BA in classics. The importance of the father’s untimely death for Hugh MacColl’s later life and career is reflected in the two novels, Mr. Stranger’s Sealed Packet [MacColl 1889] and Ednor Whitlock [MacColl 1891b] he published in 1889 and 1891 respectively. 6 In both stories the sudden death of the hero’s father occurs before the son’s professional education is finished, and deeply affects it. Mr. Stranger’s father states in his last will and testament that his son Joseph’s education ‘henceforward was to be on other lines. Classics were to be completely thrown aside, and I was to devote myself wholly to science, and especially to mathematics, astronomy, chemistry, electricity, and practical mechanical engineering—a sufficiently wide curriculum’ [MacColl 1889, 22–23]. Ednor Whitlock, on the other hand, risks not receiving an academic education at all. In both novels the loss of the father also has the consequence that the protagonist moves to another country. The sealed parcel Mr. Stranger inherits from his father contains the plan of a rocket that allows him to travel to Mars. When Ednor Whitlock and his sister lose their parents, economic reasons make them accept inferior teaching jobs. Both children seek engagements as teachers. They find work in boarding schools at Blouville, a fictitious place, somewhere on the coast of northern France. Although information on Hugh MacColl’s life is scarce, the following quotations from his second novel undoubtedly allude to the author’s personal condition: If Ednor Whitlock had gone up to Cambridge to try for a scholarship it is not unlikely that he would have succeeded, for his mathematical abilities and knowledge were considerably above the average; but a heavy and unexpected calamity destroyed the project of a university career. His father, while ministering consolation at the sick bed of a poor old woman who had been struck down by typhus fever, caught the fell disease, and died from it after a few days’ illness. His mother’s death followed soon after. Worn out by anxiety and constant attendance upon her husband, she caught the infection, and in spite of the most devoted nursing, both by her son and her daughter, passed away before their eyes. [MacColl 1891b, 9] The two novels confirm the few specific autobiographical details actually known about their author. In view of the various correspondences between factual and literary information, these novels help us grasp the problems and perspectives that have shaped his personal form of life. 6. MacColl’s literary writings and their cultural background in Victorian Britain are discussed in [Astroh 1999] and [Olsen 1999]. A survey of the life of Hugh MacColl (1837-1909) 11

After the death of John MacColl his widow moved with her children to Letterfearn and then to Ballachulish, where the children started to learn English. In 1841, Hugh’s elder brother Malcolm, then aged nine, was living alone with his aunt and grandmother, both called Anne Macrae, in Letterfearn, on the shore of Loch Duich. 7 Malcolm did so well at school that a wealthy lady paid for him to attend a seminary in Dalkeith near Edinburgh where schoolteachers were trained. He taught at Callander, Stonehaven and Perth. In 1854, at the age of 23, Malcolm was admitted to the relatively newly opened Theological College at Glenalmond near Perth. In 1856 he was ordained Deacon of the new Mission of the Episcopal Church at Castle Douglas in Kirkcudbrightshire. He was ordained priest in St Mary’s, Glasgow by the Bishop of Glasgow in August 1857. 8 At this time Malcolm was supporting his younger brother Hugh in his studies, probably preparatory to his becoming a schoolmaster. He had hopes to be able to support Hugh to study at Oxford. However, Malcolm was already the outspoken controversialist for which he achieved fame in later life. He became involved in the dispute over the proper interpretation of the Eucharist which divided the Episcopal Church in 1857-1858, refusing to sign an address of loyalty to the Bishop, and in fact was party to a ‘Remonstrance’ against the Bishop’s position. Consequently, the Bishop dismissed him from his post at Castle Douglas. He was therefore incapable of providing for Hugh as he had hoped, and too proud to accept oﬀers of help from others. Eventually, he took up the cause of Gladstone, as a pamphleteer (‘the best in England’ in Gladstone’s words), once he had overcome the barrier to ordination in the Church of England which was enshrined in law in the 18th century (at the time of the Jacobite rebellion) and not repealed until 1864. In the meantime, Hugh had taken up various posts as schoolmaster in England from 1858, 9 until for unknown reasons he left the country. In 1865 MacColl moved to France. For the rest of his life he settled at Boulogne-sur- Mer. Little is known about the kind of life he was leading there; but even less is known about his intellectual development or his personal circumstances during the ﬁrst three decades of his life.

MacColl’s Boulogne

Initially, MacColl’s decision to earn his living in northern France might have been a provisional one. But it seems to have been well thought out. Especially in Britain, Boulogne was known for its prosperity and the refined way of life it could offer to well off or, perhaps, to educated persons. No other town on the French side of the Channel adapted technological progress to its

7. Census for 1841, West Register House, Edinburgh. 8. These details and the following paragraph are drawn from Russell’s Memoir. 9. [Poggendorff et al 1863-1938]. 12 Michael Astroh et al. geographical condition in so comprehensive a manner. 10 Here, it was almost natural to learn from Britain’s industrial superiority. Since 1848 Boulogne had been connected to the French railway network. Five years earlier the South Eastern Railway had been extended to Folkestone. When MacColl lived at Boulogne it took five and a half hours to reach London; in 1880 Paris was at a distance of three hours. For decades Boulogne rivalled Dieppe and Calais. At the turn of the century after continuous improvements of its docks the seaport regained its former superiority. During these years its importance for France’s fishing industry certainly excelled its role as a traffic port. Throughout the second half of the nineteenth century—despite a number of recessions—Boulogne’s com- merce and diversified industry were growing vigorously. MacColl was living in a prospering town with close economic and cultural links with Britain: Comme le progrès venait alors souvent d’Angleterre qui avait ef- fectué sa révolution industrielle avant les autres pays, il péné- trait en France par Boulogne dans les domaines les plus divers, qu’il s’agisse des techniques textiles, de l’industrie des plumes métalliques, de la vaccination de Jenner, de l’éclairage au gaz, des améliorations nautiques, des techniques de l’hôtellerie, de la doctrine artistique néogothique de Pugin, de nombreux sports: courses de chevaux, régates, golf, football, tennis. . . [Hilaire 1998b] On both sides of the Channel Boulogne was also renowned as one of the most elegant seaside resorts and spas of the Belle Époque, much frequented by the elites of both nations. At the end of the July Monarchy about 3000 of its almost 40 000 inhabitants were British subjects. 11 But the completion of the French railway network as well as the Franco-Prussian war of 1870 led to a steady decline of their number; by the turn of the century only a third of the British had kept their French domicile. MacColl’s Boulogne was a vivid French port with a truly British flavour. At this time, English shops and pubs, various Protestant churches, English surgeons and undertakers, local newspapers and regular theatre productions in English were natural facets of a balanced and liberal form of urban life. At the peak of European nationalism, living at Boulogne could have meant that there is not only ‘nothing sacred about signs’, as MacColl used to hold, 12 but likewise about cultural identity. However, the British inhabitants of Boulogne kept up their genuine forms of interaction and communication. Merridew’s English Library and its club- like Reading and Conversation Rooms fully equipped with Britain’s major

10. For a detailed report on Boulogne’s economic, political and social history before 1914 cf. [Oustric 1998a], [Oustric 1998b] and [Hilaire 1998a]. 11. If not otherwise specified our information about Boulogne’s British community stems from [Hilaire 1998b]. 12. Cf. [MacColl 1906, § 1, 1–2]. A survey of the life of Hugh MacColl (1837-1909) 13 newspapers and reviews, like the Westminster or the Edinburgh Review, was a well known institution of Boulogne’s cultural life. Apparently, MacColl frequented the place: a letter to Bertrand Russell on the Reading Room’s writing paper has been preserved. Merridew’s [Merridew 1866] Guide to Boulogne was published in French and English, and sold on both sides of the Channel. It provided not just general tourist information but also specific references to various colleges or schools at Boulogne offering a French education under Protestant conditions. Here, already in Europe but still close to the UK, a young person of some standing could receive a schooling sufficiently French to acquire or to preserve a higher- ranking social position in British society. Even in the 1930s Boulogne’s Collège communal, then called Collège Mariette, counted 35 per cent of British pupils among its boarders [Vasseur n. d.]. MacColl came to Boulogne in order to earn his living as a teacher. Professional and intellectual aspiration let him become a part of the local, continental enclave of British society. Gradually, time turned his decision into a definite form of life. But still, and perhaps under the encouragement of his brother Malcolm’s growing political fame, Hugh hoped to return to his cultural mainland, and to receive some kind of public recognition for his pioneering contributions to logic. Even as late as 1901, MacColl when 64 years old recommended himself to Bertrand Russell as a lecturer in logic [MacColl 1901a]. Due to a University of London Parliamentary Act of 1898 some changes to the university’s teaching arrangements were made. Apparently, MacColl got to know about these changes.

Marriage and family

While still in Britain MacColl married Mary Elisabeth Johnson of Loughborough in Leicestershire. Two years younger than Hugh, she accompanied him to Boulogne where their family soon started to grow. All five children, four girls and a boy were born there. In April 1866 MacColl reported to W. J. C. Miller, then mathematical editor of the Educational Times, on the birth of his first daughter, Mary Janet [MacColl 1866b]. Martha Christina followed in October 1867, Flora in May 1869, Hugh Ernest in May 1871 and Annie Louise in November 1873 [Ville de Boulogne 1866-1873]. Hugh Ernest was to study at Christ Church College Cambridge and pass a legal career in Burma. Throughout these years the MacColls were living in the centre of Boulogne-sur-Mer at 4, rue du Temple. 13 In this area where mostly crafts- men and employees were living the family occupied a decent flat. They 13. The street is now called rue Ernest-Hamy, situated between the former rue Siblequin and the former rue des Vieillards, presently rue Faidherbe and rue Félix- Adam respectively. For general information on the streets of Boulogne cf. [Charles 1992]. 14 Michael Astroh et al. lived in the close vicinity of the original seat of the Collège communal where MacColl was teaching, at least initially. Nowadays it is the Université du Littoral-Côte d’Opale. In May 1883, C. S. Peirce intended to visit MacColl at Boulogne. It is not known whether they actually met; but in a letter written shortly before the planned visit, MacColl informed Peirce about the circumstances of his personal life, and writes: ‘We have been in Boulogne now eighteen years. We have five of a family—four girls besides the little boy already mentioned. We live a very quiet life.’ [MacColl 1883]. Meanwhile the MacColls had moved to an apartment at 73, rue Siblequin 14 even closer to the town’s commercial centre. At least during these months and during the family’s immediate future this quiet life was not a happy one. The letter to Peirce reported on a severe and long-lasting illness of MacColl’s wife: . . . You will be glad to hear that our little boy is now quite out of danger & appears to be quite well again. I wish I could say the same about my wife; still, she too is much better than she was when I wrote to you last. She went out the day before yesterday in a bath-chair—her first going out for more than eight months. I hope the coming warm weather will do much to set her up again. The beginning of her illness was a severe cold which the 15 doctor who then attended has culpably neglected as of no importance. [MacColl 1883] Mary Elisabeth MacColl died on 2 February 1884. We do not know the cause of death; nor is there any personal information on his reaction to her death and the loss of his children’s mother. Mary Janet, the eldest daughter, was 22 years old, while her youngest sister Annie Louise was still a girl of 11 years. MacColl’s published novels both tell of the death or severe illness of a major female character. In Mr. Stranger’s Sealed Packet the hero’s Martian wife dies from a lack of resistance against ‘food and air’ on earth. In Ednor Whitlock Mrs. Nora Kent, the mother of the main female character, repeatedly suffers from inflammation of the lungs. 16 Within just one year her life is twice seriously threatened. When the second attack is imminent Mr. Kent, the worried husband, says to her: ‘You are rather feverish, dear,. . . It’s probably nothing but the beginning of an ordinary cold. Still as small things may become serious if neglected, I will send at once for Dr. Karcher.’ [MacColl 1891b, 268 and likewise 178–179] In view of the quoted letter to C. S. Peirce it is highly probable that MacColl is drawing here on personal experience.

14. Presently this address is 119, rue Faidherbe. 15. MacColl scratched out ‘our’. 16. Mr. Stranger’s father dies of the same disease. A survey of the life of Hugh MacColl (1837-1909) 15

Three years after the death of his first wife, on 17 August 1887, MacColl married Mlle Hortense Lina Marchal, at Boulogne-sur-Mer. An official announcement of the event was sent to Peirce [MacColl 1887]. MacColl was now 50 years old, 12 years older than Mlle Marchal who came from Thann, Alsace. In several respects this union was a mutually thoughtful choice; the couple built up a harmonious family life never lacking a solid economical basis. Both were Protestants, and both had a keen interest in teaching [Anonymous 1918]. The marriage certificate indicates that the parents of Hortense Marchal were living on a private income, while her brother Jules Marchal, and her brother-in-law Gustave Busch, were shopkeepers at Boulogne. The fact that a marriage contract was put up might point to some prosperity to be dealt with; however, there is no evidence that MacColl ever owned property at Boulogne [Ville de Boulogne 1887]. Most likely, the families Marchal and Busch had moved to Boulogne from Alsace, which since 1871 had been possessed by Germany. In Ednor Whitlock MacColl refers extensively to this political context: Ethel Whitlock, the hero’s sister, has become an assistant teacher in northern France while her brother is still living with the family of their father’s successor: Ethel’s life at mademoiselle Lacour’s pensionnat was pretty much what she had anticipated. Her main work was to teach English to the French pupils, of whom there were altogether fifteen, including the eight externes. With the English girls she had little to do, at least during school hours. She gave them a short English dictation every day, and three lessons a week in arithmetic—that was all. Her very imperfect knowledge of French, coupled with her youth and inexperience, exposed her to some trials at first, but as she and her pupils got to understand each other better these gradually diminished. Human nature is pretty much the same everywhere, and French girls, like English girls, have their good as well as their bad points. [MacColl 1891b, 38] One day, the furious reaction of a pupil to a letter from her cousin leads to an extensive argument between this girl and the school’s German teacher, Frl. Hartmann: Among the boarders was a little, dark-eyed, fiery Alsatian, of about sixteen, named Suzanne Müller, whose parents had sold their property in Alsace for a fraction of its value, and had emigrated into the interior of France in order that their sons might not be obliged to serve in the German army. One evening, after class, this girl received an unusually bulky letter with the German stamp on it: it was from a cousin in Alsace—her father’s brother’s daughter. Mademoiselle Lacour, Ethel, the German governess, and all the boarders were present. [MacColl 1891b, 39] The argument that follows is not worth reporting. Finally, Mlle Lacour calms down the women’s excitement and restores the emotional balance on which a 16 Michael Astroh et al. pedagogical institution naturally depends. Her arguments coincide well with the narrator’s, and presumably MacColl’s own benevolent, but, of course, re- served excuse for the girls’ treatment of young and unexperienced Ethel: ‘. . . Let us . . . hear no more upon the subject. I don’t like these discussions on our different nationalities. If we read out histories, be we French, German or English we shall all find things to blush at as well as things to be proud of.’. . . Thus ended the discussion which Suzanne Müller’s letter had started. It had gone much further than the prudent maîtresse de pension had foreseen, and she made a mental resolve that nothing of the kind should happen again, at any rate in her presence. Fräulein was a necessity in her establishment. [MacColl 1891b, 46] Together with her sister, Mme Busch-Marchal, Hortense Marchal was running a well-known ‘pensionnat de jeunes filles’. Possibly, MacColl took his second wife as a model for Mlle Lacour. At any rate, MacColl’s novel depicts the social and personal context of his later life at Boulogne. He himself was teaching at the sisters’ school [Anonymous 1909]. When MacColl married Hortense Marchal the pensionnat was apparently situated in 37, rue Basse-des-Tintelleries opposite to a park. 17 Though the name of the quartier des Tintelleries refers back to an ancient cloister this area belonged to the most modern and elegant parts of the Boulogne of the late 19th century; it even had its own railway station. The building was large enough to house even all members of the new family. As the domicile’s quality indicates, MacColl’s marriage with Hortense clearly was an economic and social improvement of his condition. 18

Private teaching

During his ﬁrst years at Boulogne MacColl earned his living as a schoolmaster for mathematics and English at the Collège communal. 19 An advertisement

17. This address is now 45, rue Basse-des-Tintelleries. 18. A detailed description of the quartier des Tintelleries is given in [Tiébaut 1998]. Even a picture of the house of the boarding school is included (p. 333). 19. Cf. [Anonymous 1910]. In Merridew’s Guide to Boulogne the college’s peda- gogic profile is described as follows: This important school is supported by municipal grants, and the revenue arising from payments made by pupils; it is placed under the immediate control of a principal, a chaplain, and a large staff of professors, who are all appointed by the Minister of Public Instruction, under whom the pupils receive an educational course, comprising French, English, German, Latin, Greek, History, Drawing, Mathematics, Natural History, Natural Philosophy, Chemistry, and Rhetoric; in short, such an instruction as will enable a pupil to take a Bachelor’s degree, to enter the naval or military schools, to fill public A survey of the life of Hugh MacColl (1837-1909) 17 from 1866 in which MacColl offers private tuition confirms this employment [Merridew 1866]. But in later advertisements he never refers to himself as a ‘Professor at the Communal College’. In June 1870 the publishing company Longman advertised MacColl’s Algebraical Exercises and Problems in its Notes on Books in the following terms: By Hugh MacColl, late Mathematical Master at the Collège Communal, Boulogne-sur-Mer. [Anonymous 1870] In view of this reference to MacColl’s professional situation the letter to Peirce makes it rather likely that from about 1870 onwards MacColl’s income was essentially derived from private lessons: My income, a very fluctuating & precarious one, is derived entirely from my private teaching. [MacColl 1883] In contrast with this description of his professional situation, an obituary published in La France du Nord on 30 December 1909 depicts MacColl as a successful teacher of the Collège communal. It has not been possible to identify MacColl as one of its professors. The college’s advertisements and brochures do not mention him as being responsible for a course on any subject. However, both the English and the French edition of Merridew’s Guide to Boulogne regularly publish his private advertisements. 20 The example points to the major purpose of MacColl’s instruction at Boulogne. When the hero of his second novel moves to France, he starts teaching mathematics at the boarding school of ‘Mr. C. H. Kent, MA, of Trinity College Cambridge’. His professional condition is that of MacColl himself: This Mr. Kent . . . was what is commonly called a crammer. He prepared young men for the English competitive examinations, especially the examinations for the Army and Indian Civil Service.

situations, or to follow any of the liberal professions. If the parents or guardians desire a commercial education only, the rules of the college admit of that particular branch being exclusively taught, . .. Boarders, half-boarders, and day-scholars are received. Religion is in- culcated, and those belonging to the Protestant faith receive instruction in their own religion from proper tutors, and attend divine service at the Protestant Church in Rue du Temple every Sunday. Corporal pun- ishments are never resorted to. [Merridew 1882, 56–57]

20. One of them reads as follows: HUGH M’COLL, BA (LONDON UNIVERSITY) Gives Lessons in MATHEMATICS, CLASSICS, ENGLISH, LOGIC, with all the other subjects of the University Course, and prepares Young Gentlemen for the Naval and military Examinations. 73 RUE SIBLEQUIN 73. [Merridew 1882] This self-description was not correct. MacColl’s degree was awarded by the University of London (UoL); “London University” was the name of the private company which had launched itself in the mid 1820s, and which was renamed “University College London” in 1836 when the UoL was established as a degree-examining and -awarding body. 18 Michael Astroh et al.

He had selected Blouville for his scholastic establishment in order to give his pupils greater facilities for learning the French language, for which a good many marks were allotted in the said examinations. [MacColl 1891b, 52] In most of his writings MacColl presented himself as a serious academic with high moral standards and outspoken religious beliefs. Nevertheless, and foremost in this very context, he was able to reﬂect upon his professional condition from an ironical, if not depressing point of view: Ignorance of Nature’s laws and errors of judgement bring their punishment no less surely than deliberate crime. This is Nature’s way of teaching; and Nature, as already stated, is but another name for God—God as manifested in the working of his laws. God’s teaching through the operations of his laws is slow, sure, and thorough—the exact opposite of that of the modern crammer. He allows men to commit errors, and he allows them to commit crimes (which are the most grievous of errors) in order that the discomforts and suﬀerings which those errors and crimes sooner or later entail, here or hereafter, may in the long-run purify their souls and accelerate their progress upwards. [MacColl 1909, 56–57]

Academic misfortunes

Almost nothing is known about MacColl’s early life in Britain. It is thus difficult to know that the specific reasons for his emigration or to understand the various conditions that obstructed the intended academic career. At the very least, economic difficulties prevented him from studying as a young man. In Ednor Whitlock the name of Mrs. Kent’s doctor, Mr. Karcher, points to the possibility that MacColl’s move to France resulted from his personal con- tacts with well-established academics. In a letter from 1866 to W. J. C. Miller he writes: The words of Prof. Sylvester . . . must have had reference to the little pamphlet on Ratios which I published in 1861. I left one of these with my friend Mr. Karcher about a year ago, but I had no idea that he had shown it to Prof. S. [MacColl 1866a] By then James Joseph Sylvester (1814-1897), one of the most influential mathematicians of the Victorian era, was teaching at the Royal Military Academy at Woolwich, London. He shared a keen interest in poetry with his colleague, Prof. Théodore Karcher. At least from 1867 onwards MacColl’s friend edited a number of readers of French literature as well as courses of translation from English into French—books most valuable for a scholarly establishment like the fictitious one of Mr. Kent at Blouville. 21 21. Cf. the following examples: [Brette et al 1867], [Cassal & Karcher 1876, 1885]. A survey of the life of Hugh MacColl (1837-1909) 19

The ‘pamphlet on ratios’ from 1861 mentioned earlier is MacColl’s earliest known publication, when he was 24 years old. It has not so far been possible to locate a copy of this text. 22 At least from 1865 onwards MacColl contributed Questions, Problems and Solutions to the Educational Times. He participated in these discussions until 1872. Then, for about five years, he abandoned this public forum of mathematical investigation, most likely in order to prepare for the BA in mathematics he took in 1876 at the University of London. He gained his degree, with a ‘second division’ grade, by private study in Boulogne between 1873 (matriculation) to the degree in 1876, with an ‘Intermediate Arts’ qualification at first division en route in 1874; he did not proceed to take the Honours Examination. 23 In later terms he was an external student of the University, not fulfilling residence qualifications; he may even have taken the examinations in France [University of London 1877, 60]. Three years earlier than Hugh his older brother John MacColl could afford to study classics at Oxford University: John McColl, second son of John McColl, land steward, of Kilmarly, Inverness-shire, matriculated on 15 October 1870, aged 44; he was an ‘unattached’ or non-collegiate student. The pass list published in the Oxford University Gazette in December 1873 records that he satisfied the examiners in the School of Literae Humaniores in Michaelmas Term 1873 but was not awarded honours. The degree of BA was conferred on him in 1874. [Bailey 2001] On 22 May 1875 John MacColl, his second wife, Theresa Whitgreave, and the seven children at their responsibility emigrated to Australia where he became headmaster of St James Grammar School in Melbourne. Like Hugh MacColl himself, his character Ednor Whitlock makes a considerable effort in order to obtain an academic degree: Ednor’s probation lasted a little over four years, during which he was no sluggard. He worked hard as a teacher in the school, and still harder out of school hours as a student for the London University. He passed all his examinations creditably, taking mathematical honours in the final examination. His place on the honours list was not quite so high as he had hoped; but considering what a small margin his school duties allowed him for study, it

22. The quoted letter to W. J. C. Miller suggests that this paper was an early version of [MacColl 1866c]. 23. The degree is listed. Cf. [University of London n. d., 389]. The details are logged in the University Archives (Senate House, London) in these ledgers: ‘Matriculation I’ (ledger D3), 1873, entryno 230 (supported by ‘his brother’, presumably Malcolm, if not John MacColl); ‘First BA Examination’ (D32), 1874, entry no 129 (supported by one ‘G. C. Hope Esqr. D.D.’); and ‘Second BA Examination’ (D32), 1876, entry no 44 (examination in November). 20 Michael Astroh et al.

was no small achievement to take honours at all. He owed much of his success to the help which Mr. Kent occasionally gave him, and especially to the judicious course of reading which the latter had recommended. Left entirely to himself, Ednor would no doubt have wasted much time and energy in mastering things which seldom turn up in examinations, while he might be neglecting things which Mr. Kent knew from experience to be more paying. As soon as Ednor had taken his degree, Mr. and Mrs. Kent allowed his probationary engagement with their daughter to terminate in the way which all had hoped, namely, in the intimate, life long union of marriage. [MacColl 1891b, 339–340] In contrast with his hero, MacColl took his degree not as a younger, unmar- ried man, but in the middle of his life, and with economic responsibility for a growing family. Hence, it was for good reason that he interrupted his work for the mathematical section of the Educational Times where his academic teachers were competing with one another. In a situation like MacColl’s even a highly talented person risked failure. As the Calendar of London University reports, 17 of the 25 candidates above the age of 30 failed in the BA exams of July 1876. In October of the same year 25 of the 33 elder candidates were not successful. After his graduation MacColl joined the university’s congregation [University of London 1877]. Sporadically, its reports mention him participating in decisions. We do not know MacColl’s earliest writings. But, apparently, he could not impress influential academics like Sylvester sufficiently. The letter to Miller from 1866 quoted earlier reported the professor’s reaction to the young man’s pamphlet: The whole of his criticism is as follows: ‘The interpretation affixed by the author to trilinear or quadrilinear coordinates is not, I think, new: I scarcely remember the time when I was not acquainted with it.’ ‘The notation of angles does not seem to be very important although probably ingenious. Y X and XY are always understood to be complementary in respect to 360○: Mr. McColl seems to manage to express with precision the inclinations of two lines complementary in respect to 180○. The rule for fixing the sign absolutely of perpendiculars to lines and planes appears to me at first blush sound.’ Next time I wrote to Mr. Karcher I desired him to thank Prof. S. in my name for his criticism, and to tell him that I had brought the title & language of my paper down to my own lowered opinion of its importance . . . I have at present in my possession Professor Sylvester’s tract on the proof of Newton’s rule for discovering imaginary roots, & from the way he speaks of Dr. Young I cannot A survey of the life of Hugh MacColl (1837-1909) 21

help thinking that the worthy professor has a somewhat peppery disposition. [MacColl 1866a] It seems, however, that such experiences did not prevent MacColl from pursu- ing his, perhaps, original ideas, and defending them publicly—even if context and occasion were by no means suitable. On the other hand MacColl seems to have been supported by the Rev. Robert Harley (1828-1910). 24 From 1864 the latter was professor for mathematics and logic at Airedale College, Bradford [Poggendorff et al. 1863-1938, 588] He was a specialist in Boolean Algebra, and was a good friend of Boole. Through his work on the Stanhope Demonstrator he influenced the history of logical machines [Harley, 1879]. Sylvester was neither the only nor the most important critical reader of MacColl’s work. In 1880, four years after his graduation, MacColl wrote to W. S. Jevons: It seems to me that Mr. Venn’s method of criticism, if consistently & logically carried out, would deprive every inventor that ever lived (Boole not excluded) of all claim to originality; for all inventions are built out of and may be resolved into simple truisms which all then would know. This kind of criticism does not seem to me either just or generous. I am not surprised that with Mr. Venn for examination I was plucked in logic in 1875. [MacColl 1880] Actually, MacColl took his BA in 1876. 25 On 28 October 1876 Rev. J. Venn, MA set the BA pass examination for Logic and Moral Philosophy. Either MacColl was wrong about the year of his BA examination, or he did not refer to it. In these years not only Venn, but likewise Karcher and Sylvester were examiners of the University of London. In the appendix to his third paper on The Calculus of Equivalent Statements MacColl reports how his first period without scientific research incidentally came to an end: . . . my thoughts did not again revert to the subject till two or three months before the appearance of my article on ‘Symbolical Language’ in the Educational Times for July 1877. [MacColl 1878- 1879, 27]

24. In an Appendix to ‘The Calculus of Equivalent Statements (III).’ MacColl writes: I am also indebted to the kindness of Rev. Robert Harley for the loan of Boole’s ‘Laws of Thought’, in which, as in Jevons’s ‘Pure Logic’ in spite of a very different notation and mode of treatment, I find many points of resemblance to my method. My method, however, differs both from Prof. Boole’s and Prof. Jevons’s in three cardinal points, which a perusal of this paper will show to be so important as to necessitate an essentially different treatment of the whole subject. [MacColl 1878-1879, 27]

25. Cf. [University of London 1877, 389]. 22 Michael Astroh et al.

For the next seven years MacColl published regularly on logic and mathematics. Once again, he participated in the discussions published in the Educational Times. In a series of papers, read at the London Mathematical Society, MacColl designed a first, apparently non-classical propositional calculus. A letter of 1905 to Bertrand Russell describes this period of his development as follows: When . . . I discovered by Calculus of Limits, or as I then called it, my ‘Calculus of Equivalent Statements and Integration Limits’, I regarded it at first as a purely mathematical system restricted to purely mathematical questions . . . When I found that my method could be applied to purely logical questions unconnected with the integral calculus or with probability, I sent a second and a third paper to the Mathematical Society, which were both accepted, and also a paper to Mind (published January 1880). These involved me in a controversy with Venn & Jevons, of which I soon got tired, as I saw it would lead to no result. I sent a fourth paper (in 1884) to the Math. Soc., on the ‘Limits of Multiple Integrals’, which was also accepted. This I thought would be my final contribution to logic or mathematics . . . [MacColl 1905b] The long illness of MacColl’s first wife and her premature death in February 1884 may have contributed to his long-lasting abstention from research. During this second period without research MacColl wrote a number of literary works: probably five novels, two of which were published, 26 a short story, 27 and, anonymously, a first essay on the religious impact of evolution theory. 28 To a large extent these works are a first articulation of his metaphysical and religious beliefs. In the last few years before his death he elaborated them in a series of essays and in a book on Man’s Origin, Destiny and Duty. No earlier than 1896 MacColl returned to his former investigations. At least he says so in the already quoted letter to Bertrand Russell, . . . for . . . twelve or thirteen years, I devoted my leisure hours to general literature. Then a friend sent me Mr. Dodgson’s (‘Lewis Carroll’s’) Symbolic Logic, a perusal of which rekindled the old fire which I thought extinct. My articles since then I believe to be far more important from the point of view of general logic than my earlier ones; but unfortunately the views which they express are far more subversive of the orthodox or usually accepted principles in symbolic logic. [MacColl 1905b] This self-assessment certainly is correct, but it also mirrors the reservation of those who opposed the publication of some of his later articles. As in previous years MacColl submitted his papers to the London Mathematical Society. However, after his sixth and seventh papers on ‘The Calculus of Equivalent

26. Cf. [Anonymous 1896]. 27. Cf. [MacColl 1891a]. 28. Cf. [MacColl 1891b, 343]. So far it has not been possible to identify this paper. A survey of the life of Hugh MacColl (1837-1909) 23

Statements’ and an ‘Explanatory Note and Correction’ the Society refused to publish further contributions. An eighth paper with the familiar title was rejected in October 1898. Alfred Bray Kempe (1849-1922), a close cooperator of Sylvester remembered for his work on the four colour problem and multisets, and Alfred North Whitehead had been appointed referees. MacColl reacted to the decision with a revised and extended version of the paper. After a report on the matter by Kempe the Society’s council decided to ‘decline further communication on the subject, from Mr. M, at any rate for the present . . . ’ Between then and 1903 MacColl submitted two further articles to the London Mathematical Society. Both were declined. 29 However, the two places in which MacColl’s final views on his logical system are found are an address to the Paris conference of 1900 [MacColl 1901b] and his book Symbolic Logic of 1906. Sadly, there is evidence that even here in his last statement of his ideas, unfortunate circumstances made MacColl reduce the text by half. On 22 July 1905 he writes to Bertrand Russell: I am working hard re-casting and re-writing my book on Symbolic Logic. The ex-publisher Mr. Grant Richards, having seen my articles in the Athenaeum, asked me to write a treatise on the subject, which I did. But Mr. G. R. unfortunately failed before the work was finished, and other publishers don’t feel inclined to take the risk. I cannot afford to pay for the publication myself unless I reduce the book by at least one-half; and even so, I must expect to lose. Still, I have made up my Mind to face the loss by the publication of a small edition of about 500 copies. [MacColl 1905a] During the last 13 years of his life MacColl wrote several articles in which he steadily altered, and, perhaps, improved his systems of modal logic. Almost all of them appeared in The Athenaeum and in Mind—the journal that published his controversy with Bertrand Russell on ‘ “If” and “Imply” ’, and in which from 1912 onwards Lewis presented his initial account of strict implication [Lewis 1912].

Local recognition

Living permanently in Boulogne, Hugh MacColl acquired French citizen- ship. An obituary published in the Revue de métaphysique et de morale suggests that he did so on the occasion of his second marriage [Anonymous 1910]. However, there is no ﬁrm evidence as to when MacColl was naturalized in France. Boulogne’s social niche allowed him to participate unreservedly in a French form of life without endangering his own cultural identity. At the end of his life Hugh MacColl as well as Hortense Marchal were rather well-known citizens highly respected for their pedagogical and academic achievements:

29. For this passage cf. [The London Mathematical Society 1894-1914]. 24 Michael Astroh et al.

Quand il quitta son cours du collège, après y avoir fait un nombre incalculable d’élèves qui lui doivent leur connaissance raisonnée et pratique de la langue anglaise, M. Mac Coll [sic] continua cet enseignement ainsi que celui des mathématiques dans le pensionnat de jeunes ﬁlles si réputé que tenaient en notre ville Mmes Mac Coll [sic] et Busch-Marshal. Cette collaboration . . . à l’œuvre pédagogique entreprise chez nous par sa non moins érudite épouse et Mme Busch fut pour beau- coup dans le succès toujours croissant de cette excellente pension d’éducation dont les pensionnaires se recrutaient non seulement en Angleterre, mais encore dans toutes les parties de l’Amérique. Elle comptait en outre comme externes bon nombre de nos conci- toyennes qui en ont conservé le meilleur souvenir. En dehors du professorat proprement dit M. Mac Coll [sic] était un érudit et un linguiste du plus sérieux mérite, comme en témoignent ses divers ouvrages de critique, de philosophie et de mathéma- tique, qui jouissent encore chez nos voisins d’outre-Manche, d’une vogue des mieux justiﬁées. [Anonymous1909]

Hugh MacColl and his family in 1907 30

The family’s domicile between 1906 and 1908 at 28, rue Beaurepaire truly was a ‘maison de maître’. 31 Presumably, the boarding school had been transferred to this place. About one year before his death MacColl and his wife retired and moved to 67, rue Porte-Gayole—a house close to Boulogne’s ancient noble centre. 32

30. By courtesy of Mrs. Rosemarie Maconchy. 31. Presently this address is 53, rue Beaurepaire. The house does not exist any longer. 32. This address presently is 77, rue Porte Gayole. A survey of the life of Hugh MacColl (1837-1909) 25

Here MacColl died on 27 December 1909 [Ville de Boulogne 1909]. He was, however, working actively until a few days before his death, as is evidenced by a letter to Bertrand Russell of 18 December. His elder brother, Malcolm, predeceased him, on 5 April 1907, and his son Hugh died in London in 1924 shortly after retiring to Boulogne after his legal career in Burma. Hortense died on13 October 1918 [Ville de Boulogne 1918].

Acknowledgements

The present research was ﬁnanced by a grant from the Kultusministerium des Landes Mecklenburg-Vorpommern. At Boulogne-sur-Mer the following institutions oﬀered their co-operation: Archives Municipales, Bibliothèque Municipale, Services Techniques, Hôtel des Impôts. Some valuable information was provided by the Mairie d’Equihen-Plage and the London Mathematical Society. Especially Emanuel Buselin-Boulourde, Raymond Carpentier, Johan W. Klüwer MA, Michel de Sainte Maresville and Benoît Tuleu have actively participated in this research. We are most grateful to Rosemarie Maconchy (Houston), Dr. Fergus G. Brand (Sydney), and the families of all other descendants of Hugh MacColl for their kind and enduring help in the present investigation; and for other advice to Janet Delve, Albert Lewis and Claire Hill.

References

Anonymous 1870 Announcement: Algebraic Exercises and Problems, Notes on Books, Reading: The University of Reading, The Library, Archives & Manuscripts: Longman’s Archive. 1896 Extracts from Chatto & Windus Manuscripts Entry Books, Reading: The University of Reading, The Library, Archives & Manuscripts; Chatto& Windus, Early Contracts File. 1909 Nécrologie Monsieur Hugh MacColl, La France du Nord. 1910 Nécrologie. Hugh MacColl (1837-1909), RMM, vol. 18 (Supplément). 1918 Nécrologie Madame Mac-Holl [sic], La France du Nord. Astroh, M. & Read, S. (eds.) 1999 Hugh MacColl and the Tradition of Logic. An International Colloquium. Universität Greifswald, 29 March - 1 April 1998, The Nordic Journal of Philosophical Logic, 3, 1–2. Astroh, M. 1999 MacColl’s evolutionary design of language, in M. Astroh and S. Read, 141–73. 26 Michael Astroh et al.

Bailey, S. 2001 Letter to Michael Astroh of 24 September, Oxford: Bodleian Library. Brette, P. H. E., Cassal, C. & Karcher, T. (eds.) 1867 The Little French Reader. Extracted from ’The Modern Reader’, London: Trübner and Co. Cassal, C. & Karcher, T. (eds.) 1876 Anthology of Modern French Poetry. Junior Course (Senior Course), London: Longmans & Co. 1885 The Modern French Reader. Prose. Senior Course, London: Trübner & Co. Charles, Y. 1992 Les Rues de Boulogne-sur-Mer, Boulogne-sur-Mer: Société des Editions de la Côte d’Opale. Feys, R. 1937-1938 Les logiques nouvelles des modalités, Revue Néoscholastique de Philosophie, 517–53; 41, 217–52. Harley, R. 1879 The stanhope demonstrator, an instrument for performing logical operations, Mind, 14, 192–210. Hilaire, Y.-M. 1998a De la monarchie à la république. Un siècle de vie politique boulon- naise (1814-1914), in A. Lotin, 263–88. 1998b Loisirs et vie de l’esprit dans une cité balénaire au XIXe siècle, in A. Lottin, 289–310. Lewis, C. I. 1912 Implication and the algebra of logic, Mind, 21, 522–31. 1918 A Survey of Symbolic Logic, Los Angeles: University of California Press. Lottin, A. (ed.) 1998 Histoire de Boulogne-sur-Mer, Terres Septentrionales de France, Boulogne-sur-Mer: Editions Le Téméraire. MacColl, H. & Peirce, C. S. 1883-1906 Correspondence between Hugh MacColl - Charles S. Peirce, Cambridge, MA: Harvard College Library, Houghton Library. A survey of the life of Hugh MacColl (1837-1909) 27

MacColl, H. 1866a 4, rue du Temple, Boulogne-sur-Mer France, April 25th. Letter to W. J. C Miller. Columbia University, NY: W. J. C Miller Correspondence, Butler Library. 1866b 4, rue du Temple, Boulogne-sur-Mer France, Aprilé 4th 1866. Letter to W. J. C Miller. Columbia University, NY: W. J. C Miller Correspondence, Butler Library. 1866c Angular and linear notation. A common basis for the bilinear (a transformation of the Cartesian), the trilinear, the quadrilinear, &c., Systems of geometry, The Educational Times and Journal of the College of Preceptors, 19, 20. 1878-1879 The calculus of equivalent statements (III), Proceedings of the London Mathematical Society, 10, 16–28. 1880 73, rue Siblequin, Boulogne-sur-Mer, August 18th 1880, Letter to W. S Jevons, The John Rylands University Library, University of Manchester. 1883 73, rue Siblequin, Boulogne-sur-Mer, May 16th 1883, in [MacColl & Peirce 1883-1906], Letter to C. S. Peirce, Harvard College Library, Cambridge, MA, Houghton Library, MS CPS L 261. 1887 Boulogne-sur-Mer, le 17 août 1887, in [MacColl & Peirce 1883-1906], Marriage announcement, sent to C. S. Peirce, Harvard College Library, Cambridge, MA, Houghton Library, MS CPS L 261. 1889 Mr. Stranger’s Sealed Packet, London: Chatto & Windus. 1891a 37, rue Basse-des-Tintilleries, Boulogne-sur-Mer, December 9th 1891, Letter to Messrs. MacMillan, Publishing Company; The University of Reading, The Library, Archives & Manuscripts, item 15/33. 1891b Ednor Whitlock, London: Chatto & Windus. 1896-1897 The calculus of equivalent statements (V), Proceedings of the London Mathematical Society, 28, 156–83. 1901a 37, rue Basse-des-Tintelleries, Boulogne-sur-Mer, 10. IX. 1901, in H. MacColl, Letters to Bertrand Russell, Hamilton, Ontario: McMaster University, Russell Archives. 1901b La logique symbolique et ses applications, in [. . . ] vol. III of Bibliothèque du Ier Congrès International de Philosophie, Logique et Histoire des Sciences, Paris: Librairie A. Colin, 135–83. 1902-1903 On the validity of certain formulae (communicated from the Chair 16. 04. 1903, not printed), Proceedings of the London Mathematical Society, 35, 459. 28 Michael Astroh et al.

1905a 37, rue Basse-des-Tintelleries, Boulogne-sur-Mer, 22. VII. 1905, in H. MacColl, Letters to Bertrand Russell. Hamilton, Ontario: McMaster University, Russell Archives. 1905b 37, rue Basse-des-Tintelleries, Boulogne-sur-Mer, 17. V. 1905, in H. MacColl, Letters to Bertrand Russell. Hamilton, Ontario: McMaster University, Russell Archives. 1906 Symbolic Logic and its Applications, London: Longmans, Green and Co. 1909 Man’s Origin, Destiny and Duty, London: Williams and Norgate. Merridew, H. M. 1866 Merridews illustrated Guide to Boulogne and Environs with plan and map, Boulogne-sur-Mer: Merridew - English Booksellers. Olsen, S. H. 1999 MacColl - Victorian, in M. Astroh and S. Read, 197–229. Oustric, G. 1998a Une société originale et variée, in A. Lottin, 1998, 247–62. 1998b Un siècle de croissance économique (1814-1914), in A. Lottin, 1998, 211–46. Poggendorff, J. C., Feddersen, B. W & Von Oettingen, A. J. (eds) 1863-1938 Biographisch-literarisches Handwörterbuch zur Geschichte der exacten Wissenschaften enthaltend Nachweisungen über Lebensverhältnisse und Leistungen von Mathematikern, Astronomen, Physikern, Chemikern, Mineralogen, Geologen, Geographen u. s. w. aller Völker und Zeiten, vol. 1–6. Leipzig: Verlag von Johann Ambrosius Barth. Read, S. 1999 MacColl and strict implication, in M. Astroh and S. Read, 1999, 59–83. Russell, G. W. E. (ed.) 1914 M. MacColl, Memoirs and Correspondence, London: Smith Elder & Co. The London Mathematical Society 1894-1914 Minute Books: 11 October 1894 - 27 April 1914, vol. 7. London: Society Archives, University College London. Tiébaut, J. 1998 L’art monumental, in A. Lottin, 1998, 311–37. University of London 1877 Calendar for the Year 1877, London: Taylor and Francis. A survey of the life of Hugh MacColl (1837-1909) 29

University of London n. d. General Register. Part 3, London: University of London. Vasseur, M. n. d. Boulogne à l’heure anglaise, Archives de la Ville de Boulogne, Boulogne-sur-Mer. Ville de Boulogne 1866 Birth Certificate for Mary Janet MacColl: 28. 03. 1866. 1867 Birth Certificate for Martha Christina MacColl: 13. 10. 1867. 1869 Birth Certificate for Flora MacColl: 18. 05. 1869. 1871 Birth Certificate for Hugh Ernest MacColl: 21. 05. 1871. 1873 Birth Certificate for Annie Louise MacColl: 16. 11. 1873. 1887 Marriage Certificate for Hortense Lina Marchal and Hugh MacColl: 17. 08. 1887. 1909 Death Certificate for Hugh MacColl: 27. 12. 1909. 1918 Death Certificate for Hortense Marchal – épouse MacColl: 13. 10. 1918. Von Wright, G. H. 1951 An Essay in Modal Logic, Amsterdam: North Holland.

Outside the intellectual mainstream ? The successes and failures of Hugh MacColl

Stein Haugom Olsen Østfold University College, Norway

Résumé : En plus de son œuvre logique, MacColl a aussi écrit et publié des travaux que l’on qualifierait aujourd’hui de cultural criticism. Ces écrits concernaient deux des thèmes centraux de la période victorienne : l’avancement et l’influence grandissante de la science, et le déclin de la religion. À l’image de son œuvre logique ignorée, jusqu’à très récemment, aucune attention n’a été accordée à ses œuvres littéraires ou celles relevant du cultural criticism. Cet article examine les travaux de MacColl qui ne relèvent pas du domaine de la logique et discute la façon dont MacColl a approché et traité les deux thèmes de la science et de la religion. Il fournit ensuite une explication du peu d’attention accordée à ces œuvres malgré les talents intellectuels évidents de leur auteur. Cette explication impute l’oubli de l’œuvre de MacColl aussi bien à la façon dont il traite ces thèmes qu’à sa position d’outsider. En effet, MacColl vivait expatrié dans une ville provinciale française sans lien avec l’environnement culturel et littéraire de Londres. Par ailleurs, il était au sens véritable du terme, un auteur dont les sensibilités correspondent au milieu de la période victorienne, mais dont les œuvres paraissent tardivement à la fin de cette période et au cours de la période edwardienne, alors que les termes du débat sur la science et la religion avaient déjà changé. Abstract: In addition to his work in logic, MacColl also wrote and published a short story, a poem, and several novels as well as works that one today would refer to as ‘cultural criticism’. All of these works were concerned with two of the most central themes of the Victorian period, the growth and increasing influence of science and the decline of religion. Just as his work in logic has been ignored, until very recently, so no attention has been paid either to his literary works nor to his works of ‘cultural criticism’. This paper discusses why so little attention has been given to these works in spite of MacColl’s obvious intellectual talents. An explanation for this neglect can be found both in the way in which MacColl handles these themes and in his outsider position. MacColl was not only an expatriate living in a French provincial town without any contact with the metropolitan literary and cultural environment of London. He was also in a very real sense a mid-Victorian who produced

Philosophia Scientiæ, 15 (1), 2011, 31–54. 32 Stein Haugom Olsen his works in the late Victorian and Edwardian period when the focus of the debate about science and religion had changed.

Introduction

In 1924 Virginia Woolf gave a lecture in Cambridge entitled ‘Mr. Bennett and Mrs. Brown’ in which she made her famous statement that ‘[I]n or about December 1910 human character changed’: I am not saying that one went out, as one might into a garden, and there saw that a rose had flowered, or that a hen had laid an egg. The change was not sudden and definite like that. But a change there was, nevertheless; and, since one must be arbitrary, let us date it about the year 1910. [Woolf 1966, 320] Virginia Woolf’s choice of date was not completely arbitrary. There were a number of developments in the years just before 1910 that indicated a change in the nineteenth century conception of ‘human character’. Chekhov’s short stories appeared in English in 1909. Dostoyevsky’s novels did not begin to appear in Constance Garnett’s influential translations until 1912, but there had been earlier English versions, and he was also known through French renderings. In Vienna, Freud had already laid the foundations of psychoanalysis and though he was not yet been published in England, he and Jung had lectured, in 1909, in the United States. And in December 1910 the first London exhibition of Post-Impressionist painting, ‘Manet and the Post-Impressionists’ took place. The exhibition focused on the work of Cézanne, Van Gogh, and Gauguin, but also attempted to illustrate the development from Manet down to the most recent ‘post-impressionists’, Matisse and Picasso, who were both represented in the exhibition. The exhibition, by rejecting impressionism as old-fashioned, baffled even sympathetic observers. For most English people interested in art the exhibition was their first glimpse of the works of Cézanne, Van Gogh, Picasso, and Matisse who rejected ‘the abominable error of naturalism’ which they saw the impressionists as perpetrating. All these developments had this in common that they emphasized the individual human being, the individual sensibility, the individual reaction. The attention of artists, authors, and psy- chologists moved away from the outside world of facts and nature to the inner world of the human mind. So a change was taking place. The change in ‘human character’ that Virginia Woolf identifies led to a shift in ‘all human relations’: [T]hose between masters and servants, husbands and wives, parents and children. And when human relations change there is at the same time a change in religion, conduct, politics, and literature. [Woolf 1966, 321] As a striking illustration of the change that has taken place Virginia Woolf ask the reader to: Outside the intellectual mainstream? 33

[C]onsider the married life of the Carlyles and bewail the waste, the futility, for him and for her, of the horrible domestic tradition which made it seemly for a woman of genius to spend her time chasing beetles, scouring sauce-pans, instead of writing books. [Woolf 1966, 320–321] And she obviously assumes that her readers will agree with her in ‘bewailing’ this situation. Virginia Woolf sees the first signs of this change in Samuel Butler’s The Way of all Flesh and in the plays of George Bernard Shaw. Though The Way of All Flesh was not published until 1903, it was begun thirty years before in 1873, when Butler was 37, and he finally put it aside in 1884, having revised only the first part [Butler 1966, 7–8]. So if The Way of All Flesh did indeed signal a change in the conception of human character and in ‘all human relations’, then it seems that this change was already well under way towards the end of the 1870s. Indeed, it is arguable that the changes in human relations that Virginia Woolf lists started much earlier than that. The change in the relations between master and man was diagnosed by Thomas Carlyle in Chartism where he recalls a feudal period in which: [T]he old Aristocracy were the governors of the Lower Classes, the guides of the Lower Classes; and even, at bottom, that they existed as an Aristocracy because they were found adequate for that. Not by Charity-Balls and Soup-Kitchens; not so; far otherwise! But it was their happiness that, in struggling for their own objects, they had to govern the Lower Classes, even in this sense of governing. For, in one word, Cash Payment had not then grown to be the universal sole nexus of man to man; it was something other than money that the high then expected from the low, and could not live without getting from the low. Not as buyer and seller alone, of land or what else it might be, but in many senses still as soldier and captain, as clansman and head, as loyal subject and guiding king, was the low related to the high. With the supreme triumph of Cash, a changed time has entered; there must a changed Aristocracy enter. We invite the British reader to meditate earnestly on these things. [Carlyle 1840, 58] The industrialisation of Britain changed the relationship between master and man from a relationship which involved mutual, if unequal, rights and duties to a relationship where the soldier, clansman, loyal subject, tenant, or servant was replaced by the ‘operative’ or the ‘hand’, both metaphors that make the man less than human and where the only relationship between master and man is cash in hand for work done. This dehumanisation of what had been human relations was one of the major targets of criticism for those who became the cultural critics of the period. 34 Stein Haugom Olsen

Also the other changes in human relations mentioned by Virginia Woolf did not develop overnight and autonomously but had their background in the industrial revolution and the development of a complex and bureaucratized industrial society that opened up a host of new opportunities for women, including middle class women, to work, and that changed the very ideals of this society from an ideal of leisure to an ideal of work. The changes in ‘religion, conduct, politics, and literature’ did not really follow upon the change in human relations but were contemporaneous with them and took place over the whole of the Victorian Period. Indeed, these changes in British society and the British polity during the reign of Queen Victoria were of such a character that it makes little sense to speak of a ‘period’. ‘Had the great Victorians lived under three or four sovereigns’, said Hugh Kingsmill in 1932, [T]hey would be judged on their own merits instead of being regarded as embodiments of an epoch which owes the illusion of its spiritual unity to the longevity of a single person. [Kingsmill 1932, 684]

The outsider

Hugh MacColl was born in the year that Queen Victoria succeeded to the throne and lived through the whole of her long reign, dying in 1909. In that sense, he was a Victorian. However, it is arguable that MacColl, though he lived in the Victorian Period was not of it. Indeed, he had impeccable credentials as an outsider. He was born in January 1837 in Strontian, a vil- lage in Argyllshire (abolished as an administrative region in 1975 with other Scottish counties) in the very west of the Scottish Highlands [Astroh, Grattan- Guinness, & Read 2001, 81–98]. Even today it is remote and isolated, described on the webpage Undiscovered Scotland as ‘a white-painted oasis on an inlet in the sea-loch, Loch Sunart’. 1 When his father died in 1840, his mother moved back to Letterfearn where she came from and then to Ballachulish, where the children started to learn English. MacColl was supported at school by his older brother, Malcolm, who had to cease this support in 1857 when he was dismissed from his post in the Scottish Episcopalian Church because of a disagreement with his bishop over the proper interpretation of the Eucharist. Malcolm, however, did manage to recommend himself to William Gladstone and became in time a political ally of Gladstone’s. He also persuaded Gladstone to support Hugh at Oxford, but Gladstone set the condition that Hugh would take orders in the Anglican Church. This Hugh refused to do and consequently remained without a university education. This clearly prevented MacColl from taking the conventional path to ﬁnancial security and entry into English intellectual life. Up to the reform of the ancient universities that resulted from the reports

1. See: www.undiscoveredscotland.co.uk/strontian/strontian/index.html, accessed 30 May 2010. Outside the intellectual mainstream? 35 of the two Royal Commissions 2 ‘The standard career pattern for Oxbridge dons led from undergraduate to fellow to orders to a parish living or the bar, and resignation of the fellowship. [. . . ] [F]ellowships were regarded as steps in a career structure leading out of the university’ [Heyck 1982, 163]. From the mid-1850s another career path was added for those who were academically able: the ancient universities opened up college fellowships to merit, to be determined by competitive examinations. Both these paths would have been open to MacColl had he entered Oxford as his brother intended. However, it is doubtful if MacColl’s interest in logic would have developed in quite the same way, had he been sent to Oxford. Though logic was a required subject for anyone reading for Greats, it was a minor part of the curriculum in Literae Humaniores. Indeed, under the regulations of 1853 the Lit. Hum. school ‘was still very much a classical school, even though Aristotle was by then studied in a more philosophical manner and Plato was beginning to be taken seriously’ [Walsh 2000, 313]. In the early 1860s, says Walsh in The History of the University of Oxford, Oxford philosophy generally remained largely parochial, above all in Logic, of which subject neither then nor later was there any clear definition. To find out what ‘Logic’ comprised you had to look at past papers, a procedure that was in any case strongly commended by some tutors. [Walsh 2000, 317] William Spooner, Warden of New College from 1903 to 1925 and famous for producing ‘spoonerisms’, took Greats in 1866 and had this to say about logic as it was then taught: Our Logic we worked at principally in Mill, which many of us knew with great thoroughness, supplementing it on the critical side by reading his work on Hamilton, Mansel’s Prolegomena Logica and his appendix on Aldrich’s Logic, and Ferrier’s Institutes of Metaphysic. [Hayter 1977, 42–43] It is of course possible that MacColl should have entered the school of Disciplinae Mathematicae et Physicae but even in the early 1860s this road to an honours degree did not have quite the prestige of the Lit. Hum. school. Though MacColl’s social background and the fact that he did not enter Oxford no doubt contributed to making his position that of a social and intellectual outsider, it did not really confirm him in this role. There were a number of examples of educationally deprived Scotsmen that had come to London and made good and even become famous. Thomas Carlyle came from Ecclefechan in Dumfriesshire, and though he received four years of formal education at the University of Edinburgh, he took no degree and the Scottish universities being much more democratic and open institutions than the ancient universities of England did not offer the same kind of entrance ticket to a rewarding

2. The Royal Commission on Oxford reported in 1852 and the Commission on Cambridge in 1852-1853. 36 Stein Haugom Olsen career and intellectual respectability. Yet Carlyle became the ‘Sage of Chelsea’ producing a range of works of what we today would call ‘cultural criticism’. Closer to home, there was, of course, Hugh’s elder brother, Malcolm MacColl: Malcolm did so well at school that a wealthy lady paid for him to attend a seminary in Dalkeith near Edinburgh where schoolteachers were trained. He taught at Callander, Stonehaven and Perth. In 1854, at the age of 23, Malcolm was admitted to the relatively newly opened Theological College at Glenalmond near Perth. In 1856 he was ordained Deacon of the new Mission of the Episcopal Church at Castle Douglas in Kirkcudbrightshire. He was ordained priest in St Mary’s, Glasgow by the Bishop of Glasgow in August 1857. [Astroh, Grattan-Guinness, & Read 2001, 84] But Malcolm did not stop there. The entry in the 1911 edition of the Encyclopaedia Britannica that he earned, termed him a ‘publicist’ and ‘a political and ecclesiastical controversialist’. He came into confrontation with his bishop over his stand on the ‘real essence’ of the Eucharist, and was dismissed from his post at Castle Douglas. Persisting in writing letters to Gladstone about the threat to high church clergy in Scotland, he finally attracted Gladstone’s notice and started on a career as a pamphleteer for Gladstone who secured his introduction into and modest preferment in the Church of England. The church living that Gladstone gave him, says the Encyclopaedia Britannica, [W]as practically a sinecure, and he devoted himself to political pamphleteering and newspaper correspondence, the result of extensive European travel, a wide acquaintance with the leading personages of the day, strong views on ecclesiastical subjects from a high-church standpoint, and particularly on the politics of the Eastern Question and Mahommedanism. He took a leading part in ventilating the Bulgarian and Armenian ‘atrocities’, and his combative personality was constantly to the fore in support of the campaigns of Gladstonian Liberalism. [Anonymous 1911] What finally determined Hugh’s outsider position was his choice to live and work in France. In their presentation of ‘MacColl’s Boulogne’, Astroh, Grattan-Guinness, and Read emphasize the British influence in Boulogne, its close economic and cultural links with Britain, its cosmopolitan flavour as ‘one of the most elegant seaside resorts and spas of the Belle Époque, much frequented by the elites of both nations’ [Astroh, Grattan-Guinness, & Read 2001, 85]. However, though London was only five and a half hours away and Paris only three, Boulogne was a provincial town. The developments in art, in literature, in attitudes, mores, sensibilities, and beliefs that took place in Paris and in London in the last quarter of the 19th century and the first decade of the 20th did bypass the British community at Boulogne, or, at least, they by- passed Hugh MacColl. The changes ‘in human character’ that Virginia Woolf Outside the intellectual mainstream? 37 mentions as culminating in 1910 are not in any significant way reflected in MacColl’s literary and philosophical production. There is some evidence that MacColl himself felt how this move had marginalized him in relation to British intellectual life and prevented him from participating fully in the ongoing discussions in that world. ‘If I were a professor of logic’, he writes in 1904 in a letter to F. H. Bradley, then: I would certainly get your books and study them; but as I am only an amateur, driven by I know not what mental perversity towards abstract studies from which I can never hope to reap any material gain or benefit, I am afraid I must content myself with the few books on logic that I already posses. Though I have contributed articles on logic both to Mind and to the Athenaeum, I do not see these publications regularly. Living as I do, and as I have done since 1865, in a foreign country, my opportunities of reading English magazines or standard works on Science, Logic, or Literature are very few; and I cannot afford the luxury of a large library. [Keene 1999, 308]

The logician

There is hardly any point in branding a man ‘an outsider’ unless there is a circle of people or a community into which he for some reason fails to enter though he desires to do so. Hugh MacColl became a school teacher like Thomas Carlyle and his brother Malcolm, but unlike them Hugh remained a school teacher all his life. Carlyle and Malcolm MacColl found other roles in which to exercise their abilities and in the process of doing so they became well known public figures. Hugh, too, was looking to fill other roles, but failed to do so. ‘Hugh hoped,’ say Astroh, Grattan-Guinness, and Read, to return to his cultural mainland, and to receive some kind of public recognition for his pioneering contributions to logic. Even as late as 1901, MacColl when 64 years old recommended himself to Bertrand Russell as a lecturer in logic (MacColl 1901a). Due to a University of London Parliamentary Act of 1898 some changes to the university’s teaching arrangements were made. Apparently, MacColl got to know about these changes. [Astroh, Grattan- Guinness, & Read 2001, 86] And in MacColl’s letter to F. H. Bradley quoted at the end of the last section there is a tone of bitterness and frustration at his failure to become an academically accredited, professional logician. In the last quarter of the nineteenth century the way in which knowledge was authorized and underwritten in Britain changed. Up to the reform of the ancient universities from the 1860s onwards and the foundation of new institutions, the development of new knowledge and intellectual work generally was a matter for the individual and the authority of an individual in 38 Stein Haugom Olsen any one field of knowledge depended upon the recognition someone got for his work from his peers or, in some fields, from the general, and generally informed, public. However, as a part of the reform of the universities a career structure was introduced within the universities based on certification of merit. This development paralleled the development of what Harold Perkin has called the rise of professional society [Perkin 2002], a development which was driven by the need for certified professionals and specialists to solve in- creasingly complicated technical, social, and administrative problems. Indeed, the universities became the main providers and certifiers of the most sophisti- cated specialists and professionals. Intellectual authority in fields of knowledge was consequently institutionalized. ‘By the last two decades of the nineteenth century’, says Ian Small, [T]o give authority to a particular practice or body of knowledge required the professional status which only the university, or some similar institution, could bestow. Thus there were two ways in which the process of professionalization in the nineteenth century affected the universities: first, there was the attempt to accommodate the needs of the newly emerging professions by establishing new subjects within the universities; and second, there was the attempt to reorganize the practices of the universities themselves along more professional lines, a process which can be seen both in general university reforms, and in particular reforms in different disciplines. [Small 1991, 22] 3 A logician, then, in the last quarter of the 19th century was not merely someone doing logic, but also someone who was part of a community of certified professionals, deriving his authority in his field from belonging to this community. It is in this context that MacColl’s failure to take up a place at Oxford becomes significant. Had he entered Oxford and pursued his studies and his work as part of an academic career he would have become part of the new professional class whose authority was founded in their professional roles. He could then in Stefan Collini’s words have laid a claim to ‘a title to be heard’ [Collini 1991, chap. 6]. If one feels that the neglect of MacColl’s logical work needs an explanation, it may perhaps be found here. Perhaps. The description of the development the professional academic given in the above paragraph is based on the general outline presented by Heyck in The Transformation of Intellectual Life in Victorian England. However, in some subjects, in particular in literary studies (‘English literature’) and philosophy, the borderline between the knowledgeable and interested amateur and the professional was fuzzy well into the twentieth century. University teachers in these subjects did not necessarily have any formal qualifications for the subjects they were teaching, and the interested and 3. For a general account see [Heyck 1982], especially chapter 6: ‘The Reform of the University System’. For the development of the discipline of history and the professionalization of this subject, see [Levine 1986]; and for the discipline and professionalization of literary criticism, see [Atherton 2006]. Outside the intellectual mainstream? 39 knowledgeable amateur was listened to with as much respect as the university don. 4 The Rev. Thomas Dale, the first professor of English in England, appointed in 1828 to the Chair of English Literature and Language at the recently founded University College, London, and then, in 1835, to the chair of English Literature and History at the even more recently founded King’s College, was an evangelical clergyman. Indeed, most of the early professors of English literature came ‘from the Low Church or Dissenting Ministry, then from journalism, and later from the Oxford “Greats” School, particularly from Balliol’ [Palmer 1965, vii]. Even in the first decades of the twentieth century the appointment of teachers of English literature at the two ancient universities was a haphazard affair. In 1911 the King Edward Chair of English Literature was founded at Cambridge with money from Harold Harmsworth (the founder with this brother, Alfred, of the Daily Mail). When, after only a year the first holder of the chair, A. W. Verrall died, Sir Arthur Quiller-Couch was appointed to succeed him: [T]here was much surprise when the Crown appointed so un- academic a person as Sir Arthur Quiller-Couch to succeed him. Ridgeway told me his story of the appointment, and I give it though I have never heard it confirmed or denied. When the va- cancy occurred, Grierson had just finished his great edition of Donne and had emerged as one of the chief scholar-critics in the country. Asquith, then Prime Minister, had decided that he was the man for the chair. The matter was as good as settled, when Lloyd George got busy and said they ought to make a party appointment. Quiller-Couch had been keeping Liberalism going in Cornwall for many years, and Lloyd George knew The Oxford Book of English Verse. Asquith yielded to Lloyd George’s pressure, and Quiller-Couch was offered the chair. [Tillyard, 1958, 39] 5 And when Tillyard himself, a classical scholar specializing in Greek vases, was recruited to teach English literature at Cambridge immediately after the First World War, it was even more of a haphazard affair and came about as a result of what he calls three accidents. The story is too long to be told in all its details, but an extract from Tillyard’s account of the last of the three accidents, which include reference to the two others, will give the general idea of the process: In the summer of 1917 I was back on leave from the Salonica front. Having my home in Cambridge, I spent most of it there. All my contemporaries were away except Forbes, whose short sight ex- empted him from service; and we met repeatedly. The regulations

4. Carol Atherton chronicles the resistance among literary critics in the late nineteenth and early twentieth century both within and outside the academies to accepting that there was a professional role of academic critic which was diﬀerent from that of the knowledgeable and interested amateur [Atherton 2006]. 5. The story is given in some more detail in [Atherton 2006]. 40 Stein Haugom Olsen

for the English Tripos had recently been passed, and Forbes held forth on the virtues and the freedom of the new examination. He had already agreed to Chadwick’s demand that he should leave history and lecture for it. I was interested, but from a distance only; the pressure of the war and the prospect of a return to the front prevented any enthusiasm. However, my interest must have been sufficient for Forbes ultimately to come out with the proposal that I should lecture for the paper on literary criticism which was part of the new Tripos. [. . . ] So it was arranged that I should see Chadwick next evening. It was the first time I had met this extraordinary man, who combined extreme shyness with an uncanny power of getting his own way. After the first agonising moments when his apparent ter- ror froze any conversation I might have been good for, I quickly found myself discussing a topic very near to my heart at the moment, namely Balkan politics, and with a man who, without having travelled, knew all about them. It was a splendid talk; and it obliterated all thought of the errand I had come on. As I rose to go, Chadwick suddenly asked if I would lecture for the criticism paper of the English Tripos when the war was over; and I, hyp- notised by Chadwick’s quiet but compelling importunity, replied as suddenly that I should be delighted. [Tillyard, 1958, 16–18] And in philosophy the story was apparently not very different even as late as the nineteen fifties: In 1945, Herbert Hart was a 38-year-old London barrister who had spent the previous six years largely working in military intelligence. What could be more obvious, then, than that he should be thought the perfect candidate for a full-time teaching position at Oxford in philosophy, a subject with which he had had no sustained connection since it formed part (though only part) of his undergraduate degree sixteen years earlier? Similarly, in 1952 Hart was a 45-year-old philosophy tutor who had by that point published only three essays and two book reviews. Self- evidently, he was the ideal man to elect to Oxford’s Professorship of Jurisprudence. By the standards prevailing at the beginning of the twenty-first century, these two appointments are bound to seem scandalous, perhaps even barely intelligible. [Collini 2008, 286] Hart, of course, proved to be perhaps the most important legal philosopher of the 20th century just as Tillyard became one of its most important academic literary critics. What these anecdotes illustrate is not merely that the borderline between the amateur and the professional was, and indeed for a long time remained, Outside the intellectual mainstream? 41 fuzzy in such areas of intellectual activity as literary criticism and philosophy. It also highlights another absence in the Hugh MacColl story that contributed to making him an outsider. Quiller-Couch, Tillyard, and Hart were all ‘well connected’ though their networks of connections were of different kinds. In Quiller-Couch’s case it was political while in the case of Tillyard it was partly academic, partly social. Tillyard’s account of the three accidents that brought him into English studies are essentially an account of how he established various forms of friendships in academia at various points in time and how these friendships opened up opportunities that would not otherwise have presented themselves. And it is not irrelevant to the story that he was a Cambridge man in a double sense. Not only was he a graduate of the university, but he also had his home there. In Hart’s case the network developed as a result of his employment in military intelligence during the war, though his post there was in its turn conditioned by his social standing and background (he was, of course, an Oxford graduate). Stefan Collini points out what he calls a ‘tantalizing detail noted in [Nicola Lacey’s] biography’: [W]hen Hart was involved in delicate negotiations on behalf of MI5 during the Second World War to enable that agency and MI6 to share secret information, three of the officials in MI6 with whom he had closest dealings were Hugh Trevor-Roper, Gilbert Ryle, and Stuart Hampshire. [Collini 2008, 286] Collini points out that in the case of individuals belonging to literary and intellectual elites there is a particular problem reconciling the account of the achievements of such figures as these achievements appear ‘tangibly individual, the expression of apparently autonomous creative energies’ with an account of ‘the enabling effect of belonging to certain advantaged groups’ because in these cases the enabling effect ‘is usually more implicit or resistant to demonstration than in the case of conventional social and economic elites whose power and influence are so highly visible’ [Collini 2008, 285]. The fuzzy borderline between the amateur and the academic professional in such a discipline as philosophy did not make it any easier for just anyone to gain a hearing and recognition. MacColl’s case certainly strengthens the point that the presence of the enabling effect of belonging to an elite is necessary to gain such recognition. For, in Hugh MacColl’s case not only did his failure to get into Oxford prevent him from entering the new class of professional academics, but there was also an absence of any enabling effect of belonging to any form of elite.

The novelist

In a letter to Bertrand Russell from 1905 MacColl says that ‘[F]or . . . twelve or thirteen years, I devoted my leisure hours to general literature’. In this period (from 1884 to 1896) he published a poem [MacColl 1884], two novels [MacColl 1889, 1891a], and one short story [MacColl 1891b]. He apparently also wrote three more novels but these were not accepted for publica- 42 Stein Haugom Olsen tion. 6 It does appear then, that MacColl had the ambition to gain recognition as a novelist. Both MacColl’s published novels deal with current topics that attracted serious interest from many quarters, among them authors of fiction. Mr. Stranger’s Sealed Packet is the third novel in English to be published about Mars and one of fourteen novels about Mars published in the last two decades of the 19th century. 7 It also focuses and gives extended treatment to the topic of science and technology and the possibilities for new discoveries that the combination of these opens up. And MacColl handles the topic of science and its technological application ably. In a longish review of Mr. Stranger’s Sealed Packet in the book reviews section of Nature, he is given high praise for ‘his acquaintance with the Cosmos’: A work of fiction, founded upon scientific facts, is interesting to us, inasmuch as it may extend, to no inconsiderable degree, the scientific knowledge of its readers. Such attempts, however, to assimilate science with fiction may have an injurious effect, unless treated by one having an intimate knowledge of the phenomena which he describes, and we have to congratulate the author of this work upon his acquaintance with the Cosmos, exhibited in this account of an imaginary journey through interplanetary space. [Gregory 1889, 291] The review spelt out in some detail MacColl’s scientific ideas and it concludes: We might quote many other descriptions of phenomena all agreeing with acknowledged facts and rigid scientific principles. We refer to observations of the extreme blackness of the shadows cast by the rocks of the Martian satellite which was supposed to have been visited, the noiseless explosions of the meteor above described, the apparent motionlessness in space of the flying machine, in spite of its enormous velocity, the inferior attraction of Mars and its satellites, and the explanation of how men got transferred from the earth to Mars. Indeed, the work is as interesting to us as

6. According to Chatto & Windus Manuscript Entry Books, MacColl submitted manuscripts in 1891 (Was She his Wife?), in 1893 (Sister Joan), and in 1896 (The Search for Meerin). See the Extracts from Chatto & Windus Manuscripts Entry Books (Reading: The University of Reading, The Library, Archives & Manuscripts; Chatto & Windus, Early Contracts File, 1896). 7. The frequency and number of novels about Mars increases further in the ﬁrst decades of the 20th century. A bibliography of novels about Mars is to be found on the BookBrowser site on the Internet at http://www.bookbrowser.com/TitleTopic/mars.html. This bibliography lists novels about Mars chronologically as well as by author. An even fuller bibliography comprising all ﬁctional stories about Mars has been compiled by Gene Alloway, Senior Associate Librarian, University of Michigan (NSF/NASA/ARPA Digital Library Project), available on the Internet (http://www-personal.engin.umich.edu/ cerebus/mars/index.html#bibs). Outside the intellectual mainstream? 43

to the general reader, and as a means of disseminating scientific knowledge may be eminently useful. [Gregory 1889, 292] Ednor Whitlock deals with an equally current and also controversial theme. It is a novel of faith and doubt and as such joins a genre that includes novels like Samuel Butler’s The Way of All Flesh (written 1873-1884; published 1903); William Hale White’s The Autobiography of Mark Rutherford (1881), Mrs. Humphrey Ward’s Robert Elsmere (1888); as well as Edmund Gosse’s autobiographical Father and Son (1907). And like Mr. Stranger’s Sealed Packet it presents and endorses values entrenched among the majority of the reading public of the time. MacColl’s views of women, of religion, of education, of colonisation, and his general moral views are conservative, ‘establishment’ views that at the time would have appealed to a broad readership and offended very few. 8 Nevertheless, MacColl failed to gain recognition by his contemporaries. The two novels that MacColl did publish received only few and short notices in the periodical press, and these notices were either condescending in their faint praise or dismissive in their comments. Mr. Stranger’s Sealed Packet did receive some praise also in other periodicals for its handling of scientific ideas, but there was only condescending and faint praise for the plot and characters: Mr. Stranger’s Sealed Packet belongs to the class of ‘Voyages Imaginaires,’ the planet Mars being the goal of the journey, and the vehicle an electro-magnetic flying-machine. Most of the incidents have been used before by the various writers who have essayed this kind of composition—the flying chest, for instance, is in the story of Malek and Schirine in the Persian Tales. But there is no ground for supposing Mr. MacColl to have annexed them; rather they are such as would be apt to suggest themselves independently to persons exercising their invention in this direction. The scientific portions are very carefully and plausibly worked out, and the slight story is effective. [Anonymous 1889b] 9 [T]he voyage to Mars, which is really the subject of Mr. MacColl’s book, becomes a mere holiday excursion. Mr. Stranger’s adven- tures are well conceived and varied, very entertaining and exciting. There is, of course, a love story, not without a touch of pathos. It is, perhaps, a little astonishing to find on a page near the end of the book a sudden and hasty exposition of the whole scheme of Christianity. On the whole, Mr. Stranger’s Sealed Packet is a thoroughly readable book, and can be confidently recommended to any one who has a taste for romance of the kind. [Anonymous 1889a] 8. For a detailed discussion of MacColl’s attitudes and beliefs as expressed in his two published novels, see [Olsen 1998]. 9. The notice, like the other quoted in the following, occurs in a review of several novels altogether all dealt with in less than a page. 44 Stein Haugom Olsen

One notes the remarks about MacColl recycling standard ‘incidents’ and mo- tifs, the ‘slight story’, and the ‘thoroughly readable book’ for ‘any one who has a taste for romance of the kind’. Ednor Whitlock received harsher treatment: Whitlock was a young lad of eighteen who was working for a Cambridge entrance scholarship with a view to taking Holy Orders. One afternoon, to avoid a shower, he entered the Free Library of the town of Wishport, and, with singular taste, en- deavoured to while away the time with a copy of The Westminster Review. In that far from exhilarating periodical he was fool- ish enough to read an essay on ‘The Evidences of Christ’s Resurrection’, which converted him to Atheism on the spot. He went home and wept bitterly, and for the space of 350 pages he struggles more or less vainly with unbelief, flavoured with love. Or is it that love is flavoured with mathematics, and ethics, and metaphysics, and theology? It is hard to say, for there is such a confused blend that one is lost in wonder whether the book is meant to be a novel or a theological treatise. The sermons and the theological discussions are endless—and oh, so dull! It is reasonable to suppose that the book would never have been written but for ‘Robert Elsmere,’ and therefore Mrs. Ward must be held to some degree responsible. But it is on The Westminster Review that the most awful responsibility rests. Had it not been for that magazine Edwin Whitlock would never have wandered from the faith of his fathers, and therefore the story of his mental troubles and spiritual vagaries would never have been written. Our grievance against The Westminster is distinct. [Anonymous 1891a] Other reviews are hardly less damning and it is the didactic tone and long excursion on theology, mathematics, and metaphysics that are singled out for condemnation: The late Mortimer Collins was a mathematician as well as a ver- sifier; but certainly nobody would have guessed it from his novels. That Hugh MacColl also enjoys the unusual distinction of combining abstruse mathematics with his imaginative gifts is only too evident from his novel Ednor Whitlock. (1 vol.: Chatto and Windus). The general drift and purpose of the work is to counter- act the rationalistic influence of an article which appeared in the Westminster Review many years ago, and of which Mr. MacColl has forgotten the title. In the course of his comment, or refutation, he makes ingenious use of the doctrines of curves, making free use, he is careful to explain, in order to avoid suspicion of pla- giarism, of an article of his own in another quarterly magazine. In short, Ednor Whitlock is about as stiff and solid reading as Outside the intellectual mainstream? 45

can be desired by the least frivolous of readers—many a professed mathematical or theological treatise is twenty times as entertaining. Mr. MacColl is an admirable and suggestive controversialist, in the heavy ordnance division, in all essential and substantial matters; but he is no hand at a love story. He must needs be instructive in and out of season, and cannot even let French idioms alone. That Ednor Whitlock should achieve popularity cannot have been within its author’s wildest dreams. [Anonymous 1891c]

Kindlier reviews tend to be condescending and overbearing:

Readers of Mr. MacColl’s previous story, Mr. Stranger’s Sealed Packet, will not need to be told that he is a writer possessing a distinct individuality of his own. Whatever merits or defects his stories may reveal, they are certainly original. In his new venture, Mr. MacColl shows us a youth of great promise who passes through the dark and gloomy throes of scepticism, to emerge ﬁnally into the clear atmosphere of faith beyond. Some of the discussions on evolution and theology may seem just a little tedious, but they are ably conducted. There is a thread of romance running through the narrative which invests it with suf- ﬁcient interest for those persons who do not care for polemics. [Anonymous 1891b]

The fact that MacColl’s novels did not receive recognition by his contemporaries cannot be ascribed to his position as an outsider. ‘Novelist’, though it may be a social role, is not a profession in the way that logician or philosopher became a profession with the development of the modern research university. The ‘autonomous creative energy’ to which Collini refers is consequently much more important in the making of a novelist and the enabling effect of belonging to an elite much less important than in the case where an intellectual creative activity has developed into a profession. MacColl, in other words, had the same chance of making it as a novelist as, say, Anthony Trollope who in spite of his socially disadvantaged origins, his lack of a university education, and his many years as a post office official in Ireland became one of the most celebrated and highest earning novelists of the 19th century.

The question of contemporary recognition is, of course, different from the question of artistic merit. Marie Corelli was outstandingly successful as a novelist and one of the great earners of the writing fraternity of the Victorian period, but her novels had no artistic merit. Grant Allen called her, in the pages of The Spectator: ‘a woman of deplorable talent who imagined that she was a genius, and was accepted as a genius by a public to whose common- place sentimentalities and prejudices she gave a glamorous setting’, and James Agate represented her as combining ‘the imagination of a Poe with the style 46 Stein Haugom Olsen of an Ouida and the mentality of a nursemaid’. 10 No one today beyond those who are interested in Victorian popular culture knows even her name. And similarly, a failure to achieve recognition as a novelist by one’s contemporaries tells one nothing about the artistic merit of the novels in question. It is not unknown that novelists who received no recognition by their contemporaries and whose artistic merit was unappreciated in their own time are rediscovered by later ages. Indeed, the process of revaluation is a constant feature of literary practice. The question concerning MacColl’s status as a novelist is consequently not settled by the failure of recognition in his own time. The question which one must ask is whether his novels have sufficient artistic merit to repay attention or whether they are simply bad novels that do not repay attention and therefore should remain forgotten and unread by those who look for literary merit. Unfortunately, the answer to this question is that both Mr. Stranger’s Sealed Packet and Ednor Whitlock fail artistically. They do not fail grandly trying to realise an ambitious artistic goal. They fail trivially in many respects. To detail such failures would be tiresome and would not offer the usual rewards of critical appreciation: an enhanced experience of the work under discussion. However, elsewhere I have traced the ways in which MacColl’s novels fail in order to make the point that artistic failure is a central reason for leaving novels unread and forgotten. 11 The majority of forgotten and unread novels, MacColl’s novels among them, are forgotten and unread for good, aesthetic reasons. There is, however, no reason to repeat this argument here.

A controversialist?

One predominant feature of MacColl’s two published novels is their didactic intent: they are, as one of the reviewers has it, ‘instructive in and out of season’ [Anonymous 1891c]. The novels give a clear impression that MacColl had enthusiasms and concerns he wanted to share with a greater public and where necessary, he wanted to educate that public and persuade them to accept certain religious views and metaphysical doctrines that he felt were under attack. ‘Mr. MacColl’, says the same reviewer, ‘is an admirable and suggestive controversialist, in the heavy ordnance division, in all essential and substantial matters; but he is no hand at a love story’ [Anonymous 1891c]. In the last two years of his life he also published a number of shorter essays as well as a

10. These two judgements exactly in the form given here appear together almost every time there is a reference to Marie Correlli on the Internet. I have so far not identified the venues where they were published. 11. In [Olsen 2001], I attempt to show in some detail and through a comparison with other contemporary novels the ways in which MacColl’s novels fail artistically. The article has been reprinted in [Peer 2008, 31–51], with the title “Why Hugh MacColl is not, and will never be, part of any literary canon”. Outside the intellectual mainstream? 47 philosophical treatise [MacColl 1906-1907, 1907-1908, 1908-1909, 1909, 1909- 1910] dealing with the interrelated questions of the argument from design, the foundation of ethics, the survival of the soul, and the existence of a Christian God. 12 These themes all appear in MacColl’s novels, but are taken up in more detail in his prose writings. As the reviewer in Nature of Mr. Stranger’s Sealed Packet observed, MacColl was intimately familiar with many aspects of modern science. And as the novel shows, he was also eager to share his knowledge and, indeed, his enthusiasm with his readers. At the same time, as is abundantly clear from Ednor Whitlock, MacColl was deeply committed to defending the Christian faith, in particular in its Anglican form. For MacColl Christian faith gives life its meaning and provides a foundation for morality. ‘No stable system of ethics’, says MacColl can be constructed on mere human authority. The final authority must be superhuman, and this superhuman authority cannot be effectively appealed to till the educated and uneducated alike are firmly convinced that it really exists. [MacColl 1909, 129] In order to secure this conviction and counter the corrosive influence of science on religious belief, MacColl attempts to formulate what one can call a post- Darwinian or post-evolutionary version of the argument from design and in this way provide a rational basis for Christian faith. It is in his attempts to intervene in these religious and moral debates that MacColl’s position as an outsider is revealed most clearly. First, it is remarkable that between the publication of his two novels in 1889-1890 and the publication of his articles in The Hibbert Journal seventeen years later, MacColl’s views concerning the design argument and the foundation of morality seems to have changed hardly at all. There is no awareness in MacColl’s writing of the change in human nature that Virginia Woolf focuses on, nor any development in MacColl’s view of religion or morality. These appear to have passed MacColl by. Secondly, even by the time MacColl published his two novels the views he expressed about the relationship between science and religion and about the foundation of morality were out of step with the views held by those making up the intellectual mainstream at that time. The argument from design was losing its relevance and its adherents already in the 1850s before the publication of Darwin’s The Origin of Species (1859) and for good reason. In the third quarter of the century the realisation spread that the argument from design simply assumed what it aimed to prove. ‘Design’ is an intentional term and choosing to see something as designed is to choose a vocabulary and a perspective in which the designer is already built in. For someone who would dismiss the vocabulary of ‘design’ and instead adopt the evolutionist vocabulary provided by The Origin of Species, the argument had no force.

12. For a detailed discussion of MacColl’s views on these questions see [Cuypers 1998]. 48 Stein Haugom Olsen

But if the argument from design could not prove God’s existence, neither could the negative version of it prove that there is no God. By the time Frederick Temple, as Bishop of Exeter, came to give his Bampton Lectures in 1884, he was stating the official view of the matter when he said that the argument from design is not strong enough to compel assent from those who have no ears for the inward spiritual voice, but it is abundantly sufficient to answer those who argue that there cannot be a Creator because they cannot trace His action. [Temple 1984, 208] In 1891, when Ednor Whitlock was published, the theological function of the argument from design had long since changed from a demonstrative to a non- demonstrative one. What was required for the truly Christian, was a leap of faith. Once that was taken, problems of doubt did not arise, only problems of interpretation of the word of God. There is the same problem with MacColl’s view of morality as needing a basis in “The belief that an invisible Being or Beings take note of all we do, and can even read our most secret thoughts [. . . ]” [MacColl 1909, 149]. The view that morality ultimately needed a religious basis might still have been the view of the majority of believers in 1890, but the fear that ‘Society must fall to pieces if Darwinism be true’ was no longer widespread. Twenty years had then passed since the publication of The Descent of Man (1871) and more than thirty years since the publication of The Origin of Species (1859) and Essays and Reviews (1860), and, in spite of Darwin’s theory of evolution gaining wide acceptance not only among the generally informed public, but also among the most influential minds within the Church of England, the much feared regression of man into a mere beast and the decline of society into moral chaos had not occurred. Moreover, it was becoming accepted among Victorian intellectuals that morality could be separated from its religious basis, and that the example of Christ had its force whether or not he was divine. In Robert Elsmere (1888) Mrs. Ward called the “dissociation of the moral judgment from a special series of religious formulæ [. . . ] the crucial, the epoch-making fact of our day” [Ward 1987, 534]. And, most important of all, towards the end of the century there was a decline in orthodoxy as radical evangelicalism slowly lost its grip on the religious consciousness and moral conscience. Generally, believers became more relaxed about the way they practiced their belief, and consequently also more relaxed about the way in which Christian morality was founded. This was at least partly a generational change. It was in its extreme the replacement of the Philip Gosse of the early and mid-Victorian world with the Edmund Gosse of the late Victorian and Edwardian periods, or the Theobald Pontifexes of that world with the Ernest Pontifexes. Ernest makes his name with a book of essays apparently written in 1866 or 1867 where he has this to say about faith and the church: The writer urged that we become persecutors as a matter of course as soon as we begin to feel very strongly upon any subject; we Outside the intellectual mainstream? 49

ought not therefore to do this; we ought not to feel very strongly even upon that institution which was dearer to the writer than any other—the Church of England. We should be churchmen, but somewhat lukewarm churchmen, inasmuch as those who care very much about either religion or irreligion are seldom observed to be very well bred or agreeable people. The Church herself should approach as nearly to that of Laodicea as was compatible with her continuing to be a Church at all, and each individual member should only be hot in striving to be as lukewarm as possible. [Butler 1966, 415]

Though late Victorian attitudes to religion never became quite as lukewarm as Ernest recommends, the passage gives a description of an attitude that was developing at the time when Butler stopped his work on The Way of All Flesh in 1884. The changes briefly mentioned above may not be, as Virginia Woolf would have it, changes in human nature but they were significant shifts in attitudes, shifts that are registered neither in MacColl’s novels, nor in his essays, nor in his monograph. There was also another deep-seated change in the approach to human nature and human behaviour that made MacColl’s moral views seem out of phase with the wider intellectual climate in Britain and Europe in the last decades of the nineteenth century. Temple in his 1884 Bampton lectures speaks of ‘the inward spiritual voice’ which is also the voice of conscience. This is where morality finds its basis. This attention to the inner nature of man becomes an important feature of developments in art, in literature, and in psychology in the last two decades of the nineteenth century. All the works and developments mentioned in the first section of this paper as an illustration that Virginia Woolf’s contention that “in or about December 1910 human character changed” have this in common that they emphasize the inward nature of man, the individual human being, the individual sensibility, the individual reaction. And again, one can find this development both mentioned and illustrated in Butler’s The Way of All Flesh. At the end of chapter 5, Butler has the narrator, Overton, reflect:

I fancy that there is some truth in the view which is being put forward nowadays, that it is our less conscious thoughts and our less conscious actions which mainly mould our lives and the lives of those who spring from us. [Butler 1966, 54]

MacColl is not in any way touched by this reorientation of attention to the inner life in religion, in art, in literature, and in psychology. Of course, this reorientation of attention does not invalidate MacColl’s own metaphysical and ethical project, but the fact that at no point in his works is there any reference to such a development, does indicate that life in Boulogne did not pull MacColl into dialogue with the most recent intellectual developments. 50 Stein Haugom Olsen Conclusion

As I previously remarked, there is hardly any point in branding a man ‘an outsider’ unless there is a circle of people or a community into which he for some reason fails to enter though he desires to do so. That MacColl sought recognition for his contributions to logic and to join the newly professionalized academia seems clear. It also seems true to say that he did seek recognition as a novelist. In both areas he failed to attain recognition in his own time. Present attempts to rediscover and draw attention to MacColl’s work in logic represents a belated recognition of the importance of his work in this ﬁeld. I have argued here and elsewhere that his work in the ﬁeld of ‘general literature’ does not deserve similar attention. Hugh’s brother achieved fame as a controversialist and one might think that this would have provided a role model for Hugh and that he also aspired to recognition in this respect. There is, however, nothing in MacColl’s writings that indicates that he aspired to such a role. He had enthusiasms and concerns he wanted to share with a greater public and where necessary, he wanted to educate that public and persuade them to accept certain religious views and metaphysical doctrines that he felt were under attack. When he failed, as he did, to gain any wide hearing for his views, this was because the public debates which he wanted to join were no longer at the centre of public attention, the attitudes he represented were recognizably those of another period, and the arguments he presented were no longer current.

References

Anonymous 1889a Review of Hugh MacColl’s Mr. Stranger’s Sealed Packet (London: Chatto & Windus, 1889), The Athenæum, 3213, 25 May, 661. 1889b Review of Hugh MacColl’s Mr. Stranger’s Sealed Packet (London: Chatto & Windus, 1889), The Academy, 893, 15 June, 410. 1891a Review of Hugh MacColl’s Ednor Whitlock (London: Chatto & Windus, 1891), The Glasgow Herald, 30 April, 9. 1891b Review of Hugh MacColl’s Ednor Whitlock (London: Chatto & Windus, 1891), The Academy, 995, 30 May, 510. 1891c Review of Hugh MacColl’s Ednor Whitlock (London: Chatto & Windus, 1891), The Graphic, 17 October, 464. 1911 MacColl, Malcolm, The Encyclopaedia Britannica, 11th ed., vol. 17, 205. Astroh, Michael, Grattan-Guinness, Ivor, & Read, Stephen 2001 A survey of the life and work of Hugh MacColl (1837-1909), History and Philosophy of Logic, 22, 2001, 81–98. Outside the intellectual mainstream? 51

Atherton, Carol 2006 Deﬁning Literary Criticism: Scholarship, Authority and the Possession of Literary Knowledge, 1880-2002, Houndmills, Basingstoke: Palgrave Macmillan. Butler, Samuel 1966 The Way of All Flesh, edited by James Cochrane with an introduction by Richard Hoggart, Harmondsworth: Penguin Books (original edition in 1903). Carlyle, Thomas 1840 Chartism, London: James Fraser. Collini, Stefan 1991 Public Moralists: Political Thought and Intellectual Life in Britain 1850-1930, Oxford: Oxford University Press. 2008 Common Reading: Critics, Historians, Publics, Oxford: Oxford University Press. Cuypers, Stefaan E. 1998 The metaphysical foundations of Hugh MacColl’s religious ethics, Nordic Journal of Philosophical Logic, 3 (1), 175–196. Gregory, R. A. 1889 A journey to the planet Mars: Review of Hugh MacColl’s Mr. Stranger’s Sealed Packet (London: Chatto & Windus, 1889), Nature, 40 (1030), 291–292. Hayter, William 1977 Spooner: A Biography, London: W. H. Allen. Heyck, Thomas William 1982 The Transformation of Intellectual Life in Victorian England, London: Croom Helm. Keene, Carol A. (ed.) 1999 F. H. Bradley: Selected Correspondence: June 1872-December 1904, Bristol: Thoemmes. Kingsmill, Hugh 1932 1932 and the Victorians, English Review, 54, 681–684. Levine, Philippa 1986 The Amateur and the Professional: Antiquarians, Historians and Archaeologists in Victorian England 1838-1886, Cambridge: Cambridge University Press. 52 Stein Haugom Olsen

MacColl, Hugh 1884 The philosopher’s atom, The Spectator, 2929, 16 August, 1076. 1889 Mr. Stranger’s Sealed Packet, London: Chatto & Windus. 1891a Ednor Whitlock, London: Chatto & Windus. 1891b Mrs. Higgins’s strange lodger, Murray’s Magazine, 10, 937–976. 1906-1907 Chance or purpose, The Hibbert Journal, 5, 384–396. 1907-1908 What and where is the soul, The Hibbert Journal, 6, 158–170 & 673–674. 1908-1909 Mathematics and theology, The Hibbert Journal, 7, 916–918. 1909 Man’s Origin, Destiny, and Duty, London: Williams & Norgate. 1909-1910 Ptolemaic and Copernian views of the place of mind in the universe, The Hibbert Journal, 8, 429–430. Olsen, Stein Haugom 1998 MacColl—victorian, Nordic Journal of Philosophical Logic, 3 (1), 197–229. 2001 The canon and artistic value, British Journal of Aesthetics, 41, 261– 279. Palmer, David J. 1965 The Rise of English Studies: An Account of the Study of English Language and Literature from its Origins to the Making of the Oxford English School, London: Oxford University Press. Peer, Willie van (ed.) 2008 The Quality of Literature: Linguistic Studies in Literary Evaluation, Amsterdam: John Benjamins. Perkin, Harold 2002 The Rise of Professional Society: England Since 1880, new ed., London: Routledge (ﬁrst edition in 1989). Small, Ian 1991 Conditions for Criticism: Authority, Knowledge, and Literature in the Late Nineteenth Century, Oxford: Oxford University Press. Temple, Fredrick 1984 Apparent collision between religion and the doctrine of evolution, in T. Cosslett (ed.), Science and Religion in the Nineteenth Century, Cambridge: Cambridge University Press (originally published in 1884), 190–204. Outside the intellectual mainstream? 53

Tillyard, Eustace M. W. 1958 The Muse Unchained: An Intimate Account of the Revolution in English Studies at Cambridge, London: Bowes & Bowes. Walsh, William, H. 2000 The Zenith of Greats, in M. Brock & M. C. Curthoys (eds.), The History of the University of Oxford, Oxford: Oxford University Press, 311–326. Ward, Mrs. Humphry 1987 Robert Elsmere, edited by R. Ashton, Oxford: Oxford University Press (original edition in 1888). Woolf, Virginia 1966 Collected Essays, vol. 1, London: Hogarth Press.

Hugh MacColl and Lewis Carroll: Crosscurrents in geometry and logic Francine F. Abeles Kean University, NJ, USA Amirouche Mokteﬁ LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université & IRIST (EA 3424), Université de Strasbourg, France

Résumé : Dans une lettre adressée à Bertrand Russell, le 17 mai 1905, Hugh MacColl raconte avoir abandonné l’étude de la logique après 1884, pendant près de treize ans, et explique que ce fut la lecture de l’ouvrage de Lewis Carroll, Symbolic Logic (1896), qui “ralluma le vieux feu qu’il croyait éteint”. Dès lors, il publie de nombreux articles contenant certaines de ses innovations majeures en logique. L’objet de cet article est de discuter la familiar- ité de MacColl et son appréciation du travail de Carroll, et de comprendre comment cela l’amena à réinvestir le domaine de la logique. Rien n’indique que les deux hommes se soient jamais rencontrés ou aient échangé une correspondance personnelle. Cependant, MacColl recensa dans le journal The Athenaeum plusieurs ouvrages de Carroll, essentiellement en géométrie et en logique. Souvent, Carroll répondait aux critiques de MacColl dans les éditions suivantes de ses ouvrages. Une discussion de ces sources fournit des éclaircisse- ments intéressants sur les investigations mathématiques et logiques de MacColl et Carroll. Abstract: In a letter to Bertrand Russell, dated 17 May 1905, Hugh MacColl tells how he abandoned the study of logic after 1884 for about thirteen years and how it was the reading of Lewis Carroll’s Symbolic Logic (1896) that “rekindled the old ﬁre which [he] thought extinct.” From then onwards, he published several papers containing some of his major logical innovations. The aim of this paper is to discuss MacColl’s acquaintance with and appreciation of Carroll’s work, and how that reading convinced him to come back into the realm of logic. There is no evidence that the two men ever met or exchanged any personal correspondence. However, MacColl reviewed in The Athenaeum several of Carroll’s books, mainly on geometry and logic. Often, Carroll replied to MacColl’s criticisms in the following editions of his works. A discussion of these sources provides interesting insights to MacColl’s (and Carroll’s) mathematical and logical investigations.

Philosophia Scientiæ, 15 (1), 2011, 55–76. 56 Francine F. Abeles & Amirouche Mokteﬁ

It is well known that Hugh MacColl (1837-1909) interrupted his logical investigations around 1884 for about thirteen years, a period which he mainly devoted to literature. In a letter to Bertrand Russell (1872-1970), dated 17 May 1905, MacColl recalls this break and tells how it was the reading of Lewis Carroll’s Symbolic Logic in 1896 that led him to reinstate his logical studies: When, more than twenty-eight years ago, I discovered my Calculus of Limits [. . . ], I regarded it at first as a purely mathematical system restricted to purely mathematical questions. [. . . ] When I found that my method could be applied to purely logical questions unconnected with the integral calculus or with probability, I sent a second and a third paper to the Mathematical Society, which were both accepted, and also a paper to Mind (published January 1880). [. . . ] I sent a fourth paper (in 1884) to the Math. Soc., on the “Limits of Multiple Integrals,” which was also accepted. This I thought would be my final contribution to logic or mathematics, and, for the next twelve or thirteen years, I devoted my leisure hours to general literature. Then a friend sent me Mr. Dodgson’s (“Lewis Carroll’s” 1) Symbolic Logic, a perusal of which rekindled the old fire which I thought extinct. My articles since then I believe to be far more important from the point of view of general logic than my earlier ones. [Astroh, Grattan-Guinness & Read 2001, 93–94] The aim of this paper is to discuss MacColl’s acquaintance with and appreciation of Carroll’s work, and how that reading convinced him to come back into the realm of logic. There is no evidence that the two men ever met or exchanged any personal correspondence. Both mathematicians worked in relative isolation on the margins of the mathematical scene. They shared an interest in educational issues and solving mathematical problems. 2 Their activities made them interact face to face, however, through the London journal, The Athenaeum. Indeed, MacColl acted as the main reviewer of mathematical books for that journal in the 1880s and 1890s. 3 As such he reviewed anony-

1. “Lewis Carroll” is the pen name of Charles L. Dodgson (1832-1898). To avoid confusion, in this paper we will use the pseudonym, Carroll, even when we are concerned with works signed with his real name. However, the works signed with the real name (Dodgson) and those signed with the pen name (Carroll) are listed separately in the bibliography. 2. Both were active contributors to the “mathematical questions and solutions” section of the Educational Times [Grattan-Guinness 1992]. At least once, they were concerned with the very same question in probability [Abeles 1994, 213–218]. Later on, MacColl replied to a logical problem published posthumously by Carroll [Abeles 2010, 184–185]. 3. Reviews in The Athenaeum are anonymous. However, the editor’s marked copies (now at City University Library, London) reveal the name of the reviewers. No survey of MacColl’s reviews in this journal has yet been published. A look at the marked copies shows, however, that MacColl reviewed most mathematical books from the mid-1880s onwards. It might look surprising to see a minor mathematics H. MacColl and L. Carroll: Crosscurrents in geometry and logic 57 mously most of Carroll’s mathematical books during that period, mainly on geometry and logic. Often, Carroll replied to MacColl’s criticisms in the following editions of his works. A discussion of these connected sources provides interesting insights to both Carroll’s and MacColl’s mathematical and logical investigations. They also reveal influences on each other’s thinking in several areas that they wrote about subsequently. Carroll was the mathematical lecturer at Christ Church, Oxford. As such, he was greatly interested in geometrical teaching and participated actively in the debates of the time on the adequacy of Euclid’s Elements as a manual for teaching purposes in schools and Colleges [Moktefi 2007b]. He argued for Euclid against the new rival textbooks, but didn’t object to some changes in Euclid’s text when appropriate, as long as the order of the theorems was not violated. In 1888, however, he published A New Theory of Parallels, a book where, as its title clearly indicates, he suggested a completely new Euclidean parallel axiom [Dodgson 1888]. Two more editions of the book appeared in the immediately subsequent years [Dodgson 1889 & 1891], and a fourth appeared later in 1895 [Dodgson 1895a]. MacColl reviewed the three first editions in The Athenaeum. In the following sections, we will first explore the dominant themes that attracted his attention when he read A New Theory of Parallels: the nature of axioms, the order in which axioms and theorems are used, the meaning and use of the terms ‘infinity’ and ‘infinitesimal’, the role of Archimedes’s axiom in a geometry, and the relationship between Euclidean and non-Euclidean geometry. It is important to keep in mind the time period when MacColl was reviewing Carroll’s work on geometry (1888-1891). The non-Euclidean geometry of Nikolai Lobachevsky (1792-1856) had begun to be widely known soon after Jules Hoüel (1823-1886) published the first edition of his French translation of it in 1863. Beginning in 1882 with the work of Moritz Pasch (1843-1930) in his Vorlesungen über neuere Geometrie, and continuing with the work of David Hilbert (1862-1943) in his Grundlagen der Geometrie (1899) which was translated into French in 1900, and into English in 1902, the transition began to occur from the older view of an axiom as a generally accepted principle that is a self-evident truth to the modern view that an axiom is arbitrarily chosen and deals with indefinable concepts about arbitrarily chosen objects of thought. Thus, mathematics and formal logic each can be described as a science of pure thought.

figure living in a foreign country act as the main mathematical reviewer for such an established journal, a task that was for instance previously undertaken by the London Professor of mathematics Augustus de Morgan (1806-1871) for over twenty years (1850s-1860s). Hugh MacColl might have been recommended by his brother Malcolm (1831-1907), who had previously reviewed some theological books for the journal. He might also have benefitted from a “natural” sympathy by the editor of the journal with whom he shared Scottish origins and even the name (although we know of no direct family relation between them), for Norman MacColl (1843-1904) was the editor of The Athenaeum from 1869 to 1901. 58 Francine F. Abeles & Amirouche Moktefi The nature of axioms and how they are used

We commonly consider that the theorems of a mathematical system are understood from its axioms and the logical machinery employed to deduce them. Beginning in the second half of the 19th century the question arose of how an axiom is to be understood. One possibility was that we believe the axioms are true because some of the obvious theorems of the system can be deduced from them. Each axiom is expressing something that is true about its concepts and terms. We also trust the integrity of the mathematical system itself because the less obvious and non-obvious theorems can then be deduced from the axioms and from the obvious theorems. This is not really circu- lar thinking, but more an assertion that a psychological element is involved in understanding an axiom. The second possibility follows the earlier and stricter Euclidean approach that only the terms and concepts in the axiom matter, so that we understand an axiom only when, after suitable reflection, we have a clear idea of the meaning of those terms and concepts, i. e. the axiom expresses a self-evident truth. How then are axioms selected? In any mathematical system there are some true theorems that are not dependent on others. Because these theorems involve fundamental truths that do not need to be proved, they become the candidates of choice for the axioms of the mathematical system. These fundamental truths are not necessarily self- evident except in the Euclidean approach to understanding an axiom, where they would have to be. 4 In A New Theory of Parallels, Carroll chose as an equivalent to the Euclidean parallel axiom, an axiom, number 6, the last in his set of axioms, that states: In every Circle, the inscribed equilateral Hexagon is greater [in area] than any one of the Segments which lie outside it. [Dodgson 1888, Frontispiece] Carroll used this axiom to prove Euc. I. 32 [Abeles 1995]. MacColl objected to this axiom on several grounds and continued to criticize it even after Carroll had modified it in response to his criticisms. In his review of the first edition, MacColl questioned Carroll’s “implied assumption of the possibility of the inscribed equilateral hexagon,” that he asserts Euclid did not demonstrate until Book IV, Proposition 15 of the Elements [MacColl 1888, 557]. MacColl clarified his objection to the axiom in his review of the second edition of this book a year later, after Carroll had responded specifically, but not by name, to his objection in the preface to this new edition. In that preface, in response to another reviewer, Carroll had added at the bottom of the page containing his offending axiom that “it is assumed to be theoretically possible to inscribe an equilateral Hexagon in a Circle” [Dodgson 1889, xi]. Then answering MacColl, Carroll questioned whether the possibility needs to be demonstrated: 4. For a penetrating analysis of self-evidence, see [Shapiro 2009]. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 59

May we not assume (1) that the Magnitude ‘four right angles’ contains 6-6ths of itself; (2) that it is theoretically possible to draw radii dividing it into these 6-6ths? Once grant me this, and I ask no more. I have then the logical right to join the ends of these radii, and to prove (by Euc. I. 4) that the chords are equal. [Dodgson 1889, xi]

But MacColl still was not satisfied with Carroll’s statements in the preface to this second edition. He rejoined that Carroll’s assumption conflicts with Euclidean practice and asks him to keep within Euclid’s restrictions: We objected that it was not consistent with the spirit or practice of Euclid’s reasoning to assume the “theoretical possibility” of a regular hexagon inscribed in a circle without first proving that such a figure could be actually constructed from his three postulates. Euclid’s restrictions may be arbitrary, unnecessary, cramp- ing, vexatious, absurd—indeed, we think they deserve these and many other epithets—but there they are, and if Mr. Dodgson accepts them he is bound to keep his assumptions within the boundaries which they prescribe. [MacColl 1889, 457] The third edition of A New Theory of Parallels was published in 1890 and it was really an expanded and revised edition of the book. Carroll modified his problematic axiom, replacing the hexagon with a tetragon. In the preface to the new edition, he responded to MacColl quite sarcastically, when he described for the hexagon, the construction of an angle one sixth of four right angles in size, using Euc. I. 1 to construct an equilateral triangle; then using Euc. I. 9 to draw the bisectors of two of its angles; then post. 1 to join the points of intersection to the third vertex; then Euc. I. 4 to prove the three angles at the third vertex are equal; finally to conclude that each of these angles is one third, (and that therefore its half is one sixth) of four right angles [Dodgson 1890, xiii]. Besides objecting to Carroll’s unnecessary sarcasm, MacColl remained un- satisfied. In his much longer review of the third edition he criticized Carroll’s use of propositions, in addition to postulates, to construct the new figure, the square. He wrote:

An axiom founded on propositions, however few and elementary, strikes us as somewhat of an anomaly. [MacColl 1891, 196]

Then he gives his own development “to find a simple substitute for Euclid’s twelfth axiom” by exchanging its order with I. 30, then proving I. 29. “The twelfth axiom of Euclid could then be proved in place of his thirtieth proposition so that there would be hardly any disturbance in the accepted arrangement of matter ” [MacColl 1891, 196–197]. By way of explanation, Euclid’s fifth postulate, the parallel postulate, came to be known in England as the twelfth axiom from the well regarded and popular edition of the Elements by 60 Francine F. Abeles & Amirouche Moktefi the Scottish mathematician, Robert Simson (1687-1768), which had first appeared in 1756. Simson’s text contains three postulates and twelve axioms. Of the first ten axioms in the edition reproduced in the 19th century by Isaac Todhunter (1820-1884), only five can be attributed to Euclid. And of the last two of the twelve axioms, Simson’s twelfth axiom was Euclid’s fifth postulate. Alluding to Carroll’s sarcastic comments about his criticisms, MacColl went on to say that, “Postulates and axioms are permissions to do or to assume certain things, and are therefore, the opposite of restrictions. The restrictions of Euclid arise from the paucity of his postulates and axioms, a paucity which [he thinks] regrettable” [MacColl 1891, 197]. It seems quite clear that MacColl is a strict Euclidean in that he believes that an axiom should express a fundamental self-evident truth. Carroll, on the other hand, believes that a psychological element is involved in knowing an axiom as he explained in the preface to the first edition of his book:

I shall be told, no doubt, that this is too bizarre and unprece- dented an Axiom—that it is an appeal to the eye and not to the reason. That it is somewhat bizarre I am willing to admit—and am by no means sure that this is not rather a merit than a defect. But, as to its being an appeal to the eye, what is ‘two straight Lines cannot enclose a space’ [a fact included in the meaning of postulate 1: to draw a straight line from any point to any point] but an appeal to the eye? What is ‘all right angles are equal’ [postulate 4] but an appeal to the eye? [Dodgson 1888, xv]

What exactly did Carroll have in mind when he created his axiom 6? First, it appears that he selected a hexagon because of all the Euclidean polygonal constructions that of a regular hexagon is the simplest. 5 But there appears to be another reason. In hyperbolic geometry there is an upper limit to the area of a triangle, but there is no upper limit to the lengths of its sides. Since a polygon can be triangulated and its area equals the sum of the areas of these triangles, eventually the sides of the hexagon will become very long while its area will reach a maximum. In a suﬃciently large circle the area of a segment will exceed the area of the inscribed hexagon. From the center of the circle circumscribing the hexagon, the hexagon has a unique triangulation while for an inscribed square (which does not exist in hyperbolic geometry) there are two such triangulations. As he indicated in a diary entry dated 8 April 1888, he was initially considering a diﬀerent formulation for his axiom: My Axiom has taken many forms, the last a discovery of this evening, “if an equilateral triangle be inscribed in a circle, it cannot be made to lie partly within it, with each vertex outside, and the circle cutting each side in two points.” [Wakeling 2004, 390]

5. The construction of a square does not occur until Euc. 1. 46. And unlike the square, the hexagon construction does not require the parallel postulate. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 61

He alluded to this argument in Appendix IV of the first edition of A New Theory of Parallels where he provided greater detail, calling it a Will-o’-the- Wisp (but omitted it in the third edition). In the end he settled on the inscribed hexagon formulation but concluded that: [A]ny polygon will do, and any ratio between it and the out-lying Segment, so long as it is a finite ratio. [Dodgson 1888, 61; 1890, 74–75] MacColl may have been influenced by his exchange with Carroll on the nature of axioms. This becomes apparent in his late writings on geometry, particularly an article written at the end of 1909 and published posthumously in Mind the following year. In section II, devoted to “Axioms, Inference, Implication”, MacColl wrote: What is an axiom? No clear line of demarcation can be drawn between an axiom and any other general proposition or formula that is known and admitted to be true [. . . ] A proposition that may appear axiomatic to one person may appear doubtful to another, until he has obtained a satisfactory proof of it; after which he treats it as an axiom in all subsequent researches [. . . ] To an omniscient mind would not all true propositions be equally axiomatic? Would it not be absurd to speak of such a mind as inferring B from A? This, of course, is an extreme case, but such cases are precisely those that most effectively test the validity of a principle. [MacColl 1910, 193] This passage suggests that MacColl has given up his belief that an axiom must express a fundamental self-evident truth, and he has accepted the idea that people differ in what they view as axioms as opposed to propositions. MacColl goes on to say that, “strict Euclideans consider no proof valid [. . . ] if it takes anything for granted that is not founded on Euclid’s twelve axioms, although as a matter of fact, Euclid himself, in several of his formal proofs, tacitly assumes axioms which are absent from his given list” [MacColl 1910, 193]. These are the restrictions we referred to earlier (that MacColl had alluded to in his last review of A New Theory of Parallels). Further evidence that MacColl may have been influenced by Carroll’s views is that in the same paper he describes axioms as “fundamental formulae of appeal” [MacColl 1910, 196], a term difficult to understand except when cast in the light of Carroll’s statement that his axiom is an appeal to the eye rather than to the mind. These are the permissions that MacColl had also alluded to in this same last review.

Inﬁnity, inﬁnitesimal, the Archimedean axiom, and non-Euclidean geometry

It is interesting to note, however, that when MacColl reviewed the three editions of A New Theory of Parallels he did not analyze Carroll’s unusual for- 62 Francine F. Abeles & Amirouche Moktefi mulation of the Euclidean parallel axiom nor did he include any mention of the appendices in that book where Carroll dealt with infinities and infinitesimals. For the superiority of his axiom, Carroll claimed: While every other Theory (that I have seen), which attempts to supersede Euclid’s 12th [parallel] Axiom, introduces the ideas of Infinities and Infinitesimals, mine dispenses wholly with their aid, and deals with nothing but what is, by universal consent, absolutely within the field of Human Reason. [Dodgson 1890, xiv] Although he was silent here, MacColl discussed the issue of infinity and infinitesimal in a subsequent review of Russell’s Essay on the Foundations of Geometry (1897) that appeared in The Athenaeum in January 1898. Russell discussed the validity of non-Euclidean geometries whose principal writers he named as Lobachevsky, Carl Friedrich Gauss (1777-1855), Bernhard Riemann (1826-1866), Hermann von Helmholtz (1821-1894) and Arthur Cayley (1821- 1895) [Russell 1897]. MacColl identified as the key issue of the book, Russell’s belief that certain axioms, particularly the parallel axiom, should be “regarded as empirical laws, derived from the investigation and measurement of our actual space, and true only [. . . ] within the limits set by errors of observation” [MacColl 1898, 25]. MacColl, asserting that Russell’s view is based on his erroneous notion of an infinitesimal, writes: Those who assert the possibility of discovering error in Euclid’s axioms by a conceivably more refined method of measurement forget (as Mr. Russell appears to do) [. . . ] that the infinitesimal dx is never synonymous with zero, that it is wholly relative, and has no fixed and absolute magnitude. The infinitesimal dx has simply any magnitude, great or small, we choose to assign to it, consistently with the limitation that the variable x, with which we compare it is (when not zero) infinite in comparison. The infinitesimal dx may, in fact [. . . ] have any value, from the millionth part of the diameter of an hypothetical atom to a million times the distance of the farthest star that will ever become visible through the most powerful telescope. [MacColl 1898, 25] MacColl goes on to say that Russell’s “thoughts may be correct, but the words in which he expresses them border upon self-contradiction. [. . . ] Whence come these contradictions? [. . . ] they arise solely from the unnecessary ambiguity created by confounding the ‘infinitesimal’ with ‘zero”’ [MacColl 1898, 26]. However, MacColl’s notions about infinitesimals and infinities were somewhat odd and sometimes incorrect. He had discussed these ideas in “The calculus of limits” [MacColl 1904], and again in “Symbolic Reasoning (VIII)” (1906) where, for example, he asserted that “the symbol 0 represents [. . . ] the death of a real infinitesimal ratio h in passing from the positive to the negative state” [MacColl 1906b, 506]. In contrast to Russell, Carroll’s view of axioms is quite different. In the Supplement to Euclid and His Modern Rivals from 1885, Carroll gave what H. MacColl and L. Carroll: Crosscurrents in geometry and logic 63 amounts to his definition of an axiom: “ [It] always requests the voluntary assent of the reader to some truth, for which no proof is offered, and which he is not logically compelled to grant” [Dodgson 1885, 351]. He goes on to say that accepting any axiom that may be proposed “largely depends on the amount of truth he [the reader] has already grasped in connection with it” [Dodgson 1885, 351]. So, Carroll does not subscribe to the idea that they are empirical laws subject to verification, but rather that they must be acceptable to the faculties of reason which, by its nature, cannot apprehend infinities and infinitesimals. MacColl seems to be more in agreement with Carroll’s view. It was primarily for this purpose that Carroll chose his axiom 6 as a substitute for Euclid’s parallel axiom. It is a closed form that deals with finite area and not with lines, and is unique in this respect. In view of MacColl’s and Carroll’s lack of clear and correct ideas about the nature of infinitesimals, it is remarkable that Carroll arrived at the role the Archimedean axiom plays in a geometry. Euclid stated this axiom as the first proposition in Book X. Using Euclid’s definitions of ratio, i. e. relative magnitude, which are numbers 3 and 4 in Book V, Carroll reasoned correctly that definition 4 was meant to exclude the relation of a finite magnitude to an infinitely great or infinitely small one, where both are magnitudes of the same kind. And he saw that Euclid employed this definition in his proof of X. 1. So, Carroll was one of the few mathematicians of his time to understand that the Archimedean axiom excludes infinitesimals, and by implication that hyperbolic geometry, because it admits infinitesimals, is necessarily non-Archimedean. Carroll wrote: My conclusion, then, is that, in Book X. Prop. 1, Euclid is limiting his view to the case of two homogeneous Magnitudes which are such that neither of them is infinitely greater than the other [. . . ] that he is contemplating Finite Magnitudes only. [Dodgson 1890, 42] The importance of Carroll’s discovery became apparent in the independent work of Giuseppe Veronese (1854-1917). Just as one can exclude the parallel axiom from the rest of the axioms of Euclidean geometry to investigate what other geometries are possible, so one can exclude the Archimedean axiom and obtain the non-Archimedean geometries of which Veronese’s 1890 geometry was the first one. 6 However, Carroll’s discussion of area in the context of non-finite objects like strips and sectors was largely incorrect. He made comparisons between finite and infinite areas that led to paradoxical, actually fallacious results, because he handled infinities as if they were magnitudes having a definite size. He based his understanding of the infinite and the infinitesimal primarily on

6. Pasch was probably the first mathematician to fully understand the significance of Archimedes’s axiom [Schlimm 2010]. Russell had discussed Pasch’s book in his Principles of Mathematics [Russell 1903, 393–403], but it is doubtful that MacColl read this book. 64 Francine F. Abeles & Amirouche Moktefi the work of the Swiss mathematician Louis Bertrand (1731-1812) who used the notion of infinite area [Bertrand 1778]. Carroll came to know Bertrand’s work because Adrien-Marie Legendre (1752-1833) used it in an attempt to prove Euclid’s parallel axiom. In an unpublished letter, dated 2 April 1878, to the Oxford mathematician Wallis Hay Laverty (1847-1928), Carroll discussed Legendre’s proof [Dodgson 1878]. In spite of their disagreements, Carroll and MacColl were both Euclideans. MacColl stated clearly his strong preference for Euclidean geometry with an implicit rejection of hyperbolic non-Euclidean geometry when he wrote in his 1910 paper that: The Euclidean geometry seems to me to be the only true one, not merely because it is admittedly the simplest and most convenient, but also, and chiefly, because it is the only system that frankly accepts the customary conventions of ordinary language. [MacColl 1910, 188] MacColl had already alluded to this belief in his review of Russell’s Essay on the Foundations of Geometry when he wrote that the non-Euclideans’ “so- called straight lines are not straight lines in the ordinary sense, but two great circles of an infinite sphere, or some other curves of infinitesimal (but not quite zero) curvature [. . . ] his familiarity [. . . ] with their peculiar ways of thinking leads him occasionally [. . . ] into inconsistency of language, if not into positive error” [MacColl 1898, 26]. Carroll’s rejection of hyperbolic non- Euclidean geometry was consistent with his conception of the nature of human reasoning, i. e. the impossibility of picturing two parallel lines produced to infinity. So we see that both Carroll and MacColl agreed in their negative assessments of non-Euclidean geometry, but not entirely for the same reasons.

The barbershop problem

Despite their exchanges in print on geometry, it seems that Carroll and MacColl didn’t know each other. Indeed, the barbershop controversy that occurred in the British logical scene in the subsequent years, with no par- ticipation by MacColl at its beginning, suggests that MacColl was not per- sonally acquainted at all with Carroll or with the British logicians in the early 1890s, although he was familiar with John Venn (1834-1923), William S. Jevons (1835-1882) and Charles S. Peirce (1839-1914) in the early 1880s and with Russell in the early 1900s. The barbershop problem was Carroll’s first publication in the journal Mind. It is the transcription of a dispute which opposed him to the Oxford Professor of Logic John Cook Wilson (1849-1915). The two men debated regularly on various topics in geometry, chances, and logic. The dispute which led to the barbershop problem began around the end of 1892. Despite their exchanges during the next year, the two men never agreed. Carroll asked Wilson for H. MacColl and L. Carroll: Crosscurrents in geometry and logic 65 an accepted transcript of the problem, which he obtained in February 1894. Then, he distributed copies of the problem to his colleagues and friends, in Oxford and elsewhere, and asked them for their opinion, in order to collect and compare their solutions. 7 Recipients included: Venn, Henry Sidgwick (1838-1900), James Welton (1854-1942) and Francis H. Bradley (1846-1924). The contradictory opinions that Carroll collected from his correspondence encouraged him to publish the problem, as he explains in a note annexed to his Mind paper: The paradox, of which the foregoing paper is an ornamental pre- sentment, is, I have reason to believe, a very real difficulty in the Theory of Hypotheticals. The disputed point has been for some time under discussion by several practised logicians, to whom I have submitted it; and the various and conflicting opinions, which my correspondence with them has elicited, convince me that the subject needs further consideration, in order that logical teachers and writers may come to some agreement as to what Hypotheticals are, and how they ought to be treated. [Carroll 1894, 438] The problem was as follows: there are three men (Allen, Brown and Carr) working in a barbershop. It is known that at least one man should be in the shop, hence: (1) If Carr is out, then (if Allen is out, then Brown is in). Also, it is known that Allen can’t leave the shop without being accompanied by Brown, hence: (2) If Allen is out, then Brown is out. Suppose that Carr is out. From (1), it follows that: (3) If Allen is out, then Brown is in. The proposition (3) is considered as incompatible with (2) which is known to be true. Hence proposition (3) is false. From (1), by contraposition, it follows that Carr is in the shop (which contradicts the initial hypothesis that Carr was out). From this absurdity, one concludes that Carr can’t leave the shop and should necessarily be inside. This result is paradoxical because it is easy to show that Carr can leave the shop without contradicting any of the rules (1) and (2), as long as Allen is in the shop. The debate that followed the publication of this problem focuses on the assumption that the two propositions “If Allen is out, then Brown is in” and “If Allen is out, then Brown is out” were incompatible. The modern logician will clearly see here what is known to him as the paradoxes of material implication. Indeed, Russell used the barbershop problem in his Principles of Mathematics to illustrate his principle that a false proposition implies all others [Russell

7. On the genesis of the barbershop problem, see [Moktefi 2007a]. Different versions of the problem are reproduced in [Bartley 1986, 449–465] & [Abeles 2010, 111– 128]. 66 Francine F. Abeles & Amirouche Moktefi

1903, 18]. We do not aim to discuss the problem here. The interesting point is that the barbershop problem continued, after its publication, to be widely commented about by logicians. 8 Venn was one of the first to discuss it in print, in the second edition of his Symbolic Logic, where he writes: This particular aspect of the question will very likely be familiar to some of my readers from a problem recently circulated, for comparison of opinions, amongst logicians. As the proposer is, to the general reader, better known in a very different branch of literature, I will call it Alice’s Problem. [Venn 1894, 442] Venn confirms in this passage that the problem circulated between the British logicians of the time. It seems, however, that MacColl did not participate in this wide debate. He probably discovered it through Venn, as suggested by his first reference to the problem in 1897: To show the working of this logical calculus of three dimensions I may take the following problem, which Dr. Venn in his Symbolic Logic (second edition, p. 442) calls “Alice’s Problem,” and which I understand has (under another form) been already discussed by logicians, but with varying conclusions. [MacColl 1897, 501–502] By 1897, the barbershop problem had already been discussed by various authors in the journal Mind, but MacColl was not yet among them. This late interest is comprehensible: MacColl was not an active reader of the mathematical and philosophical journals of his time, although he continued to review books for The Athenaeum. He certainly had access to some British (and foreign) periodicals at the Merridew Library, founded in Boulogne-sur-Mer by the British librarian Henry Melville-Merridew (1826-1879) who lived there. But MacColl’s reading was not regular as he explained in a letter to Bradley on 14 December 1904 [Keene 1999, 308]. Moreover, when the Barbershop debate occurred, MacColl was no longer involved in the study of logic and was more interested in fiction, as he told Russell in the letter quoted at the opening of this paper. Of course, the historian has to deal with MacColl’s testimony with caution. Still, MacColl’s publications record confirms that he returned to the study of logic at the end of 1896, the year Carroll published his Symbolic Logic. Indeed, that year MacColl published his first logical writings since the mid- 1880s and continued in the following years to contribute actively to various mathematical and philosophical journals. Ultimately, he published in 1906 his only logic book: Symbolic Logic and its Applications, where he gave a large overview of his logical theory [MacColl 1906a]. 9 In the 1890s, MacColl was the main reviewer of mathematical books in The Athenaeum. Until 1896, however, he didn’t review books on logic. These 8. All of the responses to the barbershop problem, published between 1894 and 1905 in Mind, are reproduced in [Abeles 2010, 129–183]. 9. The book is reproduced in [Rahman & Redmond 2007, 79–226] together with most of MacColl’s other logical writings. The editor’s introduction provides an interesting overview of MacColl’s contributions to logic. On this, see also [Astroh & Read 1998]. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 67 were customarily attributed to other regular reviewers and were usually noticed in the ‘Literature’ section of the journal, contrary to the mathematics books reviewed by MacColl in the ‘Science’ section. 10 It is therefore a kind of “historical accident” to see MacColl reviewing Lewis Carroll’s Symbolic Logic in 1896, especially as Carroll signed it with his literary pseudonym, not with the real name (Dodgson) that he usually used for his mathematical works. Carroll tried his best to keep secret his real identity and denied in print any connection between the two names. However, it was probably a public secret in the 1890s that the author of the Alice tales was Dodgson, the Oxford mathematician. MacColl himself alluded to Alice in his 1891 review of the third edition of A New Theory of Parallels, although the book was signed ‘Dodgson’. It is certain that Carroll did not appreciate this allusion: [W]e strongly recommend non-mathematicians as well as mathematicians to read his witty and ingenious ‘Curiosa,’ which (if their experience agrees with ours) they will find as entertaining as little Alice found the curiosa of Wonderland. [MacColl 1891, 197] It seems likely that MacColl was asked to review Carroll’s logic book mainly because of his acquaintance with the author’s previous works. In addition to his reviews of Carroll’s successive editions of A New Theory of Parallels, discussed earlier in this paper, MacColl reviewed in 1893 another mathematical book by Carroll, Pillow Problems, a collection of seventy-two problems “solved, in the head, while lying awake at night” [Dodgson 1895b, xiii]. The last problem is particularly intriguing and was assumed by MacColl to be “more intended as a joke than as a real question admitting a definite answer” [MacColl 1893, 557]. It has since given rise to several interpretations and com- mentaries. 11 Both A New Theory of Parallels and Pillow Problems reached a fourth edition in 1895, but apparently neither was reviewed by MacColl.

Symbolic logic

MacColl’s review of Symbolic Logic appeared in October 1896. It was based on the ﬁrst edition of the book (having a preface dated January 1896). In the meantime, Carroll published two further editions (the preface of the third edition is dated 20 July 1896). A fourth and ﬁnal edition appeared later at the beginning of 1897 (but the preface is dated Christmas 1896) and contained some changes which might have been motivated by MacColl’s review, although Carroll never refers to him explicitly. It is, however, certain that Carroll read MacColl’s review carefully as he wrote a long response which unfortunately

10. There is one exception: in 1892, MacColl reviewed a logic book by Edward T. Dixon (1862-1935): An Essay on Reasoning [MacColl 1892b], together with a mathematical treatise on The Foundations of Geometry by the same author [MacColl 1892a]. Both reviews appeared in the ‘Science’ section however. 11. On Carroll’s pillow problems, and particularly the 72th problem, see [Seneta 1993] & [Dale 1999, 447–464]. 68 Francine F. Abeles & Amirouche Moktefi has not been published and did not survive [Anonymous 1898]. In his review, MacColl says that he appreciated the exposition of the book but considered that its author didn’t go beyond the existing logical theories of the time: Before offering any detailed criticism of Lewis Carroll’s methods we may state certain favourable points which his book undoubtedly possesses. It is well arranged, its expositions are lucid, it has an excellent stock of examples—many of them worked out, and not a few witty and amusing; and its arguments, even when wrong, are always acute and well worth weighing. [. . . ] On the whole, though the author has unquestionably written an interesting and useful little work, his proposed symbolic method does not appear to possess any advantages over some other methods that have preceded it; but, of course, we cannot say what important surprises parts ii. and iii. of his system may have in store for us when they make their appearance. [MacColl 1896, 520–521] MacColl’s general judgment is comprehensible as he was reviewing a book that Carroll wrote in such a way as to be understood by a wide audience. Carroll himself considered the content of the book as elementary, but warned that the second and third parts of his logic trilogy will address advanced and transcendental logic respectively. MacColl is, however, severe when he says that Carroll’s method has no advantages over existing methods. In fact, in the bulk of the review, MacColl himself recognized that Carroll developed elegant and effective symbolism and diagrams to represent relations of classes. 12 What MacColl really did reproach Carroll for is that he followed his contemporaries in working with a logic of classes, which differs from MacColl’s own propositional approach. It is well known that MacColl developed a propositional notation, which he believed would improve (and thus should supersede) the mainstream symbolic approach inherited from George Boole (1815-1864). Peirce and Ernst Schröder (1841-1902) were sympathetic with MacColl’s work. In Britain however, it had been virulently criticized by Jevons and Venn who both defended an equational notation where letters represent terms or classes. There is no evidence that Carroll ever knew about MacColl’s early work on logic, although he might have seen some references to him in logical treatises by other authors. Carroll’s notation was different from those of his predecessors as it requires the use of subscripts to express the existence (or not) of classes. Carroll did occasionally use a propositional notation, but his book expresses primarily a logic where letters represent classes, not propositions as MacColl would have wished. Indeed, in his review of Symbolic Logic, MacColl takes advantage of his anonymity and maintains that propositional calculus (and particularly his own notation for it) was better and more convenient:

12. For an overview of Carroll’s contributions to logic, see [Mokteﬁ 2008] & [Abeles 2010]. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 69

Of the various proposed notations in symbolic logic, the most convenient and symmetrical in our view is that adopted by Mr. MacColl, who (contrary, we believe, to all his predecessors in this field) insists that the simplest as well as the most effective symbolic system is that in which each single letter denotes not a class and not a property but a complete proposition, i. e. an assertion or denial. [. . . ] Nothing so well illustrates the disadvan- tages of a system of symbolic logic in which single letters represent classes or properties (instead of propositions) as the way in which logicians are sometimes obliged to twist and torture simple sentences in order to adapt them for syllogistic reasoning. For example, Lewis Carroll paraphrases the simple proposition “John is not well” into the astounding assertion that all Johns are men who are not well, as if the illness of one member of that numerous and widely scattered family must necessarily involve the illness of all! Among the many Johns of Lewis Carroll’s acquaintance is there really not one who enjoys good health? [MacColl 1896, 520] MacColl refers here to an individual proposition used by Carroll to explain how to write a given proposition in a normal form. The proposition is: “John is not well.” In the first edition of his book, Carroll explains that the subject of this proposition is the class “Johns” which includes all persons referred to by the name “John”, and then writes it in the following normal form: All Johns are men who are not well. [Carroll 1896a, 17] In the second edition however, Carroll changed his mind and followed a suggestion given to him by his colleague George Osborn (1864-1932) [Dodgson 1896]. He explains that the subject is a one Member Class shaped by the “John” referred to in the proposition. The normal form becomes: All Johns are not-well. [Carroll 1896b, 16] Carroll maintained this form in the third edition of the book [Carroll 1896c, 12]. However, in the fourth edition (which followed the publication of MacColl’s review), Carroll again changed his normal form, in accordance with MacColl’s above criticism, to make it unambiguous as to what is the subject of the proposition: Let us take, as an example, the Proposition “John is not well.” This of course implies that there is an Individual, to whom the speaker refers when he mentions “John,” and whom the lis- tener knows to be referred to. Hence the Class “men referred to by the speaker when he mentions ‘John”’ is a one-Member Class, and the Proposition is equivalent to “All the men, who are referred to by the speaker when he mentions ‘John’, are not well.” [Carroll 1897, 10] Carroll’s difficulties here come partly from his failure to distinguish between a single thing and a one-member-class [Carroll 1897, 2–3]. A class in Carroll’s 70 Francine F. Abeles & Amirouche Moktefi logic is a collection of things, not a set (as modern logicians understand it). As such, a single thing coincides with the one-member-class, and more importantly there are no classes without individuals. This conception leaves no room for a null class properly. Venn avoids the difficulty by pointing out that logic symbols represent strictly compartments, in which classes may be put, and are not classes themselves [Venn 1894, 119–120]. Consequently, there are no null classes, but merely empty compartments. Carroll and MacColl handled the problem otherwise. Both Carroll and MacColl assumed the universe to contain things which do really exist and things which do not really exist. The latter are said to be imaginary by Carroll [Carroll 1897, 11]. Hence, the classes “Things” (which is the Universe) and “Things that exist” denote different extensions. Carroll insists on the mental nature of the processes (division and classification) he uses to form classes. As such, they do not ask classes to contain existing things. All they need is to contain things, whatever be their real existence. MacColl tack- les the problem on similar grounds. He might have been inspired by Carroll’s theory, but there is no way to establish that with certainty. MacColl considers too that the Universe contains both existing and not existing things:

Let e1, e2, e3, etc. (up to any number of individuals mentioned in our argument or investigation) denote the universe of real existences. Let 01, 02, 03, etc., denote our universe of non-existences, that is to say, of unrealities, such as centaurs, nectar, ambrosia, fairies, with self-contradictions, such as round squares, square circles, flat spheres, etc., including, I fear, the non-Euclidean Geometry of four dimensions and other hyper-spatial geometries. Finally, let S1, S2, S3, etc., denote our Symbolic Universe, or “Universe of Discourse,” composed of all things real or unreal that are named or expressed by words or other symbols in our argument or investigation. By this definition we assume our Symbolic Universe (or “Universe of discourse”) to consist of our Universe of realities, e1, e2, e3, etc., together with our universe of unrealities, 01, 02, 03, etc., when both these enter into our argument. [MacColl 1905, 74] Russell objected to this approach on the ground that MacColl was confusing two meanings of existence: in philosophy and life, existence means real existence. So that: Aristotle exists while Hamlet doesn’t. In mathematics and symbolic logic, however, existence is applied only to classes which are said to exist if they contain members [Russell 1905, 398]. It follows that only the null class is considered as non-existent in logic because it contains no individuals. Carroll defined the existence of classes too, in addition to that of individuals. Contrary to Russell, however, his definitions were related to each other. A class is said to be real if it contains things existing really, and is said to be imaginary if it contains only imaginary things [Carroll 1897, 2]. Consequently, an imaginary class is not a null class. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 71

In spite of their analogous conceptions of existence, Carroll and MacColl had opposite views on the existential import of propositions. This issue forms MacColl’s second important criticism in his review of Carroll’s Symbolic Logic. Indeed, Carroll considered that universal affirmative propositions have existential import, a position severely attacked by MacColl: Hence, according to Lewis Carroll’s ruling, the assertion “All S is P ,” if correct, implies that S really exists. Now there is a certain theorem, generally (we will not rashly say universally) accepted as valid, which does not seem to accept this ruling with the meekness that it ought. The theorem is that “A is A.” It will generally be admitted, we think, that “All non-existent things are non-existent”; yet, according to the author, this proposition would imply that non-existent things really exist: a rather staggering assertion in the prosaic world of our experience, though the most fundamental of all axioms in Wonderland. [MacColl 1896 , 520] Carroll was conscious of the difficulties raised by his theory. Indeed, he limited it in the third edition of his book to propositions of relation where the terms are “two Species of the same Genus, such that each of the two Names conveys the idea of some Attribute not conveyed by the other” [Carroll 1896c, 14]. This restriction eliminates the propositions of the kind used by MacColl in his criticism, from the scope of his logic treatise. Still, Carroll had always been in trouble with the issue of existential import, even if he had maintained the same position since his early symbolic investigations in the 1880s. In his first logic book, The Game of Logic (1887), he already attributed existential import to I propositions (of the form “Some x are y”) but none to E propositions (of the form “No x are y”). Propositions of the form “All x are y” are double however: they assert that “Some x are y” and “No x are not-y”. Consequently, A propositions (of the form “All x are y”) do assert the existence of their subject [Carroll 1887, 19]. In Symbolic Logic (1897), Carroll insists at some length on the purely conventional nature of any choice regarding the existential import of propositions: [W]ith regard to the question whether a Proposition is or is not to be understood as asserting the existence of its Subject, I maintain that every writer may adopt his own rule, provided of course that it is consistent with itself and with the accepted facts of Logic. Let us consider certain views that may logically be held, and thus settle which of them may conveniently be held; after which I shall hold myself free to declare which of them I intend to hold. [Carroll 1897, 166] MacColl would certainly agree on this conventional feature, but he disagreed on what ought to be considered as convenient. Carroll paid more attention to what he called “the facts of life” and did his best to keep his theory faithful to common use in real life rather than looking for symmetry and simplicity in his notation and calculus. The fourth edition of Symbolic Logic (which 72 Francine F. Abeles & Amirouche Moktefi

followed the publication of MacColl’s review) contains, however, several minor changes and warnings which suggest that Carroll might have been on the way to changing his mind. For instance, he opens the section on existential import with the following notice: “Note that the rules, here laid down, are arbitrary, and only apply to Part I of my Symbolic Logic” [Carroll 1897, 19]. We cannot tell, however, whether Carroll followed MacColl’s advice, as the few surviving fragments of the second and third parts of Symbolic Logic are silent on this issue.

Conclusion

Lewis Carroll and Hugh MacColl shared an interest in Euclidean geometry and symbolic logic. In spite of some fundamental differences and opposed views, each benefitted from the reading of the other’s writings. From the evidence provided in this paper, it is clear that each man’s work influenced that of the other, leading both of them to adjust their arguments to make them more precise. Ultimately, MacColl’s reading of Carroll’s Symbolic Logic “rekindled the old fire which [he] thought extinct” and encouraged him to work again in the domain of logic, as he told Russell. Carroll’s literary skills certainly intrigued MacColl who enjoyed reading Carroll’s prefaces! In addition to clear exposition and the unusual style that characterize his books, there seems to be one more essential affinity that supports MacColl’s attraction to Carroll’s work. Indeed, their exchanges show that both had a deep interest in the precise use of words. And both see no harm in attributing arbitrary meanings to words, as long as the meaning is precise and the attribution agreed upon. Carroll promoted this license in most of his writings, and MacColl made thorough use of it in his later mathematical work and second-stage logical theory.

Acknowledgments

The authors express grateful acknowledgement to the librarians of City University Library (London) who helped the second author to consult The Athenaeum marked copies collection. The authors acknowledge also the two anonymous referees for their remarks and suggestions.

References

Abeles, Francine F. 1995 Avoiding ‘the bewildering region of inﬁnites and inﬁnitesimals’: C. L. Dodgson’s Euclidean parallel axiom, Proceedings of the Canadian Society for History and Philosophy of Mathematics, 8, 143–151. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 73

Abeles, Francine F. (ed.) 1994 The Mathematical Pamphlets of Charles Lutwidge Dodgson and Related Pieces, vol. 2 of The Pamphlets of Lewis Carroll, New York: the Lewis Carroll Society of North America. 2010 The Logic Pamphlets of Charles Lutwidge Dodgson and Related Pieces, vol. 4 of The Pamphlets of Lewis Carroll, New York: the Lewis Carroll Society of North America. Anonymous [probably Norman MacColl, the editor of The Athenaeum] 1898 The Rev. C. L. Dodgson, The Athenaeum, 3665, 22 January, 121. Astroh, Michael, Grattan-Guinness, Ivor & Read, Stephen 2001 A survey of the life and work of Hugh MacColl (1837-1909), History and Philosophy of Logic, 22, 81–98. Astroh, Michael & Read, Stephen (eds.) 1998 Hugh MacColl and the Tradition of Logic, special issue of the Nordic Journal of Philosophical Logic, 3 (1), December. Bartley III, William Warren (ed.) 1986 Lewis Carroll’s Symbolic Logic, 2nd ed., New York: Clarkson N. Potter.

Bertrand, Louis 1778 Développement nouveau de la partie élémentaire des mathématiques: prise dans toute son étendue, Genève : Bardin. Carroll, Lewis 1887 The Game of Logic, London: Macmillan. 1894 A logical paradox, Mind, 3 (11), July, 436–438. 1896a Symbolic Logic. Part 1: Elementary, 1st ed., London: Macmillan. 1896b Symbolic Logic. Part 1: Elementary, 2nd ed., London: Macmillan. 1896c Symbolic Logic. Part 1: Elementary, 3rd ed., London: Macmillan. 1897 Symbolic Logic. Part 1: Elementary, 4th ed., London: Macmillan. Dale, Andrew I. 1999 A History of Inverse Probability: from Thomas Bayes to Karl Pearson, 2nd ed., New York: Springer. Dodgson, Charles L. 1878 Letter to Wallace Hay Laverty, 2 April, The Alfred C. Berol Collection of Lewis Carroll, Fales Library, New York University, USA. 1885 Supplement to Euclid and his Modern Rivals: Containing a Notice of Henrici’s Geometry Together with Selections from the Reviews, London: Macmillan. 74 Francine F. Abeles & Amirouche Mokteﬁ

1888 Curiosa Mathematica. Part I: A New Theory of Parallels, 1st ed., London: Macmillan. 1889 Curiosa Mathematica. Part I: A New Theory of Parallels, 2nd ed., London: Macmillan. 1890 Curiosa Mathematica. Part I: A New Theory of Parallels, 3rd ed., London: Macmillan. 1895a Curiosa Mathematica. Part I: A New Theory of Parallels, 4th ed., London: Macmillan. 1895b Pillow Problems, 4th ed., London: Macmillan. 1896 Letter to George Osborn, 9 March, The Parrish Collection of Lewis Carroll, Princeton University Library, USA, box 8, folder 25. Grattan-Guinness, Ivor 1992 A note on The Educational Times and Mathematical Questions, Historia Mathematica, 19, 76–78. Keene, Carol A. (ed.) 1999 F. H. Bradley: Selected Correspondence. June 1872-December 1904, vol. 4 of the Collected Works of F. H. Bradley, Bristol: Thoemmes. MacColl, Hugh 1888 Review of C. L. Dodgson’s Curiosa Mathematica. Part I: A New Theory of Parallels (1st ed., London: Macmillan, 1888), The Athenaeum, 3183, 27 October, 557. 1889 Review of C. L. Dodgson’s Curiosa Mathematica. Part I: A New Theory of Parallels (2nd ed., London: Macmillan, 1889), The Athenaeum, 3232, 5 October, 457. 1891 Review of C. L. Dodgson’s Curiosa Mathematica. Part I: A New Theory of Parallels (3rd ed., London: Macmillan, 1890), The Athenaeum, 3328, 8 August, 196–197. 1892a Review of Edward T. Dixon’s The Foundations of Geometry (Cambridge: Deighton, Bell & Co., 1891), 3354, 6 February 1892, 184. 1892b Review of Edward T. Dixon’s An Essay on Reasoning (Cambridge: Deighton, Bell & Co., 1891), 3354, 6 February 1892, 184. 1893 Review of C. L. Dodgson’s Curiosa Mathematica. Part II: Pillow Problems (London: Macmillan, 1893), The Athenaeum, 3443, 21 October, 557–558. 1896 Review of Lewis Carroll’s Symbolic Logic. Part I: Elementary (London: Macmillan, 1896), The Athenaeum, 3599, 17 October, 520– 521. 1897 Symbolic reasoning (II), Mind, 6 (24), October, 493–510. H. MacColl and L. Carroll: Crosscurrents in geometry and logic 75

1898 Review of Bertrand Russell’s An Essay on the Foundations of Geometry (Cambridge: Cambridge University Press, 1897), The Athenaeum, 1 January, 25–26. 1904 The calculus of limits, Mathematical Questions and Solutions from “The Educational Times”, 5, 66–70. 1905 Symbolic reasoning (VI), Mind, 14 (53), January, 74–81. 1906a Symbolic Logic and its Applications, London: Longmans, Green, & Co. 1906b Symbolic reasoning (VIII), Mind, 15 (60), October, 504–518. 1910 Linguistic Misunderstandings. Part I, Mind, 19 (74), April, 186–199. Moktefi, Amirouche 2007a Lewis Carroll and the British nineteenth-century logicians on the barbershop problem, Proceedings of the Canadian Society for the History and Philosophy of Mathematics, 20, 189–199. 2007b How to supersede Euclid: geometrical teaching and the mathematical community in nineteenth-century Britain, in Amanda Mordavsky Caleb (ed.), (Re)Creating Science in Nineteenth-Century Britain, Newcastle: Cambridge Scholars Publishing, 216–229. 2008 Lewis Carroll’s logic, in Dov M. Gabbay & John Woods (eds.), The Handbook of the History of Logic, vol. 4: British Logic in the Nineteenth- Century, Amsterdam: North-Holland (Elsevier), 457–505. Rahman, Shahid, & Redmond, Juan 2007 Hugh MacColl: an Overview of his Logical Work with Anthology, London: College Publications. Russell, Bertrand 1897 An Essay on the Foundations of Geometry, Cambridge: Cambridge University Press. 1903 The Principles of Mathematics, Cambridge: Cambridge University Press. 1905 The existential import of propositions, Mind, 14 (55), July, 398–401. Schlimm, Dirk 2010 Pasch’s philosophy of mathematics, The Review of Symbolic Logic, 3 (1), March, 93–118. Seneta, Eugene 1993 Lewis Carroll’s ‘Pillow Problems’: on the 1993 centenary, Statistical Science, 8 (2), 180–186. 76 Francine F. Abeles & Amirouche Mokteﬁ

Shapiro, Stewart 2009 We hold these truths to be self-evident: but what do we mean by that?, The Review of Symbolic Logic, 2 (1), March, 175–207. Venn, John 1894 Symbolic Logic, 2nd ed., London: Macmillan. Wakeling, Edward (ed.) 2004 Lewis Carroll’s Diaries: the Private Journals of Charles Lutwidge Dodgson (Lewis Carroll), vol. 8, Cliﬀord, Herefordshire: The Lewis Carroll Society. Hugh MacColl’s contributions to the Educational Times James J. Tattersall Providence College, RI, USA

Résumé : Il y eut plus de 1900 contributeurs à la section mathématique du journal Educational Times durant ses 67 ans d’existence. Cette rubrique mensuelle contenait des problèmes et leurs solutions, de courts articles, et souvent de courts comptes rendus de l’assemblée la plus récente de la London Mathematical Society. Hugh MacColl (1837-1909), enseignant et tuteur, était l’un des contributeurs les plus proliﬁques à cette section. Il soumettait ses contributions depuis Boulogne-sur-Mer, France. Son expertise portait essentiellement sur la probabilité géométrique, les probabilités classiques et la logique. Après un bref aperçu de l’histoire de la section mathématique du Educational Times et de la correspondance de MacColl avec son directeur de publication, W. J. C. Miller, nous étudierons les contributions de MacColl à ce journal. Abstract: There were over nineteen hundred contributors to the mathematical department of the Educational Times during its sixty-seven-year existence. The monthly column contained problems and their solutions, brief articles, and often short accounts of the most recent meeting of the London Mathematical Society. Hugh MacColl (1837-1909), a teacher and tutor, was one of the more proliﬁc contributors to the department. He submitted his contributions from Boulogne-sur-Mer, France. His expertise was mainly in geometric probability, standard probability, and logic. After a brief history of the mathematical department of the Educational Times and some of Carroll’s correspondence with the editor, W. J. C. Miller, we highlight MacColl’s contributions to the journal.

The Educational Times

The Educational Times and Journal of the College of Preceptors (ET ) was a long-lived pedagogical journal published in London from 1847 to 1918. In the second year of its publication, a mathematical department was added to serve as an outlet for men and women to exhibit their mathematical skills. The section consisted of problems, solutions, and brief articles. William

Philosophia Scientiæ, 15 (1), 2011, 77–96. 78 James J. Tattersall

K. Clifford, professor of mathematics and mechanics at University College London, claimed that in the late nineteenth century, ET did more to encour- age original mathematical research than any other European periodical [Miller 1897]. The first publications of the British mathematician Godfrey H. Hardy, the philosopher Bertrand Russell, and the Indian mathematician Srinivasa Ramanujan appeared as solutions to problems in ET [Tattersall 2008]. Among its contributors were some of the most esteemed mathematicians in the British Commonwealth, Europe, and America including John Couch Adams, Emile Borel, Eugène Catalan, Arthur Cayley, Ernest Césaro, Augustus De Morgan, Francis Galton, Charles Hermite, Felix Klein, Magnus Mittag-Leffler, James Clerk Maxwell, Simon Newcomb, Benjamin Pierce, and James J. Sylvester. The College of Preceptors was incorporated by Royal Charter in London in 1849. Its main objectives were to promote sound learning, advance interest in education among the middle class, and provide means to raise the status and qualifications of teachers. In order to accomplish those goals, training was offered to those entering the teaching profession and periodic examinations for certification were administered to both teachers and students. The group aimed to establish education as a subject of study in colleges and universities. In addition, the organization strove to facilitate better communication between teachers and the public. ET contained notices of available scholarships, lists of successful candidates on examinations given by the College, notices of vacancies for teachers and governesses, numerous book reviews, and textbook advertisements. To many, the most singular feature of the monthly journal was the section devoted to mathematical problems and their solutions that was inaugurated in the winter of 1848. From December 1848 to December 1915, more than 18,000 problems were posed in the pages of ET. 1 The first editors of the department of mathematical questions and solutions were Richard Wilson and James Wharton, both of whom matriculated at of St. John’s College, Cambridge. When Stephen Watson of Haydonbridge and William John Clarke Miller, mathematical master and vice-principal of Huddersfield College in Yorkshire, assumed the editorship of the department in the late 1850s, the quality of the problems and their solutions rose dramatically. 2 Numbered problems first appeared in the August 1849 issue, before then about a dozen unnumbered problems had been published. Solutions were submitted to 81 percent of the problems with proposers submitting their own solutions 25.8 percent of the time. There were a remarkable number of contributions by women [Tattersall 2004]. The majority of contributors hailed from England (62.2%), India (7.7%), Ireland (7.2%), France (5.6%), and the United States (5.3%) [Tattersall 2007]. In

1. From 1915 to 1918, ET was published quarterly without a section devoted to mathematical problems. Problems numbered 18,139 to 18,702 were sent to individual subscribers on a monthly basis. Solutions to 63 percent of these problems appear in Mathematical Questions with their Solutions from the Educational Times. 2. For a comprehensive account of the early history of the Educational Times see [Delve 2003]. Hugh MacColl’s contributions to the Educational Times 79 addition, there were contributors from Australia, Austria, Belgium, Bohemia, Canada, Ceylon, Dagestan, Germany, Hong Kong, Italy, Malaysia, Malta, Mauritius, The Netherlands, New Zealand, Russia, Scotland, South Africa, Spain, Sweden, Switzerland, and Wales. 3 Under Miller’s editorship, the standard directions for submission of problems were: to make your answers as short as possible, write each question and answer on a separate sheet of paper with your name at the top of each, and remember to pay the postage in full [Miller 1886]. Space in ET was at such a premium that normally less than a page and a half was devoted to the mathematics section. Various departments vied for space. On one occasion Miller commented, “Want of space necessitated the omission of several solutions this month” [Miller 1853a]. Soon thereafter, he wrote, “In consequence of the great pressure of advertisements and reports from the College, the mathematical matter is necessarily abridged this month” [Miller 1853b]. On several occasions the mathematics section was completely omitted. In 1864, Miller took action: problems and solutions that had appeared in ET were re- published semiannually in Mathematical Questions with Their Solutions from the “Educational Times” (MQ) [Grattan-Guinness 1992]. More importantly, MQ contained solutions to problems whose solutions hadn’t appeared in ET. In 1876 Miller, Fellow of the Royal Statistical Society and a member of the London Mathematical Society, migrated to London and became registrar, secretary and statistician to the General Medical Council. He served as editor of MQ and the mathematical department of ET until illness forced him to retire in 1897 [Finkel 1896]. During his tenure, he endeavored to publish solutions at most two months after the problems had appeared in print. Many nineteenth- century mathematical textbooks did not contain pages of diverse exercises as do their modern counterparts. It was customary for teachers and students to seek out or create their own applications of theory. ET proved to be an invaluable source of practice problems to anyone interested in mathematics. Hugh MacColl was one of the more productive contributors to the ET mathematical department. His problems were well contrived, his solutions clever, and his articles innovative and informative. MacColl made 205 contributions to ET, ranking him 22nd in terms of output among the over one thousand eight hundred and ﬁfty contributors to its mathematical department, placing him just ahead of Thomas Simmons, 25th Wrangler on the 1874 Cambridge Mathematical Tripos. MacColl posed 125 problems, solved 70 problems, and submitted articles on various aspects of analysis, logic, and probability. One article described short cut methods to evaluate xm a bxn pdx [MacColl 1871f]. In another, he suggested a scientiﬁc nota- ( + ) 7 3 ∫tion whereby, for example, 23 10 was represented by 237 and 57 10− by × × 57 3 [MacColl 1880b]. Another dealt with paradoxes in symbolic logic that he − found of interest [MacColl 1903]. Another involved the determination of the

3. The author has compiled a database listing the posers, solvers, and mathematical classification of the problems for all but 35 of the relevant 816 issues of ET. 80 James J. Tattersall limits of integration of multiple integrals using analytic methods without any reference to curves or curved surfaces [MacColl 1904a]. The problems that MacColl posed dealt with geometric probability (39), logic (34), probability (15), analysis (12), geometric expectation (6), algebra (6), geometry (5), integral calculus (4), arithmetic (2), spherical trigonometry (1), and applied mathematics (1). The problems he solved dealt with geometric probability (27), logic (21), analysis (12), probability (6), combinatorics (2), and algebra (2). Up to 1885 all of his submissions with few exceptions were under the appellation H. McColl. In the majority of exceptions, he is listed as H. M’Coll. Of the 125 problems that MacColl posed in ET, 53 went unsolved. Of the remaining 72 problems, he submitted solutions to 42 of them. In 34 of those problems, his was the only solution. Very little is known about MacColl’s early life and education. He was born in Strontian, Argyllshire in the Scottish Highlands in 1837, the youngest of six children of John and Martha MacColl. His father was a shepherd and tenant farmer who died at age 45 when Hugh was three years old [Astroh, Grattan- Guinness & Read 2001, 81]. The death of his father had a life-long effect on him. His older brother Malcolm assumed the responsibility of supporting Hugh through his elementary and secondary education. However, when Malcolm was dismissed from his clerical post, Hugh abandoned any hope of a university education. He worked for a time in England as a schoolmaster before marrying Mary Elizabeth Johnson of Loughborough, Leicestershire. In 1865, they migrated to fashionable and prosperous Boulogne-sur-Mer which had a sizable English community. While in Boulogne, Hugh and Elizabeth had five children, four girls and a boy. There he taught English and mathematics at the Collège communal until 1869. From then on, he supported his family privately by tutoring mathematics, classics, English, and logic mainly to students preparing to take university and military examinations. MacColl was undoubtedly able to peruse the latest issue of ET to keep up with pedagogical activities in Britain and sharpen his mathematical skills at Merridew’s English Library in Boulogne.

The MacColl–Miller correspondence

In November of 1865, MacColl sent Miller revised solutions to problems numbered 1739 and 1761 and an article on the numerical solutions of equations that dealt with the determination of the number of real roots of polynomials over a given interval [MacColl 1866a]. I have taken great pains on the solution of 1761, endeavoring to make it so complete and general as to be in itself suﬃcient for the purposes of numerical solution; and have been as brief as I could consistently with the accomplishment of this object. I could not make my solution shorter without sacriﬁcing so much accuracy, clearness, and generality. [MacColl 1865a] Hugh MacColl’s contributions to the Educational Times 81

MacColl felt that his method of derived functions to solve polynomial equations was more efficient than Sturm’s method using successive derivatives [Sturm 1835]. MacColl included a discussion and some examples of polynomials with nearly equal roots. In his reply, Miller praised MacColl’s article. 4 MacColl replied that: I cannot sufficiently thank you for your kind letter. I had certainly bestowed great pains on the solutions which I sent last, but I was by no means prepared for so much praise of them as you have been good enough to give. I feel very pleased for your good opinion of my methods is sufficient to convince me that they are of some value, and that other mathematicians will in time see the use of them. [MacColl 1865b] MacColl added that it is beyond his comprehension how Miller can know all the mathematics necessary to edit ET and MQ. Unbeknownst to MacColl, Miller had a group of mathematicians reviewing many of the problems and solutions before they were published. We also get a glimpse of the state of MacColl’s mathematical background when he added: But as soon as I have left the subject of numerical solution, I must really apply myself to the study of mathematical subjects which I have not yet touched. Beyond analytical conic sections and the elements of differential and integral calculus I know very little. The theory of equations I only took up quite recently—about a year ago. Would it be too much of me (considering how little time you have) to ask your advice as to the course of study I ought to pursue, and the books I ought to read? [MacColl 1865b] In reference to the article he had submitted on angular and linear notation [MacColl 1866b], which described a coordinate system based on the perpen- dicular distances from a point to a set of lines. If you find that my discovery really possesses all the importance that I myself attach to it, I will in another letter, give you the history, so far as I can recollect it, of the development in my mind [. . . ] important discoveries and inventions do not always fall to the lot of the talented [. . . ] there are many treasures slightly inserted under the surface of the field of science, which an accidental tread may lay bare [. . . ] I believe that such a piece of good fortune has now happened to me. [MacColl 1866c] In March, MacColl wrote that he had just taken up the subject of modern analytic geometry and confessed that the idea for the article had originated when he encountered some difficulty in explaining to a pupil the difference between Cartesian and trilinear coordinates as found in [Todhunter 1862, 32]. He mentioned that he had sent Sylvester a copy of his discovery and, after hearing back from Sylvester, wrote:

4. For a note on correspondence between Miller and contributors, see [Grattan- Guinness 1994]. 82 James J. Tattersall

Professor Sylvester’s opinion is rather less favorable. He sums up his (confessedly hasty) criticism in the following words: “The brochure on the whole appears to me creditable to Mr. M’Coll’s ingenuity and clearness of understanding but, as far as I am able to judge, does not contain principles which will be productive of such important consequences as the author appears to contemplate. . . ”. [MacColl 1866d] MacColl realized that he had omitted to include a crucial definition that made the article unintelligible. He wrote about his plans to revise and improve the article. He added that complex analysis is beyond him and that there are a number of algebraic terms that are unfamiliar to him. He concluded by asking Miller’s opinion as to whether his knowledge of analytical geometry and the calculus was sufficient to do relevant mathematical research. Perhaps in an effort to further his knowledge of applied mathematics he added: If you would kindly advise me as to the course of study I might pursue and the books I ought to read, I should feel very much obliged. I long to study physics—especially astronomy—of which at present I know next to nothing, but first I think it would be advised to make myself better acquainted with analytical geometry and the calculus. [MacColl 1866d] The next month he wrote: When I first mark what I conceive (rightly or wrongly) to be an important discovery, I feel tempted, like many other mathematical tyros to adopt a tone of exultation which is hardly becoming. [MacColl 1866e] His revised version of the article on angular and linear notation was published in MQ later that year [MacColl 1866f]. In a subsequent letter to Miller, he reiterated the disturbing comment in Sylvester’s review of his article: The interpretation affixed by the author to trilinear or quadralin- ear coordinates is not, I think new; I scarcely remember the time when I was not acquainted with it. [MacColl 1866g] In June, MacColl wrote to Sylvester to thank him for his frank criticism adding that he had lower his opinion of the importance of his article. However, after consulting several books on modern analytic geometry, he wrote Miller that: In spite of what Prof. Sylvester said about ‘my interpretation of trilinear & quadrilinear coordinated not being he thought new’, I still cannot help think it is new [MacColl 1866h].

Geometric probability

The a priori deﬁnition of the probability that an event will occur is the number ways that the event can occur divided by the number of all possibilities. Hugh MacColl’s contributions to the Educational Times 83

In geometric or local probability, the definition of the probability of an event is given by the area pertaining to the event divided by the total possible area. This is normally expressed as the quotient of two double integrals, where the limits of integration of the numerator delineate the area pertaining to event E and the limits of integration of the denominator the total possible area S as given by dxdy P E = ∬E [ ] dxdy ∬S The history of geometric or local probability can be traced back to 1777 and Buffon’s needle problem: Given a large plane area ruled with equidistant parallel lines and a slender needle to determine the probability that if thrown down the needle will fall across a line [Buffon 1777]. Interest in this type of problem was rekindled when the editor of ET asked readers to determine the probability that if a line segment is divided at random into three pieces that a triangle can be formed from the three segments [Miller 1859]. Soon thereafter, W. S. B. Woolhouse, a multitalented London actuary, asked readers on three occasions with three distinct hypotheses to determine the probability that three random points form an acute triangle [Woolhouse 1862; 1863a; 1863b]. Woolhouse’s problem piqued the interest of several readers including J. J. Sylvester, professor of mathematics at the Royal Military Academy [Sylvester 1866a], and W. C. Crofton, an instructor at the Academy who would succeed Sylvester in 1870. In the autumn of 1865, Crofton provided a clever solution to Woolhouse’s problem when the points are taken on a sphere [Crofton 1866a]. He then posed a problem concerning the average distance between two given points within a circle [Crofton 1866b]. Sylvester devised a series of interesting geometric probability problems and discussed them at a meeting of the British Society for the Advancement of Science [Sylvester 1866b]. One of the problems he highlighted was: Given four points at random in the plane determine the probability that one of the points lies inside the triangle formed by the other three [Sylvester 1864]. The problem caused quite a stir among ET readers leading to six different interpretations of the phrase ‘at random in the plane’ that in turn led to six different answers [Pfeifer 1989]. It was Crofton who while working on the problem devised an ingenious method to attack geometric probability problems bypassing the integration techniques normally employed [Crofton 1867; 1868; 1885]. 5 After retiring from teaching at the Collège communal and getting settled as a private tutor, MacColl became fascinated with geometric probability problems. His calculus skills were more than adequate to evaluate the multiple integrals that he encountered. His solutions to such problems are impressive.

5. For an account of Sylvester and Crofton’s contributions to the subject, see [Kendall & Moran 1963], [Solomon1978], and [Senta, Parshall & Jongmans 2001]. 84 James J. Tattersall

They require much ingenuity in setting up the problem and a solid grasp of integral calculus to obtain the answer. Between February 1871 and February 1872, MacColl solved a dozen problems in ET including a succinct solution to the problem that Miller had posed: If a straight line is divided at random into three segments, show that the chance that these three segments will form an acute- angled triangle is 3ln2 2 and three times out of four no triangle can be formed. [Miller− 1859] MacColl began by considering two points x and y with x less than y on the interval 0,a . Thus, the three segments have lengths x,y x, and a y. In order for[ the] three lengths to form a triangle, the triangle i−nequality must− be satisﬁed. That is, the sum of two sides must be greater than the third side. Hence, x < a 2,a 2 < y, and y x < a 2. Thus, the probability of a triangle being formed~ is given~ by: − ~

a a 2 a ~ dxdy a y dy 2 a∫2 y ∫a 2 a∫2 ( − ) a 8 ~ − ~ = ~ = ~ = 1 4 a y a a2 2 ~ dxdy ydy ~ ∫0 ∫0 ∫0 For a triangle with an obtuse angle, the square of one side must be greater than 2 2 2 the sum of the squares of the other two sides. Hence, x > y x a y 2 ( − ) + ( − ) or equivalently x > y a a . Thus, the probability is given by: − + 2y 2 a a a 2 ~ 3a a dxdy 2 y 2y dy 2 2 2 a a∫2 y a ∫a 2y a∫2 − − 3a 8 2 ln2 ~ − +( ~ ) = ~ = ~ − = 3 4 ln2 a y a a2 2 ~ − dxdy ydy ~ ∫0 ∫0 ∫0 Since the three cases in which an obtuse triangle is formed are mutually exclusive with similar answers, the probability that an obtuse triangle is formed is 3 3 4 ln2 = 9 4 3ln2. Thus, the probability that an acute triangle is formed( ~ − is given) ~ by−1 4 (the probability that a triangle is formed) minus 9 4 3ln2 (the probability~ that an obtuse triangle is formed) yielding the desired( ~ ) − answer, 3ln2 2. Obtaining the solution required a clear knowledge of the mathematics− involved, the skill to obtain an analytic description of the area under consideration, and the ability to evaluate double integrals [MacColl 1871e]. Below are several geometric probability problems posed by MacColl. The ﬁrst was solved by MacColl and Stephen Watson; the second by MacColl and George Shoobridge Carr, author of A Synopsis of Elementary Results in Pure Mathematics, the most inﬂuential book in Ramanujan’s early mathematical education; the third by James Heber Taylor, a private tutor in Cambridge. For the remainder, although interesting, no solutions were published in ET or MQ: Hugh MacColl’s contributions to the Educational Times 85

If P and Q be random points within a circle, what is the probability that the circle of which P is the centre and PQ the radius will lie wholly within the given circle? [MacColl 1871c] Three random points P , Q, R are taken in the perimeter of a square. Find the respective chances of the straight lines PQ, QR, RP forming an acute-angled triangle, and obtuse-angled triangle, or no triangle. [MacColl 1885a] Four random points P , Q, R, S are taken within a circle; ﬁnd the chance that the straight lines PQ and RS (produced if necessary) will intersect within the circle. [MacColl 1885b] A point is taken at random in the diameter of a square, and from this point three straight lines are drawn in random directions to meet the perimeter of the square; ﬁnd the chance that these three lines can be the sides of a triangle. [MacColl 1891a]

Four points P , P ′, Q, Q′ are taken at random within a circle. What is the chance that the circles of which P and Q are the centers and P P ′ and QQ′ the radii will intersect? [MacColl 1891b] If x, y,z be each taken at random between 0 and 1, show that the chance that the fraction z 1 x y 1 y yz will also be between 0 and 1 is 5 4 ln2. [MacColl( − − )~( 1899]− − ) ~ − Given that x and y are each taken at random between 0 and 1, what is the chance that 2x y 3a is negative, the constant a being between 0 and . [MacColl+ − 1901a] ∞ Given that 2 x y, 3y 4x 6,y 2x 7 are all three negative. What is the chance− − that−y is− negative?+ − [MacColl 1901b] What is the chance that xy is greater than 5; ﬁrstly, on the assumption that x and y are each taken at random between 1 and 5; secondly, on the assumption that x is taken at random between 1 and 5; and then y at random between 1 and x? [MacColl 1904b]

Other mathematical interests

MacColl introduced an innovative integral notion to handle certain geometric probability problems [MacColl 1871a; 1871b]. It was clever and useful but not a powerful as that of Crofton for handling intricate geometric probability problems. He let the symbol f x px represent the limiting value of f x ∆x, as ∆x is diminished without∫ ( limit) and the number of terms in- creased∑ ( ) without limit, subject to two conditions: (1) the positive values of f x greater than unity are to be considered as unity, and (2) any negative values( ) of f x are to be taken as zero. In essence, what he proposed was a real-valued( ) function f x deﬁned on the open interval a,b that can be ( ) ( ) 86 James J. Tattersall expressed as follows:

0 f x ≤ 0 b ⎧ ⎪ b ( ) ⎪ f x px = ⎪ f x dx 0 < f x < 1 ( ) ⎨a ( ) ( ) Sa ∫ ⎪ ≥ ⎪b a f x 1 ⎩⎪ − ( ) For example,

9 5 5 x2 5 x px = 4 1 5 x dx 0 = 3 5x = 3 1 2 2 S1 ( − ) ( − ) + S4 ( − ) + +W − W4 ~

MacColl applied this novel notation to solve the following problem that, quite inexplicably, he would later propose: 2 In the quadratic equation ax bx c, the numerical values of the coeﬃcients a, b, c are each taken− + at random between 1 and 10. What is the chance that the equation has two real roots between 1 and 10? [MacColl 1872] He reasoned that according to the quadratic formula, the roots of the equation 2 2 are real and unequal if b 4ac > 0 or equivalently c < b 4a. The probability of this is given by the length− of the assigned limits of c over~ the length of the 2 1 b interval or 9 4a 1 . Treating b as the variable and using MacColl’s notation, − 1 2 the average value of b 1 as b goes from 1 to 10 is given by: 9 4a − 10 1 1 b2 1 pb 9 9 4a S1 −

If a is less than 5/2 we have that:

10 √40a 1 1 b2 1 1 1 b2 1 pb = 10 √40a 1 db 0 = 9 S 9 4a − 9 ( − ) + 9 S 9 4a − + 1 √4a 10 4 3 1 10 2 1 a 2 = f1 a 9 − 243 − ( ) When a is greater than 5/2 we obtain:

10 10 1 1 b2 1 1 b2 1 pb = 1 db 0 = 9 S 9 4a − 9 S 9 4a − + 1 √4a 2 3 125 2a 2 15a = f2 a 243a + − ( ) Hugh MacColl’s contributions to the Educational Times 87

Treating a as the variable and taking it at random between 1 and 10 the ﬁnal answer is given by:

5 2 10 1 ⎛ ⎞ 1500ln 2 160√10 468 f1 a pa f2 a pa = ( ) + − 9 ⎜ ⎟ 6561 ⎜S1 ( ) + S5 ( ) ⎟ ⎝ 2 ⎠ [MacColl 1871d]

Another of MacColl’s interests was inverse probability or Baysian analysis problems. According to [Dale 1999, 583], MacColl was one of the first, and perhaps the first, to devise a symbolic notation to represent conditional probability, denoting the probability of A given B first as AB [MacColl 1880a] and A later as B [MacColl 1897]. MacColl’s work in inverse probability is highlighted in [Dale 1999, 501–504]. Another innovative probability notation that MacColl devised dealt with series of events [MacColl 1871a; 1871b]. He defined p r to denote the probability of the occurrence of the rth event given in a specific( ) ordering of possibilities and p r denoted the nonoccurrence of the rth event. Hence, p r p r = 1. He denoted(∶ ) the probability of the occurrence of the mth and (nth) +events(∶ ) and the nonoccurrence of the rth event by p m.n r and ( th ∶ ) th extended it to p ma.n c re which signifies the occurrence of the m and n ∶ and nonoccurrence( of the∶ r)th events imply the occurrence of the ath and eth and nonoccurrence of the cth event. Several years later, MacColl proposed yet another probability notation [MacColl 1877; 1878]. In 1906, he published a handbook on symbolic logic [MacColl 1906] highlighting this notation and promoting the use of “+” for “or”, juxtaposition for “and”, a colon for implication, and a prime for negation. Hence, a bc′ d symbolized the logical statement that a or, b and not c, imply d(. Some+ of∶ the) logic problems in ET posed by MacColl are:

Suppose it to have been ascertained by observation (1) that, whenever the events A and B happen together, they are invariably followed by the event C, and also by either the event D or the event E; and (2) that, whenever the events D and E both happen, they have invariably been preceded by the event A, or else by both the events B and C. When may we conclude (from the occurrence or non-occurrence of the events A, B, C, D)—(1) that E will certainly happen; (2) that E will certainly not happen? [MacColl 1879] Given (1) that whenever the statements a,b,x are either all three true or all three false, then the statement c is false and y is true, or else c is true and y is false; (2) that whenever d,e,y are either all three true or all three false, then the statement a is false and x true, or a is true and x false, When may we infer from these premises that either x or y is true? [MacColl 1880c] 88 James J. Tattersall

When, from the three implications, (1) a′b ab′ dx, (2) ax by c, (3) cd y, may we conclude that either x or+y is true,∶ but not+ both?∶ [MacColl∶ 1880d] Show that fifteen valid syllogisms of the common logic (excluding the four defective syllogisms, Darapti, Felapton, Fesapo, Bramantip.) may be converted into mathematical syllogisms as follows: Considering each letter as representing a mathematical quantity (positive or negative, integral or functional, real or imaginary), for every proposition of the form “all α is β” substitute “α is a multiple of β” (that is “α β is a real integer”) for “no α is a β”, put “αβ is integer”; to “some~ α is not β,” put “α is not a multiple of β”; and for “some α is a β” put “αβ is not an integer”. What is the weakest premise that must be added to the two given premises in each of the four defective syllogisms (in their logical or mathematical form) to make them also valid? (Definition x is said to be the stronger and y the weaker proposition when x implies y but x is not implied by y). [MacColl 1887] MacColl submitted solutions to the first problem, the last two went unsolved. The second was solved by Elizabeth Blackwood who, making use of MacColl’s notation expressed the premises as: abx a′b′c′ c′y cy′ dey ( + ∶ + )( + d′e′y′ a′y ay′ . Hence, x′y′ a b c d e a or equivalently ∶ + ) ∶ ( + + )( + + ) x′y′ a b c d e . Thus, by contraposition a′ b′c′ d′e′ x y. Among the geometric∶ + ( + )( probability+ ) problems that Blackwood( propose+ )d ∶ and+ MacColl solved are: A point is taken at random in a window consisting of nine equal square panes, and through this point a line is drawn in a random direction; find the respective chances of the line so drawn cutting one, two, three, four, or five panes. [Blackwood 1871a] A point is taken at random in a given quadrant of a circle, and a random line is drawn through it. Find the chance of its cutting the arc. [Blackwood 1871b] If A, B, C, D are random points in the perimeter of a square; find the chance that the straight lines AB, CD will intersect within the square. [Blackwood 1877] Blackwood’s origin, educational background, and profession remain a mys- tery. She contributed to ET from London from 1872 to 1874, from New York from 1875 to 1876, and from Boulogne-sur-Mer from 1877 to 1898. Her name does not appear in Boulogne city directories. She may have been using her maiden name when contributing to ET. There were other contributors from Boulogne; Miss Myra Greaves sent in solutions to problems from there from 1871 to 1877, and Mr. J. J. Sides, a teacher, contributed from 1871 to 1879. In the summer of 1900, MacColl, renewing his interest in the subject, solved a logic problem that had been proposed by Charles L. Dodgson (Lewis Carroll). Namely, Hugh MacColl’s contributions to the Educational Times 89

It is given that (1) if C is true, then, if A is true, B is not true; and (2) if A is true, B is true. Can C be true? What difference in meaning, if any exists between the following propositions? — (1) A,B,C cannot all be true at once; (2) if C and A are true, B is not true; (3) if C is true, then if A is true, B is not true. [Dodgson 1899] An analysis problem that MacColl proposed and solved went as follows: Let f x be any function of x, and let f a,b express (according as a is( greater) or less than b) a superior( or inferior) limit to f x for all values of x between a and b; show that if f a,b has( the) same sign as b a, no real root of the equation f (x =) 0 exists between a and b−. [MacColl 1865c] ( ) MacColl’s proof is straightforward. He assumes the hypotheses and supposes that f a,b and b a are positive; then since a < b, f a,b is an inferior limit of f x ( for) all values− of x between a and b. Thus, when( ) x is thus restricted f (x )> f a,b > 0. Let f a,b and b a be negative. Since b < a, f a,b is a superior( ) limit( of)f x for all( values) of x− between a and b. Thus, when( x) is thus restricted f x (< f) a,b < 0. Hence, the result is established. ( ) ( ) An elegant Euclidean geometry problem that MacColl proposed in ET for a secondary school geometry class is as follows: Show that in any triangle the bisector of the less[er] of two angles is greater that the bisector of the greater. [MacColl 1898] The next examples show that MacColl was not averse to the use of poetry to pose problems. The first describes a 3-dimensional geometric problem: Some shapeless solids lie about, No matter where they be; Within such solid or without, Let’s take a centre C. From Centre C, in countless hosts, Let random radii run, And meet a surface, each at P , Or, maybe, meet with none. Those shapeless solids, far and near, Their total prove to be The average volume of the sphere Whose radius is CP . The sphere, beware, is positive, When out at P they fly; But, changing sign, ’tis negative, If entrance there you spy. One caution more, and I have done, The sphere is nought, if P there’s none. [MacColl 1884] 90 James J. Tattersall

The problem being: From the center C of a shapeless solid, straight lines are drawn to the surface meeting it at P , the number of such rays increasing without limit. If we consider CP as the radius of a sphere, the problem, which went unsolved, was to determine the average volume of all the spheres. In another, he posed an intriguing applied mathematics problem. To put a question somewhat queer, Suppose the earth an airless sphere, Through which is bored, from pole to pole, A bottomless cylindrical hole; Right o’er the centre let there fall A smooth and polished marble ball. Dear Reader, may I ask you when That marble would come back again? [MacColl 1889] A solution to the second problem was submitted by Daniel Biddle, a London physician, Fellow of the Royal Statistical Society, who in 1898 replaced Miller as editor of the ET mathematical department.

Conclusion

MacColl subscribed to MQ from 1865 until his death in 1909. He submitted problems to the editor of the ET mathematical department on an irregular basis. He contributed a set of eight analysis and algebraic problems that, beginning in the spring of 1865, appeared semi-regularly over the following eighteen months. He submitted sixteen problems that dealt mainly with geometric probability, geometric average, and probability problems that, beginning in the fall of 1870, appeared monthly over the next two and a half years. A smaller set of similar problems appeared in the spring of 1877, the fall of 1877, the fall of 1884, and the spring of 1890. A set of thirty-eight geometric probability, logic, and regular probability problems that MacColl submitted began appearing monthly in June of 1896. The last set of ten problems dealt mainly with logic and appeared from March 1902 to January 1910. Before 1870, his solutions were mainly to analytic problems, from 1870 to 1872 to geometric probability problems, and after 1900 to logical problems. Little or no contributions from MacColl were published from 1866 to 1870, 1872 to 1876, 1886 to 1889, and 1891 to 1896. Adjusting to life in Boulogne and his responsibilities at the Collège communal may have been the cause of the ﬁrst hiatus. From 1872 to 1876, he was concentrating on his studies for the University of London external examination for a BA degree in mathematics, which he completed in 1876. The third break could very well have been caused by the death of his ﬁrst wife and his marriage three years later to Hortense Lina Marchal in 1887. Finally, the fourth break occurred during the period when MacColl was more interested in literature than mathematics and logic. Hugh MacColl’s contributions to the Educational Times 91

The mathematical proﬁciency, logical innovations, and number of contributions exhibited by MacColl in ET were impressive. From his intricate solutions to problems in geometric probability, inverse probability, analysis, and logic, we see that he was more than just an average mathematician and logician. While his problems and articles were clever, well-thought out, and often innovative, they were not of the caliber of Sylvester, 2nd Wrangler on the 1837 Cambridge mathematical tripos, or Crofton who had taken the highest mathematical honors when graduating from Trinity College, Dublin in 1848. MacColl had remarkable mathematical potential, and if fate had been kinder and oﬀered him the opportunity of a normal university education, he may well have become one of the leading mathematicians of the late nineteenth century.

Acknowledgments

The author wishes to thank the referees for their helpful suggestions on revising the article and to Amirouche Mokteﬁ for supplying copies of the MacColl-Miller correspondence.

References

Here ET is used to denote the Educational Times and Journal of the College of Preceptors and MQ the Mathematical Questions and Their Solutions from the ‘Educational Times’. Astroh, Michael, Grattan-Guinness, Ivor, & Read, Stephen 2001 A survey of the life of Hugh MacColl (1837-1909), History and Philosophy of Logic, 22, 89–98. Blackwood, Elizabeth 1871a Problem # 3408, ET, 24, 216. 1871b Problem # 3561, ET, 24, 164. 1877 Problem # 5313, ET, 30, 69. Buffon (Comte de), Georges-Louis Leclerc 1777 Essai d’arithmétique morale, in Histoire naturelle, générale et parti- culière, servant de suite à l’histoire naturelle de l’homme, Supplément, Tome IV, Paris : Imprimerie Royale, 46–148. Crofton, Morgan William 1866a Solution to Problem # 1333, MQ, 4, 61–62. 1866b Problem # 1829, MQ, 4, 77–79. 1867 Note on local probability, ET, 20, 40. 1868 On the theory of local probability, applied to straight lines drawn at random in a plane, the methods used being also extended to the proof of 92 James J. Tattersall

certain new theorems in the integral calculus, Philosophical Transactions of the Royal Society of London, 158, 181–199. 1885 On local probability, Encyclopedia Britannica, 9th ed., Cambridge: Cambridge University Press, 768–788. Dale, Andrew I. 1999 A History of Inverse Probability from Thomas Bayes to Karl Pearson, 2nd ed., New York: Springer. Delve, Janet 2003 The College of Preceptors and the Educational Times: changes for British mathematics and education in the mid-nineteenth century, Historia Mathematica, 30, 140–172. Dodgson, Charles L. 1899 Problem # 14122, ET, 52, 93. Finkel, Benjamin Franklin 1896 Mr. W. J. C. Miller, American Mathematical Monthly, 3, 159–163. Grattan-Guinness, Ivor 1992 A note on the Educational Times and Mathematical Questions, Historia Mathematica, 19, 76–78. 1994 Contributing to The Educational Times, Letters to W. J. C. Miller, Historia Mathematica, 21, 204–205. Kendall, Maurice G. & Moran, Patrick A. P. 1963 Geometrical Probability, London: Charles Griﬃn. MacColl, Hugh 1865a Letter to W. J. C. Miller, 2 Nov., Rare book & manuscript Library, Columbia University, New York, USA. 1865b Letter to W. J. C. Miller, 15 Nov., Rare book & manuscript Library, Columbia University, New York, USA. 1865c Problem # 1709, ET, 18, 140. 1866a On the numerical solution of equations, MQ, 4, 105–107. 1866b Angular and linear notation, ET, 19, 20. 1866c Letter to W. J. C. Miller, 22 Feb., Rare book & manuscript Library, Columbia University, New York, USA. 1866d Letter to W. J. C. Miller, 12 March, Rare book & manuscript Library, Columbia University, New York, USA. 1866e Letter to W. J. C. Miller, 16 March, Rare book & manuscript Library, Columbia University, New York, USA. Hugh MacColl’s contributions to the Educational Times 93

1866f Angular and linear notation: a common basis for the bilinear (a transformation of the cartesian), the trilinear, the quadrilinear, &., systems of geometry, MQ, 5, 74–77. 1866g Letter to W. J. C. Miller, 25 April, Rare book & manuscript Library, Columbia University, New York, USA. 1866h Letter to W. J. C. Miller, 17 June, Rare book & manuscript Library, Columbia University, New York, USA. 1871a Probability notation, ET, 23, 230–231. 1871b A probability notation: no 2, ET, 24, 111–112. 1871c Problem # 3515, ET, 24, 140. 1871d Probability notation, MQ, 15, 20–22. 1871e Solution to Problem # 1100, MQ, 15, 35–36. 1871f An easy way of remembering the formulæ of reduction in the integral calculus, both as regards their investigation and their application, MQ, 14, 43–45. 1872 Problem # 3771, ET, 25, 44. 1877 Symbolical or abbreviated language with an application to mathematical probability, ET, 30, 91–92. 1878 Symbolic language: no 2, MQ, 28, 100. 1879 Problem # 5861, ET, 32, 22. 1880a The calculus of equivalent statements (fourth paper), Proceedings of the London Mathematical Society, 11, 113–121. 1880b Arithmetical notation, ET, 33, 111. 1880c Problem # 6206, ET, 33, 23. 1880d MacColl # 6235, ET, 33, 68. 1884 Problem # 7897, ET, 37, 356. 1885a Problem # 7983, ET, 38, 30. 1885b Problem # 8026, ET, 38, 69. 1887 Problem # 13386, ET, 50, 39. 1889 Problem # 10119, ET, 42, 223. 1891a Problem # 11017, ET, 44, 239. 1891b Problem # 11235, ET, 44, 411. 1897 The calculus of equivalent statements (sixth paper), Proceedings of the London Mathematical Society, 28, 555–579. 1898 Problem # 14030, ET, 51, 472. 1899 Problem # 14296, ET, 52, 337. 1901a Problem # 14781, ET, 54, 36. 1901b Problem # 14807, ET, 54, 69. 94 James J. Tattersall

1903 Paradoxes in symbolic logic, MQ, n.s., 4, 34–36. 1904a The calculus of limits, MQ, n.s., 5, 66–70. 1904b Problem # 15588, ET, 57, 313. 1906 Symbolic Logic and Its Applications, London: Longmans, Green and Co. Miller, William John Clarke 1853a Note to mathematical correspondents, ET, 6, 87. 1853b Note to mathematical correspondents, ET, 6, 111. 1859 Problem # 1100, ET, 12, 141. 1886 Note to mathematical correspondents, ET, 39, 227. 1897 Note to mathematical correspondents, ET, 50, 331. Pfeifer, Richard E. 1989 The historical development of J. J. Sylvester’s four point problem, Mathematics Magazine, 62, 309–317. Seneta, Eugene, Parshall, Karen Hunger & Jongmans, François 2001 Nineteenth-century developments in geometric probability, in J. J. Sylvester, M. W. Crofton, J.-E. Barbier, and J. Bertrand, Archive for History of Exact Sciences, 55, 501–524. Solomon, Herbert 1978 Geometric Probability, Philadelphia: Society for Industrial and Applied Mathematics. Sturm, Jacques Charles François 1835 Mémoire sur la résolution des équations numériques, Mémoires présentés par divers savants étrangers à l’Académie royale des sciences, 6, 271–318. Sylvester, James Joseph 1864 Problem # 1491, ET, 17, 20. 1866a Note on Question # 1829, MQ, 4, 78–79. 1866b On a special class of questions in the theory of probability, Report of the Thirty-ﬁfth Meeting of the British Association for the Advancement for Science held at Birmingham in September 1865, Notices of Abstracts of Miscellaneous Communications to the Section of Mathematics and Physics, London: John Murray, 8–9. Tattersall, James J. & McMurran, Shawnee L. 2004 Women and the ‘Educational Times’, Proceedings of the History and Pedagogy of Mathematics Conference, Uppsala, Sweden: Uppsala University Press, 407–417. Hugh MacColl’s contributions to the Educational Times 95

2007 A note on submissions from India to the mathematical sections of the Educational Times and the Journal of the Indian Mathematical Society, Ganita Bharati, 29, 69–79. 2008 Ramanujan’s ﬁrst publication: a solution to a problem in the Educational Times, Ganita Bharati, 30, 115–121. Todhunter, Isaac 1862 A Treatise on Plane-Coordinate Geometry as Applied to the Straight Line and the Conic Sections, 3rd ed., Cambridge: Macmillan. Woolhouse, Wesley Stoker Barker 1862 Problem # 1333, ET, 15, 214. 1863a Problem # 1350, ET, 15, 235. 1863b Problem # 1361, ET, 15, 283.

MacColl’s inﬂuences on Peirce and Schröder Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University-Purdue University at Indianapolis, USA

Résumé : Les contributions à la logique de MacColl et Charles Sanders Peirce (1839-1914) ont été les deux plus profondes inﬂuences sur le travail de Ernst Schröder (1841-1902) en logique algébrique. Dans son Vorlesungen über die Algebra der Logik, Schröder a cité MacColl comme l’un de ses précurseurs les plus importants. Schröder a comparé les travaux de Peirce avec les pre- mières parties de la série d’articles intitulés « The calculus of equivalent statements » que MacColl publie entre 1877 et 1880. Schröder a attribué la priorité à MacColl pour avoir anticipé les résultats de Peirce. Pour Schröder, MacColl était une phase préliminaire à l’algèbre de la logique de Peirce. Abstract: The contributions to logic of MacColl and Charles Sanders Peirce (1839-1914) were the two most profound inﬂuences upon the work of Ernst Schröder (1841-1902) in algebraic logic. In his Vorlesungen über die Algebra der Logik, Schröder referred to MacColl as one of his most important pre- cursors. Schröder compared Peirce’s considerations with the early parts of MacColl’s series of papers “The calculus of equivalent statements” (published between 1877 and 1880), and he attributed to MacColl priority for having anticipated Peirce’s results. For Schröder, MacColl’s calculus was a preliminary stage of Peirce’s algebra of logic.

Introduction

Hugh MacColl (1837-1909) was originally a mathematician, who gradually turned towards logic. He contributed to propositional logic, to nonclassical logic, especially modal logic, and the theory of inference. He suggested propositions and implication as the basis for logic, being among the ﬁrst, if not the ﬁrst, to articulate the modern conception of implication, introducing the terms “implication” and “non-implication” in the second installment of “The Calculus of equivalent statements” [MacColl 1877-1878b, 181]. Towards the end of his career, he engaged in debate with Bertrand Russell (1872-1970) in the pages of Mind on the existential import of propositions. In his review of MacColl’s Symbolic Logic [Russell 1906], Russell wrote:

Philosophia Scientiæ, 15 (1), 2011, 97–128. 98 Irving H. Anellis

He is primarily concerned with implication, not with inclusion; his formulæ state that one statement implies another, not (directly) that one class is contained in another. The relation of inclusion between classes is for him derivative, being in fact the relation of implication between the statement that a thing belongs to the one class, and the statement that it belongs to the other. He was, I believe, the first to found symbolic logic on propositions and implication, and in this respect he seems to me to have made an important advance upon his predecessors. [Russell 1906, 255] And in his manuscript on “Recent Italian work on the foundations of mathematics” (1901), Russell, contrasting the conception of the algebraic logicians with that of MacColl and Gottlob Frege (1848-1925), wrote that: Formal Logic is concerned in the main and primarily with the relation of implication between propositions. What this relation is, it is impossible to define: in all accounts of Peano’s logic it appears as one of his indefinables. 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 implication. [Russell 1993, 353] MacColl carried out his work in isolation and independence from the leading logicians of his day. Christine Ladd-Franklin (1847-1930), a former student of Charles Sanders Peirce (1839-1914), remarked in “On some characteristics of symbolic logic,” for example, that “Nothing is stranger in the recent history of Logic in England, than the non-recognition which has befallen the writings of the author [H. MacColl]” [Ladd-Franklin 1889, 562]. This does not mean that his work was entirely unknown, however. There were references to his work by a few leading logicians; Giuseppe Peano (1858-1932) in his Arithmetices Principia, Nova Methodo Exposita [Peano 1889, ivn] and Formulaire de mathématique acknowledged a debt to MacColl’s work in propositional logic, in particular employing results [Peano 1901, v, 13–14, 16, 19–21, 226] from MacColl’s “Calculus of equivalent statements” [MacColl 1877-1878a, 1878-1879]. This led Russell to write that “Peano, like McColl, at first regarded propositions as more fundamental than classes. . . ” [Russell 1903, 13], and Louis Couturat (1868-1914) worked on his propositional logic, without, however, taking account of either MacColl’s modal logic or probability logic [Couturat 1899, 621]. MacColl is indubitably best known to historians and philosophers of logic for his exchanges with Bertrand Russell in the pages of Mind on the problem of the existential import of propositions. He has also received attention from those interested in the history of nonclassical logics because of his criticisms, related to the question of existential import, of the accepted definition of material implication. Norbert Wiener (1894-1964), for example, had strongly praised Clarence Irving Lewis (1883-1964) for the courage to challenge in print MacColl’s influences on Peirce and Schröder 99

Russell’s use of material implication in Principia Mathematica [Wiener 1916, 656], while Lewis himself criticized all logicians for adhering to the classical conception of implication as material implication, while explicitly exempting only MacColl [Lewis 1912]. 1 MacColl’s definition of the syllogism as an inference defined in terms of implication, however, was perhaps his more significant contribution to the over-all development of logic. This translation of the syllogism as an implication was a crucial step in establishing propositional logic in its modern form, and was adopted by such widely scattered logicians as Peirce and Russell. Whereas George Boole (1815-1864) had tended to limit the interpretation of his logic to the calculus of classes, MacColl’s first contribution to logic was to admit not only a class interpretation, but a propositional interpretation. He indeed gave preference to the propositional interpretation because of its generality, and called it “pure logic.” Most of MacColl’s work proceeded in an extensive series of journal articles. But in Symbolic Logic and its Applications (1906), he published the final version of his logic(s). There, propositions are qualified as either certain, impossible, contingent, true or false. The contributions to logic of MacColl and Peirce were the two most profound influences upon the work on Ernst Schröder (1841-1902) in algebraic logic, and in his Vorlesungen über die Algebra der Logik [Schröder 1890-1905]. Volker Peckhaus, who counted the number of references to Peirce and MacColl in the Vorlesungen, noted that, whereas Peirce was cited most often, MacColl was the second most cited author [Peckhaus 1998, 20]. So far as Schröder was concerned, MacColl’s calculus held, in Peckhaus’s words, “the status of a preliminary stage of Peirce’s algebra of logic” [Peckhaus 1998, 20]. Peckhaus has already dealt in substantial detail with Schröder’s views of MacColl, and of MacColl’s impact upon Schröder’s work in logic [Peckhaus 1998]. I shall, therefore, concentrate my discussion on Peirce’s views of MacColl, and of the influences which MacColl’s work exerted upon Peirce. It is, however, worth repeating, as [Peckhaus 2004, 584] also noted, that Schröder explicitly expressed his agreement with John Venn (1834-1923) regarding the close similarity between Peirce’s and Schröder’s work and MacColl’s.

MacColl’s inﬂuences on Schröder

Schröder’s own techniques were primarily modiﬁcations of Peirce’s, and the logical system of the Algebra der Logik can be understood as a uniﬁca- tion, systematization, and development of Peirce’s logic and an extension of

1. [Read 1998, 59–60] notes that for the later edition of his Survey, Lewis [Lewis 1960] excised nearly all of the material discussing his early work on systems of strict implication, which included the early references to the work of MacColl and re- marking on his indebtedness to MacColl. He argued by way of justiﬁcation for the wholesale abridgment that the early Lewis-Langford logic of strict implication had been superseded. 100 Irving H. Anellis

Peirce’s algebra of relatives—and, Schröder would also claim, improvement, of Peirce’s logic. Schröder, Peckhaus reminds us, “always compared Peirce’s considerations with the early parts of MacColl’s series of papers “The Calculus of Equivalent Statements” published between 1877 and 1880, and he conceded to MacColl priority when he had anticipated Peirce’s results” [Peckhaus 1998, 20]. Taking this seriously, we must examine the extant evidence to determine the extent to which MacColl is to be understood as a precursor of Peirce and the nature of the influences which MacColl’s work exerted on Peirce’s own work in logic. Peckhaus identifies three aspects in which Schröder was influenced by MacColl, either directly or through Peirce [Peckhaus 1998, 20–21]. (1) In investigating the comparative abilities of various calculi to eliminate terms and resolve logical formulas, Schröder compared his own procedure, adapted from Boole’s, with Peirce’s. Peirce’s was taken as the natural way to op- erate, and MacColl’s method, which Schröder investigated in extenso, was considered to be a “preliminary stage” of Peirce’s. (2) Schröder explicitly acknowledged MacColl’s priority in formulating propositional logic, terming it the “MacColl-Peircean propositional logic.” Schröder was, however, critical of their purported efforts to base logic upon the calculus of propositions, arguing that his founding of the propositional calculus upon the calculi of domains (Denkbereiche) and classes was more general. It must be noted that Schröder’s criticism applies to Peirce only to the extent that he undertook to generalize Boole’s efforts to formulate the categorical syllogisms of Aristotle as chains of propositions whose fundamental connective was implication. But once Peirce undertook to treat his primary connective, illation, the so-called “claw,” as interpretable as any transitive, reflexive, and asymmetric relation, holding between either terms, propositions, relations, classes, or sets, depending upon the context in which the formulas occurred, this criticism could no longer, strictly, apply. For quantified formulas, only the Boolean part of formulas could be treated strictly as propositions. (3) Awarding priority to MacColl and Peirce for defining material implication, Schröder adopted that definition for his own propositional calculus. Setting aside the question of priority, Schröder after proving his exposition of MacColl’s work [Schröder 1890-1905, I, 589–592], expresses his agreement with John Venn that MacColl’s method is “practically identical with those of Peirce and Schröder” [Venn 1881, 414; 1894, 492]. To appreciate Schröder’s positive attitude towards MacColl, it is perhaps sufficient to recall Schröder’s testimony, in the first volume of the Vorlesungen über die Algebra der Logik [Schröder 1890-1905, I, 560] that he accounted only MacColl’s and Peirce’s methods equal to his own in their proficiency. The advantage that Schröder identified in the logical systems of Peirce and MacColl over those of Venn and William Stanley Jevons (1835-1882), is the greater naturalness and simplicity of the former [Schröder 1890-1905, I, 573]. His quibble with the “MacColl-Peircean propositional logic” was its lengthiness, and he undertook in the pages that followed [Schröder 1890-1905, I, 574–584] to repair that defect by modifying Peirce’s methods as presented in Peirce’s MacColl’s influences on Peirce and Schröder 101

“On the algebra of logic” [Peirce 1880, 37–42]. Schröder’s changing attitude towards MacColl’s work is summed up by Shahid Rahman and Juan Redmond as follows: “Ernst Schröder, who in his famous Vorlesungen über die Algebra der Logik quotes and extensively discusses MacColl’s contributions, first had quite a negative impression of MacColl’s innovations though later he seems to have changed his mind, conceding that MacColl’s algebra has a higher degree of generality and simplicity in particular in contexts of applied logic. What Schröder definitively rejects is MacColl’s propositional interpretation of the Aristotelian Syllogistic” [Schröder 2008, 538]. Peckhaus [Peckhaus 1998, 30] is led to think that Schröder learned of MacColl’s work indirectly, through references by Peirce, but notes as well that Schröder, treating MacColl primarily as historical, consistently acknowledges MacColl nonetheless as a predecessor of Peirce’s and his own work. One of the principal and most severe failures of MacColl’s system is the limitation imposed by his failure to introduce a fully fledged apparatus for quantifiers into the system, rendering his system far less flexible than Peirce’s and Schröder’s, and ensursing that it would remain a propositional logic only. 2

MacColl’s inﬂuences on Peirce

MacColl developed a propositional logic within a general Boolean framework in the second installment of his paper “The calculus of equivalent statements” [MacColl 1877-1878b], and provided an example and exposition of what he called “tabular logic” (“logique tabulaire”) in his 1902 article of that title in the Revue de métaphysique et de morale [MacColl 1902a]. In his interpretation, as expressed in the opening of the first installment of “The calculus of equivalent statements,” A = 1 means that the statement A is true; A = 0 that it is false [MacColl 1877-1878a, 9]. The second installment opens with the explanation that the expression A B means that statement A implies statement B, i.e. in any case if A is true,∶ then B is true [MacColl 1877- 1878b, 177]. In the third installment, MacColl enumerated what he considered to be the three principal differences between his development of logic and Boole’s and Jevons’s [MacColl 1878-1879, 27]. They are that (1) every letter, as well as every combination of letters, signifies a statement; (2) that he introduces the symbol ‘ ’ to denote that the statment following it is true, provided the statement preceding∶ it is true; and (3) he introduces a special symbol for negation, which may be applied, depending upon circum-

2. Whether modal operators, with MacColl’s ‘quasi-modal’ values of certain, impossible, contingent, true or false, could or should have been reconstituted as fully fledged quantifiers, in line with Russell’s interpretation of universally quantified propositions as necessary and existentially quantified propositions as possible, is critical, but beyond the scope of our present concern, and it is, moreover, moot, since MacColl did not adopt that option; see [Anellis 2009, 154] and [Rahman & Redmond 2008, 543–544]. 102 Irving H. Anellis stances, to any number of terms, or even a complex statement. For our particular purposes, the second innovation is crucial for distinguishing the classical Boolean algebra from the symbolic logic of MacColl in its more significant aspect because it introduces implication as the principal connective of the system (along with negation), and can be seen in this sense as closely related to Schröder’s subsumption [Peckhaus 2004, 584]. It differs significantly, however, from Peirce’s (and Schröder’s) introduction of implication into the algebra of logic, given that, as here defined by MacColl, it amounts to strict implication, whereas Peirce had adopted material implication, together with negation, as his main connectives. Although remembered today primarily, when at all, for his discussions with Russell on the nature of propositions and their existential import—to the extent that Max Fisch and Atwell Turquette thought that Peirce might have been led to work out a triadic logic by the debate between MacColl and Russell on the nature of implication [Fisch & Turquette 1966, 82–83]—MacColl was in his day a respected member of the logical community, taking his position in the competition for the best (most effective) logical system. Schröder’s comparison of the procedures for solving logical problems provided by different logical systems, in particular the different ways of solving Boole’s famous “Example 5,” is discussed. MacColl’s work demonstrates the importance of the organon aspect of symbolic logic. We have: Ex. 5. Let the observation of a class of natural productions be supposed to have led to the following general results. 1st, That in whichsoever of these productions the properties A and C are missing, the property E is found, together with one of the properties B and D, but not with both. 2nd, That wherever the properties A and D are found while E is missing, the properties B and C will either both be found or both be missing. 3rd, That wherever the property A is found in conjunction with either B or E, or both of them, there either the property C or the property D will be found, but not both of them. And conversely, wherever the property C or D is found singly, there the property A will be found in conjunction with either B or E, or both of them. Let it then be required to ascertain, first, what in any particular instance may be concluded from the ascertained presence of the property A, with reference to the properties B, C, and D; also whether any relations exist independently among the properties B, C, and D. Secondly, what may be concluded in like manner respecting the property B, and the properties A, C, and D. [Boole 1854, 146] In an entry into his Logic Notebook for 14 November 1865 (see MS 114 of Peirce’s Logic Notebook of 1865-1909, published in [Peirce 1982, 337]), Peirce MacColl’s influences on Peirce and Schröder 103

first declared that there is “no difference logically between hypotheticals and categoricals,” and that “[t]he subject, is a sign of the predicate, the antecedent of the consequent,” thus virtually, if not yet actually, equating the copula of predication with implication and opening the way to eventually represent the syllogism “All S are M, All M are S, Therefore all S are P ” as S ⊃ M & M ⊃ P ⊃ S ⊃ P . This translation is, as we noted, strongly suggested( in) [Pe( irce 1870],) ( and also,) independently, ten years later, by Hugh MacColl in his 1880 paper on “Symbolical Reasoning,” where he wrote in his Definition 3 that:

The symbol , which may be read “implies,” asserts that the statement following∶ it must be true, provided the statement preceding it be true. Thus, the expression a b may be read “a implies b,” or “If a is true, b must be true,”∶ or “Whenever a is true, b is also true.” [MacColl 1880a, 50–51]

MacColl followed this up in “Symbolical Reasoning IV,” when he explains that: from two implicational premisses A B and B C draw implicational conclusion A C.∶ That is to∶ say [. . . ] A B B C A C . [MacColl∶ 1902b, 367] ( ∶ )( ∶ ) ∶ ( ∶ ) MacColl made this reading more fully explicit, in case it still needed to be by this time, where he wrote that, in the case, for example, of the Barbara syllogism “All A are B, All B are C, therefore All A are C,” In this form the syllogism is true whether premisses or conclusion be true or false [. . . ] and must, therefore, be classed amongst the formal certainties. Now a statement is called a formal certainty when it follows necessarily from our formally stated conventions as to the meanings of the words or symbols which express it; and until a language has entered upon the propositional stages those conventions (or deﬁnitions) cannot be formally expressed and classiﬁed. [MacColl 1902b, 368] For MacColl, there is no distinction between categorical and hypothetical propositions. 3 This is one of the essential features permitting MacColl, Peirce, and Venn, and later Bertrand Russell, to understand the syllogism of the form “All A are B. All B are C. Therefore all A are C” as the implication A ⊃ B B ⊃ C ⊃ A ⊃ C . Venn in Symbolic Logic objected to MacColl’s notation[( )( regarding)] implication,( ) and in particular the appellation “implication,” however, principally on the ground that MacColl applies the term “implication” to hypotheticals; “implication,” Venn argued, “implies” that it is known that there is a connection between the terms, and this is a stricter condition than the “if. . . then” relation [Venn 1881, 376–378]. Thus Venn’s objection regards not the treatment of the logical relation which we call “implication,” but rather his calling it by that name. Ladd-Franklin considered it one of MacColl’s greatest

3. See, e.g. [Rahman 2000]. 104 Irving H. Anellis accomplishments to define Aristotle’s A-proposition in terms of implication, writing that “The logic of the non-symmetrical affirmative copula, “all a is b,” was first worked out by Mr. MacColl” [Ladd-Franklin 1889, 562]. She then explains that: Nothing is stranger, in the recent history of Logic in England, than the non-recognition which has befallen the writings of this author. The fact that his contributions appeared in a journal which logicians were not in the habit of referring to (his brief article in Mind did not do his method justice), the fact that he was not acquainted with the writings of Boole, and the further accident that he considered it a matter of importance to read “all a is b,” which he wrote a b, in the words, “the statement that a thing is a implies the statement∶ that it is b,”—all doubt- less contributed to making him seem foreign to writers trained in the usual schools of logic. The fact that the nature of the connection between the terms in “all x is y,” and between the propositions in “a is b is-always-followed-by c is d,” is exactly the same, and must be exhibited in the same formal rules of procedure, has nothing to do with the words in which the proposition and the sequence may be expressed. But if it had not been for this accidental misfortune, it seems incredible that English logicians should not have seen that the entire task accomplished by Boole has been accomplished by MacColl with far greater conciseness, simplicity and elegance; and, what is an interesting point, in terms of that copula which is of by far the most frequent use in daily life. [sic] [Ladd-Franklin 1889, 562] MacColl’s preference for multiple-valued, especially triadic logic over bivalent logic was the subject of criticism by Platon Sergeevich Poretskii (1846-1907), 4 and Jevons [Jevons 1881, 486] among those who challenged his definition of implication. Nevertheless, MacColl’s contributions to non-classical logic were more consequential than his work in two-valued logic. His definition of inference in terms of implication, and his concept of strict implication, aimed at avoiding the ex falso sequitur quodlibet of the accepted definition of material implication which he himself had framed, is a precursor to the introduction by Clarence Irving Lewis (1883-1964) of strict implication [Lewis 1912; 1913]. Whether Peirce had MacColl in mind among those of whom he wrote that: “The algebra of logic was invented by the celebrated British mathematician George Boole, and has subsequently been improved by the labors of a number of writers in England, France, Germany, and America,” [Peirce 2010, 63], can only be conjectural, since, in the manuscript “Boolian Algebra” of circa 1890 in which Peirce made this comment, no individual names, other than Boole’s,

4. Styazhkin asserts, without supplying citations, that MacColl’s “tendency toward augmenting two-valued logical formalism with a third truth value. . . evoked many objections from P. S. Poretskiy” [Styazhkin 1969, 214]. MacColl’s influences on Peirce and Schröder 105 are mentioned. 5 Neither does he clearly make explicit in this context the specific improvements that he had in mind, perhaps because the manuscript in question was intended to serve as an introductory logic textbook rather than as either a critical or an historical exposition. Instead, we must go to the manuscript “Boolian algebra—elementary explanations” [Peirce 2000, 5] of 1886 to achieve an explicit reference by Peirce to MacColl’s work. There he compares his so-called “claw” of illation as roughly equivalent to MacColl’s representation, as introduced by MacColl in [MacColl 1877-1878b, 177], of the connective of implication expressing the relation between antecedent and consequent. Peirce writes: In my different publications I have used a sign like Y turned over on its side, , to signify the relation between the antecedent and consequent of an hypothetical proposition. It is a very convenient sign, but as I have no such sign on this typewriter, I shall use a colon for the same purpose, according to the practice of Mr. Hugh MacColl. [Peirce 2000, 5] Peirce had already referred to MacColl’s work in logic a decade earlier, in his paper “On the algebra of logic” of 1880. This was, in fact, Peirce’s earliest published reference to MacColl, and it is again, specifically in connection with MacColl’s use in [MacColl 1877-1878b, 183] of the sign for implication several times over in a single proposition. Peirce stresses the role of implication in connection precisely to MacColl’s Def. 12 [MacColl 1877-1878b, 177]. The reference to MacColl there occurs in a footnote [Peirce 1880, 24], in which Peirce writes: “Mr. Hugh McColl (Calculus of Equivalent Statements, Second Paper, 1878, 183) makes use of the sign of inclusion several times in the same proposition. He does not, however, give any of the formulæ of this section,” concluding Peirce’s explanation for the use of his illation in defining the categorical propositions A, E, I, and O, their Aristotelian transformations, and the determination of the truth-values of the basic four forms of categorical propositions. In the same paper, Peirce notes that four different algebraic methods had been devised for solving problems in the logic of non-relative terms, namely those of Boole, of Jevons, of Schröder, and of MacColl, to which he himself added a fifth [Peirce 1880, 37]. Peirce nevertheless finds in a footnote that MacColl, “apparently having known nothing of logical algebra except from a jejune account of Boole’s work in Bain’s Logic [1870] published several papers on a Calculus of Equivalent Statements” [Peirce 1880, 32]. 6

5. We do not know for certain whether Peirce would have accounted MacColl among the British, the French, or British residing in France. In any event, since Peirce here gives no names, he could possibly have had in mind either Louis Liard (1846-1917), whose work included studies of Jevons’ and Boole’s contribution to logic [Liard 1877a; 1877b], as well as a wider survey of British work in algebra of logic [Liard 1878], or the Belgian Joseph-Rémi Léopold Delboeuf (1831-1896), author of “Logique algorithmique” [Delboeuf 1876]. 6. The footnote in question runs half a page long. Only the ﬁnal two sentences refer to MacColl, and explicitly to [MacColl 1877-1878b]. In that note, Peirce wrote in 106 Irving H. Anellis

In fact, MacColl testified in the second installment of his “The calculus of equivalent statements” [MacColl 1877-1878b, 178] that he had not read Boole’s work. Peirce in a footnote likewise noted that, as a consequence of his unfamiliarity with the latest work in algebraic logic beyond that gleaned from the sophomoric account by Alexander Bain (1818-1903), MacColl’s contributions to logic there was based upon “nothing but the Boolian algebra, with Jevons’s addition and a sign of inclusion” [Peirce 1880, 32]. It was not until after his “Bainsian” treatment of Boole’s logic that MacColl became directly acquainted with Boole’s work, and in particular with Boole’s Laws of Thought (1854). This came about through the good offices of Robert Harley (1828-1910), who loaned him a copy of that work [MacColl 1878-1879, 27]. 7 Schröder, for his part, suggested that the five methods referred to by Peirce could, in fact, be reduced to three, because his own was a modification of Boole’s, which thereby became obsolesent [Schröder 1890-1905, I, 559], while Peirce’s and MacColl’s were essentially sufficiently similar to permit their reduction to, or combination into, one, the “MacColl-Peircean” [Schröder 1890- 1905, I, 589]. MacColl, on the other hand, claimed the superiority of his methods over that of the “Boolian Logicians” for solving certain logical and mathematical problems, specifically those that involve questions of probability, and he indicated that his logic was developed explicitly in response to questions about probability [Rahman & Redmond 2008, 556]. 8 The explanations for MacColl’s lack of direct familiarity with the work of Boole and the other algebraic logicians, and for the relatively insubstantial knowledge which he had of Boole’s logic were, first, his comparative isolation from the small circle of investigators in algebraic logic, 9 his limited linguistic knowledge, on account of which he was unable to read much of the work of his contemporaries and immediate predecessors [Grattan-Guinness 1998, 10], and his insubstantial mathematical background relative to that shared by the majority of algebraic logicians of that period, achieving only a B.A. in mathematics from the University of London in 1876. Despite this, and in particular, we full: “Later in the same year, Mr. Hugh McColl, apparently having known nothing of logical algebra except from a jejune account of Boole’s work in Bain’s Logic, published several papers on a Calculus of Equivalent Statements, the basis of which is nothing but the Boolian algebra, with Jevons’s addition and a sign of inclusion. Mr. McColl adds an exceedingly ingenious application of this algebra to the transformation of definite integrals” [Peirce 1880, 32]. 7. Interestingly, and perhaps not merely coincidently, Harley performed a similar service for Jevons, by reporting [Jevons 1877, xxi] that Leibniz had anticipated Boole’s laws of logical notation, and noting additionally that Boole had been informed about Leibniz’s work by Robert Leslie Ellis (1817-1859) a year after Boole published his Laws of Thought. 8. See [MacColl 1881] for his negative appraisal of Peirce’s conception of probability. 9. Thus, in a letter of 1905 to Bertrand Russell, MacColl complained: “I feel myself an Ishmael among logicians, with my hand against every man, and every man’s hand against me” (quoted in [Astroh 1998, 142]). MacColl’s influences on Peirce and Schröder 107 might add, despite MacColl’s scanty mathematical background, Peirce judged, indirectly referring to [MacColl 1877-1878a] that “Mr. McColl [sic] adds an exceedingly ingenious application of the algebra to the transformation of definite integrals” [Peirce 1880, 32]. MacColl devised his techniques so as to apply his logical calculus to the theory of limits in an effort to expand and develop the logical system as a proper analog of algebra. Specifically, as [Grattan-Guinness 1998, 7] noted, referring to 35 pages of his 1906 Symbolic Logic, MacColl sought to establish the analogy between truth-values in compound propositions and the laws of signs in algebra. For example, the conjunction True and False is False in logic, and positive times negative is negative in algebra. MacColl, as [Grattan-Guinness 1998, 7] also notes, used superscripts, with “P ” and “N” to define appropriate propositions such as

x 3 P = x > 3 , and x 3 N = x < 3 ( − ) ( ) ( − ) ( ) with “=” serving as equality by deﬁnition—in addition to its use as arithmetical equality and equivalence between propositions [MacColl 1906, 107]. But, while change of limits in multiple integrals, where inequalities such as in (1) were used to state that a variable lay between given values, and as Peirce said, MacColl’s procedures were “exceedingly ingenious”; but for some of the more elementary needs, such as obtaining the roots of a quadratic equation [MacColl 1906, 112–113], were far more complicated than necessary. 10 MacColl’s variables, as presented in the ﬁfth installment of “The calculus of equivalent statements” [MacColl 1896, 157), are similar in several respects to Peirce’s assertion of possibility. In the more mature development of his theory of modal propositional logic, as found in the sixth installment of his

10. The fact however that MacColl’s innovations with integrals on this score actually are less efficient than the standard procedures, and the confusions which he exhibited with regard to questions of defining the use of zero in expressions involving differentials as ratios df dx between a function f and its respective variable x, indicate that, whereas MacColl~ was well within the French tradition of Lagrange of treating the theory of limits algebraically, he was not familiar with the rigorous Cauchy-Weierstrassian reformulation of the theory of limits in terms of ε δ notation, in which a careful distinction is made between [absolute] zero and a very− small ε approaching, but not attaining zero, in relation to the change, from either the positive or negative side, of δ where the limit of f x as x approaches a is L, i.e. lim f x L if for every number ε 0, there exists( a) corresponding number δ 0 x a ( ) = > > such→ that f x L ε whenever 0 x a δ, despite Weierstrass’s students helping to disseminateS ( )− thisS< across Europe from

“Symbolic reasoning,” these variables became for MacColl existentially defined assertions; there MacColl wrote that: Firstly, when any symbol A denotes an individual; then, any intelligible statement φ A , containing the symbol A, implies that the individual represented( ) by A has a symbolic existence; but whether the statement φ A implies that the individual represented by A has a real existence( ) depends upon the context. Secondly, when a symbol A denotes a class, then the intelligible statement φ A containing any symbol A implies that the whole class A has( a) symbolic existence; but whether the statement φ A implies that the class A is wholly real, or wholly unreal, or partly( real) and partly unreal, depends upon the context. [MacColl 1905, 77] Bain’s conception of Boole’s principal contribution to the syllogistic logic is that, with the addition of the axiom that equals added to equals yield equal sums, and that, if A is greater than B and B is greater than C, then A is greater than C, he “draws the Syllogism under the axiom that suffices for the reduction of equations,” he “assumes that the analogy of the logical method and the algebraical is sufficiently close to allow of the substitution” [Bain 1870, 164]. Boole’s additions to the syllogism, then consist in drawing the details of the algebraic representation of logical propositions; these are adumbrated by Bain in a few lines short of seventeen pages [Bain 1870, 190–207]. Examining the details of MacColl’s major productions, we find that the first two papers on “The calculus of equivalent statements” [MacColl 1877- 1878a-b] and his first paper on “Symbolical reasoning” [MacColl 1880a] present a calculus of propositions which has essentially the properties of Peirce’s, albeit without Π and Σ operators to serve, as they did for Peirce (and then Schröder), as logical product and logical sum, doing duty for the universal and existential quantifiers respectively. It is, therefore, in fact a calculus of propositions, like the Boolean (two-valued) algebra of logic with which we are familiar. Lewis [Lewis 1918, 108] believed that the date of these papers indicates that MacColl arrived at their content independently of Peirce’s work along the same lines. In his Symbolic Logic and its Applications (1906), MacColl’s attention is di- rected to truth-functional logic, and hence the fundamental symbols represent propositional functions rather than propositions. It is also here that MacColl examines a trivalent logic. Instead of just the two traditional truth values true and false, we also have certain, impossible and variable (not certain and not impossible). These are indicated by the exponents τ,ι,ε,η and θ respectively. The result is a highly complex system, the fundamental ideas and procedures of which suggest somewhat Lewis’s system of strict implication to be set forth in Chapter V of his Survey of Symbolic Logic (1918). The logical theory of MacColl’s Symbolic Logic is neither mathematically nor logically equivalent to the Boole-Schröder calculus. We will return to a consideration of the question of the relation between Peirce’s trivalent logic and MacColl’s four-valued, “three-dimensional” logic. MacColl’s influences on Peirce and Schröder 109

One of the questions of priority which would appear to be problematic is whether Peirce or MacColl first proclaimed that Boole’s calculus was amenable to alternative interpretations. It is well known that Peirce allowed his “claw” of illation to function either as material implication or as class inclusion. Indeed, one of the criticisms which both Peano and Russell leveled against the Boole- Schröder calculus was that it neither offered nor provided a basis for distinguishing between classes and propositions. Russell’s criticism of the equivocal use by Peirce and Schröder of one connective for both inclusion and implication occurred first in his “Sur la logique des relations avec des applications à la théorie des séries” [Russell 1901], and recurs, in somewhat different guise, in the Principles of Mathematics [Russell 1903, 24]. Frege’s similar criticsm of Schröder occurs in his review [Frege 1895] of Schröder’s Vorlesungen über die Algebra der Logic. Likewise, in his Arithmetices Principia, Nova Methodo Exposita [Peano 1889, V], Peano laid stress on the difference between set el- ementhood (or class membership) on the one hand and material implication on the other. From the perspective of Russell and Peano, it accrues to the disadvantage of MacColl that, although he placed his greater emphasis on the logic of propositions, he nevertheless allowed for the interpretation of his calculus both with respect to propositional logic and the class calculus. By the same token, it accrued to his advantage, from the perspective of Peirce and Schröder, that he imposed no constraints between these interpretations. For Peirce’s illation and Schröder’s equivalent Subsumption (e), they more probably initially had in mind for this connective general inclusion, that is, any relation that is transitive, reflexive, and asymmetrical, rather than material implication specifically [Houser 1987, 431, 437 n. 6]. However, its interpretation, as either inclusion or implication, could vary, according to the specific context in which it occurred. The potential for multiple interpretations of the kind made by Peirce for his illation and Schröder’s subsumption could be justified, prior to Cantor’s development of set theory by the pre-Cantorian treatment of collections, in terms of the part-whole relationship since it is only after the development of the concepts of set membership and class inclusion and the consequent distinction between the two notions that it made sense to distinguish between these two kinds of relation or that such a distinction had a purpose. 11 MacColl was among those who, like Peano and Russell, criticized Peirce’s illation and Schröder’s Subsumption for conflating implication and inclusion [MacColl 1906, 78]. But this is reflective of MacColl’s inability to appreciate that algebraic logic is concerned with the general algebraic structure of equations as relations, rather than with propositional logic specifically, which was MacColl’s concern. Whether, however, MacColl’s myopia in this regard is rooted in his comparative isolation from the mathematical community working in algebraic logic, or something else, is perhaps moot. What is salient

11. For a philosophical approach to the history (and “pre-history”) of set theory and the difference between the Aristotle-inspired conception of infinity, based on the whole-part relation, and concept of infinity in Cantorian set theory as based upon the element-set relation, see, e.g. [Tiles 1989]. 110 Irving H. Anellis is that this is indicative of the primary difference between the “Booleans” on the one hand and MacColl on the other. Nathan Houser [Houser 1989, l] has noted that Peirce reported on 14 December 1880 that he had received copies of papers from a number of authors “on various logical and psychological subjects,” and that among those whose works he received was MacColl. What Houser does not specify is which of MacColl’s writings he had received, or when, but we may certainly conclude that Peirce must at the very least have known enough about MacColl’s “The calculus of equivalent statements” by 1880 to permit his reference to it in his own “On the algebra of logic” and explicit citation of the second installment of that series. Houser [Houser 1989, lix] also notes that “Peirce had made plans to visit with Hugh MacColl in Boulogne-sur-Mer, which he probably did near the beginning of his stay,” which would indicate the occurrence of a personal meeting between Peirce and MacColl some time in mid-May 1883 [MacColl 1883]. In 1883, Studies in Logic [Peirce 1883] appeared, edited by Peirce and containing contributions by Peirce and his Johns Hopkins University students. In the “Preface,” Peirce noted that between 1864 and 1877, Boole in the Laws of Thought, Jevons in Pure Logic [Jevons 1864], Robert Grassmann (1815-1901) in his Die Formenlehre oder Mathematik [Grassmann 1872a], and, presumably more explicitly, in his Begriffslehre oder Logik [Grassmann 1872b], Schröder in his Operationskreis [Schröder 1877], MacColl in the “The calculus of equivalent statements,” and Venn all, independently of one another, proposed the use of the addition sign for combining different terms into an aggregate, meaning that, whether they disagreed upon whether logical addition should be interpreted as inclusive or exclusive disjunction, they all nevertheless agree that logical addition is a disjunction [Peirce 1883, iii]. Boole and Venn alone among them, he notes, argue in favor or logical addition as exclusive disjunction [Peirce 1883, iv]. Peirce also adds that both he and MacColl also “find it absolutely necessary” to introduce as well a new sign to represent existence. Boole’s notation permits assertions that “some description of a thing does not exist,” but does not allow the possibility of asserting that anything does exist [Peirce 1883, iv]. Moreover, the use of the sign of equality, as used by Boole, in expressions such as xy = 0 of the E-proposition “No xs are ys” was reformulated in the Laws of Thought [Boole 1854, 61–64] as y = ν 1 x , asserting with “No xs are ys” and being converted to “All xs are not y(s.”− But,) as Augustus De Morgan [De Morgan 1842a, 13], [De Morgan 1842b, 288] pointed out equality is a complex relation, so that the expression “A = B” asserts both that all As are Bs and that all Bs are As. For these reasons, Peirce wrote, he and McColl “make use of signs of inclusion and of non-inclusion” [Peirce 1883, iv]. Peirce writes “Griffin Aquatic” to mean that some animals are not aquatic, or that a non-aquatic animal does exist. McColl’s notation is not, he tells us, essentially different. At this point, Peirce reiterates a common complaint of logicians of the day, that there is neither a uniform system of MacColl’s influences on Peirce and Schröder 111 notation nor a uniform determination of the most basic propositions and the respective relations or logical connectives holding between them. The contributions to Studies in Logic by Christine Ladd-Franklin [Ladd- Franklin 1883] and Oscar Howard Mitchell (1851-1889) [Mitchell 1883], operating with simplifications of Boole’s logic by Schröder and MacColl, helped, in Peirce’s estimate to improve the Boolean logic further, and found it “surprising to see with what facility their methods yield solutions of problems more intricate and difficult than any that have hitherto been proposed” [Peirce 1883, v]. Ladd-Franklin [Ladd-Franklin 1883, 23], in considering the copula for her algebra of logic, noted that whereas Jevons considered the two propositions “The sea-serpent is not found in the water” and “The sea-serpent is not found out of the water” to be mutually contradictory, MacColl, Venn, and Peirce were led to consider the existential import of propositions, asserting that a universal proposition can be contradicted only by a particular proposition, and a particular proposition contradicted only by a universal propositions, on the basis of which they concluded that the two propositions “The sea-serpent is not found in the water” and “The sea-serpent is not found out of the water” taken together allowed the inference that sea-serpents do not exist. It is in turn on this basis that Peirce held that, from the formal algebraic perspective, there is no distinction to be made between hypothetical and categorical propositions, and that the Barbara syllogism can be reformulated as S M & M P S P . Another difference between Jevons on the one(( hand) and (MacColl)) (and( Peirce)) on the other that Ladd-Franklin [Ladd-Franklin 1883, 24] makes is that, while for Jevons, a letter c is a representation of an indefinite class symbol, every letter used by MacColl denotes a distinct proposition, and as MacColl’s a b states that if some object is an a, it also expresses that any statement a implies∶ statement b, as does Peirce’s a b. The singular advantage that Ladd-Franklin claims for Peirce’s “ ” over MacColl’s “ ” is that the former is, in its graphic design, asymmetrical, and therefore more∶ adept at designating the asymmetrical nature of the relation which it is designed to express. Ladd-Franklin does not address the historical question of priority, or even the broader question of the direction of influence between Peirce and MacColl in “On the algebra of logic,” but is chiefly concerned to explore the comparative merits of the offerings of the contemporary algebraic logicans, and to set forth her own contributions in accordance with what she was learning in Peirce’s Johns Hopkins courses. (Indeed, the ahistoricality of mathematicians engaged in leading-edge research is, proverbially, nearly universal.) In her discussion on the methods for elimination of superfluous or redundant terms form logical equations, 12 Ladd-Franklin notes [Ladd-Franklin 1883, 39] that there is a significant difference between the Boolean and classical squares of oppositions, and in particular addresses the question of the existential import of propositions that was a major issue in MacColl’s published corpus (as well as an issue in his exchanges on the subject with Bertrand

~~12. For the historical background and exposition of elemination of terms in formulas of algebraic logic, see [Green 1991]. 112 Irving H. Anellis~~

Russell). As we know, the modern or Boolean square of opposition and the determination of which inferences are valid rests upon the acceptance of empty sets, whereas the traditional (Aristotelian) square and the determination of the validity of inferences presupposes that there are no empty classes involved. In explaining that: “Those syllogisms in which a particular conclusion is drawn from two universal premises become illogical when the universal proposition is taken as not implying the existence of its terms,” [Ladd-Franklin 1883, 39], Ladd-Franklin cites both Peirce’s “On the algebra of logic” [Peirce 1880] and MacColl’s “Symbolical reasoning” [MacColl 1880a]. In this particular instance, the two articles being nearly contemporaneous, no inference whatever is ventured by Ladd-Franklin concerning possible influence of either author on the other. What is patently clear is that Ladd-Franklin regarded Peirce and MacColl as intellectual companions, sharing many of the same concepts and methods in their respective, if presumably independent, development of logic, and both superior to the techniques devised by Jevons. With this evaluation in mind regarding the techniques and their applications to the problems which she considers, Ladd-Franklin adopts those, in her own work, of MacColl, and, finding MacColl’s superior to that of Jevons, but more complicated than the alternative that she offers in its place, she writes: Complicated problems are solved with far more ease by Mr. McColl than by Mr. Jevons; but that is not because the method of excluded combinations is not, when properly treated, the easiest method. A method of implications, such as that of Mr. McColl, is without doubt more natural than the other when universal premises are given in the affirmative form, but the distinction which it preserves between subject and predicate introduces a rather greater degree of complexity into the rules for working it. [Ladd-Franklin 1883, 51] Likewise, Mitchell remarks that, apart from the notational difference between Peirce’s “ ” over MacColl’s “:” if symbolizing the copula, they both “have given algebraic methods in logic, in which the terms of these propositions are allowed to remain on both sides of the copula” [Mitchell 1883, 96]. Mitchell adds the additional information that: Mr. Hugh McColl, in his papers on logic in the “Proceedings of the London Mathematical Society” [Vol. IX, et seq.], has been using a notation for the copula identical in meaning with that of Mr. Peirce. He uses a colon to denote implication, instead of . Mr. Peirce has recently told me that Mr. McColl justifies his use of the colon by its mathematical meaning as a sign of division. Thus Barbara and Celarent are: m p m p¯ ∶ ∶ s m s m ∶ ∶ s p s p¯, ∴ ∶ ∴ ∶ and the analogy to division is obvious. [Mitchell 1883, 101] MacColl’s influences on Peirce and Schröder 113

Peirce’s choice of notation is dictated by the breakdown of the analogy upon which MacColl’s selection of notation depends. As Mitchell explains: [T]his analogy exists only in the two universal moods of the first figure. Thus Cesare and Festino are p m¯ p m¯ ∶ ∶ s m s m¯ ∶ ÷ s p¯ s p, ∴ ∶ ∴ ÷ where is the negative copula. And the analogy to division is wanting.÷ [Mitchell 1883, 101–102] By far the best known of MacColl’s work is his advocacy, and development, of nonclassical logic, that is, his three-dimensional, four-valued logic. There are arguments as to whether MacColl’s extension of propositional logic should be considered as a modal propositional logic or as a trivalent propositional logic, since it is open to question whether his classifications true and false, certain, impossible and variable were intended to be truth values or modal operators. MacColl’s mature system of modal logic was developed in the years 1895-1905 [Astroh & Read 1998, ii] but can be traced back to at least as far back as MacColl’s series of papers in Mind on “Symbolical reasoning” [MacColl 1880a, 54], in which MacColl presents the denial of a′ b as the equivalent to ab′ as a definition of certainty or actuality (see [Goldblatt+ 2006, 4]). Fania Cavaliere argued that, working with his operators τ,ι,ε,η,θ to express true, false, certain, impossible, and variable respectively, and defining “p implies q” as“p and not-q is impossible,” MacColl developed a modal propositional logic, thereby anticipating enroute C. I. Lewis’s definition of strict implication [Cavaliere 1996]. Peter Simons [Simons 1998] argues, against Nicholas Rescher [Rescher 1969, 4], that MacColl’s logic cannot be considered a many-valued logic. It is more properly understood as a modal probability logic. 13 MacColl for his part accounted the failure by Boole and Peirce (and Schröder) to distinguish between an argument (or its contents) and its probability as a serious flaw [Hailperin 1996, 132–134]. But this effectively reduces proofs from distinctions between validity and invalidity to degrees of validity. And while it is indeed crucial to distinguish the contents of an argument and its truth-value, a distinction which is already present in Peirce when he introduced truth functional analysis (beginning in 1884 and perhaps stimulated by Christine Ladd-Franklin [Ladd-Franklin 1883, 62ff.] and her study of Jevons [Jevons 1877, 162ff.]), 14

13. In more recent terms, we would be more inclined to suggest that many-valued logics, modal logics, and probability logics all share the same algebraic structure, as generalized Galois logics (ggl), in which the main connective is a fusion of implication and some other specialized reflexive, transitive, and asymmetrical relation, with the generalized connective being such that a b a b for any ggl. See [Bimbó & Dunn 2008]. ○ ≤ ⇔ ○ 14. [Houser 1991, 20] suggests that Peirce’s study of truth-functional logic may have arisen from consideration of dilemmatic reasoning and truth-functional analysis of material implication, in 1884 (MS 527, Part II of “algebra of logic”; published in 114 Irving H. Anellis by restricting the elements of his logic to propositions, MacColl does not have the capability of interpreting the elements of his system as either terms, sets (or classes), or propositions in the same way that Peirce and Schröder could. Likewise, he does not have the capability to take his basic connective, depending upon context, as Peirce and Schröder were able to do, as either the copula, or class inclusion (or set membership), or implication (or inference). 15 What is not in dispute is that, as we have already noted, MacColl placed the emphasis in logic upon propositions, and insisted that the basic logical relation is implication between propositions, rather than class inclusion. He investigated the details in his two series of papers “The calculus of equivalent statements” and “Symbolic reasoning,” devising a “pure [propositional] logic.” The same questions could perhaps be raised concerning the trivalent logics devised by Peirce, as to whether there is a modal content to the third truth- value. What is evident is that truth-values were paramount, regardless of how those truth values might be modally interpreted. In discussing Russell’s criticisms of MacColl’s definition of variables, Max Harold Fisch (1900-1995), who devoted much of his professional career to the study of Peirce, and Atwell Rufus Turquette, noted that MacColl’s variables are similar to Peirce’s concept of possibility [Fisch & Turquette 1966, 80]. For our purposes, however, the primary question is not concerning the interpretation of the classification of propositions or the issue of priority, but rather the issue of the possible influence on Peirce of MacColl’s work on non- classical propositions and their logic. Bertrand Russell, as is too well known to record here in any detail, engaged in debates with MacColl on these issues (as well as many others) in the first few years of the twentieth century. It is debatable whether or not Russell countenanced non-classical logics. The general consensus is that he did not. What is probably indisputable is that, whatever consideration (if any), whether positive or negative, that Russell gave to non-classical logics, was stimulated in all probability by his encounters on the matter with MacColl, rather than through the offices of any other source. 16 Fisch and Turquette wrote that: “It would be interesting to know exactly what circumstances stimulated Peirce’s successful solution of the problem of

[Peirce 1993, 107–109]), along with his rejection of syllogistic. In “On the algebra of logic: A contribution to the philosophy of notation” [Peirce 1885] Peirce used his connective ( ) for material implication and had introduced truth-functional analysis. 15. For example, in [Pierce 1870], Pierce introduced the same sign, “illation” ( ) to represent both implication and class inclusion. He was criticized for this by Russell [Russell 1903, 187]; and Schröder too, following Peirce, used the same symbol, subsumption (e), for both class inclusion and implication, for which he, in turn, was criticized by Frege [Frege 1895]. 16. See in particular [Anellis 2009] on the sources from which Russell encountered nonclassical logic. See [Anellis 2009, 153–218], especially § 11: “Peirce again, and MacColl and Bradley” [Anellis 2009,187–193] and § 12: “More on MacColl and Bradley” [Anellis 2009,193–195]. MacColl’s influences on Peirce and Schröder 115 triadic logic” [Fisch & Turquette 1966, 82]. In the absence of concrete extant evidence, Fisch was left to conjecture whether, and if so, when and where, his thinking in terms of the possibility of developing a non-classical logic may have been stimulated by MacColl. Thus, Fisch and Turquette wrote that: For example, could it have been that Peirce had been following the controversy in Mind between Bertrand Russell and Hugh MacColl, prompted by Russell’s (1906) review of MacColl’s (1906) Symbolic Logic and MacColl’s (1907) reply? If so, late in 1908 or early in 1909, Peirce might have seen MacColl’s (1908) note entitled “ ‘If’ and ‘Imply” ’ which had appeared in January 1908. 17 In this note, MacColl considers the difference between his and Russell’s treatment of implication. He indicates that for “nearly thirty years” he has been “vainly trying to convince” logicians of the errors involved in equating “implication” with what is now called “Russell’s material implication.” MacColl then asks the following question: Is it too much to hope that this test case will at last open the eyes of logicians to the necessity of accepting my three-fold division of statements (ε,η,θ) with all its consequences? [Fisch & Turquette 1966, 82–83] Fisch and Turquette go on to suggest that it would be speculative to attempt to determine the direction of influences between Peirce and MacColl with respect at least to Peirce’s triadic logic and MacColl’s three-dimensional logic [Fisch & Turquette 1966, 83]. They can confidently assert that they were interested in one another’s work for a considerable period of time, and they quote a letter to Peirce dated May 16, 1883 in which MacColl wrote that: “It will be a great pleasure indeed to me if you can stay a little while in Boulogne on your way to England. It is not often that I have the opportunity of making the personal acquaintance of my correspondents in logic and mathematics” [MacColl 1883]. Much later, in a draft of a letter dated November 16, 1906, Peirce writes to MacColl that: Although my studies in symbolic logic have differed from yours in that my aim has not been to apply the system to the working out of problems, as yours has, but to aid in the study of logic itself, nevertheless I have always thought that you alone, so far as I know, except myself, have understood how the matter ought to be treated by making the elements propositions or predicates and not common nouns. [Peirce 1906] Peirce concludes by noting that he has learned of a new book just published on symbolic logic which was reviewed in the November first issue of Nature [Anonymous 1906b], and hints that he should like to receive a copy. This

17. Fisch and Turquette are clearly referring here to the ﬁrst of two parts of MacColl’s [MacColl 1908]. 116 Irving H. Anellis letter is also cited by Michael Astroh [Astroh 1998, 170] as evidence that Peirce was in agreement with MacColl that the traditional account by logicians of subject and predicate in terms of general nouns was erroneous when reﬂected in logical form. 18 The book to which Peirce refers, is, of course, MacColl’s Symbolic Logic [MacColl 1906]. The account of MacColl’s book begins: “WHETHER Mr. MacColl is the Athanasius of symbolic logic or only its Ishmael, the fact remains that he seems unable to come to an agreement with other exponents of the subject. But he contends that his system in the elastic adaptability of its notation bears very much the same relation to other systems (including the ordinary formal logic of our text-books) as algebra bears to arithmetic” [Anonymous 1906b, 1]. 19 After noting that its contents had been previously published in journals, the reviewer continues that points on which MacColl lays

18. In light of the claims by Francis Herbert Bradley (1846-1924), referring to his Principles of Logic [Bradley 1883], as expressed in his letter to Bertrand Russell of 27 September 1914, that “I always have believed that in 1883 in the Logic I pointed out a number of inferences which fell outside the category of subject and attribute and pointed out again that there was nevertheless a form in every possible inference,” quoted in [Keen 1971, 10]. As Francisco A. Rodríguez-Consuegra made patently clear, Bradley, as much as Russell, was “an enemy of this [subject-predicate] mode of analysis,” and he explains that, for Bradley, the subject-predicate analysis “was the basis for traditional logic, and therefore to be rejected,” and that his writings “contain many examples of this rejection,” because, judgments as relational, the only distinctions that apply between subject and predicate are grammatical [Rodríguez- Consuegra 1992, 95]. See also [Chalmers & Griffin 1997]. While a discussion of interactions of MacColl and Bradley, MacColl and Russell, and Bradley and Russell are quite tangential to the discussion of MacColl’s influences on Peirce and Schröder, I suggest that it might nevertheless be useful to also examine what influence MacColl’s work in logic may have had on Bradley, and to investigate Bradley’s studies of MacColl’s work. Exchanges of correspondence between Bradley and MacColl, and between Russell and Bradley have been published in [Keene 1999]. Regardless of the issue of the connections between Bradley and MacColl, these discussions between Peirce and MacColl, and Bradley and Russell, ought, when taken collectively, dispell the claims, perpetuated by much of twentieth-century historiography of logic, that it was Frege who, single-handedly, broke the Aristotelian tradition of treating logical propositions exclusively in subject-predicate form. This does not, of course, contradict the role or significance of Frege in replacing the subject-predicate syntax with the function-theoretic syntax of mathematical logic. An anonymous reviewer of MacColl’s Symbolic Logic in the Educational Times [Anonymous 1906a] remarked on the significance of MacColl’s opposition to the subject-predicate form in logical syntax. It would be interesting, if perhaps also fruit- less, to speculate whether this anonymous author might actually have been Peirce, or Bradley, or Russell. 19. MacColl’s book was considered in that review along with The Development of Symbolic Logic [Shearman 1906] of Arthur Thomas Shearman (1866-1937); An Introduction to Logic [Joseph 1906] of Horace William Brindley Joseph (1867-1943); and Thought and Things, or Genetic Logic [Baldwin 1906] of James Mark Baldwin (1861-1934). MacColl’s influences on Peirce and Schröder 117

“considerable stress, and in which he does not command the uniform ascent of the other symbolic logicians,” are: (a) that he takes statements and not terms to be in all cases and necessarily the ultimate constituents of symbolic reasoning; (b) that he goes quite beyond the ordinary notation of the symbolists in classifying propositions according to such attributes as true, false, certain, impossible, variable; (c) that in regard to the existential import of propositions, while other symbolists define the null class o as containing no members, and understand it as contained in every class, real or unreal, he, on the other hand, defines it as consisting of the null or unreal members o1, o2, o3, &c., and considers it to be extended over every real class. . . [Anonymous 1906b, 1] The anonymous reviewer for Nature does not name the “other symbolic logicians” from whom MacColl differs on these three issues, although, since a criticism by Shearman is mentioned by the reviewer in the context of the discussion of Shearman’s Development of Symbolic Logic, we can be assured that he, at least, is one of the dissenters had in mind by the reviewer, it being noted that “the doctrines of Prof. Jevons and Mr. MacColl are subjected” by Shearman “to some severe criticisms. . . ” [Anonymous 1906b, 1]. It is noted in particular that Shearman “rejects all attempts to deal with any but assertoric propositions, and holds that if Mr. MacColl wishes to work with such data as probable and variable he should introduce new terms” [Anonymous 1906b, 1]. Fisch and Turquette conclude, with respect to their work on nonclassical logic, that: “As far as it is now known, however, there is not sufficient concrete evidence available to make possible an accurate account of the exact relationship between MacColl’s work in three-dimensional logic and Peirce’s investigations of triadic logic. The solution to this problem will have to await future historical research” [Fisch & Turquette 1966, 83]. We are obliged then to turn, on this matter, to the findings of Rahman and Redmond [Rahman & Redmond 2008] and others to glean what we can, or to hope that new extant evidence will appear. The question of MacColl’s influences on Peirce leading to Peirce’s consideration of the possibility of devising a nonclassical logic, and then to his efforts to do so, are indirect at best. We look in vain for an instance in which Peirce states that he was led in this direction by reading MacColl’s work. And, as questions of influence can be shadowy, the best we can do is examine the extant written evidence. We know that Peirce had already considered paradoxes of self-reference in “Grounds of the validity of the laws of logic” [Peirce 1869], in particular dealing with the ramifications of the proposition “This proposition is false,” and offering an analysis closely akin to that of Paul of Venice (Paulus Venetus; Paolo di Venezia; 1369-1429), but giving a proof of assumptions that Paul merely assumed, and demonstrating that the analysis of the underlying premise is in- 118 Irving H. Anellis adequate, namely that it states merely that the proposition is false, whereas, in reality, the underlying premise also includes the presupposition that the statement is true. Here, perhaps, we can see the roots of the trivalent logic that Peirce eventually devised beginning in the 1890s.

We also know that Peirce also raised the question of the possibility of developing non-classical logics as early as the 1890s. He wrote that by 1895 he had entertained the idea of toying with, and altering, the laws of logic and even of doing without some of them, as he suggested in a letter to Judge Francis Calvin Russell (1838-ca.1920) [no relation to Bertrand Russell], quoted by Paul Calvin Carus (1852-1919) in [Carus 1910a, 44–45]. In that letter, Peirce wrote that “before I took up the general study of relatives, I made some investigation into the consequences of supposing the laws of logic to be different from what they are.” He did not consistently, persistently or thoroughly follow through, although he left manuscript notes in which his “tinkering” appears. On the other hand, Peirce left no doubt, as Carus assured his readers in “Non-Aristotelian logic” [Carus 1910b], that he did not, as some readers of the original Carus paper [Carus 1910a, 44–45] might have been led to surmise, consider the possibility of non-Aristotelian logics to be in any wise “lunatic.” To the contrary, Peirce indicated that had he deigned to pursue the issue, he might have gained some insight into features of logic that might be overlooked (Peirce quoted at [Carus 1910b, 158]). Even earlier, George Boole, in his Laws 2 2 of Thought, after proving on the previous page that x = x, showed that x = x is impossible, and added, in the attached footnote, that, had it been accepted, it could have opened the possibility of a non-bivalent logic in which “the law of thought might have been be different from what it is” [Boole 1854, 50]. Peirce was certainly familiar with Boole’s Laws of Thought, and could have been thinking of Boole’s footnote. His own comment (as quoted by Carus [Carus 1910a, 45]) of entertaining the consequences of “supposing the laws of logic to be different from what they are,” is clearly an echo of Boole’s similar remark. This leaves open the possibility that Boole, together with MacColl, or even possibly Boole alone, could have led Peirce to take seriously the possibility for constructing non-classical logics.

Consider as examples of Peirce’s “tinkerings” with the laws of logic the fragments of Peirce’s manuscript Logic Notebook (1865-1909), in particular those dating from at least early 1909 if not somewhat earlier (MS 339:440– 441, 443; see [Peirce 1849-1914]), which had been examined by Fisch and Turquette On the basis of this work. Fisch and Turquette [Fisch & Turquette 1966, 72] concluded that by 23 February 1909 Peirce was able to extend his truth-theoretic matrices to three-valued logic, thereby anticipating both Jan Łukasiewicz (1878-1956), in “O logice trójwartosciowej” [Łukasiewicz 1921], and Emil Leon Post (1897-1954), in “Introduction to a general theory of elementary propositions” [Post 1921], by a decade in developing the truth table device for triadic logic and multiple-valued logics respectively. MacColl’s inﬂuences on Peirce and Schröder 119 Peirce’s inﬂuences on MacColl

The influence which Peirce and Schröder had upon MacColl, on the other hand, was comparatively tenuous. In particular, as Ivor Grattan-Guinness [Grattan-Guinness 1998, 11] noted, MacColl used Peirce’s symbol for implication in a passage of his book, where he also noted Schröder’s [MacColl 1906, 78–80], but he did not discuss or use their systems. We may attribute this at least in part to MacColl’s comparative isolation from the mainstream of the logical research community and from his unfamiliarity with many of the languages in which some of the work of active researchers was being carried out. Thus, for example, Grattan-Guinness [Grattan-Guinness 1998, 10] noted that lack of Italian left MacColl unable to read the work of Peano and his school, while his lack of German rendered him unable to take advantage of the work of Schröder, although MacColl attended and delivered a talk [MacColl 1901] at the same International Congress of Philosophy in Paris in 1900 at which Schröder also spoke [Schröder 1901]. 20 Thus, in a letter to Russell of 28 June 1901 (quoted in [Russell 1993, 239–240]), MacColl professes his ignorance of the logic of relations and, alluding to the first two volumes of the Vorlesungen über die Algebra der Logik [Schröder 1890-1905], his inability to read the work of Schröder because he cannot read German. The congress was MacColl’s best, and perhaps last, opportunity to get his work before an influential group of mathematical logicians, and his talk, “La logique symbolique et ses applications,” provided them with a survey of his logical system. MacColl made no reference in his own work to the work of Schröder, nor to Schröder’s congress paper. The influences which Peirce and Schröder had on MacColl were, at best, minimal. He was aware of their work, at one point using Peirce’s notation for implication in his own Symbolic Logic [MacColl 1906, 78–80] and taking note of the existence of Schröder’s work, but failing even at that juncture to provide his own exposition, or even summary, of that work. MacColl indeed went to great lengths to differentiate his work from that of Peirce and Schröder. The entirety of his “A note on Prof. C. S. Peirce’s probability notation of 1867” [MacColl 1881] is devoted to establishing that any similarity between his own notation for implication and Peirce’s illation is purely coincidental; thus, the Secretaries of the London Mathematical Society assert, in MacColl’s name, that MacColl, referring to his article “The calculus of equivalent statements (fourth paper)” [MacColl 1880b], has shown that: the apparent coincidence of notation, in some few particulars, between himself and Prof. Peirce, was entirely accidental, and that

20. [Lovett 1900-1901] is a summary of the the talks on mathematics presented at the Congress. MacColl’s and Schröder’s talks and discussions are summarized at [Lovett 1900-1901, 166–168], with Couturat raising questions for MacColl regarding the diﬀerences between Boole’s treatment of probability and MacColl’s [Lovett 1900- 1901, 176–177]. If there were any discussions between MacColl and Schröder, or even whether either attended the other’s talk, is not recorded in [Lovett 1900-1901]. 120 Irving H. Anellis

Mr. McColl was not at that time acquainted with Prof. Peirce’s paper. In fact, the revised fourth paper (“On Probability Notation,” Proceedings of the London Mathematical Society, Vol. xi., no 163) was communicated to the Society about nine months before the Author read Prof. Peirce’s paper. [MacColl 1881] The assumption that we may infer is that, whereas Peirce and Schröder were willing to acknowledge debts to MacColl, this willingness was not, evidently, reciprocated. Alternatively, we might just as easily take MacColl at face value and allow the possibility that he was largely uninfluenced by either Peirce or Schröder, or, at the very least that, if he was influenced by either or both, he failed to recognize it because his conception of logical systems and their underlying structures were significantly different from theirs, namely, that whereas he considered implication to be fundamental to logical form and the basic syntactic elements of logic were propositions, Peirce and Schröder considered illation and subsumption respectively, understood as generalized reflexive, transitive, assymmetrical relations, to be fundamental to the algebraic structure of logical systems and the basic syntactic elements to be the relata, i.e. the elements of the relations, interpretable as either terms, sets, classes, or propositions, depending upon the particular requirements of the system in its application.

~~Acknowledgments~~

~~Thanks to Amirouche Mokteﬁ and the anonymous referees for comments and suggestions for improvement.~~

~~References~~

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1901 La logique symbolique et ses applications, Bibliothèque du Ier Congrès international de philosophie, Paris: A. Colin, 135–183. 1902a Logique tabulaire, Revue de métaphysique et de morale, 10, 213– 217. 1902b Symbolic reasoning (IV), Mind, 11, 352–368. 1905 Symbolic reasoning (VI), Mind, 14, 74–81. 1906 Symbolic Logic and Its Applications, London: Longmans, Green, & Co. 1907 Symbolic logic (a reply), Mind, 16, 470–473. 1908 ‘If’ and ‘Imply’, Mind, n. s., 17, 151–152; 453–455. 1910 Linguistic misunderstandings, Mind, 19, 186–199, 337–355. Mitchell, Oscar Howard 1883 On a new algebra of logic, in C. S. Peirce (ed.), Studies in Logic, Boston: Little, Brown & Company, 72–106. Peano, Giuseppe 1889 Arithmetices Principia, Nova Methodo Exposita, Torino: Bocca. 1901 Formulaire de mathématique, III, Torino: Bocca. Peckhaus, Volker 1998 Hugh MacColl and the German algebra of logic, Nordic Journal of Philosophical Logic, 3, 17–34. 2004 Schröder’s logic, in D. M. Gabbay & J. Woods (eds.), The Rise of Modern Logic from Leibniz to Frege, vol. 3 of Handbook of the History of Logic, Amsterdam: North-Holland (Elsevier), 557–609. Peirce, Charles S. 1849-1914 The C. S. Peirce Papers, Manuscript collection in the Houghton Library, Harvard University, USA (Photocopies at the Peirce Edition Project, Indiana University – Purdue University at Indianapolis, USA). 1869 Grounds of the validity of the laws of logic, Journal of Speculative Philosophy, 2, 193–208. 1870 Description of a notation for the logic of relatives, resulting from an ampliﬁcation of the conceptions of Boole’s calculus of logic, Memoirs of the American Academy, 9, 317–378 (reprinted in: C. J. W. Kloesel (ed.), Writings of C. S. Peirce: A Chronological Edition, 4, Bloomington – Indianapolis: Indiana University Press, 1989, 359–429). 1880 On the algebra of logic, American Journal of Mathematics, 3, 15–57 (reprinted in: C. J. W. Kloesel (ed.), Writings of C S. Peirce: A Chronological Edition, 4, Bloomington – Indianapolis: Indiana University Press, 1989, 163–209). 126 Irving H. Anellis

1885 On the algebra of logic: a contribution to the philosophy of notation, American Journal of Mathematics, 7, 180–202. 1906 Letter to Hugh MacColl, November 16; Peirce Archives, Widener Library, Harvard University, USA (Robin catalog #MS L 261). 1982 Writings of C. S. Peirce: A Chronological Edition, 1: 1857-1866, edited by M. H. Fisch, Bloomington: Indiana University Press. 1993 Writings of C. S. Peirce: A Chronological Edition, 5: 1884-1886, edited by C. J. W. Kloesel, Bloomington – Indianapolis: Indiana University Press. 2000 Writings of C. S. Peirce: A Chronological Edition, 6: 1886-1890, edited by N. Houser, Bloomington – Indianapolis: Indiana University Press. 2010 Writings of C. S. Peirce: A Chronological Edition, 8: 1890-1892, edited by N. Houser, Bloomington – Indianapolis: Indiana University Press. Peirce, Charles S. (ed.) 1883 Studies in Logic, Boston: Little, Brown & Company (reprinted with an introduction by M. H. Fisch and a preface by A. Eschbach, Amsterdam: John Benjamins Publishing Co., 1983). Post, Emil Leon 1921 Introduction to a general theory of elementary propositions, American Journal of Mathematics, 43, 163–185. Rahman, Shahid 2000 Hugh MacColl and George Boole on hypotheticals, in J. Gasser (ed.), A Boole Anthology: Recent and Classical Studies in the Logic of George Boole, Dordrecht: Kluwer Academic Publishers, 287–310. Rahman, Shahid, & Redmond, Juan 2008 Hugh MacColl and the birth of logical pluralism, in D. M. Gabbay & J. Woods (eds.), British Logic in the Nineteenth Century, vol. 4 of: Handbook of the History of Logic, Amsterdam: North-Holland (Elsevier), 533–604. Read, Stephen 1998 Hugh MacColl and the algebra of strict implication, Nordic Journal of Philosophical Logic, 3, 59–83. Rescher, Nicholas 1969 Many-Valued Logic, New York: McGraw-Hill. MacColl’s inﬂuences on Peirce and Schröder 127

Rodríguez-Consuegra, Francisco A. 1992 Review of Elizabeth Ramsden Eames’s Bertrand Russell’s Dialogue with his Contemporaries (Carbondale and Edwardsville: Southern Illinois U. P., 1989), Russell, n. s., 12 (1), 93–104. Russell, Bertrand 1901 Sur la logique des relations avec des applications à la théorie des séries, Revue de mathématiques/Rivista di Matematiche, 7, 115–148. 1903 Principles of Mathematics, London: Cambridge University Press (2nd ed., London: Allen & Unwin, 1937; 2nd American ed., New York: W.W. Norton, 1938). 1906 Review of Hugh MacColl’s Symbolic Logic and its Applications (London: Longmans, Green, & Co, 1906), Mind, n. s., 15, 255–260. 1993 Towards the “Principles of Mathematics”, 1900-1902, vol. 3 of The Collected Papers of Bertrand Russell, edited by G. H. Moore, London – New York: Routledge. Schröder, Ernst 1877 Der Operationskreis des Logikkalkuls, Leipzig: Teubner (reprinted at: Darmstadt, Wissenschaftliche Buchgesellschaft, 1966). 1890-1905 Vorlesungen über die Algebra der Logik (Exakte Logik), 3 vols., Leipzig: B. G. Teubner (reprinted at: New York: Chelsea Publishing Co., 1966). 1901 Sur une extension de l’idée d’ordre, Bibliothèque du Ier Congrès international de philosophie, Paris: A. Colin, 235–240. Shearman, Arthur Thomas 1906 The Development of Symbolic Logic: A Critical-historical Study of the Logical Calculus, London: Williams and Norgate. Simons, Peter 1998 MacColl and many-valued Logic, Nordic Journal of Philosophical Logic, 3, 85–90. Styazhkin, Nikolai Ivanovich 1969 History of Mathematical Logic from Leibniz to Peano, Cambridge, MA. – London: MIT Press. Tiles, Mary 1989 The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise, Oxford-Cambridge, MA: Blackwell (reprinted at: Mineola, N.Y.: Dover Publications, 2004). 128 Irving H. Anellis

Venn, John 1881 Symbolic Logic, 1st ed., London: Macmillan and Co. (2nd ed., 1894; reprinted at: Bronx, N.Y: Chelsea Publishing Co., 1971). Wiener, Norbert 1916 Mr. Lewis and Implication, The Journal of Philosophy, Psychology and Scientiﬁc Methods, 13, 656–662. Some arguments for propositional logic: MacColl as a philosopher J.-M. C. Chevalier Université Paris-Est (Val-de-Marne) & LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université, France

Résumé : L’article examine les raisons philosophiques, plutôt que mathé- matiques ou logiques, pour lesquelles MacColl a pu vouloir développer une logique propositionnelle. Nous trouvons des éléments de réponse dans le refus discret d’endosser une logique des choses, dans l’anti-psychologisme nuancé de MacColl, et dans une subtile épistémologie de la certitude liée à un usage méthodique de la grammaire. Abstract: The paper considers the philosophical, rather than mathematical or logical, reasons why MacColl decided to develop a propositional logic. We ﬁnd some answers in a discrete refusal to countenance a logic of things, in MacColl’s qualiﬁed anti-psychologism, and in a subtle epistemology of certainty linked to the methodical use of grammar.

Before the end of the 19th century, logic developed without any conscious need of a speciﬁc calculus of propositions. After more than twenty centuries of Aristotelian syllogistic, a formal system was eventually created whose atomic formulæ are propositional variables. Łukasiewicz emphasized Frege’s decisive role in that birth:

~~[. . . S]uddenly, without any possible historical explanation, modern propositional logic springs with almost perfect completeness into the gifted mind of Gottlob Frege. [Łukasiewicz 1967, 84]~~

It is nowadays well known that it was developed quite independently in the late 1870s by MacColl [MacColl 1877], Frege [Frege 1879] and Peirce [Peirce 1880]. As for the absence of any possible historical explanation, it has also been qualiﬁed by the study of its sources, in Bolzano and Boole in particular. It generally results in the assumption that the death knell for Aristotelian logic was given for “technical” reasons, that is chieﬂy in order to allow more convenient procedures for calculus, especially in the wake of the development of algebras of logic.

~~Philosophia Scientiæ, 15 (1), 2011, 129–148. 130 J.-M. C. Chevalier~~

The object of the present paper is to take a glimpse at some possible reasons MacColl considered for developing a logic of propositions without either an impulse from mathematics or by reason of logical requirements proper, but rather on a philosophical basis. “Philosophical” is here meant to include epistemology, metaphysics and psychology. Can MacColl seriously be held not only as a great logician but also as a great philosopher of logic? 1 To be sure, he advocated his calculus of equivalent statements as a mere mathematical tool for problems such as handling the limits of multiple integers. Nevertheless, given that MacColl aspired to produce essays “of a real philosophic depth” [MacColl 1910, 186], one would expect his reasons to be more profound than mere technical skills to propel the revolution of modern logic in which he took part. Propositional logic is not only a new calculus or a mere study of logical operators, it supposes a new, truth-preserving semantics, a concept of grammar, a clariﬁcation of such fundamental notions as inference and substitution, more generally a new approach of the mental, and perhaps even a philosophy of space and time. The search for such a proper philosophical rationale in MacColl’s writings is the aim of the present paper.

~~MacColl’s propositional calculus~~

Before investigating the philosophical motivations, if any, for MacColl’s logic of propositions, one must clarify its role and status in his general system of logic. The reﬂections about MacColl’s propositional logic, during his lifetime and ever since, have sometimes failed to distinguish various senses of propositional logic. The most obvious one is a logic of propositions as opposed to a logic of terms. It maps the distinction between classes (of things or events) and truth-value bearers. Propositional logic thus presents a more suitable alternative to Peripatetic logic. 2 But it can also refer to the axiomatic formalization of the propositional variables in terms of inference rules and truth-functions (Frege, Russell) as opposed to an implicit interpretation, called by Alonzo Church “the older point of view” (whose adherents included MacColl, Peirce, Schröder and to a certain extent Peano) [Church 1956, 157]. The axiomatic propositional calculus then opposes syllogistic logic with its intuitive interpretation. Furthermore, to the supporters of an algebra of logic, propositional logic can designate the precedence of equivalence over equality and the central position of the implication relation. It thus stands in contrast with the algebraic syntax of Boolean logic. 3

1. At least John R. Spencer recognized MacColl as a philosopher of logic in [Spencer 1973]. 2. Roughly speaking, propositional logic opposes Aristotelian logic, though the former corresponds more or less to the hypothetical syllogism in Aristotelian and medieval logic [Castrillo 1994]. 3. Not to mention that a logic of propositions diﬀers from a logic of judgments, the former only belonging to formal logic (cf. Bolzano). Some arguments for propositional logic: MacColl as a philosopher 131

One can argue whether or not MacColl had a propositional logic according not only to the previous contrasts, but to the importance given to the following criteria. There is first a question of chronological priority, which is no doubt minor and need not seriously distract historians. More interesting is the foun- dational problem of logical priority: does propositional logic provide a basis for a logic of classes, or does it go the other way round? The problem is that such a question is biased: looking for a foundation, one is likely to discover logicism as a solution which denies the logical primacy of propositions over sets. Hence, a third question is to be asked: is logic only a convenient tool, or does it reveal some truth about reality? For instance, does a logic based on events require an ontology of events? Does a logic of propositions need a philosophy of propositions? In which of the previous senses did MacColl develop a propositional calculus? Whatever be the exact chronology of the articles published, he thought of himself as a pioneer: Alone, or nearly so, among logicians, I have always held the opinion, and my recent studies have confirmed it, that the simplest and the most effective system of Symbolic Logic is that whose elementary constituent symbols denote—not classes, not properties, not numbers, ratios, regions, or magnitudes, not things of any kind—but complete statements. [MacColl 1899, 108–109] In writing that he was nearly alone, MacColl may have had in mind his colleague Peirce, with whom he was long acquainted, and who, without mention- ing Frege’s Begriffsschrift, attributed to himself and MacColl alone the idea of making “the elements propositions or predicates and not common nouns” [Peirce 1906]. But in including predicates in his list, Peirce clearly misses part of the import of MacColl’s innovation. 4 Such seems to be Couturat’s judgment: in presenting the principal systems of symbolic logic (including those of Jevons, Peirce, MacColl and Peano) as quite dissimilar, he makes that of MacColl alone consist “in considering propositions alone as the elements of reasoning, and in assigning to them three distinct values” [Couturat & Ladd- Franklin 1902, 645]. However, MacColl’s and Peirce’s systems share an important feature: they ground the whole of logic on propositional calculus. If an interpretation of MacColl’s formal system in terms of classes is still possible, the calculus of statements is more basic. As for Peirce’s alpha graphs, which are at least “isomorphic to the theory of propositional logic” [Pietarinen 2006, 348] in representing Peirce’s “calculus of first intention,” they are prior to first-order predicate logic (represented by beta graphs): it proves their fundamental nature. It is noteworthy that Schröder reproached such “MacColl-Peirce’schen

4. Nevertheless, MacColl does not make a big distinction between the predicative B form (A is B) and the adjectival form (the A which is B); that is, AB assumes A [MacColl 1897, 495]. 132 J.-M. C. Chevalier

Aussagenlogik” for putting the cart before the horse, since for him propositional calculus is viewed as a specialization of his calculus of domains [Peckhaus 1998, 29]. It calls attention to the fact that, like Peirce, MacColl played a role in the birth of modern propositional logic not only because of its systematic development, but in constantly arguing for its fundamental place at the base of logic. Indeed, the rather solitary position of MacColl does not mean that other logicians ignored the calculus of propositions, but often rather that they explicitly rejected its relevance. 5

~~A mere emphasis?~~

MacColl often affirms that letters should only represent statements. As early as 1878, he insists that with him “every single letter, as well as every combination of letters, always denotes a statement” [MacColl 1878, 27]. This early position nevertheless needs to be qualified in order to respond to some obvious objections: symbols (and especially letters) often stand for numbers, ratios, subjects or predicates. That letters represent propositions should not be a “cast-iron system of notation” [MacColl 1907a, 472]. Hence, it could appear that MacColl’s originality mainly rests upon the emphasis he repeatedly placed upon the need for a propositional interpretation, among others, of the algebraic calculus. In other words, his logic of propositions would be nothing but a defense of pluralism. What is more, even in his early years MacColl still had an ambiguous position vis-à-vis propositional logic. Though he claims that the ultimate elements of his system are statements, he nevertheless treats inference as holding between sentences or more atomic terms. After noticing that the relation between subject and predicate is analogous to the relation between antecedent and consequent, he indeed analyses a syllogism as a complex implication whose premises are expressed in an implicational form. It maintains a confusion in identifying “the ‘physical’ combination of propositions into a system with the ‘chemical’ combination of subject and predication into a proposition” [Shearman 1906, 11]. In short, in 1880, inference is said to take place between statements, but is used between a subject and a predicate (or more precisely, between two attributions of predicates). It shows that MacColl did not from the start have a clear conception of the difference between first-order and “zeroth-order” logic.

~~A parting from Boolean algebra?~~

MacColl’s life shows a deep striking break: although he never stopped (re)publishing mathematical puzzles for The Educational Times and meanwhile produced two books of literature, he put his logical work to rest between 1883 or so and 1896. He referred to this temporary abdication from logic as

5. See [Ladd-Franklin 1883], [Mitchell 1883], [Schröder 1891], [Peano 1896-1897], [Shearman 1905]. For an opposite position, see [Johnson 1892]. Some arguments for propositional logic: MacColl as a philosopher 133 a long pause in his mental life. MacColl himself stressed the difference of his new, 20th century views from his 1877-1883 period: [T]he principle which underlies my method. . . did not appear (in my earlier papers) to lead to any essential difference in the symbolic processes. That this is no longer the case my recent papers in MIND and in the Proceedings of the Mathematical Society will show; but the new development is still further removed than the old from the allied algebras which it has been the great aim of Mr. Whitehead to unite into one general comprehensive system. [MacColl 1899, 109] If his first system was accidentally analogous to Boole’s logic, his second is deliberately free from any Boolean influence. As compared with the other systems, the outward “resemblances of mere form hide important differences in matter, method, and limits of application” [MacColl 1903, 364]. Propositional logic, MacColl claims, is not equivalent to equational logic, even if the al- gebraists “found that, in all the numerous cases they had examined, the for- mulæ of the logic of Pure Statements had their exact analogues in the logic of Concrete Quantity, and they inferred, very naturally but quite erroneously, that this must also hold in the infinite number of cases which they had not examined” [MacColl 1899, 111]. Jevons is one of those who argued for equations against propositions and explicitly criticized MacColl’s choice: Even his letter-terms differ in meaning from mine, since his letters denote propositions, not things. Thus A B asserts that the statement A implies the statement B, or that∶ whenever A is true, B is also true. . . It is difficult to believe that there is any advantage in these innovations. [Jevons 1880, xv] MacColl intends to replace the syntactic operator of equality with the grammatical notion of synonymy within language: I think I may predict that synonyms are destined to play an important part in the future development of symbolic logic. [MacColl 1897, 500] Two words, two propositions are not synonymous for all eternity, because signification depends on context: As new needs and new ideas arise with the growth of civilisation and the general advance of humanity, do we not often find that words which were at first synonyms gradually differentiate and, while still remaining synonymous in some combinations, cease to be so in others? [MacColl 1897, 500] That is why the relation of equality has too much strength and fixity. MacColl’s recurrent principle, “perhaps the most important principle underlying [his] system of notation” [MacColl 1902, 362], that there is “nothing sacred or eternal about symbols” [MacColl 1906a, 1], can even be interpreted in the sense that there are no fundamental relations in logic: 134 J.-M. C. Chevalier

I see no reason why we should treat our symbols of relation, , , =, indices, fractional forms, etc., with more respect than we+ ∶ do the ordinary letters of the alphabet. [MacColl 1899, 113] Does he criticize simply our use of the graphical marks, or of the relations themselves? Perhaps the second alternative, for “as even the so-called ﬁxed stars are only found to be relatively ﬁxed, so our so-called constant symbols of relation need only be relatively constant” [MacColl 1897, 495]. The syntactic framework of algebra is too narrow for such a semantics.

~~A rejection of terms and things?~~

Jevons claims that his letters denote things, not propositions. It is that for which MacColl reproaches Boole. Boole is indeed mainly concerned with terms, which, treated extensionally, can be reduced to a logic of classes. 6 MacColl probably did not want to rid logic of classes, which would be nonsense. Perhaps it would be more exact to say that in emphasizing the role of propositions, he in fact aspired to create a calculus of statements that would transcend the traditional partition of various logics. MacColl’s propositional logic possesses the advantage of homogeneity of matter; it “is one simple homogeneous system which comprises (either directly or as easy deductions), all the valid formulæ of their two divisions,” namely, “the logic of class inclusion and the logic of propositions” (as well as many other valid formulæ) [MacColl 1903, 355]. To a large extent, this opposition has been viewed as the necessary choice between placing either inclusion or implication at the heart of logic. Peirce is notoriously guilty of a careless shift of his “scorpion tail” operator from implication to inclusion, for he turned out to consider that transitivity always depends upon inclusion [Peirce 1897]. 7 MacColl was aware of such a conﬂation of class inclusion with implication, reproaching some logicians for “us[ing] their symbol in one sense in their logic of class inclusion, and in quite a diﬀerent

6. Nevertheless, Boole writes: Instead of classes of things, we shall have to substitute propositions, and for the relations of classes and individuals, we shall have to consider the connexions of propositions or of events. [Boole 1854, 162] Boole was allegedly about to develop a viable logic of propositions in his early Mathematical Analysis of Logic, though the project remained uncompleted, and was replaced by another approach in the later Laws of Thought [Hailperin 1984]. On the other hand, MacColl sometimes tends to minimize the diﬀerence between class and statement: he acknowledges that AB is a statement while AB should be viewed as a class, inviting us not to give so much importance to this( diﬀerence.) In conclusion, the status of sentential logic as compared to class logic can be read as rather vacillating and not too strict. 7. At least Peirce intended to subject this single operator to multiple interpretations such as the copula, class inclusion or material implication. For a thorough study of this issue, see [Anellis 1995]. Some arguments for propositional logic: MacColl as a philosopher 135 sense in their logic of propositions” [MacColl 1903, 356]. He insists that not only is this operator unnatural as a sign, but it “is not (virtually) equivalent to A B, i. e. the statement A implies the statement B” [MacColl 1906a, 78]. 8 ∶ As a consequence of replacing things by propositions, MacColl insists that “though this system does not necessarily exclude metaphysical considerations,” it is theoretically and practically independent of such considerations and is all the more reliable for that reason [MacColl 1880, 58]. It would probably be far- fetched to argue that MacColl wanted to get rid of the ontology of substance, as the pragmaticist maxim would do: Peirce’s logic of propositions no doubt has deeper philosophical roots. But at least, no ontology is required by pure logic. Thus, one only has to investigate the relations between signs:

[. . . ] we should, in my opinion, ﬁrst investigate the mutual relations of these statements, representing each by its own independent symbol, and call this process of investigation Pure Logic. [MacColl 1899, 109]

“The logic of functions or relations” is mentioned in MacColl’s fourth “Symbolic reasoning” article in 1902. He there writes that the logic of relations does not belong to the “logic of statements” but is a development of it in a particular direction, exactly in the same way as the theory of functions is a particular development of pure mathematics.

~~For this reason, in order to mark the analogy, the logic of relations should rather be called the logic of functions. [MacColl 1902, 362]~~

If MacColl does not consider the function-theoretical approach as pure logic, it is because he fails to see the interest of a logic of relations, and to grasp the concept of a function that would not depend on the theory of language. Hence the failure to give to propositional functions the place they deserve in logic. 9

~~8. For instance η : ε is a formal certainty, η ε a formal impossibility. Another example is that (A : B) (B : C) : (A : C) makes sense, in contrast to (A B) (B C) (A C). 9. Indeed, MacColl wrote:~~

What Mr. Russell calls a propositional function, I should prefer calling a functional proposition; but whatever locution we adopt, it must, from my point of view, be classed as a proposition. [MacColl 1907a, 470]

~~Shearman criticised this position as follows:~~

Mr. MacColl allows his symbols indiscriminately to represent propositions and propositional functions; but, in so far as he has done so, he has not assisted in producing the newer Symbolic Logic, for in this it is a matter of fundamental importance to draw a clear line of distinction between the two uses. [Shearman 1906, 151–152]. 136 J.-M. C. Chevalier An investigation of the laws of mind

~~The laws of thought~~

One of the reasons for adopting a logic of propositions rather than of predicates paradoxically seems to lie in a certain conception of the mental. Of course MacColl defends an unpsychological view of logic. He argues “that formal logic should not be mixed up with psychology—that its formulæ are independent of the varying mental attitudes of individuals” [MacColl 1906a, 60]. And his formal system also shows evidences of anti-psychologism (for instance in embracing the paradoxes of strict implication). Meanwhile, MacColl is in search of the “fundamental rules of our ordinary reasoning” [MacColl 1880, 48], and deﬁnes (symbolic) logic as “the science of reasoning by the aid of representative symbols” [MacColl 1897, 493]. But surprisingly, when one remembers the psychological quest for man’s intellectual powers which motivated Boole’s logical works, the “second” MacColl, while harshly discarding the attempts of the Booleans to perfect an algebra of logic, extols Boole’s philosophical insights. No one can admire Boole’s Laws of Thought more than I do. As a philosophical and speculative work it is brimful of profound thought and original suggestions, while its style is charmingly lucid and attractive. [MacColl 1897, 504] MacColl claims Boole’s application of algebra to probability did not succeed, though his attempt has a philosophical interest. One can compare Boole and Shakespeare: Both authors possessed a remarkable analytical insight into the workings of the human mind; the one of its secret motives and passions; the other of the subtle laws of its intellectual operations; yet both—the one judged by his plays, the other by his Laws of Thought—showed but little constructive ability. [MacColl 1897, 505] Boole’s mistake was to endeavour to make the multiplicity of human thought ﬁt into the given rules of algebra, “to squeeze all reasoning into the old cast- iron formulæ constructed specially for numbers and quantities” [MacColl 1897, 505]. Boole indeed alleges that there is an exact adequacy in the laws which conduct the operations of reasoning and of algebra [Boole 1854, 6]. In other words, the actual reasoning does not bear on symbols, be they mathematical or, as we will see, thought-signs; but the major fact Boole clearly understood is that it relies on transitiveness. Paradoxically, while dismissing psychologism, MacColl grants that an inquiry in the depths of logic nevertheless requires an acknowledgement of our mental processes—of Peirce’s so-called “anthropological facts” which “have a great bearing upon logic” [Peirce 1982, 362]: Some arguments for propositional logic: MacColl as a philosopher 137

Though in purely formal or symbolic logic it is generally best to avoid, when possible, all psychological considerations, yet these cannot be wholly thrust aside when we come to the close discussion of ﬁrst principles, and of the exact meanings of the terms we use. [MacColl 1906a, 82] It goes with MacColl’s interest for more psychical and psychological matters at the end of his life. It would be erroneous to believe, MacColl writes, “that formal logic is absolutely independent of psychology—that they have nothing to do with each other”, for mind is a “pre-condition” of a working logical or mathematical formula, be it human or “superhuman” [MacColl 1910, 191]— a kind of “cosmical reason,” for Peirce. Such an apparent contradiction in MacColl is due both to a chronological evolution and, from the start, to his qualiﬁed and subtle anti-psychologism.

~~Inference vs. implication~~

Contrary to what the Russellian posterity could make one expect, the debate over propositional logic did not focus on the inference vs. inclusion opposition as much as on an inference vs. implication contrast at the time MacColl wrote. The priority of the concept is a postulate that separates Boole from some of his most important followers [Vassallo 1995, 89–91]. The former considered that a logic of terms should be first because the analysis of the laws which govern the conception should precede the analysis of those governing judgment and reasoning. It corresponds to the first step of the threefold division of logic that has been in use ever since Aquinas at least, namely a doctrine of terms related to the concept, a doctrine of propositions related to judgment, and a doctrine of inferences or syllogism related to reasoning. Without going into MacColl’s technical notion of implication, 10 suffice it to recall his “law of implication,” according to which “the sole function of the reason is to evolve fresh knowledge from the antecedent knowledge already laid up in the store-house of the memory” [MacColl 1880, 52]. It echoes Peirce’s 1864 assumption that every judgment is an inference, since it relates something known to the unknown—save for the Kantian “I think” [Peirce 1982, 152]. Nevertheless, many logicians insisted that in contrast to ampliative reasoning, deduction does not lead from a known truth to a truth unknown, because its conclusion is a mere clarification of what was already admitted in the premises [Peirce 1982, 152]. Yet, MacColl paraphrases John Stuart Mill in writing that

10. In particular the distinction between logical and causal implication, the cases in β which α+β′ are not synonymous with α , the paradoxes of material implication, etc. Neither will we discuss the existential import of propositions, that is, the diﬀerence between A ∶ B and A ∴ B, which justiﬁes the conviction that “Not one syllogism of the traditional logic—neither Darapti, nor Barbara, nor any other—is valid in the form in which it is usually presented in our text-books, and in which, I believe, it has been always presented ever since the time of Aristotle” [MacColl 1906a, 47]. 138 J.-M. C. Chevalier

“[w]e must reason from the known to the unknown whatever be the subject of investigation” [MacColl 1880, 58]. That is probably why thirty years later he would emphasize that implication is not inference, for “[a]s commonly understood, inference involves psychological considerations; implication does not” [MacColl 1910, 191]. Inference is about obtaining knowledge, whereas implication refers to the impossibility that a certain premise does not lead to a certain conclusion. It is noteworthy that, to him, symbolization reflects the inferential process, not the actual reasoning. Writing with symbols (and the logical relation of the truth of premisses to the truth of conclusion) does not accurately represent the process of implication. The words if and therefore are examples of terms for which we need psychological considerations [MacColl 1906a, 82]. And of course “it is convenient in scientific researches to consider the two as far as possible apart” [MacColl 1910, 191]. Inference is a temporal relation. Thus, contrary to Boole, MacColl refuses to view logic as concerned with temporal portions. It is a fundamental reason why MacColl denies that Boole developed a propositional logic or any calculus of pure statements. Its elementary constituents indeed never represent statements, “they always represent the fractions of time (referred to some understood arbitrary whole unit) during which the various statements in question are true; and it is quite clear from Boole’s language that he considered this convention as a necessary and fundamental principle” [MacColl 1899, 111]. The Laws of Thought indeed vacillates between two interpretations of signs, either representative of things or of operations of the human intellect. It in both cases should refer to the laws of reasoning holding for the “time for which the proposition is true.” Thus, Boole’s embryonic propositional logic is based on time [Godart-Wendling 2000]. It is reminiscent of the Stoics, who, though they developed a kind of propositional logic in a very different sense from what we mean, nevertheless did not ground their logic on the Aristotelian apophansis (the subject-predicate structure) but on the axioma, that is, a proposition stating events. Therefore, whereas Aristotelian reasoning is a system of inclusion of concepts, reasoning for Chrysippus bears on implications of temporal relations. MacColl’s rejection of time as a fundamental conception of logic shows the modernity of his notion of proposition. As a comparison, Peirce puts it that “a succession of time among ideas is obviously presupposed in the conception of a logical mind” [Peirce 1986, 106]. It goes with the Stoic idea that “[t]he letters must denote the objects of logical thought; that is facts or supposed facts” [Peirce 1866].

~~Logic and mathematics~~

Neither is logic concerned with parts of space. Not only is the calculus of pure statements not a set of mathematical operations, but it differs from Some arguments for propositional logic: MacColl as a philosopher 139 algebra as much as algebra differs from geometry [MacColl 1899, 111]. For it is an error to believe that: [T]o every valid formula in Symbolic Logic, no matter how abstract the region of thought, and no matter what the elementary symbols may be defined as representing, there always corresponds an analogous formula (symbolically identical) of which the elementary symbols represent spatial magnitudes; so that any argument referring to the non-spatial region of thought has its symbolic double (so to speak) in the spatial region, and vice versâ [MacColl 1899, 110–111]. MacColl’s assertion very nearly rejects Peirce’s diagrammatical approach to reasoning. Statements, MacColl claims, have sui generis characters, which cannot be reduced to mathematical properties. He defends a linguistic conception of logic, completely at odds with Peirce’s system of existential graphs, which “is essentially different from language” [Peirce 1910] and “expresses everything with a precision that no human tongue can approach” [Peirce 1905]. On the contrary, MacColl assumes that “the great advantage of this ‘Calculus of Limits’ is that it is independent of all diagrams” [MacColl 1906a, 140]. In short, MacColl views thought as conveyed by propositional entities that have no proper duration nor any situation or location. Having refused to ground logic in time or space, MacColl cannot say that logic is a branch either of arithmetic or of geometry. Propositional logic is not mathematical, as Peirce would perhaps have held when stating that mathematics is logic rather than that logic is mathematics [Peirce 1897]. 11 MacColl refuses to trust some more or less superficial or fancied analogies between logic and mathematics. The “law of symbolic coincidence” between pure (that is, propositional) logic and algebra should no longer be relied on. But on the other hand, MacColl does not draw such a sharp line between his mathematical puzzles and logic, not afraid, as was Peirce, that logic “is in danger of degenerating into a mathematical recreation” [Peirce 1883, 143]. MacColl’s main interest indeed lies in what Peirce calls the logica utens, that is, logic as a tool, especially for problems of probability and limits. Peirce draws the attention of his fellow logician to this difference: [. . . ] my studies in symbolic logic have differed from yours in that my aim has not been to apply the system to the working out of problems, as yours has, but to aid in the study of logic itself. [Peirce 1906] MacColl in fact states that the question of the primacy of logic or of mathematics over the other is pointless, for they do not have the same domain: mathematics deals with things, logic with signs. The question of which science subsumes which is meaningless, argues MacColl, who “would like to contribute

11. We of course do not intend to discuss Peirce’s relation to logicism. On this matter, see e.g. [Houser 1993]. 140 J.-M. C. Chevalier his humble share as a peacemaker between the two sciences, both of which he profoundly respects and admires” [MacColl 1880, 47].

~~An investigation of the laws of signiﬁcation~~

~~A logic of statements or of propositions?~~

Whereas mathematics works on things and human thought processes ideas, symbolic logic deals with signs. MacColl has no inclination for mental symbolics, for “[i]n mental reasoning we dispense entirely with symbols” [MacColl 1880, 48]. It goes with an explicit rejection [MacColl 1910, 350] of William James’s thesis that “the thoughts themselves are the thinkers” [James 1892, 83], and also boldly contradicts Peirce’s famous thesis that “every species of actual cognition is of the nature of a sign” [Peirce 1986, 76]. MacColl needs a definition of the basic symbol, namely the statement. To this end, a basic theory of signs and language founded on natural evolution is developed. He does not precisely define a sign, and seems to identify it with what he calls a symbol. The examples he unvaryingly gives are sounds like “the ‘caw’ of a sentinel rook” or visual symbols like “the national flag of a passing ship.” When a sign is likely to bear a meaning, it can be interpreted as a statement. A statement is what Peirce would simply call a sign, the indication of intelligible information or of a possible meaning for the logician. What meaning do they convey? What does the “Caw” of the sentinel rook perched on the branch of a commanding tree say to the others on the ground busily feeding on the farmer’s property? To one of these it may say, “A man is coming with a gun”; to another it may say, “A boy is coming with a catapult”; to all it says, “Danger approaches,” though their respective ideas as to the precise danger may be vague and varied. [MacColl 1902, 365] In other words, every single interpretation of a given sign describes a particular “immediate object,” as Peirce calls it; immediate objects are fated to converge into the final, real object. The rook’s “caw” refers to a particular fact, but its meaning is indeterminate, that is, vague and general. That is why it would be a mistake to reserve the name “propositions” to those which convey a real, determinate meaning, as opposed to propositional forms [MacColl 1910, 192]. As a statement is any sign which communicates information to ear or eye, it need not have a specific form. A statement which, in regard to form, has such a subject-predicate structure, is called a proposition. 12 The threefold partition of MacColl’s classification of propositions puzzled many logicians:

12. In fact, MacColl’s use is hardly rigorous. Furthermore, it gets blurred by the author’s bilingualism: under MacColl’s pen, the French word “proposition” often stands for the English “statement”. Some arguments for propositional logic: MacColl as a philosopher 141 how can a proposition be a variable, that is, sometimes true and sometimes false? Here is MacColl’s reply: I do not say that the same information may be sometimes true and sometimes false, nor that the same judgment may be sometimes true and sometimes false; I only say that the same proposition— the same form of words—is sometimes true and sometimes false. [MacColl 1907a, 470] Thus, a proposition is simply a conventional arrangement of words or other symbols employed to convey information or express a judgment. In this respect, MacColl disagrees with Russell, who “does not consider a propositional form of words, such as ‘He is a doctor,’ a real proposition till it is actually employed to give information” [MacColl 1908, 453]. Nevertheless, the apparent simplicity of the concept of proposition still hides a major diﬃculty. For if in pure logic statements are represented by single letters, then a pure statement is, for instance, S, or P , that is, a subject (e.g. the national ﬂag), or a predicate (e.g. being blue, white and red). In consequence, the logic based on a letter standing for “S is P ” is not pure, although it is what we mean by propositional logic. In the same spirit, while MacColl asserts that a pure proposition is represented by a single letter, that is, is an indecomposable, atomic sign, he also explains that “AB is a pure proposition,” as opposed to being a modal proposition [MacColl 1906a, 94]. In other words, included among the statements are the terms of a proposition, so that the calculus of equivalent statements should be both a logic of classes and a logic of propositions so as to exclude the possibility of the calculus dealing with classes. MacColl’s logical pluralism turns out to be rather embarrassing.

~~Logical classiﬁcation and grammatical clariﬁcation~~

For MacColl, the task of logic is to classify statements according to attributes as true, false, certain, impossible, variable. Pure logic is just a classification of propositions: [W]hatever be the subject of research, in order to articulate itself all reasoning requires propositions. Hence, in order to make our reasoning perfectly general and our formulæ universally applicable, the first goal of our research must be the classification of the different kinds of propositions and the rela- tionships between them, and we have to call this work Pure Logic. [MacColl 1901, 135] 13 MacColl writes that the first step in his classification is to specify the difference between A and “A is true” [MacColl 1906b, 513]: Aτ is of a “higher degree” than

~~13. Quoted from the English translation in [Astroh 1998, 147]. 142 J.-M. C. Chevalier~~

A [MacColl 1906a, 17]. 14 To be true, truth values are not always clearly distinguished from modalities, predicates and sets of propositions. For instance, ε is both a modal predicate of necessity and a class statement, namely the class of all the necessary propositions. MacColl nevertheless lays the foundations for a hierarchical typology of propositions based on their order of predication (AB being a pure proposition, ABC a modal proposition of the first order, ABCD of the second order, etc.). He even lays the basis of an epistemic logic when suggesting that his three- dimensional system of logic (including the values of necessity, impossibility and uncertainty) could be completed with a “new and more subjective scheme” [MacColl 1897, 508]. Such values as “known to be true,” “known to be false” and “doubtful” are intended to better fit the ordinary facts of our consciousness and experience. Unfortunately, MacColl refuses to consider it seriously because he fears they might raise psychological questions. He would later even consider that the modality of certainty itself has an epistemic value: Aτ is said to be a revision and confirmation of A [MacColl 1906b, 515], which seems to explain logical necessity in terms of revision of beliefs, a dangerous method. The division of statements into certainties, impossibilities and variables should be “wholly independent of psychology” [MacColl 1907a, 473], though logic should indeed be concerned with ideas of certainty, not only with the true/false dichotomy. “All such attempts to surround symbolic logic by a Chinese wall of exclusion are futile” [MacColl 1906b, 515]. This difficulty is the reason why A. T. Shearman assumes that MacColl in fact quits the domain of pure logic when developing his modal logic: Pure Logic can take account of the uncertainties that such data occasion, but the propositions dealt with will then denote not the relation of the respective letters to x, but the relation of the thinker to each implication. [Shearman 1906, 30] MacColl’s originality in this discussion derives from the fact that he does not regard the enterprise of clarification as logical but linguistic. As evidence, the difference between “it rains” and “it is true that it rains” is present within our natural tongues. Grammar is concerned with the relations within a proposition, logic with the relations between propositions: that is why a logic of predicates is not a real logic. Logic could focus on the links between propositions rather than on the content of every proposition. The clarification of ordinary language through grammar is thus requisite for propositional logic. Grammar in this sense is not reduced to the meaning of words in a sentence but implies a whole context: The mere word, and sometimes the complete grammatical proposition, when considered apart from context, may be meaningless or misleading. [MacColl 1907b, 162]

14. It corrects the previous assertion that “[t]he symbol A is short for Aτ , and may be read ‘A is true’; the exponent τ being often left understood, like the sign + in common algebra” [MacColl 1900, 75]. Some arguments for propositional logic: MacColl as a philosopher 143

MacColl’s philosophical insights are modest. His defense of propositional logic remains ambiguous: his attempt at reduction to one formalism conﬂicts with his hatred for dogmatic symbolization; his claim that symbols are only a question of economy [MacColl 1897, 493] 15 contradicts the fundamental role of propositions; and his mathematical use of logic does not accord with the rejection of an algebraic syntax. Perhaps his best achievement does not consist so much in his logical reduction to a symbolic system of propositions as in his conviction that propositions are the true logical atoms—which would open the road to the traditional problems of logical atomism, such as the unity, ob- jectivity and bipolarity of propositions. Such a view can nevertheless be said to have philosophical motivations: a discrete refusal to countenance a “metaphysical” logic of things, his qualiﬁed anti-psychologism (nonetheless refusing, like Frege and contrary to Peirce, to ground logic in the forms of the understanding), and a subtle epistemology of certainty linked to the methodical use of grammar, are the basic traits of MacColl’s “philosophy” of logic.

~~Acknowledgments~~

I am most grateful to Amirouche Mokteﬁ for kindly providing documen- tation and advice, and also wish to express my thanks to my two anonymous referees, being very indebted in particular to the minute reading of the one who made so many tremendously accurate corrections and inspiring suggestions.

~~References~~

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15. A weak argument for propositions, for one can easily argue that a logic of classes is more convenient. Sherman, for instance, writes: As regards the question of economy of space in the solution of problems, the evidence seems to show that the class interpretation is to be preferred. [Shearman 1906, 21]. 144 J.-M. C. Chevalier

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~~Some remarks on Hugh MacColl’s notion of symbolic existence Shahid Rahman STL (UMR 8163), CNRS-Université Lille 3, France~~

Résumé : L’approche la plus influente de la logique des non-existences est celle provenant de la tradition Frege-Russell. L’un des plus importants dissidents à cette tradition, à ses débuts, était Hugh MacColl. C’est en relation avec la notion d’existence et avec les arguments impliquant des fictions, que le travail de MacColl montre une grande différence avec celui de ses contemporains. En effet, MacColl fut le premier à implémenter dans un système formel l’idée qu’introduire des fictions dans le domaine de la logique revient à fournir un langage muni de sous-domaines avec différents types d’objets. Dans cet article, nous avançons quelques remarques sur la portée de la logique de MacColl sur les non-existences. Plus précisément, nous suggérons qu’il y a un lien concep- tuel fort entre la notion de subsistance chez Russell et la notion d’existence symbolique chez MacColl. Abstract: The most influential approach to the logic of non-existents is certainly the one stemming from the Frege-Russell tradition. One of the most important early dissidents to that tradition was Hugh MacColl. It is in relation to the notions of existence and arguments involving fictions that MacColl’s work shows a deep difference from the work of his contemporaries. Indeed, MacColl was the first to attempt to implement in a formal system the idea that to introduce fictions in the context of logic amounts to providing a many- sorted language. The main aim of the paper is to add some brief remarks that should complete the scope of MacColl’s logic of non-existence. More precisely, I will suggest that there seems to be a strong conceptual link between Russell’s notion of subsistence and MacColl’s notion of symbolic existence.

~~Introduction~~

The most inﬂuential approach to the logic of non-existents is certainly the one stemming from the Frege-Russell tradition. The main idea is relatively simple and yet somehow disappointing, to reason with ﬁctions is to reason with propositions which are either (trivially) true, because with them, on

~~Philosophia Scientiæ, 15 (1), 2011, 149–162. 150 Shahid Rahman~~

Russell’s view, we deny the existence of these very fictions, or otherwise they are (according to Russell) false or (according to Frege) lack truth-value in the same trivial way. One of the most important early dissidents to that tradition was Hugh MacColl. 1 It is in regard to the notions of existence and arguments involving fictions that MacColl’s work shows a deep difference from the formal work of his contemporaries. Indeed, MacColl was the first to attempt to implement in a formal system the idea that to introduce fictions in the context of logic amounts to providing a many-sorted language. Interesting is the relation between Bertrand Russell’s criticisms of Alexius Meinong’s work and Russell’s discussions with MacColl on existence. Recent scholars of Meinong such as Rudolph Haller [Haller 1972] and Johan Marek [Marek 2003] and modal Meinongians such as Graham Priest [Priest 2005], Richard Routley [Routley 1980] and Edward Zalta [Zalta 1988] make the point that Russell’s Meinong is not Meinong. An interesting historical question is to study how Russell’s criticism of Meinong in their debates from 1904 to 1920 could have been influenced by Russell’s discussion with MacColl. Notice that the main papers on this subject by Russell, Meinong and MacColl, were published between 1901 and 1905. MacColl’s work on non-existents resulted from his reaction to one lively subject of discussion of the 19th century, namely the existential import of propositions. This topic was related to the traditional question about the ontological engagement or not of the copula that links subject and predicate in a judgement. J. S. Mill [Mill 1843] introduced to the discussion the work of Franz Brentano who published in 1874 his theory on the existential import of the copula and on how to define away the alleged predicate of existence [Brentano 1874, chap. 7]. However, most of the British traditional logicians did not follow Brentano and the opposition between them and the “Booleans”, who also charged the copula with existential import, triggered a host of papers on that subject. 2 The early Russell of the Principles and MacColl defended the idea that there is a real and a symbolic existence, that seems to be close to Russell’s use of subsistence—a notion that Russell rejects later on, namely, in his notorious “On denoting” [Russell 1905b]. 3 MacColl’s example, probably borrowed from Mill, targeted the meaning of the copula “is” in expressions such as “the non-existent is non-existent”. Unfortunately, the example hinges on the ambiguity of the copula as identity and as predicative expression. Nevertheless, MacColl’s development is—though sometimes puzzling—exciting and could be seen as providing the semantic basis for what nowadays we call free logic.

1. For an overview of MacColl contributions to logic, see [Rahman & Redmond 2008]. Most of MacColl’s logical writings, including his 1906 book, are collected and reprinted in [Rahman & Redmond 2007]. 2. [Land 1876] spelled out the position of the traditionalists and triggered the discussions in Mind on the existential import of propositions. 3. For a thorough discussion of Russell’s grounds for rejecting his early endorsement of Meinong’s theory of objects, see [Farrell Smith 1985]. Some remarks on Hugh MacColl’s notion of symbolic existence 151 MacColl’s Logic of Non-Existence

~~MacColl’s logic of non-existence is based on a two-fold ontology and one domain of quantiﬁcation, namely: – The class of existents – MacColl calls them reals:~~

Let e1, e2, e3, etc. (up to any number of individuals mentioned in our argument or investigation) denote our universe of real existences. [MacColl 1905a, 74] [T]hese are the class of individuals which, in the given circumstances, have a real existence. [MacColl 1906, 42] – The class of non-existents:

Let 01, 02, 03, etc., denote our universe of non-existences, that is to say, of unrealities, such as centaurs, nectar, ambrosia, fairies, with self-contradictions, such as round squares, square circles, flat spheres, etc., including, I fear, the non-Euclidean geometry of four dimensions and other hyperspatial geometries. [MacColl 1905a, 74] [T]he class of individuals which, in the given circumstances, have not a real existence [. . . ] It does not exist really, though (like everything else named), it exists symbolically. [MacColl 1906, 42] In no case, however, in fixing the limits of the class e, must the context or given circumstances be overlooked. [MacColl 1906, 43] – And the domain of quantification, the Universe of discourse, containing the two preceding classes:

Finally, let S1, S2, S3, etc., denote our Symbolic Universe, or “Universe of Discourse”, composed of all things real or unreal that are named or expressed by words or other symbols in our argument or investigation. [MacColl 1905a, 74] As expected, individuals, that are elements of the Universe of discourse, might be elements of the ﬁrst two classes:

We may sum up brieﬂy as follows: Firstly, when any symbol A denotes an individual; then any intelligible statement φ A , containing the symbol A, implies that the individual represented( ) by A has a symbolic existence; but whether the statement φ A implies that the individual represented by A has real existence( ) depends upon the context. [MacColl 1905a, 77]

~~Moreover, predicates might be interpreted by the means of classes containing reals, unreals or both of them: 152 Shahid Rahman~~

Secondly, when any symbol A denotes a class, then, any intelligible statement φ A containing the symbol A implies that the whole class A has( a)symbolic existence; but whether the statement φ A implies that the class A is wholly real, or wholly unreal, or(partly) real and partly unreal, depends upon the context. [MacColl 1906, 77] When the members A1, A2, &c.; of any class A consist wholly of realities, or wholly of unrealities, the class A is said to be a pure class; when A contains at least one reality and also at least one unreality, it is called a mixed class. [MacColl 1906, 43] Notice that MacColl actually speaks of the existence of the class. I think that we should understand it as talking about the existence of the elements of the class (see below on his rejection of interpreting hunger independently of a hungry person). Let us read MacColl’s own words on the symbolic universe more closely. On one hand it sounds as we might do logic in such a universe abstracting away whether objects are or not existent. On the other hand, MacColl, in his reply to Russell 4 and to Arthur Thomas Shearman, insists that the distinction between existent and non-existents within the symbolic universe is crucial for his logic: The explanation from my point of view is, that the confusion is solely on their side [Shearman’s and other symbolists’ side], and that it arises from the fact that they (like myself formerly) make no symbolic distinction between realities and unrealities [. . . ] With them, ‘existence’ means simply existence in the Universe of Discourse, whether the individuals composing that universe be real or unreal [. . . ] Once anything (real or unreal) is spoken of, it must, from that fact alone, belong to the Symbolic Universe S, though not necessarily to the universe of realities e. [MacColl 1905b, 579] With some hindsight, we might add two kinds of existential quantifications or at least two kinds of existential statements, namely: – One kind of existential statements has as scope the whole symbolic universe. In other words, it is about quantification over all the objects of the domain, realities and unrealities. In this sense, everything (in the universe of discourse) exists at least symbolically. Now, MacColl’s reading of this kind of quantification that ranges over all the domain is not really congenial to Meinongianism, since the analogous Meinongians quantifiers have a reading of sentences such as There are lots of things which do not exist, in which the objects in the range of the quantifier do

4. “This sense of existence [the meaning in which we enquire whether God exists] lies wholly outside Symbolic Logic, which does not care a pin whether its entities exist in this sense or not” [Russell 1905a, 398]. Some remarks on Hugh MacColl’s notion of symbolic existence 153

not exist. 5 MacColl’s notion of existence seems to be closer to that of the early Russell than to that of Meinong. Indeed, Meinong had also three ontological domains: (i) the domain of existents—signified by the verb Existieren, e.g. concrete objects in space time; (ii) the domain of subsis- tents such as abstract objects, e.g. propositions, events—which do not exist but have the kind of being Meinong called Bestehen; (iii) the domain of non-existent objects which possess no being of any kind, such as chimeras and other fictional entities. However, Meinong’s notion of non- existence excluded any kind of being, even subsistence, while MacColl’s symbolic existence and Russell’s version of subsistence included existents and non-existents. In fact, MacColl and (the early) Russell think that even non-existents have a kind of being—that the later Russell calls (in- appropriately) subsistence and MacColl symbolic existence. Compare, e.g., once more MacColl’s 1902 remark: Take, for example, the proposition “Non-existences are non-existent”. This is a self-evident truism; can we affirm that it implies the existence of its subject non- existences? [. . . ] In pure logic the subject, being always a statement, must exist—that is, it must exist as a statement. [MacColl 1902, 356] with the Russell of the Principles: Whatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term [. . . E]very term has being, i.e. is in some sense. A man, a moment, a number, a class, a relation, a chimaera, or anything else that can be mentioned, is sure to be a term. [Russell 1903, 43] MacColl and Russell make the point that everything named must have some kind of being. This point of theirs might be seen as an ontologically charged reading of Aristotle’s remark: Even non-existents can be signified by a name. [Aristotle 1989, 199] Probably because of this doctrine Russell reduced Meinong’s domain (iii) to the domain (ii). Thus, when Russell ascribed being to Meinong’s non-existents (round-squares, golden mountains), sometimes tacitly in the 1905-1907 period and later explicitly in 1918, and 1943, he was missing a crucial distinction in Meinong. It is striking that Russell’s reduction of Meinong’s ontology is already pre-figured in MacColl’s notion of symbolic existence. Notice that, according to the quotations above, one of MacColl’s points about the use of symbolic existence is that it allows the drawing of inferences by the use of the rule nowadays known as particularization. E.g. Derive: 5. This point has been stressed—and rightly so—by an anonymous referee. 154 Shahid Rahman

“Something is (symbolically) non-existent”, from: “Joseph Cartaphilus is (symbolically) non-existent”. Perhaps the following paraphrase, though awkward, might be closer to MacColl’s own analysis: “The non-existent Joseph Cartaphilus exists symbolically”. Hence, “Something non-existent exists symbolically”. There might be also some room to think dynamically about the interaction between symbolic and real existence. The real existence might come into play once the precise constitution of the universe of discourse has been specified. Juan Redmond and Mathieu Fontaine are developing a dialogic that does justice to this dynamics from an epistemic point of view: symbolic existence will be assumed so long as we do not know about the ontological constitution of our universe of discourse. 6 I shall not be discussing this approach here. – The second kind of existential statement ranges over the domain of realities. Accordingly, the following use of particularization yields a non-valid inference: “Joseph Cartaphilus is (symbolically) non-existent”. Hence, “Something is really non-existent” (“Something non-existent really exists”). Certainly, formulations such as “the non-existent Joseph Cartaphilus exists symbolically” mentioned above sound strange and motivated, later on, Russell’s rejection of the approach and the very well known criticism of Quine [Quine 1948]. If we think this in the context of two different sorts of particular and universal statements (implemented with the help of two kinds of quantifiers), and we liberate the symbolical pair of any ontological commitment then MacColl’s proposal is as plausible as modern Meinongian interpretation of positive free-logic is. 7 Now, this reading of such a kind of quantified expressions free of any ontological commitment is not really congenial with MacColl’s approach. On my view, as I will suggest below, it is possible to defend the very notion of symbolic existence in a more congenial way, though, unfortunately, this goes far beyond MacColl’s own formal and conceptual framework.

6. Cf. [Fontaine & Rahman 2010], [Fontaine, Redmond & Rahman 2011]. 7. Meinongian positive free logic allows singular terms of the language to refer to non-existent objects. The domain might contain existent and non-existent elements. The result is that the identity axiom holds in any such logic extended with equality. That is, there might be identity of non-existent objects. Furthermore, in positive free logic we might introduce two pairs of quantifiers: ontologically committed quantifiers and ontologically not committed quantifiers. Cf. [Bencivenga 1983], [Lambert 1997]. Some remarks on Hugh MacColl’s notion of symbolic existence 155

A different source of puzzlement might relate to ontological questions. What are those objects that are non-existent? Did MacColl come to a conception close to some kind of realism that, despite the differences discussed above, shares some metaphysical tenets with Meinongianism? Some arguments in favour of an affirmative answer are the following: 1. MacColl’s two notions of existence (the real and the symbolic existence) seem to have been conceived of as predicates. Indeed, in MacColl’s notation existence, when applied to an individual or to (the members of a) class, is signified by an exponential. Now, in general, leaving to one side the many changes and hesitations of his notational system, exponentials are used in principle to express a predicative role. In fact, the basic expressions of MacColl’s formal language are expressions of the form: HB where H is the domain and B a predicate. For instance: H: the domain of horses B: brown HB: The horse is brown: all of the elements of H (horses) are brown. The same can be said of the use of the predicates of symbolic, real existence and non-existence: He: The horse is real or has a real existence: all of the 0 elements of H (horses) are really existent. H : The horse is an unreality: all of the elements of H (horses) are not really existent. HS: The horse has a symbolic existence: all of the elements of H (horses) are symbolically existent. 2. Recall that according to Meinong we should distinguish the Sein of objects—their existential status—from their Sosein, their having— certain—features or properties. Thus, Meinongians claim that an object can have a set of properties even if it does not exist. This is the so-called Principle of independence: Pegasus, Ulysses, and Joseph Cartaphilus can be said to have properties without the propositions involved becoming false. MacColl’s ontological approach seems to assume such a principle. However, MacColl’s notion of symbolic existence (and Russell’s subsistence), that, as mentioned above, assumes a kind of being, even for non-existents, waters down Meinong’s principle of independence. Now, endorsement of the principle of independence pushes one towards the thesis that any singular term denotes an object, existent or not. This holds in particular for (definite and indefinite) descriptions, that is, noun phrases of the form “the/an object with such-and-such properties”. We therefore have what we may call, following [Parsons 1980], and by analogy with the principle of comprehension in naïve set theory, an Unrestricted Characterization Principle (UCP) for objects. The idea 156 Shahid Rahman

behind UCP is that we specify an object via a given set of properties, such as (1) is a horse, is ridden by Don Quijote, (2) has a philosophical discussion with Sancho Panza’s donkey, n . Take the conjunction of the relevant predicates expressing all of( the) relevant properties 1 , 2 . . . n , then, according to UCP, an object is described by precisely( ) ( this) conjunction,( ) namely the one called Rocinante by Cervantes. In his mature work, Russell raised deep objections 8 against two of the main features of UCP: (i) Can we deploy UCP to describe objects with contradictory properties? So far nothing prevents using UCP in this way. (ii) Can we deploy UCP to produce some kind of ontological argument for anything whatever? Indeed nothing prevents us from doing so if we combine it with the fact that existence is taken to be a property. Take the properties of being a Cyclops, having one eye, being son of Poseidon and being existent. If we apply UCP, we have that an object called Polyphem has all the properties mentioned above including that of existence. What about MacColl? In relation to contradictions, MacColl does not seem to be scared oﬀ by them: they are denizens of their own domain. It is not easy to see what the logic will be. Presumably the following is a valid inference in MacColl’s system: “This round square is (symbolically) round and square”. Hence, “Something is (symbolically) round and square”. Nowadays we might embed this kind of inference in a paraconsistent framework. 9 Does MacColl endorse some form of characterization principle? In fact it rather looks that MacColl would like to use a restricted form of the characterization principle. The classes of reals and unreals are disjoint classes and thus the corresponding predicates being real and being unreal—in contrast with other predicates such as being round and being square—cannot be predicated at the same time of the same object. The modern reader might invoke the distinction between nuclear and non- nuclear properties. 10

8. Cf. [Farrell Smith 1985, 313–319]. 9. A logic is said to be paraconsistent if not every contradiction entails triviality. Cf. [Routley & Meyer 1976], [Heintz 1979]. 10. Meinong’s student, Ernst Mally (1879-1944) suggested distinguishing, what have been later called, nuclear from non-nuclear properties. Nuclear properties are those that allow safe uses of the comprehension principle (not any more unrestrictedly). When applied to ﬁctional characters, this device is deployed to assert that they have all those nuclear properties that the relevant story attributes to them. [Parsons 1980] comes to the following list: Nuclear predicates (“is blue”, “is tall”, “is golden”, “is Some remarks on Hugh MacColl’s notion of symbolic existence 157

3. MacColl’s approach and Meinongianism (including Russell’s early Meinongianism) share the same unease about created objects. Indeed, since both MacColl and Meinongians assume that objects are always there, though they do not exist, they thus cannot be said to have been created. Accordingly, they are non-existents, and the creation of a fictional character cannot mean that it has been brought into existence (in the strong ontologically loaded sense). 4. MacColl provides some examples of the intentional application of his logic of non-existents [MacColl 1905a, 77–78], such as: “The man whom you see in the garden is really a bear.” “The man whom you [thought to] see in the garden is not a bear.” MacColl takes the point of view of an observer, who asserts the above propositions and studies what happens with the ontological assumption implied by them. MacColl concludes that the ontological status of the individual man is that of not-existent in the first example and existent in the second. There also are some very brief remarks on the ontological status of abstract objects, such as hardness, thoughts and feelings: There can be no hunger without a hungry person or animal; there can be no hardness without some hard substance [. . . ]. Similarly, I cannot conceive of a thought apart from a thinker, or of a feeling or sensation without a soul or feeler. [MacColl 1910, 349–350] Unfortunately, MacColl does not develop an explicit theory of intentionality or intentional objects nor does he systematically link those remarks to his ontological framework. Nevertheless, as suggested by Juan Redmond [Redmond 2010, chap. 6], this might provide a basis for his concept of symbolic existence. Indeed, take the domain of unreals as the domain of ontologically dependent objects (roughly, in the same sense that a thought is ontologically dependent upon a thinker and a fictional character upon a copy of the book that describes it), the domain of reals as the domain of ontologically independent objects, then symbolic existence can be defined as the domain that contains both, ontologically dependent and ontologically independent objects. 11 Certainly this goes far beyond MacColl’s a mountain”, etc.); and Non-nuclear predicates: Ontological (“exists”, “is mythical”, “is fictional”, etc.), Modal (“is possible”, “is impossible”, etc.), Intentional (“is thought about by Meinong”, “is worshipped by someone”, etc.), and Technical (“is complete”, “is consistent”, etc.). The nuclear/non-nuclear distinction is formally characterized in [Jacquette 2009, section 5.1]. Unfortunately the precise content of the lists is difficult to establish. Priest suggests a way out that allows the unrestricted version of the comprehension principle [Priest 2005, 82–85]. 11. The key to this approach to fictional objects lies in acknowledging them a full ontological status—considering them as denizens of our world like armchairs, 158 Shahid Rahman

~~own developments since he lacked both the notion of quantiﬁer and a thorough study of the notion of intentional object.~~

~~Conclusion~~

The main aim of the paper is historical, namely the relation between MacColl’s theory of non-existents and Russell’s early endorsement and later criticism of Meinong. To state this clearly, though MacColl was aware of the discussions that took place in the British milieu on the existential import of propositions, triggered by the work of Brentano and Meinong, it is doubtful that MacColl ever read these authors. However, while reading MacColl’s approach to this issue it is tempting to understand Russell’s version of Meinong’s notion of subsistence as an adaptation of MacColl’s symbolic existence to the Meinongian framework—a misleading understanding of Meinong, some might say. Another, less contentious, way to see the emergence of the notions of symbolic existence and subsistence is to understand both concepts as the result of an interaction of MacColl with Russell and other members of the British logical community of those days on the existential import of propositions. In relation to the emergence of these notions, I hope the paper will motivate further and wider historical studies. 12 More generally, in this context we can understand MacColl’s conceptions as the exploration of new territories in the philosophy of logic, despite the fact that he hadn’t the right instruments to develop such incursions more thoroughly. Those attempts, in their time, not only announced a new refreshing wind in philosophy of logic but also aimed to take up anew the old philosophical tradition. I am certainly happy to acknowledge my respect for his brave insights, here, at the northern part of France that oﬀered him a second home.

~~Acknowledgments~~

Many thanks to Amirouche Moktefi for his unflagging patience and encouragement and to the very helpful remarks, suggestions and corrections of two anonymous referees. elephants or galaxies. On the one hand, fictional objects are artifacts like tables or buildings. On the other hand, they are abstract creations like marriages, universities and theoretical entities postulated by physical theories. Fictional objects are tied to the everyday world by their dependence on readers, authors and copies of texts. Cf. [Thomasson 1999]. 12. See, in this volume, F. F. Abeles and A. Moktefi’s paper on MacColl’s exchanges with Lewis Carroll, notably on the notion of existence. Some remarks on Hugh MacColl’s notion of symbolic existence 159 References

Aristotle 1989 Posterior Analytics: Topica, Cambridge (Mass.): Harvard University Press. Bencivenga, Ermanno 1983 Free logics, in D. M. Gabbay, & F. Guenther (eds.), Alternatives to Classical Logic, vol. 3 of the Handbook of Philosophical Logic, Dordrecht: Reidel, 373–426. Brentano, Franz 1874 Psychologie vom empirischen Standpunkt, Leipzig: Duncker & Humblot. Farrell Smith, Janet 1985 The Russell-Meinong debate, Philosophy and Phenomenological Research, 45 (3), March, 305–350. Fontaine, Matthieu & Rahman, Shahid 2010 Fiction, creation and fictionality: an overview, Methodos: Savoirs et Textes (electronic journal), 10. http://methodos.revues.org/2343 (consulted on 28/09/2010). Fontaine, Matthieu, Redmond, Juan & Rahman, Shahid 2011 Être et être choisi : vers une logique dynamique de la fiction, in J. Dubucs & B. Hill (ed.), Fictions : Logiques, Langages, Mondes, London: College Publications (forthcoming). Haller, Rudolf (ed.) 1972 Jenseits von Sein und Nichtsein: Beiträge zur Meinong-Forschung. Graz: Akademische Druck- u. Verlagsanstalt. Heintz, John 1979 Reference and inference in fiction, Poetics, 8 (1-2), 85–99. Jacquette, Dale 2009 Logic for Meinongian object theory semantics, in D. M. Gabbay & J. Woods (eds.), Logic from Russell to Church, vol. 5 of the Handbook of the History of Logic, Amsterdam: North-Holland, 29–76. Lambert, Karel 1997 Free Logics: Their Foundations, Character, and Some Applications Thereof, Sankt Augustin: Academia Verlag. Land, J. P. N. 1876 Brentano’s logical innovations, Mind, 1 (2), 289–292. 160 Shahid Rahman

MacColl, Hugh 1902 Symbolic reasoning (IV), Mind, 11 (43), 352–368. 1905a Symbolic reasoning (VI), Mind, 14 (53), 74–81. 1905b The existential import of propositions, Mind, 14 (56), 578–580. 1906 Symbolic Logic and its Applications, London: Longmans, Green, & Co. 1910 Linguistic misunderstandings (II), Mind, 19 (75), 337–355. Marek, Johann C. 2003 On self-presentation, in C. Kanzian, J. Quitterer & E. Runggaldier (eds.), Persons: An Interdisciplinary Approach, Wien: öbv & hpt, 163– 173. Mill, John S. 1843 A System of Logic: Ratiocinative and Inductive, 2 vols., London: J. W. Parker. Parsons, Terence 1980 Nonexistent Objects, New Haven: Yale University Press. Priest, Graham 2005 Towards Non-Being: The Logic and Metaphysics of Intentionality, Oxford: Clarendon Press. Quine, Willard V. O. 1948 On what there is, The Review of Metaphysics, 2 (5), 21–38. Rahman, Shahid & Redmond, Juan 2007 Hugh MacColl: an Overview of his Logical Work with Anthology, London: College Publications. 2008 Hugh MacColl and the birth of logical pluralism, in D. M. Gabbay & J. Woods (eds.), The Handbook of the History of Logic, vol. 4: British Logic in the Nineteenth-Century, Amsterdam: North-Holland (Elsevier), 533–604. Redmond, Juan 2010 Logique dynamique de la ﬁction : pour une approche dialogique, PhD dissertation, Lille : Université Lille3. Routley, Richard 1980 Exploring Meinong’s Jungle and Beyond, Canberra: Australian National University. Some remarks on Hugh MacColl’s notion of symbolic existence 161

Routley, Richard & Meyer, Robert K. 1976 Dialectical logic, classical logic and the consistency of the world, Studies in Soviet Thought, 16, 1–25. Russell, Bertrand 1903 The Principles of Mathematics, Cambridge: Cambridge University Press. 1905a The existential import of propositions, Mind, 14 (55), 398–401. 1905b On denoting, Mind, 14 (56), 479–493. Thomasson, Amie L. 1999 Fiction and Metaphysics, Cambridge: Cambridge University Press. Zalta, Edward N. 1988 Intensional Logic and the Metaphysics of Intentionality, Cambridge (Mass.): MIT Press.

~~MacColl’s Modes of Modalities Fabien Schang Technische Universität Dresden, Institut für Philosophie, Germany LHSP, Archives H. Poincaré (UMR 7117), Nancy-Université, France~~

Résumé : Hugh MacColl est présenté d’ordinaire comme un pionnier des logiques modales et multivalentes, suite à son introduction de modalités qui vont au-delà de la simple vérité et fausseté. Mais un examen plus attentif montre que cet héritage est discutable et devrait tenir compte de la façon dont ces modalités procédaient. Bien que MacColl ait conçu une logique modale au sens large du terme, nous montrerons qu’il n’a pas produit une logique multivalente au sens strict. Sa logique serait comparable plutôt à une « logique non-fregéenne », c’est-à-dire une logique algébrique qui eﬀectue une partition au sein de la classe des vérités et faussetés mais n’étend pas pour autant le domaine des valeurs de vérité. Abstract: Hugh MacColl is commonly seen as a pioneer of modal and many- valued logic, given his introduction of modalities that go beyond plain truth and falsehood. But a closer examination shows that such a legacy is debatable and should take into account the way in which these modalities proceeded. We argue that, while MacColl devised a modal logic in the broad sense of the word, he did not give rise to a many-valued logic in the strict sense. Rather, his logic is similar to a “non-Fregean logic”: an algebraic logic that partitions the semantic classes of truth and falsehood into subclasses but does not extend the range of truth-values.

~~Modalities and many-valuedness~~

A preliminary attention to the notation is in order. MacColl’s Logic (hereafter: MCL) resorts to a symbolic language that is in accordance with the algebraic style of George Boole or Ernst Schröder. Mathematical signs are used to characterize operations between any propositions AB and CD. Thus, AB CD stands for their sum, ABCD (or AB CD, or AB.CD) for their product,+ AB CD for the implication from AB to ×CD, AB = CD for their equivalence ∶

~~Philosophia Scientiæ, 15 (1), 2011, 163–188. 164 Fabien Schang~~

B D D B B B (synonymous with (A C )(C A )), and (A ′ for the denial of A . MCL is thus a logic of predication∶ that∶ consists in propositi) ons subsuming a class under another one: in AB, the first term A is the subject term while the second term B is the predicate term whose application results in a proposition such as “A is B”, or “The A is a B”. A first terminological distinction is that between the occurrence of propositions as terms or factors and as exponents. AB and CD occur as terms in the sum AB CD and as factors in the product ABCD. Also, A and C are the subjects of the+ predications AB and CD; while B and D occur as exponents, because they are predicates to which A and C respectively belong. Another distinction is that between the adjectival and predicative use: B is used ad- B C jectivally in AB , and predicatively in A , so that the whole proposition AB means that the A which is a B is also a C. On the one hand, this symbolic device helps to convey information about quantity by introducing integers into the adjectival and predicative uses. Thus A1, A2 ..., An specify n different individuals in the class A, and the superscript 0 0 gives an expression to quantification in terms of the null class: AB means that -0 no A is B (“the A that is B belongs to the null class”), and AB that some A is 0 B (“the A that is B does not belong to the null class”); A-B means that every -0 A is B (“the A that is not B belongs to the null class”), and A-B that some A is not B (“the A that is not B does not belong to the null class”). On the other hand, the later distinction between metalanguage and object-language doesn’t appear in MCL: the truth-values are not confined to a separate higher- order language and are treated on a par with any other term of the symbolic logic. The result is a class of semantic predicates that go beyond truth and falsehood, as stated by MacColl:

The symbol Aτ only asserts that A is true in a particular case or instance. The symbol Aε asserts more than this: it asserts that A is certain, that A is always true (or true in every case) within the limits of our data and deﬁnitions, that its probability is 1. The symbol Aι only asserts that A is false in a particular case or instance; it says nothing as to the truth or falsehood of A in other instances. The symbol Aη asserts more than this; it asserts that A contradicts some datum or deﬁnition, that its probability is 0. Thus Aτ and Aι are simply assertive; each refers to one case, and raises no general question as to data or probability. The symbol Aθ (A is a variable) is equivalent to A-ηA-ε; it asserts that A is neither impossible nor certain, that is, that A is possible but uncertain. In other words, Aθ asserts that the probability of A is neither 0 nor 1, but some proper fraction between the two. [MacColl 1906, 7]

MacColl supplements the five preceding semantic predicates with four other elements, i.e. π for possibility, p for probability, q for improbability, MacColl’s modes of modalities 165 and u for uncertainty [MacColl 1906, 14]. Most of these predicates (except for p and q) behave like modalities, since they are reminiscent of the Aristotelian logic and its isomorphic oppositions within quantified and modal propositions: certainty (“every A is B”) and impossibility (“every A is not B”) correspond to universal affirmation and negation, while possibility (“some A is B”) and uncertainty (“some A is not B”) correspond to particular affirmation and negation. As to the remaining cases of truth, falsehood, variability, probability and improbability, they require an extension of the Aristotelian square: variability (“some A is B , some A is not B”) corresponds to two-sided possibility or con- tingency, and such a concept finds its rightful place in the first logical hexagon of Robert Blanché [Blanché 1953]; truth (“this A is B”) and falsehood (“this A is not B”) correspond not to universal or particular, but singular propositions, whose rightful place requires further extension from the second logical hexagon of Tadeusz Czeżowski [Czeżowski 1955] to the logical octagon of Jan Woleński [Woleński 1998] (see figures [1–4]). 166 Fabien Schang

The result is the following list of 28 logical oppositions and their five kinds of opposition: 1 Contraries (5): ε, η ; ε,θ ; η,θ ; η, τ ; ε,ι { } { } { } { } { } Subcontraries (5): π,u ; π, -θ ; u, -θ ; π,ι ; u, τ { } { } { } { } { } Contradictories (4): ε,u ; η,π ; -θ,θ ; τ,ι { } { } { } { } Subalterns (10): ε,π ; η,u ; θ,π ; θ,u ; ε, -θ ; η, -θ ; ε, τ ; η,ι ; τ,π ; ι,u { } { } { } { } { } { } { } { } { } { } Mere non-contradictories (4): -θ,ι ; -θ, τ ; ι,θ ; τ,θ { } { } { } { } The 28 oppositions above exhaust the set of logical relations between MacColl’s modalities, excluding the peculiar cases of p and q: the latter are probable values that cannot be located within a polygon of oppositions, given that they represent any of the indefinitely many values between full truth (or certainty: 1) and full falsehood (or impossibility: 0). The dotted lines stand for contradictory relations; the arrows indicate the entailment relation between an antecedent A and its consequent B, such that A B is a theorem. Here is a sample of those theorems from MacColl [MacColl∶ 1906, 8]: (T12) Aτ Aι expresses the law of excluded middle between contradictory terms, (T14)( + A)ε Aη Aθ ε means that three modalities exhaust the semantic class of MCL,( while+ + the implications) (T15) Aε Aτ and (T16) Aη Aι correspond to subalternation relations. ∶ ∶ 1 and 0 are usually used to symbolize truth and falsehood, in algebraic semantics; but it has been frequently claimed by the commentators on MacColl (including [Simons 1998]) that he thought of truth-values in probabilistic terms of truth-cases. If so, then any of the logical values of MCL should be reformulated within the closed interval [0,1] of an infinitely-valued logic. This should not be a proper characterization of MacColl’s view of modalities, however: the difference between particular and singular judgments cannot be rendered within such a probabilistic line, and the above oppositions depict logical relations that are qualitative rather than quantitative (as witnessed by the absence of p and q from the polygons). At any rate, it may seem strange to state logical oppositions between concepts expressing truth-values: if an opposition is about the possibility for two terms to be both true or false, what does it mean for τ and ι not to be true or false together? A plausible answer is that these alleged truth-values are not that but, rather, modal operators attached to propositions. Hence the ensuing problem is to specify the logical status of the elements that compose the semantic class. In other words: is MCL a modal logic, or a many-valued logic?

1. For every polygon of opposition with n vertices, the number of logical oppositions is n − 1 + n − 1 − 1 + ... + 1. Thus a square has n 4 vertices and 4 − 1 + (3 − 1 )+ 1 ((6 logical) oppositions.) We do not count four= but ﬁve kinds of (opposition) ( in the) above= octagon: the so-called “mere non-contradictories” are neither a case of subalternation nor a case of subcontrariety, as witnessed by the pair θ, τ which does not express any entailment relation between its terms. See [Smessaert{ } 2009] about the deﬁnitional link between subalternation and entailment. MacColl’s modes of modalities 167

To begin with, only five elements are brought out in the semantic class because only these are independent and cannot be reduced to each other; conversely, possibility is the negation of impossibility π = -η and uncertainty is the negation of certainty u = -ε . For this reason, we( restrict) our attention to the nature of the five elements( of) the semantic class Five = ε,η,τ,ι,θ . { } Let A be a proposition of a language and v a valuation function that maps A onto an element x of Five, so that v(A) = x. We are then faced with two alternatives. Either MCL is a modal logic, in which case the semantic predicates are not elements of Five but unary operators © that are attached to propositions and form modal propositions of the form A = ©p. Or MCL is a many-valued logic, and the semantic predicates are not operators but the resulting values x of an operation. Or can they be both, in one and the same symbolic logic? There have been such cases after MacColl’s death, in the realm of many-valued logics: in Jan Łukasiewicz’s three-valued system Ł3, the modality of possibility occurred both as a modal operator © = M and a third value x = 1 2 [Łukasiewicz 1920]; the same obtains in Dmitri Bochvar’s three-valued logic of~ assertion, where truth and falsehood occur both as truth-values and unary operators meaning “It is true that” or “It is false that” [Bochvar 1938]. Whatever the case may be, it can be said more generally that MCL is a logic of modalities, without specifying the mode in which these modalities are introduced within the formal system. The five semantic predicates occur as terms, exponents or factors, but never in the specific manner of truth-values or unary operators: these characterizations belong to Gottlob Frege’s modern or functional logic, and MacColl never adhered to this logical turn as his most famous opponent Bertrand Russell did. Despite this theoretical discrepancy, MCL is closely related to Bochvar’s three-valued logic. Firstly, MacColl and Bochvar insisted equally upon the connection between affirmation and truth, denial and falsehood. MacColl claims that any proposition A is equivalent with its being true, so that A = Aτ . Bochvar explicitly renders this by attaching an operator of affirmation * to A, so that A∗ means the same as “It is true that A”. But just as not every occurrence of the proposition A entails its being true, MacColl argued that A and Aτ are equivalent without being synonymous with each other:

It may seem paradoxical to say that the proposition A is not τ ι quite synonymous with A , nor A′ with A ; yet such is the τ fact. Let A = It rains. Then A′ = It does not rain; A = it is true that it rains; and Aι = it is false that it rains. The two propositions A and Aτ are equivalent in the sense that each implies the other; but they are not synonymous, for we cannot always substitute the one for the other. In other words, the equivalence (A = Aτ ) does not necessarily imply the equivalence φ A = φ Aτ . [MacColl 1906, 16] ( ) ( ) 168 Fabien Schang

This is characteristic of what has been repeatedly blamed in MacColl’s logic, namely: the equivocal use of its symbols. For A and Aτ are equivalent when the term A occurs as a statement (an interpreted function) uttered by a speaker, while A and Aτ are not synonymous when the term A occurs as a propositional function that is not stated and unrestrictedly ranges over the elements of Five. For if A is uttered, then A is claimed to be true by its speaker and every such proposition is trivially certain: if A = τ then Aτ = τ τ = ε, τ τ and if A = ε then A = ε = ε. Now if A′ is uttered, then A is claimed to be false by its speaker and every such proposition is trivially impossible. At the same time, if A is not stated and occurs as an unasserted form of words (proposition) then A can range over any element of Five. Let A = θτ and ε ε ε τ τ ε τ ε ε φ A = A ; hence φ A = A = θτ = η, while φ A = A = θτ = ε = ε. ( ) ( )τ ( ) ( ) ( ) ( ) Therefore, φ A ≠ φ A for some values of A. Likewise, let A = θι; hence ε( ) η ( η ) ι ι ε ι ε ε φ A′ = A′ = A = θι = η, while φ A = A = θι = ε = ε. Therefore, ( ) ( )ι ( ) ( ) ( ) ( ) φ A′ ≠ φ A for some values of A. Unlike Bochvar, MacColl thus assumed an ( ) ( ) implicit occurrence of the statements A and A′ as truth- and falsehood-claims; τ ι whereas the fact that A and A (or A′ and A ) are non-synonymous betrays their explicit occurrence as propositional functions. Secondly, MacColl foreshadowed Bochvar’s subsequent distinction between external and internal negation. Such a terminology refers to the scope of functions, unlike MacColl’s logic. But it makes sense in MCL, however: either its larger scope applies to the whole proposition AB, and negation is thus said to be external; or its narrower scope applies to the subject term A only, and negation is thus said to be internal. This diﬀerence has already been exem- pliﬁed in the preceding lines: in the upper vertex of the logical oppositions, -θ in A-θ depicts an external negation meaning that the proposition A is not θ contingent, while A′ means that the denial of A is contingent. MacColl motivated this distinction( ) with reference to the higher degree statements:

The symbol ABC means AB C ; it asserts that the statement AB belongs to the class C, in( which) C may denote true, or false, or possible, &c. Similarly ABCD means ABC D, and so on. From this deﬁnition it is evident that Aηι is( not necessarily) or generally equivalent to Aιη, nor Aει equivalent to Aιε. [MacColl 1906, 7]

In an external negation (EN) like A-η, the denial applies to the factor (or predicate) and is symbolized by a hyphen; in an internal negation (IN) η like A′ , the denial applies to the term A and is symbolized by a single inverted( ) comma. The logical diﬀerence between both is obviously seen with the following equivalences, where the place of negation does crucially matter: ιη ι η η ε ιε ι ε ε η ιθ θ θ (IN) A A A′ A ; A A A′ A ; A A′ A ηι= ( )η ι= ( ) η = ε =θ ( )ει = ( ε) ι = ε = ( η ) =θ θι (EN) A A A ′ A + A ; A A A ′ A + A ; A Aθ ι A=θ ( A) ε =+ A ( η ) = = ( ) = ( ) = = ( ) = ( )′ = MacColl will relevantly call for internal negation in order to establish a good number of theorems in MCL, assuming the deﬁnition of implication in MacColl’s modes of modalities 169

η terms of denial and impossibility: A B = AB′ . He argues for this by relying upon the usual meaning of denial:( ∶ ) ( ) ε By the “denial of a certainty” is not meant A ′, or its synonym A-ε, which denies that a particular statement( )A is certain, but (Aε ′ or its synonym Aε′ , the denial of the admittedly certain ) statement Aε. This statement Aε (since a suffix or subscriptum is adjectival and not predicative) assumes A to be certain; for both Ax and its denial assume the truth of Ax. Similarly, the -π “denial of a possibility” does not mean A but Aπ′ , or its syn- π onym (A ′, the denial of the admittedly possible statement Aπ. [MacColl 1906,) 15] A modern way to put this point is to say that denial doesn’t apply to a modal operator ( for necessity, say) but to its propositional content: the internal negation of◻ A is ∼A, rather than ∼ A. But again, any comparison between MCL,◻ modal◻ logic and many-valued◻ logic should be made with caution. Against the view that a logical system could be equally taken to be modal or many-valued (as Łukasiewicz did), it has been proved by James Dugundji [Dugundji 1940] that: There exists no finitely many-valued logic that is characteristic of any of the Lewis systems S1 to S5, because any finitely many- valued logic will contain tautologies that are not theorems of S5 (and a fortiori not of S1 to S4 either). [Rescher 1969, 192] This means that either MCL is a (finitely) many-valued logic, so that it is a weaker system than the Lewis systems S1-S5; or MCL is a modal logic that can be characterized by a modal system between S1 and S5, in which case it is not a many-valued logic. Although MCL is variously presented as a modal and (finitely) many-valued logic, it cannot be both in the light of Dugundji’s proof. So which one should it be? At any rate, the preceding entails that MCL may be seen as a modal logic only in the broad, informal sense of a logic for modalities: a modality is viewed to be a mode of being true or false, irrespective of how it is regimented in a logical system, and MacColl follows this line by arguing that certainty or impossibility are the same as being true or false in every case. Let us consider first the modal features that are commonly associated with MacColl’s system.

~~A modal logic?~~

Non-truth-functionality is usually taken to be an essential criterion of a modal logic: not every value of such a composed proposition is to be determined by the value of its components, as the case is with A with respect to its component A. Although such a criterion doesn’t hold◻ with some many- valued translations of modal logics, we have seen with Dugundji’s proof that 170 Fabien Schang no modal system between S1 and S5 can be characterized by a many-valued matrix. A truth-functional many-valued logic could be appropriate for modal systems weaker than S1, consequently. At any rate, modal logic is not merely specified by a set of logical theorems. For instance, Georg Henrik von Wright proposed a classification that departs from MacColl’s view of modalities. Thus he writes: We shall distinguish between truth-concepts or truth-categories and modal concepts or modal categories. The logic of truth- concepts we shall call truth-logic, and the logic of modal concepts we shall call modal logic. The basic truth-categories are the two so-called truth-values, viz. truth and falsehood. Further examples of truth-categories are the concept of a truth-function and the instances of such functions: negation, conjunction, disjunction, (material) implication, (material) equivalence, tautology, and contradiction. It is of some importance to observe that the words tautology and contradiction are used in this essay as names of truth-functions exclusively. The words in question are sometimes used as synonyms for certain modal words. [von Wright 1951, 1] Given this distinction, MCL appears as a conflation of truth- and modal concepts, and MacColl is one of these logicians that would use tautology and contradiction as synonyms for the modalities of certainty and impossibility. The meaning of modal logic is not definite, according to him, but a question of choice for different sakes. Thus we find in MacColl: In the traditional logic any proposition AB of the first degree is called a pure proposition, while any of my propositions ABC or ABCD, &c., of a higher degree would generally be considered a modal proposition; but upon this point we cannot speak with certainty, as logicians are not agreed as to the meaning of the word ‘modal’. For example, let the pure proposition AB assert that “Alfred will go to Belgium”; then ABε might be read “Alfred will certainly go to Belgium”, which would be called a modal proposition. Again, the proposition A-B , which asserts that “Alfred will not go to Belgium”, would be called a pure proposition; whereas ABι, or its synonym (AB ι, which asserts that AB is false, would, by most logicians, be considered) a modal proposition. [MacColl 1906, 94] Furthermore, truth and falsehood could be defined as ‘null’ modalities similar to the redundant operator of affirmation in the Fregean logic. Thus[ a modern translation of Aτ and Aι would result in the modal formulas A and ∼A. Once this assumption is accepted, it remains to see which sort of[ modal logic[ MCL is. Various answers have been given to this respect: Storrs McCall [McCall 1967] identifies MCL with S3 and Shahid Rahman [Rahman 1997] argues for S2 MacColl’s modes of modalities 171 or S3. Stephen Read [Read 1998] claims that the stronger system T (developed after Clarence Irving Lewis by Robert Feys and von Wright) is characteristic of MCL – given the theorems (T15) Aε Aτ and (T16) Aη Aι while discarding any even stronger system like S4 or S5.∶ Against McCall,∶ Read− argues that:

[T]he characteristic axiom of S3 does not ﬁgure in the nine theses McCall attributes to MacColl—indeed, if it did, then since MacColl’s logic is normal, as shown above (i.e. ηπ = η), there would ensue reduction theses such as aππ = aπ, characteristic of S4, since S4 is the union of S3 and T. Since MacColl explicitly endorses normality and denies any reduction laws, his logic is T. [Read 1998, 74]

Whatever the case may be, the initial question of a characteristic logical system for MCL does not happen to make good sense from a MacCollian point of view. For one thing, a modal system is said to be stronger (or weaker) than another whenever the former set of logical theorems includes (or is included in) the latter. But such an inclusion relation can be established only for modal systems and interpreted within a Kripkean structure, where inclusion concerns the relations of accessibility (reflexivity, transitivity, and the like) between possible worlds. Following Dugundji’s result, again, MCL cannot be seen with the modern glasses of modal logic and its model structures because it proceeds as a finite many-valued logic. Moreover, Rahman [Rahman 1997] also claimed that MacColl advocated a logical pluralism such that no unique logical system should be characteristic of MCL. In this sense, the latter is not so much a logical system in the modern sense of the word, i.e. a closed set of theorems; rather, it is a symbolic language whose theorems depend upon the applied context of discourse. According to Rahman, Aristotle’s modal syllogistic is akin to connexive logic and closely related to the system S2; but other modal discourses are conceivable and should require alternative sets of modal theorems. Finally, MCL is all the more distinct from modern modal logics in that their iteration and reduction theses don’t proceed in the same way. Thus a crucial difference is to be made between iterated modalities and higher degrees statements. The reason is that MacColl’s modalities occur as inclusion relations, while the modern modalities stand for quantifiers. For any sequence of modalities with n modal terms, the nth and ultimate (right-sided) modality occurs as a predicate while the n 1th preceding modalities are parts and parcels of the predicated subject. Given− the inclusion relation that holds from left to right between any sequence of exponents, the law of reduction from every such higher-degree to a first-order statement requires one to know whether the primary (left-sided) predicate is included into its (right-sided) successor. 172 Fabien Schang

~~Let us take the statement Aηιε as an example. According to MacColl:~~

~~The symbol Aηιε may be read “It is certain that it is false that it is impossible that A is true”; which may be abbreviated into “A is certainly possible”. [MacColl 1900, 75]~~

As a third degree statement, it amounts to Aηι ε and means that it is certain that it is false that it is impossible that (A.) Given that Aηι = Aε Aθ (see [MacColl 1901, 140]), it follows that Aηιε = Aε Aθ ε and( a modern+ ) translation of this statement matches with the preceding( + re)sult. Using for possibility, Aηιε would be rendered by ∼ ∼ A , i.e. A. (Aε Aθ εÅand A hold equally, since they both mean◻ that( AÅ is) certainly◻Å possible.+ However,) ◻Åthe reduction laws of iterated modalities in the modern modal systems don’t hold in MCL: A ↔ A in S4, whereas Aε ≠ Aεε; and A ↔ A in S5, whereas Aεπ ≠ ◻Aε. ◻ ◻ ◻Å Å Consequently, MCL can be viewed as a logic of modalities in the broad sense of containing modal expressions; but not as a logical system including modal operators, given the very diﬀerent nature of its operations upon modalities as predicates. The set-theoretical aspect of these operations is connected with an algebraic reading of modalities; and since algebraic operations ( , , , =) apply to values including a supremum 1 and an inﬁmum 0, we should be+ naturally× ∶ led to conclude that MCL is a many-valued logic whereby modalities stand for a set of truth-values. Let us scrutinize this widespread opinion among the commentators.

~~A many-valued logic?~~

MCL usually appears as an extension from a two-valued to a five-valued logic: the two classical truth-values of truth and falsehood are supplemented with certainty, impossibility, and variability. But why five values, rather than any other number? The cases of five-valued logics are very rare, given their uncommon cardinality; but we can mention in this respect the recent system of Arnon Avron [2008]. This logical system is a paraconsistent logic including a set of five values: necessary and consistent truth (T), necessary and consistent falsehood (F), contingent and consistent truth (t), contingent and consistent falsehood (f), and inconsistency (I). Although MCL has nothing to do with paraconsistency, it shares with Avron’s system the common idea to extend the set of truth-values by combining further properties with the initial values of truth and falsehood. Just as Avron’s non-classical truth-values appear as special cases of truth and falsehood, MacColl emphasized the process of partition into the basic sets of truth and falsehood: Toutes les propositions intelligibles peuvent être divisées en deux classes, les vraies t et les fausses ι . Toutes les propositions intelligibles peuvent( être) aussi divisées( ) en trois classes : les certaines MacColl’s modes of modalities 173

(ε1,ε2,ε3, etc.), les impossibles (η1, η2, η3, etc.) et les variables (θ1,θ2,θ3, etc.). [MacColl 1901, 138] A way of reformulating this valuation is to split each basic concept of truth (x = 1) or falsehood (x = 0) into four modes of being so: plainly (x.3), always (x.4), never (x.5), and sometimes (x.6). The result is a new class of eight combined elements: Eight ={1.3, 1.4, 1.5, 1.6, 0.3, 0.4, 0.5, 0.6}. While noting that two pairs of these elements are synonymous and reducible to each other, i.e. 1.4- 0.5 and 0.4-1.5, MacColl’s modalities correspond to the following subclasses of elements: ε 1.4 ∶ { } η 0.4 ∶ { } τ 1.3, 1.4, 1.6 ∶ { } ι 0.3, 0.4, 0.6 ∶ { } θ 1.6, 0.6 ∶ { } This numerical translation of the modalities helps us see that MacColl’s five elements overlap each other: the basic set of truth (1) includes both plain truth (τ for 1.3, 1.4, 1.6 ) and certain truth (ε for 1.4 ) as its elements, just as the{ basic set} of falsehood (0) includes plain{ }falsehood (ι for 0.3, 0.4, 0.6 ) and certain falsehood (η for 0.4 ). In symbols: 1 corresponds{ to 1}.3, 1.4, 1.5, 1.6 , and 0 to 0.3,{0.4, 0}.5, 0.6 . MCL would thus refer to{ the following} semantic class{ which includes} five subclasses of Eight, each of these corresponding to a modality from the set Five: 1.4 , 0.4 , 1.4, 1.5, 1.6 , 0.4, 0.5, 0.6 , 1.6, 0.6 . {{ } { } { } { } { }} Does the cardinality of Five entail that MCL is a many-valued logic, however, in the sense that it contains more than two elements (values)? Against this view, Peter Simons argues that three preconditions have to be required for a semantic class to be “essentially many-valued” [Simons 1998, 86]. The first (MV1) is: to contain at least one other value besides truth and falsehood. The second (MV2) is: for all and only all the elements to be pairwise exclusive and jointly exhaustive. The third (MV3) is: for the connectives to be value-functional. Simons concludes against Nicholas Rescher [Rescher 1969] that MCL is not an essentially many-valued logic, because no one of these three criteria are respected by it. On the one hand, it fails to satisfy (MV1) insofar as the three additional values are not introduced into Five besides truth and falsehood: ε 1.4 belongs to τ 1.3, 1.4, 1.6 within the truth-class 1, since whatever({ is true}) is so plainly (without({ qualification),}} necessarily (always), or contingently (sometimes, but not always); the same goes for η and ι, with respect to the falsehood-class 0. On the other hand, not all the elements of Five are needed to exhaust the semantic class: τ and ι are sufficient, as witnessed by the theorem (T12) Aτ Aι ε. This is because truth and falsehood are subclasses whose joint elements( + exhaust) all the modes of truth. And finally, Simons contends that MCL is not value-functional because the operations θ.θ and θ θ don’t have any determinate result [Simons 1998, 87]. ∶ 174 Fabien Schang

Let us review each of Simons’s objections to the uncritical view that MCL is a many-valued logic. About MV1, firstly. The preceding reformulation helps to bring out some inferential relations between the five predicates. Thus, the theorems (T15) Aε Aτ and (T16) Aη Aι state nothing but an inclusive relation between∶ the antecedent and∶ its consequent: 1.4 ⊂ 1.4, 1.5, 1.6 and 0, 4 ⊂ 0.4, 0.5, 0.6 . But this is not a sufficient condition{ } to{ consider the} re- {sulting} set{ of inferences} as really different from classical logic, Simons claims. He gives an example of a four-valued matrix for negation and conjunction, where each element of the corresponding semantic class {(1),(2),(3),(4)} comes from the Cartesian product {0,1} {0,1} of the classical truth-values 1 (for truth) and 0 (for falsehood). Thus× 1 = 11, 2 = 10, 3 = 01, and 4 = 00. On the basis of this matrix, Simons( rightly) notes( ) that( there) is no difference( ) between the resulting set of logical theorems and that of classical logic, which means that both logics are one and the same, despite the contrary appearances of the valuations. It follows from it that MCL is not many-valued if, by a many-valued logic, we mean a deviant system that does not contain every theorem of classical logic; given that excluded middle, non-contradiction and all other classical laws are preserved, we agree with Simons and claim that MCL could be viewed rather as a modal extension of classical logic. At any rate, the criterion (MV1) makes use of a technical device: product systems, i.e. this branch of (allegedly) many-valued logic where each logical value is the Cartesian product of classical values. Such a technique has been used by Prior [Prior 1955] to evaluate tensed propositions and is strikingly reminiscent of MacColl’s modalities. Let W = w, w be a set of two different states of affairs (or truth-cases) w and w .{ Then∗} the same valuations as above can be produced to give an intuitive∗ interpretation of necessity and possibility: whatever is necessary is true in every “possible world” w and w , while whatever is possible is true in at least one of these. Apart from the∗ probabilistic tone of MacColl’s modalities (assuming a finite set W of elements that enables one to assess the ratio between true cases and false cases), the same view of modalities occurs in Prior’s logic and MCL. This will be reviewed in a more detailed way in the next section. Although the failure of (MV1) is a sufficient ground for Simons to settle the problem and claim that MCL is not many-valued, let us consider the two other points: they should throw some new light upon MacColl’s modalities, whether they are many-valued or not. About (MV2), secondly. Simons is also right to affirm that truth and falsehood are exhaustive, given that they cover all the modes of being true and false as their particular cases. Hence the validity of the theorems (T15) and (T16), again; but this result contains a further subtlety that is not mentioned by Simons: any proposition A is a theorem in MCL if and only if it is certain, i.e. A = ε. Now this certainty is not the same as the certainty that is included MacColl’s modes of modalities 175 among the different modes of being true. To give an expression to this symbolic ambiguity, MacColl makes a difference between two ways of being certain:

A proposition is called a formal certainty when it follows necessarily from our definitions, or our understood linguistic conventions, without further data; and it is called a formal impossibility, when it is inconsistent with our definitions or linguistic conventions. It is called a material certainty when it follows necessarily from special data not necessarily contained in our definitions. Similarly, it is called a material impossibility when it contradicts some special datum or data not contained in our definitions. [MacColl 1906, 97]

It appears from this distinction that a theorem is formally certain because it is a tautology, i.e. its denial is made inconsistent by the definitions of MCL; by contrast, a formula is materially certain because it is not tautological but yields only truth-cases on the basis of empirical data. If one introduces a symbolic change to bring out this difference between logical and non-logical (physical) necessity, e.g. εf for formal certainty and εm for material certainty, theorem ε εm τ (T11) should be restated as A A′ f , and the theorem (15) as A A . ( + ) ∶ Such a precision is not gratuitous: it is essential to make sense of the way in which MacColl thought of implication. Given that A B and AB can be interchanged, this means that B is implied by A if and only∶ if the class A is included in the class B. But if so, how could the class of certainty be included in the class of truth (by (T15)) and at the same time include it (by (T11))? Returning to our preceding reformulation of the modalities, this entails that a further set-theoretical definition should be introduced for formal certainty: assuming that ε = 1.3 and τ = 1.3, 1.4, 1.6 ,εf should be a set that includes the set for τ. We will{ pursue} this{ line in the} next section. About (MV3), thirdly. The logical matrices for MCL are functionally incomplete, according to Simons, given that some of the operations for conjunction and implication are left undetermined. The two cases mentioned by Simons are gathered from counterintuitive examples. Thus, Simon concedes that Aθ Bθ is certainly true whenever A and B are one and the same proposition, for∶ the reason that any such proposition is certainly self-identical; but it is taken to be contingently true, if both propositions are different and not related by a logical or causal relation. The same is said about Aθ.Bθ, on the other hand. Nevertheless, two replies could be given to these objections about (MV3). On the one hand, that implication yields counterintuitive results is not specific to MacColl’s implication and need not lead to a non-value-functional matrix for MCL. No such matrix is explicitly given in MacColl’s writings, unfortunately; but a recursive definition of his operations should be still in order, if we assume the set-theoretical process of inclusion between classes as a general pattern for his symbolic logic. 176 Fabien Schang

On the other hand, the alleged indeterminacy of Aθ.Bθ is not justified but merely conjectured by Simons, and we suspect him of confusing variability with probability in this respect: if the probability of A is between 1 2 and 1, A is probable but its conjunction with another probable proposition~B could weaken the final ratio under 1 2; if so, then A = B = p and A.B = q. But again, the indeterminacy of p.p~ concerns probability and does not affect the conjunction rules for variability. As a general result, (MV1) seems to be the most plausible objection against the statement that MCL is many-valued: the given list of logical theorems is not deviant from classical logic, and the set of truth-values doesn’t include any new element independent from truth and falsehood. However, a difference between the bivalent set of classicists and MCL lies in the process of partition. Actually, such a division of the classes of truth and falsehood into a number of subclasses explains the view defended by Rescher [Rescher 1969] that MCL is both many-valued and modal: a partition augments the cardinality of the initial set beyond two elements, and the additional elements stand for different modes of truth. Irrespective of the proper criteria for many-valuedness or modality, let us see how to streamline MCL in such a way that its various theorems could be derived from general and recursive principles.

~~A non-Fregean logic!~~

MacColl appeared as a member of the algebraic school in logic, in the sense that he attempted to algebraize logic as Boole or Schröder did before him. Another symptom of this was the fact that MacColl regarded logic as a useful instrument, rather than a universal language for correct thought. It also turns out that every many-valued logician naturally subscribes to such an relativist or goal-dependent view of logic, by contrast to the universalist line defended by Russell, Frege or the early Wittgenstein. One corollary of this duality between relativists and universalists is the famous controversy between Russell and MacColl. According to Russell, every proposition is either true or false and cannot be anything else, unlike the previous partition of truth and falsehood into several modes. Russell defends his point with reference to MacColl’s semantic predications: Either of these is a propositional function; but neither is a proposition [. . . ] Thus we shall say that true and false are alone applicable to propositions, while certain, variable and impossible are applicable to ambiguous forms of words and to propositional functions. [Russell 1906, 257] Woleński [Woleński 1998] recalled that Russell reduced modalities to quantifiers, while MacColl made a symbolic difference between quantification 1 0 0 A ,A ,A- and modality Aε,Aη,Aπ . But whether such a reduction is ( ) ( ) MacColl’s modes of modalities 177 advisable or not is unclear: why should a strict bivalence be defended, and doesn’t it restrict the expressive power of logic? If we assume the Platonist view that propositions are abstract objects whose truth (or falsehood) means the occurrence (or failure) of a one-one correspondence with facts, bivalence naturally emerges from this alternative. Otherwise, Russell’s position needs to be defended in practical terms of convenience: a logic with only two truth- values should be favored because of its being equally informative and more economical. But MacColl seemingly contests this point: What chiefly led me to this decision was the discovery that in dealing with implications of the higher degrees (i.e. implications of implications) a calculus of two dimensions (unity and zero) is too limited, and that for such cases we must adopt a three- divisional classification of our statements. [MacColl 1897, 496] MacColl does so in order to emphasize the way in which a proposition may be true, i.e. certainly or variably. But this is not a sufficient reply to Russell’s objection: to turn a modal proposition in MCL into a propositional function in Frege-Russell’s logic makes the latter a convenient symbolism for MacColl’s modalities. At the same time, this is a sufficient reply if one wants to distinguish formal from material certainty: the latter distinction cannot be rendered by Russell’s formalism, so let us consider it in detail. Many-valuedness usually relates to a process of partition into the set of “truth-values”. MacColl also talks about a number of “dimensions” related to this process. There are three dimensions in MCL, i.e. three modes of being true or false, while only two dimensions are explored in a two-valued logic. But to call MCL a three-dimensional logic seems strange, if five semantic predicates can be predicated of a term. The reason for such a cardinality is that, according to MacColl:

Comme termes et comme facteurs, τ et ι sont équivalents re- spectivement à ε et η; mais pas toujours comme exposants. Car comme terme ou facteur (puisque A veut dire Aτ ), τ = τ τ = ε, et ι = ιτ = η. [MacColl 1901, 140]

Once the predicate terms are thus restricted to three irreducible elements, MacColl suggests a generalization from 3- to 3n-dimensional logics [MacColl 1897, 509]. There are further irreducible modes for the modes of being true, including these predicates that anticipate the modern epistemic modal logic: known to be true (κ), known to be false (λ), or doubtful (µ), i.e. neither known to be true nor known to be false. Thus a proposition A can be known to be true certainly (Aεκ), doubtfully impossible (Aηµ), etc., within a larger set of 3 3 = 9 higher-order modalities combining n = 2 groups of alethic (ε,η,θ) and× epistemic (κ, λ, µ) modalities. While such an increasing partition would have been accused of psychologism by Russell, it clearly appears that MCL does not have a fixed number of truth-values but makes room for a family of multiple-valued languages: it is not a many-valued logic, or a deductively 178 Fabien Schang closed language where new truth-values would be added to truth and falsehood; rather, it is an open language in which the set of semantic predicates includes more than two elements and can expand indefinitely the basic values of truth and falsehood. We want now to clarify this open language as a product system, in order to see which sort of deductively closed logic MCL could amount to. For this purpose, let us introduce a non-Fregean logic of modalities: an algebraic logic in which the values that interpret the propositional variables are not Fregean “truth-values”. In other words: the following logical values won’t be “truth-values” including truth and falsehood but, rather, ordered n-tuples of answers R to corresponding questions Q about the propositions. Such a device is closely related to “many-valued” systems (or, better, multiple- valued, if one takes (MV1) into account) that had been elaborated before the rise of possible-world semantics, especially by Prior [Prior 1955; 1957] or Łukasiewicz [Łukasiewicz 1920]; but it also occurs in Nuel Belnap’s four-valued logic FDE, where two main questions are asked about the state of information of a computer [Belnap 1977]. Let W = w1, w2 be a set of two states of affairs. Then two questions are { } asked about modalities, i.e. the mode of truth of a proposition A, namely: q1 = “Is A true in w1?”, and q2 = “Is A true in w2?”. Assuming that each question results in a yes-no answer (with 1 for “yes” and 0 for “no”), there are two questions and two possible answers for each question; we thus obtain a set of 2 2 = 4 ordered answers within a general framework of mn-valued logics (where n is the cardinality of Q and m is the cardinality of R), that is: (11), (10), (01), and (00). Most of MacColl’s modalities can be rendered by these new logical values: R ε = 11 , R θ = 10 or 01 , and R η = 00 . The trouble is that such a valuation( ) ( is) both( ) incomplete( ) ( and) non-value-function( ) ( ) al: variability is given two synonymous values instead of only one; truth and falsehood cannot appear in this valuation, given that they are singular predications about actual states of affairs. This establishes the fact that a probabilistic understanding of MCL is not sufficient to account for the meaning of MacColl’s modalities: the mode of a truth differs from its frequency. An alternative set of questions-answers can be found in [Smessaert 2009], giving rise to an alternative semantics (see [Schang 2011]) and characterizing modalities as generalized quantifiers. In order to express the modes of truth and falsehood, generalized quantification proceeds by subsuming an indefinite number of truth-cases under four subclasses. The ensuing questions are: q1 = “Is A always false?”, q2 = “Is A actually (but not always) false?”, q3 = “Is A actually (but not always) true?”, and q4 = “Is A always true?”. For every pair of propositions A and B, the various operations of MCL can be performed upon their values R A and R B within a Boolean algebra ( , ,⊂,–,1,0). Thus: ( ) ( ) ∩ ∪ 1 y = y and 0 y = 0; 1 y = 1 and 0 y = y ∩r = 0 if and∩ only if r =∪ 1, and ∪r = r −( ) − − ( ) MacColl’s modes of modalities 179

For every value R A = r1 A ,r2 A ,r3 A ,r4 A and every binary operation ∈ , , ⊂ : ( ) ( ( ) ( ) ( ) ( )) ○ (∩ ∪ ) R A R B = r1 A r1 B ,r2 A r2 B ,r3 A r3 B ,r4 A r4 B R(AB)○ =(R)A ( R( B)○ ( ) ( )○ ( ) ( )○ ( ) ( )○ ( )) R(A )B =(R )A ∩ (R )B ( + ) ( ) ∪ ( ) R A′ = r1 A , r2 A , r3 A , r4 A . ( ) (− ( ) − ( ) − ( ) − ( )) Then each of MacColl’s five modalities can be reformulated as follows: R ε = 0001 , R η = 1000 , R τ = 0011 , R ι = 1100 , and R θ = 0110( ) .( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The above questions do not serve to characterize the probabilistic values p and q, however: these need a questioning about all the n particular cases that are not sorts of cases, so that R A = r1 A ,...,rn A for p and q. ( ) ( ( ) ( )) These valuations also coincide with the set-theoretical definition of implication as inclusiveness. Hence the following Boolean definition (where R A ≠ R B ): ( ) ( ) R A B = R A ⊂ R B , i.e. R A R B or R A R B . ( ∶ ) ( ) ( ) −( ( ) ∩ − ( )) (− ( ) ∪ ( )) This calculus entails two further things. Firstly, implication is such that every yes-case of the antecedent is also a yes-case of the consequent, following the set-theoretical view of implication as total inclusion. That is: A B holds if and only if R A R B = R A and R A R B = R B . ( ∶ ) ( ) ∩ ( ) ( ) ( ) ∪ ( ) ( ) Secondly, the difference between material and formal certainty clearly appears in the above valuation: R εm = 0001 , and R εf = 1111 . Recalling a statement of the previous section,( ) this( makes) apparent( tha) t( a theorem) is certain in the sense of including only yes-answers in its logical value, and not in the sense of being a special mode of truth. Taking theorems (T11) and (T13) again, these respectively mean that the sum or the product of two contradictory terms A and A′ is formally certain or impossible for any logical value of A: R A A′ = 1111 and R AA′ = 0000 for any R A . ( + ) ( ) ( ) ( ) ( ) If this non-Fregean valuation does justice to MacColl’s theorems and its view of theoremhood as formal certainty, it also means that the preceding set Five is largely incomplete: the number of questions and possible answers is 2 such that there should be a total number of mn = 4 = 16 modalities, including the five or three elements MacColl usually brought out in his writings. A presentation of these values can be made by means of the following Hasse diagram [Fig. 5], where each value is classified according to its number of yes-cases. The lines of the Hasse diagram form implicational relations A B from bottom to top, such that the consequent B is above its antecedent ∶A. This diagram also shows that logical truth (⊺) is implied by everything and logical falsehood (⊥) implies everything, together with a number of other theorems like (T20) ε A = Aε, (T21) A η = Aη, (T22) Aε = A and (T23) Aη = η. ∶ ∶ 180 Fabien Schang

0000 R τ ∩ R ι R ηf R 1001 −R θ R -θ (0001)= R(ε) ( )= ( )= (⊥) (1010)= R η( )=∪ R (τ-ε) R η + τ-ε (0010)= R(τ)-ε (1100)= R( η) ∪ R( ιη )= R(η + ι-η ) ( )= ( ) ( ) = R(ι ) ( ) = ( ) = 0100 R ι-η 0111 R(θ) ∪ R ε R θ+ε R π (1000)= R(η ) (1011)= R(η) ∪ R( τ)= R( η +)=τ ( ) (0011)= R(τ)-ε ∪ R ε R τ-ε + ε (1101)= R(ι ) ∪ R (ε )=R (ι + ε ) ( )= R(τ ) ( )= ( )= ( )= ( ) ( )= ( ) 0101 R(ι-)η ∪ R ε R ι-η + ε 1110 − R ε R -ε R u ( )= ( ) ( )= ( ) ( )= ( ( )) = ( )= ( ) 0110 − R η ∪ R ε −R η ∩ 1111 R τ ∪ R ι R εf R ⊺ ( ) = −(R (ε ) R θ( )) = ( ) ( )= ( ) ( )= ( )= ( ) ( )= ( )

Turning again to the logical oppositions between MacColl’s modalities, the corresponding logical values give rise to an algebraic characterization of the four Aristotelian oppositions. Namely: Contrariety: A and B are contrary to each other iff R A R B = 0000 and R A R B ≠ 1111 ( ) ∩ ( ) (Contradiction:) ( A) ∪and( B) are( contradictory) to each other iff R A R B = 0000 and R A R B = 1111 ( ) ∩ Subcontrariety:( ) ( ) A and( )B ∪ are( subcontrary) ( ) to each other iff R A R B ≠ 0000 and R A R B = 1111 ( ) ∩ Subalternation:( ) ( ) B is( subaltern) ∪ ( ) to A( iff )R A R B ≠ 0000 and R A R B ≠ 1111 2 ( ) ∩ ( ) ( ) ( ) ∪ ( ) ( ) In the light of this exhaustive list of ordered answers, it clearly appears that some of the sixteen modalities are produced by meeting or joining some more basic ones: despite their symbolic appearance of simplicity, τ and ι are not basic modalities (with only one yes-case) because they equate respectively with the sum of merely actual truth and necessity or merely actual falsehood and

2. Strictly speaking, this algebraic deﬁnition of subalternation also includes the merely non-contradictory cases that do not form either a subalternation or a subcontrariety relation (see footnote 1). MacColl’s modes of modalities 181 impossibility. Moreover, this complete representation entails that Woleński’s octagon is partial and should be completed by the tetraicosahedron of Alessio Moretti [2009] and its 120 logical oppositions 3 [Fig. 6] (only subalternations and some of the contradictory relations are drawn here, for sake of clarity):

Contraries (18): ε,τ-ε ; ε, ι-η ; ε, η ; ε,θ ; ε, ι ; ε, η τ-ε ; τ-ε, ι-η ; τ-ε, η ; τ-{ε, ι-η }ε ;{ τ-ε,} ι ; {τ-ε,}-θ {; ι-}η, η{; ι}-η,τ{ ; ι+-η, η} τ{-ε ; ι-}η, -{θ ; η,τ} {; η, ι-η + ε}; {η,θ } { } { } { } { + } { } { } { + } { } Subcontraries (17): τ, u ; τ, ι ε ; ι-η ε, u ; θ, η τ ; θ, ι ε ; ι, π ; ι, η τ ; η τ-ε,π ; {η }τ-{ε, ι +ε }; {-θ,π+ ; }-θ,{ u ; +π,} η { τ ;+ }π, ι{ ε}; {π, u+; }η {τ,+ ι ε ; }η { τ,+ u ; ι +ε,} u { } { } { + } { + } { } { + + } { + } { + } Contradictories (8): ε, u ; η,π ; -θ,θ ; τ, ι ; τ-ε, ι ε ; ι-η ε, η τ-ε ; η τ; ι-η ; ⊺, ⊥ { } { } { } { } { + } { + + } { + } { } Subalterns (65): ε,τ ; ε, ι-η + ε ; ε, -θ ; ε,π ; ε, η + τ ; ε, ι + ε ; τ-ε,τ ; τ-ε,θ ; τ-ε,{ η +}τ-{ε ; τ-ε,π} ;{ τ-ε,} η +{ τ ;} τ{-ε, u ; }ι-η,{ ι-η + ε}; {ι-η,θ }; {ι-η, ι ;} ι{-η,π ; ι-η,} ι+ε{ ; ι-η,} u{ ; η, ι ; }η, η{+τ-ε }; {η, -θ ; η, η}+ {τ ; η,} ι +{ε ; }η,{ u ; τ,π} {; τ, η +}τ {; ι-η +}ε,π{ ; } ι-{η + ε, ι + ε} ; { θ,π }; {θ, u ; ι,} ι{+ ε ; ι,} u{ ; η}+{τ-ε, η}+{τ ; η +}τ{-ε, u ; -θ,} η{+ τ ; -θ, ι +}ε{; ⊥}, X{ and} {X, ⊺ }(for{ every} { other modality} {X) } { } { } { } { } Mere non-contradictories (12): τ, ι-η + ε ; τ,θ ; τ, η + τ-ε ; τ, -θ ; ι-η + ε,θ ; ι-η + ε, ι ; ι-η + ε, -θ ; {ι-η + ε, η +} τ {; θ,} ι {; ι, η + τ-ε} ; { ι, -θ}; {η + τ-ε, -}θ { } { } { } { } { } { } { } The diﬀerence between internal and external negation is also made apparent by the oppositions of non-Fregean values: by reference to Piaget

~~3. Note that the modalities ⊺ and stand in a contradiction and subalternation relation at once. ⊥ 182 Fabien Schang~~

[Piaget 1949], these negations correspond respectively to the two distinct operations of reciprocity and inversion in Piaget’s INRC Group. Let R AX = X X X X ( ) r1 A ,r2 A ,r3 A ,r4 A be the general form of the logical value of (a proposition( ) ( A)X . Then( ) we( have) the following characterizations of internal X -X negation A′ and external negation A : ( ) X X( ) X X X Internal negation: R A′ = r4 A ,r3 A ,r2 A ,r1 A ( -X) ( ( )X ( ) X ( ) X( )) X External negation: R A = r1 A , r2 A , r3 A , r4 A ( ) (− ( X) − ( ) − ( ) − ( )) Let us take the example of certainty A = ε, where r1 ε = r2 ε = r3 ε = ( ) ( ) (X ) 0 and r4 ε = 1. Then its internal negation is an impossibility: R A′ = ( ) ( ) r4 ε ,r3 ε ,r2 ε ,r1 ε = 1000 = R η ; whereas its external negation is ( ( ) ( ) ( ) -(X)) ( ) ( ) an uncertainty: R A = r1 ε , r2 ε , r3 ε , r4 ε = 1110 = R u . It is worth noting that( internal) (− negation( ) − ( corresponds) − ( ) − to( a))contrariety( ) -forming( ) operator, whereas external negation is a contradiction-forming operator. This logical difference will be reviewed later, since MacColl made an essential use of internal negation to validate its theorems. However useful this non-Fregean calculus may be with its Boolean understanding of MacColl’s modalities, it gives rise to problems in two other respects: certainty, and denial. The first difficulty concerns the ambiguity of MCL about certainty: although MacColl didn’t make any symbolic distinction between the material and formal sense of certainty, the theorems (T22) Aε = A and (T23) Aη = η cannot be validated with the material sense if one sticks to our non-Fregean valuation. For if so, then R Aε = R A R ε = R A 0001 = 0000 whenever ( ) ( )∩ ( ) ( )∩( ) ( ) r4 A = 0, so that Aε ≠ A; and R Aη = R A R η = R A 1000 = 0000 ( ) ( ) ( )∩ ( ) ( )∩( ) ( ) whenever r1 A = 0, so that Aη ≠ η. Nevertheless, the theorems are preserved if ε is equated( with) formal certainty, and it should be the case from an intuitive standpoint: the above theorems (T22) and (T23) suggest that each product takes the lowest value of its factors, and each sum the greatest; this is in accordance to our preceding Hasse diagram, where the supremum is not a material but a formal certainty. This does not involve any cancellation of material certainty, of course, given that MacColl repeatedly enhanced the opposition between two modes of certainty. A way to reconcile our formal reading of the theorems with the occurrence of material instances is to read the former as a case of relative necessity. Recall that MacColl endorsed the reduction of five to three main modalities (ε,η,θ) whenever the modalities occur as terms of factors. It is because a statement is given to be certainly true if given as true (τ = τ τ = ε), certainly false or impossible if given as false (ι = ιτ = η), and variable if given as variable (θ = θτ = θ). Again, the equivalence between A and Aτ stems from the nature of statement as a truth-claim. But the resulting certainty of τ τ has nothing to do with the probabilistic meaning of material necessity; rather, it has to do with the formal certainty for a statement to be claimed to be true when taken to be true, or the formal impossibility for a statement to be claimed to be true when taken to be false. Thus MacColl argues that: MacColl’s modes of modalities 183

If at the moment the servant tells me that “Mrs. Brown is not at home” I happen to see Mrs. Brown walking away in the distance, then I have fresh data and form the judgment Aε, which, of course, implies Aτ . In this case I say that “A is certain”, because its denial A′ (“Mrs. Brown is at home”) would contradict my data, the evidence of my eyes. [MacColl 1906, 19] It is not the datum that is certainly true, but the linguistic rule to the effect that I state a corresponding proposition to be true once I have evidence for its truth. This rule implicitly obtains behind MacColl’s symbolic language, where no clear reference is made about when a proposition occurs as a neutral propositional function or a statement taken to be true. Our context-sensitive account of certainty also argues for a formal reading of ε, since R τ τ = R τ τ = R τ R τ = 0011 0011 = 1100 0011 = 1111 .( ) ( ∶ ) − ( ) ∪ ( ) −( ) ∪ ( ) ( ) ∪ ( ) ( ) The second difficulty is an ambiguity in the meaning of denial. Does MacColl make an external or internal use of negation, in his definition of implication? Departing from his contemporary symbolic logicians (including Schröder) who view implication as a total inclusion of the antecedent into the consequent, MacColl said that A implies B if and only if it is impossible for η A to be affirmed and B to be denied. In symbols: A B = AB′ . Although the denial of B should refer to the external negation,∶ as it will( be) in the later version of modal logic by Lewis, MacColl makes an essential use of internal negation to establish some of his theorems. An example is the paradox of strict implication, according to which certainty is implied by everything. Thus for η η η every A in MCL, A ε = Aε′ = Aη = η = ε. The question is: why ( ∶ ) ( ) ( ) ( ) does one have ε′ = η, rather than ε′ = -ε = u? While MacColl patently needs an internal use of denial to validate this theorem, he argues for it for vernacular reasons: Some persons might reason, for example, that (. . . ) the denial of a possibility is not merely an uncertainty but an impossibility. A single concrete example will show that the reasoning is not correct. The statement “It will rain tomorrow” may be considered a possibility; but its denial “It will not rain tomorrow”, though an uncertainty is not an impossibility. [MacColl 1906, 15] Two objections can be made to this explanation. On the one hand, Woleński [Woleński 1998] rightly notes that MacColl here confuses possibility and variability: the statement “It will rain tomorrow” is neither impossible nor certain, hence variable (and uncertain, since θ u); its denied form “It will not rain tomorrow” is equally variable, because∶ this statement claims something for which no present datum is available at the time of its utterance. While it is admittedly more difficult to find a natural sentence that expresses possibility without being variable, the point is that the making of a negative statement naturally leads to an internal use of negation: “2+3 equals 5” is a certain statement and “2+3 does not equal 5” is an impossible statement, in the sense that the former is a truth-claim of something given as certain and the latter 184 Fabien Schang is a truth-claim of something given as impossible (by mathematical definition). Therefore, the context-sensitive interpretation of MacColl’s modalities implicitly favors an internal use of negation; but this does not mean that every proposition of MCL occurs as a statement rather than a propositional function without any underlying truth-claim. Rather, a fair definition of implication should be such that no special use of negation should be preferred to the other one for merely vernacular reasons. On the other hand, we suspect MacColl of assuming a material interpretation of certainty in the following objection to implication as total inclusion:

If the statement A B be always equivalent to the statement AB, the equivalence must∶ hold good when A denotes η, and B denotes ε. Now, the statement η ε, by deﬁnition, is synonymous with η ∶ ηε′ , which only asserts the truism that the impossibility ηε′ is (an impossibility) [. . . ] But by their deﬁnition the statement η ε asserts that the class η is wholly included in the class ε . . . Thus, η ε is a formal certainty, whereas η ε is a formal impossibility.( ) [MacColl∶ 1906, 78]

Not only does MacColl’s “truism” rely upon an exclusively internal use of negation such that ε′ = η; but his conclusion is wrong so long as ε is read as formal certainty while denial is applied externally. For then the validity of η ε trivially states that Rη R ε = 1000 1111 = 0111 1111 = 1111∶ . We find here a reason− to∪ adopt( ) a−( purely)∪ set-theoretical ( ) ( logic) ∪ ( of modalities) ( in) terms of class inclusion: MacColl wanted to introduce another definition of implication in order to avoid a supposedly formal impossibility, but he did so by assuming internal negation and for a partial motivation, i.e. the vernacular use of denial upon context-sensitive statements. Rather, our non-Fregean device validates the theorems of MCL by restoring the external use of denial in the definition of implication and favoring a formal sense of certainty for the same reasons as MacColl made a privileged use of internal negation: statements are given as certainly true in a context relative to given data, whereas the material sense of certainty still holds for other theorems of MCL (as with the T-characteristic theorems (T15) Aε Aτ and (T16) Aη Aι). It is at this price of a relative use of denial and modalities∶ that MCL can∶ be streamlined into a genuinely algebraic logic of classes.

~~Conclusion: an algebraized logica utens~~

This paper attempted to throw some new light on MacColl’s modes of modalities, i.e. the way in which the various modes of being true (or false) proceeded in his logical writings. Let us recapitulate our six main theses. 1. MCL is not so much a logic as a symbolic language: following a distinction of Peirce’s, it is not a logica docens, or abstract logic without implicit MacColl’s modes of modalities 185 assumptions, but a logica utens whose results rely upon a pre-theoretical use of its symbols. MacColl’s logic was not so much a closed set of theorems as a list of various implications whose explanation is not given by a totally “blind” calculus but supported by linguistic considerations. The reduction from five to three modalities witnessed this pre-theoretical use of statements as context- sensitive truth-claims. 2. MCL is a modal logic, in the sense that it mainly deals with modes of truth. But MacColl’s modalities don’t proceed like quantifiers: they occur as inclusion relations, and such a modal logic cannot be formalized in the same way as modern modal logics. The difference between iterated modalities and higher degree statements witnessed this crucial difference between modal predicates and modal operators. 3. MCL is not a many-valued logic if, by ‘many-valued’, we mean a non- classical set of theorems that includes further elements beyond truth and falsehood. Rather, it proceeds by a partition within these two basic elements and results in an enlarged set of multiple modes of being true or false. A better name for this logic of partitioned classes could be multiple-valued logic, whose theorems are a modal extension of classical logic but whose non-bivalent valuation is akin to many-valuedness. 4. MCL could be viewed as a non-Fregean logic, where the logical values are not truth-values but ordered answers about generalized quantifiers. The result is a multiple-valued system, doing justice to the usual view that MacColl was a father of both modal and many-valued logic. It also helps: to locate the five initial modalities within a range of sixteen logical values, to introduce a Boolean calculus for the demonstration of MacColl’s theorems and, finally, to take seriously the difference between formal and material certainty as an explicit difference in their corresponding values: (1111) and (0001). 5. Our non-Fregean reconstruction showed that MacColl’s definition of implication in terms of denial and impossibility was superfluous in two respects: on the one hand, a set-theoretical definition in terms of total inclusion fills the bill of obtaining the ensuing theorems of MCL; on the other hand, the use of internal negation can be avoided and replaced by external negation under the proviso that impossibility is read formally. 6. If we are right, then MacColl’s modes of modalities can be streamlined into a pure algebraic logic. Such a result should be in perfect harmony with the theoretical background of MacColl, namely the work of Boole and Schröder, as well as that of John Venn and Louis Couturat. It is commonly said that history is written by its winners; this is equally true for the history of logic, where the victory of Russell’s ideas has largely eclipsed MacColl’s writings but does not annihilate their explanatory value. The same holds in modal logic, where the victory of Kripke’s possible-worlds semantics has totally eclipsed the algebraic writings about modalities by Tarski, McKinsey, Thomason or Lemmon. We hope to have recalled their 186 Fabien Schang

~~relevance in this paper, through the logical legacy of MacColl as a peculiar father of algebraic modal logic.~~

~~Acknowledgements.~~

~~I am very grateful to the three anonymous referees for their helpful comments on and corrections to my present paper.~~

~~References~~

Avron, Arnon 2008 5-valued non-deterministic semantics for the basic paraconsistent logic mCi, Studies in Logic, Grammar and Rhetoric, 14, 127–136. Belnap, Nuel. D. 1977 A useful four-valued logic, in J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, Dordrecht: D. Reidel Publishing Co., 8–37. Blanché, Robert 1953 Sur l’opposition des concepts, Theoria, 19, 89–130. Bochvar, Dmitri A. 1938 On a three-valued calculus and its application to analysis of paradoxes of classical extended functional calculus, Matematičeskij Sbornik, 4, 287–308. Czeżowski, Tadeusz 1955 On certain peculiarities of singular propositions, Mind, 64, 392–395. Dugundji, James 1940 Note on a property of matrices for Lewis and Langford’s calculi of propositions, The Journal of Symbolic Logic, 5, 150–151. Łukasiewicz, Jan 1920 O logice trojwartosciowej, Ruch Filozoﬁczny, 5, 170–171 (English translation: “On three-valued logic”, in M. Borkowski (ed.), Selected Works, Amsterdam: North-Holland, 1970, 87–88). MacColl, Hugh 1897 Symbolic reasoning (II), Mind, 6, 493–510. 1900 Symbolic reasoning (III), Mind, 9, 75–84. 1901 La logique symbolique et ses applications, Bibliothèque du Ier Congrès international de philosophie, 3, Paris : A. Colin, 135–183. MacColl’s modes of modalities 187

1906 Symbolic Logic and its Applications, London: Longmans. McCall, Storrs 1967 MacColl, in P. Edwards and A. Pap (eds), The Encyclopedia of Philosophy, 4, London–New York: Collier-Macmillan, 545–456. Moretti, Alessio 2009 The Geometry of Logical Opposition, PhD dissertation in logic, University of Neuchâtel. Piaget, Jean 1949 Traité de Logique: Essai de Logistique Opératoire, Paris : Armand Colin. Prior, Arthur N. 1955 Many-valued and modal systems: an intuitive approach, The Philosophical Review, 64, 626–630. 1957 Time and Modality, Oxford: Clarendon Press. Rahman, Shahid 1997 Die Logik der zusammenhängenden Behauptungen im frühen Werk von Hugh MacColl, habilitation thesis in philosophy, Universität Saarbrücken. Read, Stephen 1998 Hugh MacColl and the algebra of strict implication, Nordic Journal of Philosophical Logic, 3 (1), 59–83. Rescher, Nicholas 1969 Many-Valued Logic, New York: McGraw-Hill. Russell, Bertrand 1906 Review of Hugh MacColl’s Symbolic Logic and its Applications (London: Longmans, 1906), Mind, 15 (58), 255–260. Schang, Fabien 2011 Questions and answers about oppositions, in J.-Y. Béziau & G. Payette (eds.), New Perspectives on the Square of Opposition, Bern: Peter Lang (forthcoming). Simons, Peter 1998 MacColl and many-valued logic: an exclusive conjunction, Nordic Journal of Philosophical Logic, 3 (1), 85–90. Smessaert, Hans 2009 On the 3D Visualisation of logical relations, Logica Universalis, 3, 212–231. 188 Fabien Schang von Wright, Georg Henrik 1951 An Essay in Modal Logic, Amsterdam: North-Holland. Woleński, Jan 1998 MacColl on modalities, Nordic Journal of Philosophical Logic, 3 (1), 133–140. Was Hugh MacColl a logical pluralist or a logical monist? A case study in the slow emergence of metatheorising Ivor Grattan-Guinness Middlesex University Business School, London, UK

Résumé : Dans la seconde moitié des années 1900, Bertrand Russell et Hugh MacColl échangèrent sans s’entendre sur les questions de l’implication et de l’existence, dans le cadre d’un débat plus général sur la nature de la logique. Il est tentant de voir dans cet échange une opposition entre le moniste logique Russell et le pluraliste MacColl. Dans cet article, j’affirme que cette inter- prétation est inexacte, et que les deux hommes étaient tous deux monistes, bien qu’ayant des allégeances différentes. La transition du monisme au pluralisme ne s’effectue en réalité qu’à partir du début des années 1910, peu après la mort de MacColl en 1909. Les premiers signes de cette transition sont à trouver particulièrement chez le philosophe américain C. I. Lewis, le mathé- maticien néerlandais L. E. J. Brouwer, et le logicien polonais Jan Łukasiewicz. Ces auteurs sont des exemples de l’avènement graduel de la métalogique. Abstract: In the mid- and late 1900s Bertrand Russell and Hugh MacColl had a non-discussion about implication and existence, as parts of a dispute over the nature of logic. We are tempted to see this debate in terms of logical monist Russell against logical pluralist MacColl, but I argue that this interpretation is inaccurate; each man was a logical monist, but with different allegiances. The transition from monism to pluralism began to occur from the early 1910s onwards, soon after MacColl’s death in 1909; early traces will be found especially in the American philosopher C. I. Lewis, the Dutch mathematician L. E. J. Brouwer, and the Polish logician Jan Łukasiewicz. They form examples of the gradual rise of metalogic.

~~Explicanda~~

The word ‘logic’ refers here to the theory (or theories) of correct and incorrect reasoning, and valid and invalid deductions. ‘Logical monism’ is the claim that only one theory, L, deserves to be called logic: all other candidate theories either are misconceived in some way, or are legitimate bodies of knowledge

Philosophia Scientiæ, 15 (1), 2011, 189–204. 190 Ivor Grattan-Guinness but not logic, or are special cases of L. ‘Logical pluralism’ is the stance that allows for many logics, although the pluralist may prefer some over others. Both positions are to be understood to be very general, not just, for example, concerning the difference between a word and its referent. The presence or absence of symbolism in a logic under consideration is a separate issue. The history of logic up to the early 20th century shows that logical monism has usually reigned supreme, with L assigned to some version or other of classical true-or-false time-independent logic (hereafter, ‘C’), usually involving predicates and quantification over variables as well as propositions. Examples include (post-)Boolean algebras, and a mathematical logic including a logic of relations as practised by Russell. MacColl was a pioneer among logicians in wishing to break clear from the dominance of C. Some commentators have seen him as a logical pluralist; they include [Rescher 1974], [Grattan- Guinness 1998], and [Rahman & Redmond 2008] (who elaborate MacColl’s logic in an impressive way, and also achieve the no mean feat of supplying a full bibliography for him). However, I shall argue here that he was a logical monist, adhering to an L that was wider than C. For recent historical commentary, see [Cavaliere 1996], further articles in [Astroh & Read 1998], and [Anellis 2009]. 1 I restrict the account to the last five years of MacColl’s life, including the publication of his logic book [MacColl 1906a] and his discussions with Russell in the philosophy journal Mind. These included Russell’s review [Russell 1907] of the book there with the reply [MacColl 1907], and notes on existence ([MacColl 1905b; 1905c; 1905d], [Russell 1905]) and on implication ([MacColl 1905a; 1906b; 1908a; 1908b], [Russell 1908]). 2 He and Russell also corre- sponded quite extensively during the 1900s, and 26 of MacColl’s letters are preserved in the Russell Archives at McMaster University, Canada [MacColl 1901-1909]. The main points are made several times over in their written and published exchanges; usually one typical reference is given below. While Russell was rather dismissive of MacColl to other correspondents, such as Louis Couturat [Schmid 2001, 499–500] and Philip Jourdain [Grattan-Guinness 1977, 101; 119], at least he kept these letters, which was not always his normal practice with correspondence at that time (for example, Jourdain’s).

~~Russell’s classical position~~

During the period under consideration here Russell was busily working with A. N. Whitehead on the exposition in Principia Mathematica of their logicist programme for deriving (some) mathematics from their L = C while

1. Among histories of logic covering that period, the appropriate parts of [Mangione & Bozzi 1993] and [Grattan-Guinness 2000] are useful for context, but [Haaparanta 2009] is far too brief. 2. [Russell 1905] and four of MacColl’s pieces are reprinted in [Russell 1973]; one hopes that in some year or decade Russell’s papers for the period 1906-1908 will reappear in volume 5 of his Collected Papers. Was Hugh MacColl a logical pluralist or a logical monist? 191 avoiding the paradoxes. Part of the preparation was the formalisation of C itself; Russell reported on progress, especially concerning the propositional calculus, in a paper on ‘The theory of implication’ published as [Russell 1906]. While he emphasised the role in his theory of the logic of Gottlob Frege, which he had studied in detail only from 1902, his approach was still dominated by his chief mentor Giuseppe Peano. In this context the inﬂuence was not a happy one; this paper is the most unclear one by Russell that I have ever read, on any subject. There are two main reasons: that for Russell logic is an all-embracing theory, so that there is no “room” anywhere to talk about it, nor was room sought; and that implication, inference, entailment, consequence, reasoning and deduction tended to be merged, although at least in [Russell 1908] he used implications between names of propositions. Russell followed Frege in taking implication and negation as the primitive connectives, and stressed that the former connective held between propositions of whichever truth-value: ‘Therefore is distinguished from implies by being only applicable to implications between true propositions’ [Russell 1906, 165]. One major consequence, which will come under attack from MacColl, stated that for propositions p and q,

~~‘p implies q’ is logically equivalent to ‘not-p or q’, where the disjunction is inclusive. (1)~~

In fact Russell took (1) in the form of defining ‘p or q’ as ‘not-p implies q’ [Russell 1906, 176]. Here are some further examples of his exegesis. ‘Propositional function’ referred to logical combinations of propositions, such as (p and p) [Russell 1906, 163]; this use was additional to that of referring to predicates fx of individual variables x. The first ‘primitive proposition’, itself a conflation of axiom and rule of inference, stated that ‘Anything implied by a true proposition is true’ [Russell 1906, 164]. His use of ‘anything’, not ‘any proposition’ was deliberate; for him ‘Aristotle implies Aristotle’, for example, was not badly formed or meaningless, but false. Truth-values were “included” in the conception of a proposition; for example not-p was expressed by ‘p is not true’ [Russell 1906, 164]. However, ‘p implies p’ meant ‘for any value of p, p implies p’ [Russell 1906, 163]. The law of excluded middle was ‘p or not-p’; in words, ‘Everything [sic] is true or not true’ [Russell 1906, 167; 177]. The law of contradiction was handled similarly [Russell 1906, 178]. No explicit statement of inference was made; for example, the phrase ‘modus ponens’ never appeared. A closely related proposition was: ‘If p is true, then, if p implies q, q is true’. [Russell 1906, 169] (2) Another candidate is this consequent drawn several pages later, ‘an important principle of inference, which I shall call the “principle of assertion”’ [Russell 1906, 180]: 192 Ivor Grattan-Guinness

‘p is true, and if p is true then q is true, then q is true’. (3) There was plenty here to puzzle the readers (and the audience of Principia Mathematica from 1910 onwards), although at that time some of these conﬂa- tions were also committed by other logicians, including by MacColl. However, his reaction against the status that Russell gave to C was sweeping.

~~MacColl’s claims~~

MacColl’s two main points of disagreement concerned existence, and affirming or rejecting (1) about implication [MacColl 1908b]; other features that Russell affirmed and MacColl at least doubted included the need for propositional functions, relations and quantification in logic, treating a proposition as a truth-bearer rather than just as a form of words, and the legitimacy of non- Euclidean geometries ([MacColl 1907, 1901-1909, letters of 26 January 1905, 3 February 1905]). Let us note some main features. MacColl insisted that propositions should not be classifiable only as true and false ones but also be divisible into necessary and possible ones. This brought him into territory that we now recognise as modal logic, and consti- tutes his principal claim to fame. He wished to restrict implication between propositions to those that exhibited some semantic or circumstantial connection; this position also drew him towards relevance logic. In [MacColl 1908a] he used as an example the propositions: p := ‘He is a doctor’ and q := ‘He is red-haired’ (4) and denied that ‘p implies q’ could obtain between them. Russell did not deny implication, since under the conception of logic that he inherited from Peano and found backed in this respect by Frege implication satisfied (1) and so could be asserted between any two well-formed propositions whether or not they exhibited any connection. In [Russell 1908], he offered to MacColl the option of asserting that formal implication held between the corresponding propositional functions by treating ‘he’ as a variable; but this move involved another point of dissent, since for MacColl ‘he’ was a parameter, and propositions such as those given in (4) were sometimes true and sometimes false. He showed no appreciation of quantification over individuals, propositional functions, relations or propositions, all of which were key to Russell’s logic. The two propositions in (4) are also examples of MacColl’s fifth category, ‘variable’ propositions, which are possible but uncertain; an example was: ‘Mrs. Brown is not at home’. (5) He associated this kind of proposition with probability theory [MacColl 1906a, 7], in that one might assess the likelihood of Mrs Brown’s residence at any given time. It also brought him towards what we now regard as epistemic logic, for he stressed that such propositions would have a truth value but we did not know which ([MacColl 1906a, 19] for (5)). For [Russell 1907], (5) Was Hugh MacColl a logical pluralist or a logical monist? 193 should be converted to the propositional function ‘Mrs. Brown is not at home at time x’. Later he defined a propositional function fx to be necessary or possible according as it is satisfied respectively by all pertinent values of x or by at least one of them [Russell 1919, 163]; he used modality in a few other contexts, though always in the confines of C [Dejnozka 1999]. MacColl missed an opportunity to strengthen his case against L = C by also interpreting (5) as belonging to temporal logic, even though he mentioned that (5) could be true in the morning and false in the afternoon [MacColl 1907, 470]! Thus he helped to sustain an enormous oversight in the history of the fight for logics beyond C. To explain, take one frequently encountered context that C cannot handle: we often make statements about taking sequences of actions and decisions in some order in time, with ‘and’ meaning ‘and then’. For example, if I fulfil my promise to the court to tell the truth, the whole truth and nothing but the truth, and I assert of the plaintiff that ‘he walked down the stairs and opened the door’, I definitely do not assert that ‘he opened the door and walked down the stairs’. Now this difference belongs to temporal logic, which differs from C in not being commutative. The long history of over-looking temporal logic still has to be recorded; we note here the irony that MacColl belongs to it. 3 MacColl and Russell also differed on the theory of collections, in that Russell used Cantorian set theory while MacColl seemed to draw upon traditional part-whole theory [MacColl 1905d]; the main reason for Russell’s advocacy of set theory was its central role in logicism, whereas MacColl had no special plan or place for mathematics in his exposition of logic. Their disagreements did not hinge upon this difference of aim. More problematic for Russell, and indeed also for Shearman and every reader, was MacColl’s handling of classes: for example, misattributing the predicate ‘existent’ as a class in [MacColl 1905b]. Also unfortunate was his use of ‘0’ to identify the class of ‘unrealities’, that is, of individuals that do not exist, such as mermaids ([MacColl 1905a] & [MacColl 1906a, 42–43]); for ‘0’ had long been identified with the empty class (or set). Russell’s reaction is quite reasonable [Russell 1905], though compromised by his own overuse of ‘existence’ as a predicate, with a variety of senses that sometimes contradict each other: for him an individual could exist 1) as in existential quantification, or 2) as the referent of a denoting phrase (an important special case of 1)); and a Cantorian set could exist as in 3) the existential quantification of sets, or 4) if associa- ble with a propositional function, or 5) in being non-empty! In particular,

3. On the history of temporal logics see [Øhrstrom & Hasle 2006a] & [Øhrstrom & Hasle 2006b]. We distinguish temporal logics from the expression within C of temporal order of events in terms of ordered sets—an exercise of which, curiously, is pursued in [Russell 1936]. W. V. Quine, a famed logical monist for C, seems to fail to make the distinction in his treatment of tense in [Quine 1986, 30–31], the only attention paid to time in this book on the ‘philosophy of logic’. His criticisms elsewhere of use-mention muddles infecting the understanding of implication are often justiﬁed independently of the issue of monism or pluralism. 194 Ivor Grattan-Guinness the empty set existed in the ﬁrst two senses but not in the third [Grattan- Guinness 1977, 70–74].

~~Assessments~~

The exchanges attracted a little attention from others at the time. Shearman was critical of MacColl, though not very profitably ([Shearman 1906a] & [Shearman 1906b, chap. 5]). Jourdain was judicious in his appraisal of the bearing upon mathematics of MacColl’s contributions to logic, which MacColl read and accepted shortly before his death in 1909; however, Jourdain had quoted from Russell much more than from MacColl [Jourdain 1912]. Couturat took Russell’s side in a letter of March 1906 to Peano [Luciano & Roero 2005, 102–103], who even omitted MacColl from his bibliography of logicians [Peano 1908, xxiv–xxxvi]. At this distance of time it looks as if MacColl was foraging into various non-classical logics; but there is a fundamental difference between his and modern conceptions of such activity. Today logical pluralists see themselves as shopping around in the various stores of the logics market [Beall & Restall 2005]; but I am sure that MacColl was extending his preferred logic L into a department store, with one floor occupied especially by modal logic and other floors by some forms of C, relevance and probability logics. Presumably he saw the danger that his jumbo logic would be inconsistent, but felt safe if its constituent sub-logics were kept apart. While in his letters to Russell he hoped for reconciliation of their views, in his criticisms he did not see Russell prosecuting a logic different from his own but doing some things wrong in logic: ‘the whole subject of formal logic needs recasting’ [MacColl 1908b, 454], not supplementing with alternatives to C. He was seeking for replace- ments within L and additions to it, not attempting to choose between C and competitor logics. As they both recognised, the chief difference between the two men lay in the area of deduction, and rested on the status of modality. The centre of his department store was the boutique for the necessary and the possible. In the cases of both ‘implication’ ‘If A then B’ between propositions A and B and ‘inference’ (logical consequence) between them, he saw the conjunction of A and ‘the denial of B’ as an impossibility, not just as a falsehood as Russell read it (for example, [MacColl 1906a, 7–8; 80–83]). Such a stance is not pluralism as we now understand it but a monism, albeit of a scope wider than Russell’s where there is a pares among which modality is primus. Further, the clarity of discussion was handicapped on both sides by the absence of a hierarchy of theories and languages in which the various issues and stances can be sited. When, then, did logical pluralism emerge, and with whom? Was Hugh MacColl a logical pluralist or a logical monist? 195 The emergence of logical pluralism

While MacColl and Russell were disagreeing in the late 1900s, C. I. Lewis was preparing a doctoral thesis in philosophy at Harvard University under the direction of Josiah Royce; he completed it in 1910 and soon secured a post at the University of California at Berkeley. He had taken Royce’s lecture course in logic, and from 1912 he started quickly to produce papers in the subject, publishing in Mind or in American journals in philosophy. His chief target was the newly published first volume of Principia Mathematica, of which Royce had given him a copy; like MacColl he disliked the treatment there of implication. 4 The opening sentence of his first paper, published in Mind, referred to ‘two somewhat startling theorems’ in ‘the algebra of logic’, as he called the traditional approaches based upon C: ‘(1) a false proposition implies any proposition, and (2) a true proposition is implied by any proposition’ [Lewis 1912, 522]. He named as ‘extensional disjunction’ the logically equivalent formulation (1) of implication in terms of negation and disjunction, and contrasted it with its ‘intensional’ counterpart, for which (1) did not hold; in order to highlight the difference he named as ‘strict’ the intensional kind [Lewis 1912, 524; 526]. These moves sound like pluralism, but in fact he was monist, for he made this analogy with geometry: ‘The present calculus of propositions is untrue in the sense in which non-Euclidian geometry is untrue’ [Lewis 1912, 530], which was not the pluralist understanding of geometries then normally held by mathematicians. In [Lewis 1913a], he was still more severe about C: ‘Not only does the calculus of implication contain false theorems, but all its theorems are not proved’ [Lewis 1913a, 242]. Clearly Lewis was in MacColl territory; presumably he got there independently, for only in [Lewis 1913b, 430] did he note MacColl’s work in a short footnote. Later he mentioned MacColl’s ‘highly complex system’, where for him ‘the fundamental symbols represent propositional functions rather than propositions’ [Lewis 1918, 108]; MacColl would not have been pleased. A new paper for Mind is crucial for our concerns [Lewis 1914b]. Lewis began by recalling from its predecessor the analogy with Euclidean and non- Euclidean geometries, in the same monistic spirit. But in the next paragraph he suddenly went pluralist: ‘The relation of these two [calculi of propositions] sufficiently resembles that of a Euclidean and a non-Euclidean geometry to make the analogy worth bearing in mind. Like two geometries, material and strict implication are equally self-consistent mathematical systems; but they apply to different worlds’ [Lewis 1914b, 240–241], with the tradition relegated to non-Euclidean status and preference given to strict implication, which ‘has a wider range of applications’ [Lewis 1914b, 241]. Thereafter Lewis remained pluralist; in particular, in his textbook on logic he presented the chapter on ‘the calculus of strict implication’ as concerned with ‘two entirely different

4. Interestingly, Lewis did not question the role of C in logicism in his admiring review [1914a] of the second volume of Principia Mathematica (1912). 196 Ivor Grattan-Guinness meanings of “implies”’ [Lewis 1918, 292]. In his own reminiscence of those early years as a logician he recalled a reluctant pluralist, although he did make an analogy with Henri Poincaré choosing a geometry; interestingly, he used the names ‘ “metalogics” or “pseudo-logics” ’ as synonyms, a known though infrequent use of the former word [Lewis 1930, 41–43]. 5 Another attack on the dominance of C had been made at this time in the doctoral thesis of L. E. J. Brouwer [Brouwer 1907]. On rather mysterious grounds embedded in a metaphysics about the flow of time and the importance of languagelessness he banned the law of excluded middle from mathematics and worked his own version of constructive mathematics (and logic), which he called ‘intuitionism’. In terms of his later definitive position on logic and mathematics this was an indeterminate position in which, for example, the infinite numbers of Cantor’s second number-class were admitted but the class as such not. As his context was quite separate from the concerns of MacColl and Russell ([van Dalen 1999, chap. 3] & [Hesseling 2003, chap. 2]), the details need not delay us here; but it is worth noting the coincidence in timing of another case of logical pluralism—or, perhaps for Brouwer himself in his most polemical passages, logical monism affirming intuitionistic logic as L. Very gradually a few non-classical logics began to develop during the 1910s and the 1920s. Nicolai Vasiliev introduced his ‘imaginary logic’ from 1910, Brouwer continued with intuitionism in his own eccentric way, Lewis formulated several different modal logics and compared them with each other from 1912, Jan Łukasiewicz helped to launch many-valued logics from 1917 [Mangione & Bozzi 1993, 465–487]; elsewhere, for example, quantum mechanics was to breed a logic of its own. Opposition was quite vigorous, and for a very long time; 6 but how could a discussion about logic(s) take place at all? The ground floor of the home of logics was extending with the construction of new rooms; but did not this bungalow need an upper storey?

~~The slow recognition of metalogic~~

Another feature of young Brouwer’s thesis is the subject of this final section: in order to discuss logical monism and pluralism properly we need metalogic as a “place” in which we can talk. 7 As we have seen, such mouthroom 5. On Lewis’s logic see [Parry 1968] & [Murphey 2005, chap. 3]; however, neither author explicitly raises the issue of monism and pluralism. On Lewis’s next stage concerning possible worlds, see [Sedlar 2009]. In 1920 he came back to Harvard University, where he had Whitehead as colleague for two decades from 1924; some aspects of Whitehead’s philosophy influenced him, but they do not seem to have interacted over logics [Murphey 2005, chaps. 4–7]. 6. For a personal witness of the expression of dissent by logical monists over [Rescher 1974] on MacColl, see [Grattan-Guinness 1998, 11–12]. 7. It is not practical to cite all the original sources in this brief survey. The appropriate parts of [Grattan-Guinness 2000], [Haaparanta 2009] & [Gabbay & Woods 2009] contain much pertinent information and many further references. Was Hugh MacColl a logical pluralist or a logical monist? 197 was painfully lacking in both MacColl and Russell, and Lewis did not (yet) perceive the need for it either. However, Brouwer explicitly formulated ‘mathematics of the second order, which consists of the mathematical consideration of mathematics or of the language of mathematics’ [Brouwer 1907, 61]; the corresponding extension of logic is not far away, for his metamathematics itself was classical. Brouwer was partly influenced by the first phase of David Hilbert’s programme of metamathematics, which lasted from 1898 to around 1905. The place of metatheorising was actually one of its weaker sides, for he had got his arithmetic and his logic somewhat intertwined. But the second phase, which started in 1917, was far more thrusting, and highlighted metathinking about mathematical theories. Throughout he relied upon C for his logic. Another important source of metalogic lay in Polish logic. Already in 1913 Łukasiewicz published, in German, a short book on the philosophical foundations of probability theory. Seeking an objective interpretation of probability, he plumped on the truth-values of propositions determined from a propositional function (or ‘indefinite proposition’) fx by assigning specific values to x. The proportion of true judgements that result gave the probability value; in particular, if only true judgements are obtained, then the indefinite proposition is true; if never, it is false. Thus, as an offshoot of this approach to probability, a propositional calculus was developed in which truth values of propositions played a central role, as a ‘calculus of truth-values’ [Łukasiewicz 1913, 16–18; 20–23]. This was a somewhat marginal though nevertheless direct example of the importance of metalogic for the propositional calculus. In the following years Łukasiewicz and some other Polish logicians (for example, Stanislav Lesniewski and Leon Chwistek) became sensitive to metalogic, seemingly via logics themselves, through model theory, and/or sorting out syntax from semantics in natural or formalised languages. The most famous manifestation was to be Alfred Tarski’s semantic definition of the truth of a proposition of a formal language in terms of satisfaction in a metalanguage (his term), which was published in the early 1930s. Tarski lectured in Vienna in February 1930, and seems to have alerted his friends in the Vienna Circle to the fundamental importance of metatheory. 8 Thus in 1931 Rudolf Carnap took over the word ‘metalogic’ from the normal meaning that we saw Lewis use at exactly this time, and referred it instead to the logic of logic, the sense that soon became customary. Carnap had been motivated by Kurt Gödel’s newly proved first

8. Despite substantial historical literature such as [Wolenski 1989] on the Poles, [Luschei 1962] on Lesniewski, [Feferman & Feferman 2004] on Tarski, [Stadler 2001] on the Vienna Circle, and [Szaniawski 1989] & [Wolenski & Köhler 1999] on the links between the two communities, the histories during the 1920s of their recognitions of metalogic/theory/language as central concerns are still somewhat obscure; in giving priority to the Poles I am somewhat inﬂuenced by [Menger 1994, chap. 12] and the letters in [Tarski 1992]. 198 Ivor Grattan-Guinness

theorem on the incompletability of first-order arithmetic, where the distinction between logic and its metalogic plays an essential role. The absence of Russell from these developments is ironic. We saw above that the distinction was lacking from his logic; however, when in 1921 he wrote his introduction to Ludwig Wittgenstein’s Tractatus he rejected the distinction made there between showing and saying and proposed instead ‘that every language has, as Mr. Wittgenstein says, a structure concerning which, in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit’ [Russell 1922, xxii]. This was one of the highlights of his philosophical career; sadly, he did not notice, and in particular did not apply his insight to perceive the need for a corresponding hierarchy of logics and their metalogics. So it played no part in his preparation in 1923 and 1924 of the new material for the second edition of Principia Mathematica; hence, when the new material appeared in 1925, the Harvard logician Henry Sheffer justifiably pointed in his review to the ‘logocentric predicament’ that in talking about logic one had to use logic, a conundrum that he could not resolve [Sheffer 1926]. 9 Similarly, Russell himself never understood Gödel’s first theorem, always misstating it as applicable to all mathematical theories and especially thinking that his hierarchy of languages solved the ‘puzzle’ that he thought it posed rather than being required to allow the theorem to be stated in the first place. 10 He always remained the philosopher of one all-embracing logic that he had been in the 1900s when disagreeing with another logical monist, Hugh MacColl.

~~References~~

Anellis, Irving N. 2009 Russell and his sources for non-classical logics, Logica Universalis, 3, 153–218. Astroh, Michael & Read, Stephen (eds.) 1998 Hugh MacColl and the tradition of logic, special issue of the Nordic Journal of Philosophical Logic, 3 (1), Dec. Beall, J. C. & Restall, Greg 2005 Logical Pluralism, Oxford: Oxford University Press.

9. Sheﬀer’s colleague Lewis did not raise any issues about metalogic, or even about implication, in his extensive review of the second edition of Principia Mathematica [Lewis 1928]. 10. For evidence from his nineties see [Russell 1971], and his letter of 1963 published in [Grattan-Guinness 2000, 592–593] (see also [Grattan-Guinness 2000, 542; 565]). Russell seems to have ignored Gödel’s second theorem. Was Hugh MacColl a logical pluralist or a logical monist? 199

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Mangione, Corrado & Bozzi, Silvio 1993 Storia della Logica da Boole ai nostri Giorni, Milan: Garzanti. Menger, Karl 1994 Reminiscences of the Vienna Circle and the Mathematical Colloquium, edited by L. Golland, B. F. McGuinness & A. Sklar, Dordrecht: Kluwer. Murphey, Murray G. 2005 C. I. Lewis: The Last Great Pragmatist, Albany: State University of New York Press. Øhrstrom, Peter & Hasle Per 2006a A. N. Prior’s logic, in D. M. Gabbay & J. Woods (eds.), Logic and the Modalities in the Twentieth Century, Amsterdam: North-Holland, 399–446. 2006b Modern temporal logic: the philosophical background, in D. M. Gabbay & J. Woods (eds.), Logic and the Modalities in the Twentieth Century, Amsterdam: North-Holland, 447–498. Parry, W. T. 1968 The logic of C. I. Lewis, in P. A. Schilpp (ed.), The Philosophy of C.I. Lewis, La Salle, Illinois: Open Court, 15–54. Peano, Giuseppe (ed.) 1908 Formulario Mathematico, 5th ed., Turin: Bocca. Quine, Willard V. 1986 Philosophy of Logic, 2nd ed., Engleﬁeld Cliﬀs, NJ: Prentice Hall. Rahman, Shahid & Redmond, Juan 2008 Hugh MacColl and the birth of logical pluralism, in D. M. Gabbay & J. Woods (eds.), British Logic in the Nineteenth Century, Amsterdam: North-Holland, 533–604. Rescher, Nicholas 1974 Russell and modal logic, in N. Rescher et al. (eds.), Studies in Modality, Oxford: Blackwell, 85–96 (reprinted in G. Andrews (ed.), The Bertrand Russell Memorial Volume, London: Allen & Unwin, 1979, 139–149). Russell, Bertrand A. W. 1905 The existential import of propositions, Mind, 14, 398–401. 1906 The theory of implication, American Journal of Mathematics, 28(2), 159–202. 202 Ivor Grattan-Guinness

1907 Review of Hugh MacColl’s Symbolic Logic and its Applications (London: Longmans, Green and Co., 1906), Mind, 15, 255–260. 1908 ‘If’ and ‘imply’: a reply to Mr. MacColl, Mind, 17, 300–301. 1919 Introduction to Mathematical Philosophy, London: Allen & Unwin. 1922 Introduction, in L. Wittgenstein, Tractatus Logico-Philosophicus, London: Allen & Unwin, 7–23. Quoted from the 2nd ed., London: Routledge & Kegan Paul, 1961, ix–xxii. 1936 On order in time, Proceedings of the Cambridge Philosophical Society, 32, 216–228. 1971 Addendum to my “Reply to criticisms”, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell, 4th ed., La Salle, Illinois: Open Court, xvii–xx. 1973 Essays in Analysis, edited by D. P. Lackey, London: Allen & Unwin; New York: Braziller. Schmid, Anne-Françoise (ed.) 2001 Bertrand Russell: Correspondance sur la philosophie, la logique et la politique avec Louis Couturat (1879-1913), 2 vol., Paris: Kimé. Sedlar, Igor 2009 C. I. Lewis on possible worlds, History and Philosophy of Logic, 30, 283–291. Shearman, Arthur T. 1906a Mr. MacColl’s views on logical existence, Mind, 15, 143–144. 1906b The Development of Symbolic Logic: a Critico-Historical Study of the Logical Calculus, London: Williams & Norgate. Sheffer, Henry M. 1926 Review of A. N. Whitehead & B. Russell’s Principia Mathematica (1, 2nd ed., Cambridge: Cambridge University Press, 1925), Isis, 8(1), 226–231. Stadler, Friedrich 2001 The Vienna Circle, Vienna, New York: Springer. Szaniawski, Klemens (ed.) 1989 The Vienna Circle and the Lvov-Warsaw school, Dordrecht: Kluwer. Tarski, Alfred 1992 Drei Briefe an Otto Neurath, in R. Haller (ed.), Grazer philosophis- che Studien, 43, 1–32. Was Hugh MacColl a logical pluralist or a logical monist? 203 van Dalen, Dirk 1999 Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, 1, Oxford: Oxford University Press. Wolenski, Jan 1989 Logic and Philosophy in the Lvov-Warsaw School, Dordrecht: Kluwer. Wolenski, Jan & Köhler, Eckehart (eds.) 1999 Alfred Tarski and the Vienna Circle, Dordrecht: Kluwer.

~~MacColl’s elusive pluralism John Woods University of British Columbia, Vancouver, Canada & King’s College, London, UK~~

Résumé : MacColl a été récemment l’objet de trois intéressantes thèses. D’abord, il serait le probable père du pluralisme en logique. Ensuite, son pluralisme porterait un instrumentalisme sous-jacent. Enfin, les deux thèses précédentes expliqueraient l’oubli dans lequel il serait tombé après 1909. Bien qu’il soit à la fois pluraliste et instrumentaliste à certains égards, je suggèrerai qu’il est difficile de trouver dans les écrits de MacColl un pluralisme qui puisse satisfaire les trois thèses précédentes en apparaissant pour la première fois chez MacColl, en trouvant ses sources dans un instrumentalisme adapté à la logique, et en étant l’explication à l’oubli dans lequel était tombé son auteur. Abstract: MacColl is the recent subject of three interesting theses. One is that he is the probable originator of pluralism in logic. The other is that his pluralism expresses an underlying instrumentalism. The third is that the first two help explain his post-1909 neglect. Although there are respects in which he is both a pluralist and an instrumentalist, I will suggest that it is difficult to find in MacColl’s writings a pluralism which honours the threefold attribution of having been originated by him, having been rooted in an instrumentalism adapted to logic, and being the occasion of his neglect.

The writer of this paper would like to contribute his humble share as a peacemaker between the two sciences, both of which he profoundly respects and admires. He would like to deprecate all idea of aggression or conquest. [. . . ] Why should not logicians and mathematicians unite [. . . ] under some common appellation? [MacColl 1880, 47] In Logic, there are no morals. Everyone is at liberty to build his own logic, i.e., his own form of language, as he wishes. [Carnap 1937, 52] We are pluralists about logical consequence. [. . . ] We hold that there is more than one sense in which arguments may be deductively valid, that these senses are equally good, and equally deserving of the name deductive validity. [Beall & Restall 2001, 1]

~~Philosophia Scientiæ, 15 (1), 2011, 205–234. 206 John Woods The birth of pluralism?~~

In their authoritative “Hugh MacColl and the birth of logical pluralism”, Shahid Rahman and Juan Redmond observe that “MacColl’s philosophy is a kind of instrumentalism in logic which led him to set the basis of what might be considered to be the first pluralism in logic” [Rahman & Redmond 2008, 539–540], thus echoing a suggestion of Ivor Grattan-Guinness to the same effect [Grattan-Guinness 1998]. 1 Pluralism has become a hot-button issue in the philosophy of logic. Not only was it the official theme of a meeting not too long ago of the Society for Exact Philosophy 2 and of the Tartu Conference in 2008, 3 but approving monographs now flow from the best university presses [Beall & Restall 2006]. If there were a better time than now to reflect on MacColl’s importance for logical pluralism, I would be hard-pressed to name it. Seen Rahman and Redmond (henceforth: R & R)’s way, MacColl’s pluralism embeds a philosophical component and a linguistic component. The philosophical component is logical instrumentalism. Instrumentation also has two parts. On the one hand, it is a goal-sensitive doctrine according to which the tenability of a system of logic is a matter of how well it produces the out- comes for which it was designed. On the other hand, there are different yet equally legitimate purposes which a logic can be made to subserve. The linguistic part of MacColl’s pluralism is captured by the idea that a logic’s formalism will contain classes of expressions bearing variable interpretations that facilitate the realization of logic’s shifting goals. Thus we read that: there is nothing sacred or eternal about symbols; [. . . ] all symbolic conventions may be altered when convenience requires it, in order to adapt them to new conditions or to new classes of problems. [MacColl 1906, 1] With MacColl we have one of those occasions with which intellectual history is littered. In his life-time, MacColl’s work was known to most of the major logicians of his day, with a number of whom he had a profitable correspondence. But with his death in 1909, recognition of the importance—indeed of the fact—of his writings ceased. What explains this regrettable collapse, this consignment to oblivion? R & R offer two suggestions. One is that since MacColl lacked the technical tools that lifted early twentieth century mathematical logic to its first maturity, he was unable to make good on many of

1. I. Grattan-Guinness now recants MacColl’s pluralism, as witness his contribution to this volume. 2. See the proceedings of the meeting published as [Woods & Brown 2001]. See in particular four of its chapters: [Beall & Restall 2001], [Priest 2001], [Woods 2001], and [Sher 2001]. 3. Conference on Logical pluralism, University of Tartu, August 27-31, 2008, with a bevy of notables, ranging from Beall to Westerståhl. Papers from the conference are now starting to appear in print. See, for example, [Field 2009]. MacColl’s elusive pluralism 207 his most important insights. The other is that he “was probably the first to explicitly extend conventionalism and instrumentalism to logic” [Rahman & Redmond 2008, 534], which, they say, is “the basis of what might be considered to be the first pluralism in logic” [Rahman & Redmond 2008, 539–540]. And they add: “Unfortunately most of [MacColl’s] ideas were not developed thoroughly and many others remained in a state which makes them difficult to grasp” [Rahman & Redmond 2008, 540]. Accordingly, we have from R & R three interesting claims to keep an eye on. THE PLURALISM THESIS: MacColl is the probable originator of pluralism in logic. THE INSTRUMENTALIST THESIS: This pluralism is embedded in and is an expression of MacColl’s adaptation of instrumentalism to logical theory. THE NEGLECT THESIS: MacColl’s neglect by the post-1909 waves of modern symbolic logic, even the branches to which he made pioneering contributions, is explained, in part at least, by this instrumentalism and the pluralism which inheres in it. These observations of R & R’s provide me with the wherewithal to box my own compass and to chart my own course. This is the course of interpretation and assessment. I want to know how to understand the Rahman and Redmond- theses (henceforth: RR-theses) and I want to know whether they are true. It will not be clear sailing. Trouble threatens, partly MacColl’s doing and partly R & R’s. The first difficulty is occasioned by the fragmentary and ill- explained state of MacColl’s own pronouncements. The other flows from the fact that MacColl’s pluralism and instrumentalism are not self-ascribed but rather are attributed by R & R. They say that MacColl originates our concept of pluralism and instantiates our concept of instrumentalism. There is a further challenge to be quickly considered. It arises from the so-called (and misnamed) principle of charity [Scriven 1976]. Charity bids me to interpret the RR-theses in ways that give them a fair shot at being true. But the RR-theses are claims about MacColl. Charity also requires that we read MacColl in ways that give his views a fair shot of being true. Where, then, does my first duty lie? Must I read MacColl with a view to making MacColl right or with a view to making R & R right? Let it not be supposed that these are always the same readings. For if that were the case, the only ways in which it would be permissible to read MacColl would be ways that give pluralism and instrumentalism themselves a fair chance of being true. But that’s too much front-loading for serious consideration here. What I want to know is not whether MacColl’s views are correct, but rather whether R & R’s characterizations of them are correct. This tells us something interesting about the charity principle. It triggers a tension when pairs of texts or parties are involved, with the one interpreting the other. 4

~~4. Of course, one of the main challenges to the charity principle is determining how much slack to cut in order to give an utterance or text a fair shot at coming 208 John Woods Modus operandi~~

Here is how I propose to proceed. I will try to determine whether there are versions of present-day pluralism and instrumentalism which offer some promise of confirming the appropriately adjusted RR-claims. That is, I want to see whether there is a modern pluralism shaped by a modern instrumentalism which would have been neglected by the very people (whoever they turn out to be; see below) who, according to the Neglect Thesis, were the actual neglectors of MacColl’s own instrumentally shaped pluralism. I will then try to determine whether grounds exist for attributing that pluralism and that instrumentalism to MacColl. I don’t want to minimize the difficulties that lie ahead. Our pluralism and our instrumentalism are not “free on board”. They are themselves unruly families of not always compatible views, ranging in interest, plausibility and complexity from not very to very very. There are more ways of being a pluralist about logic, and an instrumentalist too, than I have space for here. I want, if I can, to subdue some of this abundance by operating at levels of abstraction consistent with the achievement of the paper’s goals. The pluralistic and instrumentalistic themes that I will sound in the coming sections will be discernible for the most part to anyone familiar with the 20th century developments in logic. So, to the extent possible, I will flesh them out with a minimum of specific attribution. While the principal focus is present-day pluralism, at various places I will cut to MacColl, or segue to instrumentalism, when doing so will assist in testing the RR-claims. A slightly more stand-alone discussion of instrumentalism will occupy us in the “Instrumentalism” section below. In the next section I’ll be mainly concerned with house-cleaning. I want to sweep away some of the forms of pluralism which, while not devoid of interest, are not of much moment for present purposes. For recall, we are searching for a pluralism that MacColl invented and which expresses—or is shaped by—an instrumentalism of a kind that would explain its author’s neglect.

~~Many-one pluralism~~

MacColl’s contributions to logic are striking. He is an original and protean thinker, multi-tasking outside the box and beyond the reach of his technical powers. His pioneering work ranges far and wide, and his writings anticipate some of the most important developments in the work that was yet to come. In his writings we have the modal systems of strict implication, we have many- valued logic, relevant logic, and connexive logic, and we have probability logic. In one sense, this is already a kind of pluralism—one that answers Graham out true. There isn’t time for that here. For a recent discussion of how the charity principle does not work, see [Paglieri & Woods 2009]. MacColl’s elusive pluralism 209

Priest’s question, “Logic: one or many?” with “Many” [Priest 2001] . MacColl as we might say, is a many-one pluralist. Less clear is whether this is the pluralism intended by the passage on the alteration of symbolic conventions, quoted above from MacColl’s Symbolic Logic. Less clear than that is the reality of the connection to instrumentalism. Less clear still is the purported bearing on MacColl’s neglect. Consider a case. There is a well-known dispute in logical theory as to when a formalism achieves the status of logic. For most of his philosophical career, Quine was of the view that nothing is a system of logic beyond first-order quantification theory with identity [Quine 1970]. 5 Accordingly, the S systems of modal “logic” might be linguistic or philosophical theories of necessity, but they are not logics; and the FDE family of relevant “logics” might be theories of inference or belief revision but they aren’t logics either. 6 The divide between classical and non-classical logics is often marked by a dispute about which classes of expressions qualify as logical particles. 7 This in turn gives rise to a certain kind of pluralist/unitarian tension. If we hold with Quine that only the usual logical particles 8 under their classical interpretations are genuinely logical terms, then we are logical-term unitarians. But if we find against Quine, if we allow that the alethic modal ‘ ’ and the epistemic modal ‘K’ are also logical particles or that ‘∼’, ‘ , ‘ ’, ‘⊃◻’ and ‘≡ ’ non-classically interpreted also qualify, then we are logical-term∧ ∨ pluralists. Does being a unitarian about logic require that we reject instrumentalism? Does it require that we eschew the idea that “symbolic connections may be altered . . . in order to adapt them to new conditions or to new classes of problems”? Quine’s unitarianism is a substantial discouragement of these suggestions. Quine’s unitarianism arises from the thesis that there is just one task for logic legitimately to perform, namely, to provide a philosophically coherent accommodation of the language of science. If logic had other good purposes—such as the formal analysis of necessity—then Quine’s instrumentalism would have bade him to get on with it. Quine’s unitarianism flows not from an anti-instrumentalism but rather from the philosophical conviction that there is nothing in what logic is needed for beyond the provisions of first-order quantification theory. Similarly, Quine’s concomitant pledge to the canonicity of first-order formalisms counts for nothing against the point that formalisms

5. Let’s be careful. Quine’s views on logic allow for this possibility. But his philosophical views are not much at ease with it. On the other hand, the present position is explicitly held by Harman [Harman 1972]. 6. For different versions of the view that implication and inference require different treatments, see [Woods 1968], [Woods & Walton 1972], [Harman 1986, chap. 1], [Apostoli & Brown 1995] and [Brown 2002]. 7. Concerning which, even among the classically-minded, there is a notable defi- nitional uncertainty. See [Tarski 1956], [Peacocke 1976], [Hacking 1979] and [Gabbay 1994]. 8. Or some one of them to which the others are reducible by way of the functional completeness metatheorem. 210 John Woods can be adjusted when the conditions warrant it. It proceeds rather from the philosophical conviction that, as it happens, conditions don’t warrant it. 9 What now of the suggestion that MacColl is the (probable) originator of pluralism in logic? As we see, there is a perfectly good sense of pluralism, which MacColl’s work certainly instantiates, for which the originality thesis won’t hold up. In this sense, Aristotle himself was logic’s first pluralist, what with his non-modal syllogistic, his logic of immediate implication 10 and his modal syllogistic. Accordingly, NOT ORIGINAL: There is a sense of pluralism in logic—the many-one sense—according to which it is possible to be an instrumentalist without being a pluralist, and for which the thesis of a MacCollian nativity fails. This leaves the reverse question. Can you be a many-one pluralist without being an instrumentalist? One wonders. Take Aristotle again. Aristotle would say that adjusting one’s notation in ways that facilitate an analysis of necessity is a legitimate and necessary thing for a logician to do. 11 The goodness or bad- ness of modalizing the language of the syllogistic is to be judged by the extent to which the modal analysis actually comes off or at least is achievable in principle. 12 If this is what instrumentalism is, then Aristotle is an instrumentalist. If this is what instrumentalism is, then even realists are instrumentalists, and we have an answer to our second question: NOT MACCOLL’S: The present sense of instrumentalism is not the instrumentalism which scholars think is peculiar to MacColl. Of course, it is trivial that for instrumentalism in this sense the Neglect Thesis also fails. MacColl’s own words suggest an attraction to the following view: Anything that facilitates the attainment of a good end without falling foul of some sufficiently bad end inherits the good end’s goodness. More briefly, the goodness of ends is imbibed by their means. This is hardly instrumentalism in any sense seriously deserving of the name. But let’s cut it some slack and give it the name of means-end instrumentalism. Means-end instrumentalism stands to philosophically robust instrumentalism as white chocolate stands to chocolate. Accordingly, although MacColl is a many-one pluralist and a means-end instrumentalist, neither of these positions originates with him, and neither is a plausible basis for his neglect. This leaves us some further work to do, not

9. Of course, this is what we might call vintage Quine. Late in his career, he softened his resistance to intuitionistic logic and abandoned it altogether for quantum logic, albeit with regret [Quine 1992]. 10. Misleadingly called immediate inference. 11. For example, Aristotle insisted that syllogisms be built of propositions only—or what we would call categorical propositions, that is, statements of the A, I, E, O forms. 12. Or, perhaps, initiates a research programme that ultimately succeeds. MacColl’s elusive pluralism 211 least concerning the sense in which MacColl really was an instrumentalist— that is, a philosophically robust one. I want, as I say, to postpone the more important parts of the discussion of the instrumentalism question until later. For the present, let me call a brief time-out for a few words about formalisms. There is nothing especially exotic about the observation that logics with different vocabularies can be put to quite different uses—think here of the language of classical first-order logic and the language of (say) KT 45. 13 Neither is there much of interest in the claim that a symbol of one language could also be given the interpretation customarily possessed by a different symbol of a different language—think here of the decision to give to the box symbol ‘ ’ of S5 the interpretation standardly possessed by the ‘K’ symbol of the KT 45◻ adaptation of it. Regarding the first of these claims, we could say that different logical symbols can express different concepts. Regarding the second claim, we could say that a given logical symbol can be given different interpretations which express different concepts. Thus in the first instance, ‘ ’ expresses the necessity operator and ‘K’ the knowledge operator; and in◻ the second, ‘ ’, which carries the standard interpretation of necessity, is given the customary◻ interpretation of ‘K’ as a non-standard interpretation. This is not yet all there is to MacColl’s pluralism—not anywhere close. So we sum up. Many-one pluralism is as interesting as the unitarianism it rejects and the logics to which it extends its hospitality. But it bears repeating that in the present context it is hardly interesting at all. For as we see, there is nothing original about MacColl’s many-one pluralism and nothing in it that requires an instrumentalist foundation; and so nothing that would induce its neglect on that basis.

~~Multiplicities~~

The founding datum for the more interesting pluralisms of present-day logic is multiplicity. There are, as the saying has it, more logics than you can shake a stick at. It is an unruly plenitude—crowded, chaotic and conﬂicted. 14 It is a discipline excited into cellular excess by a misbehaving auto-immune system. It is a turbulence that over-runs its own quality controls. The pluralisms I now want to discuss embed a strategy for saving face. In its most general form, it is a strategy which sanctions modern logic’s multiplicities while dissipating the bad odour they exude. It is a proposal to make of logic’s ceaseless pullulations a teeming prosperity, 15 une richesse sans l’embarras. The two most problematic features of logic’s multiplicity are in-

13. KT 45 is an epistemic adaptation of S5. See, for example [Auman 1976]. 14. For a classification of this multiplicity, and the suggestion that instrumentalism is not implied by pluralism, see [Haack 1978]. 15. To borrow Quine’s words about quite another thing. (Of course!) 212 John Woods compatibility and blindness. 16 The problem with incompatibility is falsity. For every cluster of pairwise incompatible logics, at least half are false. The problem with blindness is want of subject matter, the absence of something to be about. There are two ways of annulling these difficulties, each a form of denial. In its first form, which I’ll call “existential denial”, it is the claim that these logics aren’t in fact in conflict with one another and aren’t in fact blind. In its second form, which I’ll call “even-so denial”, the incompatibilities and the blindness are acknowledged, but their offensiveness is not. Seen the first way, the properties are bad but not present. Seen the second way, they are present but not bad. Of these two forms, existential denial has a much larger provenance in the literature. Since time presses, I shall confine myself to it, reserving even-so denial for another occasion. Consider now a very simple case. M is a family of propositional modal logics spanning the period from 1918 to 1951. They are the eight strict implication logics of Lewis, S1 to S8, Brouwer’s system B, Gödel’s basic system G, Feys’ system T , and von Wright’s system M. What makes M an interesting cluster is that it has a structure, as in the figure below, 17 which attracts some typical pluralist responses.

~~Gödel’s Basic System Fey’s System Von Wright’s M~~

~~Brouwer’s System~~

~~A S4 S5~~

~~S1 S2 S3~~

~~S6 S7 S8~~

~~The arrows and dotted lines have the following signiﬁcance:~~

1. Ð→ expresses containment. 2. Systems above line A have the Gödel rule RL (if ⊢ ϕ then ⊢ ϕ). ◻ 3. Systems below line A don’t have RL. 4. Systems below line B have ⊢ ϕ. ÅÅ 5. Systems above line B don’t have ⊢ ϕ . ÅÅ 6. Systems above line A are incompatible with ⊢ ϕ . ÅÅ 7. Systems below line B are incompatible with RL.

16. These are not only diﬀerent problems, but it is not easy to see how a cluster of logics could be aﬄicted by them concurrently. See below, the “The vernacular” section. 17. Which I adapt from [Follesdal 2004]. MacColl’s elusive pluralism 213

Perhaps M isn’t much of a multiplicity. It is not nearly as big as the chart—let’s call it M′—of normal logics in [Hughes & Cresswell 1968, 346], and the even larger chart M′′ in [Hughes & Cresswell 1996, 18]. But even M’s comparatively modest multiplicity is enough to stir the pulse of pluralism. To see how, let me begin with a brief word about blindness. There are several logics in this cluster. This alone is offensive to unitarian leanings. There aren’t, we might think, enough different things for these different logics to be logics of. It is a Kantian complaint: Theories without subject matters are blind. What, then, are we to do to restore to M the desired respectability?

~~Subconcept pluralism~~

Talk of subject matters in logic is loose talk. We should try to tighten up. Perhaps concepts are the way to go. It is widely agreed that the central target for logic is the concept of consequence. It is also generally supposed that when a logic’s analysis of implication proceeds by way of its provisions for the logical particles, then their denotata are also targets for analysis. One way of specifying the subject matter of a logic is to equate it with its target concepts. In the case of the members of M, their subject matter is implication, possibility and, say, negation and conjunction. 18 Thinking of subject matters in this way solves the blindness problem for M. But it does so in a way that appears to land us in the incompatibility problem. Consider, for example, S5 and S2. While it may be true that they have this same subject matter, isn’t it problematic that concerning ϕ they make incompatible provision for it? ÅÅ One way of reconciling difference of treatment to sameness of subject matter, and of doing so in a way that solves M’s incompatibility problem, is to identify the sameness with a common generic concept and the differences with different specific subconcepts of it. Accordingly, the concept of implication would stand to its subconcepts in M as genus stands to species. The distinction between generic concepts and their specific concepts helps us see the compatibility between GENERIC UNITARIANISM: A class of logics is generically unitarian to the extent that they have the same generic target concepts. and 18. There is currently a good deal of philosophical turbulence about concepts— stretching from the sceptical view that there aren’t any (or its slighter version that no philosophical good could be got from them if there were any) to the more generous view according to which the problem isn’t that there aren’t any concepts but rather that philosophers haven’t yet come close to getting at the philosophical good that is in them. Concerning the first of these positions, see for example [Machery 2009] and, concerning the second, [Margolis & Laurence 1996]. I have considerable respect for the problems thrown up by an ontology of concepts. But unless I am much mistaken, my invocation of them here will not require the disarming of those difficulties. 214 John Woods

SPECIFIC PLURALISM: A class of logics is specifically pluralistic to the extent that its target concepts include different subconcepts of common generic concepts. We have it that while M is generically unitarian it is also specifically pluralistic. Accordingly, SUBCONCEPT PLURALISM: M exhibits subconcept pluralism with respect to, say, implication, if and only if implication is one of its generic concepts, and its different treatments of implication are treatments of implication in their respectively different subcon- ceptions. COROLLARY: Sets of logics exhibiting subconcept pluralism with respect to their key targets are not in conflict wit one another. I say again that this would seem to solve M’s incompatibility problem. It does so in the manner of existential denial. Since the apparently conflicting sentences of the apparently conflicting systems in M are about different subconcepts of some commonly targeted generic concept, the incompatibilities are only apparent. 19 In a good many approaches, the concept-subconcept arrangement is understood semantically. When G is a generic concept of which G1, ..., Gn are subconcepts, then the term ‘G’ (and its cognates) is n-wise ambiguous as between the meaning of ‘G’, the meaning of ‘G1’, the meaning of ‘G2’, and so on. When this happens, we may say that specific senses are tied to generic senses by a relation of meaning-subsumption. Specific meanings of ‘implies’ imbibe its generic meaning as a necessary but not sufficient condition. This, too, appears to solve the incompatibility problem. If ‘implies’ is ambiguous as between ‘implies’1 , ‘implies’2, and so on, then logics making apparently incompatible provisions for implication are in fact making perfectly compatible provision for different senses of “implies”. Whereupon, we have a further specimen of pluralism, also of the existential denial sort: AMBIGUITY PLURALISM (1): M exhibits ambiguity pluralism with respect to, say, implication if ‘implies’ is n-wise ambiguous as between the meanings of ‘implies’1, ‘implies’2 ...‘implies’n and the meanings of the ‘implies’i subsume the generic meaning of ‘implies’. COROLLARY: Sets of logics exhibiting ambiguity pluralism with regard to its key terms do not conflict with one another. Our generic unitarianism and the ambiguity version of subconcept pluralism are clearly present in some of the pluralisms of the present day, notably in Beall and Restall (henceforth: B & R). B & R [Beall & Restall 2001, 1] are unitarians about the generic concept of validity (or equivalently, implication and consequence):

19. Hartry Field dislikes this solution. He points out that the word ‘electron’ differs in meaning in the theories of Thomson and Rutherford, but “Rutherford’s theory disagrees with Thomson’s despite this diﬀerence in meaning.” [Field 2009, 345] (emphasis added). MacColl’s elusive pluralism 215

i. An argument is valid when there is no case in which the premises are true and in which the conclusion is not true. 20 But since they think that there are different species or subconcepts of this generic one, they are also subconcept pluralists about validity. Moreover, since they hold that subconcept pluralism arises from the ambiguity of “case” in the generic definition, their pluralism is of the ambiguity kind. Do we find this same pluralism in MacColl? There is reason to think that we do. In his battle with Russell over the place of material “implication” in logic, MacColl is stoutly unitarian about the generic notion of implication. ii. ϕ implies ψ if and only if it is in no sense possible both that ϕ and ∼ ψ. 21 But MacColl is also wholly at home with the idea that there are different subconcepts of implication—relevant implication, connexionist implication, many- valued implication, and so on. 22 There is also reason to think that MacColl would be drawn to the ambiguity version of subconcept pluralism. Accordingly, MacColl’s Unitarianism about (deductive) implication can be expressed as follows: iii. ϕ implies ψ iff there is some sense of “possible” for which the conjunction of ϕ and ∼ ψ is not in its extension. For as R & R rightly observe, MacColl tried to reflect the distinctions of natural language in his formal system rather than the other way around, as Frege would. [Rahman & Redmond 2008, 546] How could the distinctions between strict, relevant, connexionist and many- valued implication be present in natural language without it being the case that “implies” is ambiguous in just those ways? As easy as it is to find these pluralisms in MacColl, it is at least as difficult to see them as originating with him. The idea that there are various kinds of implication containing a common core notion is as old as Aristotle. Aristotle distinguished between the core notion of necessitation and the particular cases of syllogistic implication and immediate implication, with still further sub- divisions looming in the fragmentary work on modal syllogisms. So I don’t

20. Field disputes whether generic implication is in fact captured by (i). This is an important demurral and a can of worms. Field maintains that as long as (i) is bound by the seemingly faultless condition that “all cases” in (i) include all actual cases, then ‘implies’ can’t even be “extensionally captured” by it. It is interesting to muse about MacColl’s likely reaction to it. But since he didn’t react to it—he wasn’t aware of the problem—I won’t pursue the matter beyond this point. See [Field 2009, 349–353]. 21. Concerning Field’s objection to B & R’s (i), MacColl embraced his (ii) with all the innocence of B & R’s embracing of their (i). What MacColl would have made of the objection is anyone’s guess. 22. For certain many-valued systems, it is necessary to recognize an even more general notion of implication as providing for designated-value preservation. 216 John Woods yet think that it can be said that we have found a pluralism which MacColl invented and which forms the basis of his post-1909 neglect.

~~Diﬀerent-concept pluralism~~

Conceptual difference matters for pluralism in ways that range beyond the generic concepts to which subconcepts are linked. Consider again a point to which we made fleeting reference in the “Many-one pluralism” section above. Since the late 1960s there has been a (contested) body of opinion among modern logicians to the effect that truth conditions on implication cannot serve as rules of inference or belief revision. 23 Still, whether distinct concepts or just one, it is widely agreed that logic’s principal targets include implication and inference. If we turn now to the heated dispute between strict implicationists and relevantists that livened things up in the 1960s and 70s, we see that if implication and inference are indeed different concepts, they offer hope for the dissolution of the stand-off between these two camps. The strategy in question is Reconciliation. 24 Reconciliation offers warring parties the prospect of win- win. It offers different things for conflicting intuitions to be true of. In the case at hand, it proposes that what relevantist intuitions are true of is inference and what “strictist” intuitions are true of is implication. Ex falso quodlibet is a case in point. Reconciliation proposes that even if every sentence were indeed a consequence of a contradiction, it is far from true that every one of those consequences should actually—or ideally—be drawn. 25 In the present example, conceptual difference performs its conflict- dissolving role independently of any link to subconcepts. True, there may be different subconcepts of implication and different concepts of inference, but these linkages are not what does the work here. Accordingly, we have yet another breed of existentially denying pluralism. DIFFERENT-CONCEPT PLURALISM: Systems S and R exhibit different-concept pluralism if and only if C1 and C2 are different target concepts shared by S and R and, with respect to some claim ϕ over which S and R disagree, ϕ is true when construed as an S-claim and false when construed as an R-claim, and ∼ ϕ is true when construed as an R-claim and false when construed as a S-claim. AMBIGUITY PLURALISM (2): Ambiguity pluralism (1) generalizes to fit different-concept pluralism.

23. Again, the distinction goes back as far as Aristotle and is implicit in the syllogistic by way of the distinction between what a set of premisses necessitates and what may be syllogistically inferred from them. 24. So-called in my Paradox and Paraconsistency [Woods 2003]. 25. See also [Woods 1989]. MacColl’s elusive pluralism 217 A brief reprise

There is reason to think that MacColl is a means-ends instrumentalist both about the goals of logic and the wherewithal for their fulﬁllment. He is drawn to the idea that one of logic’s tasks is to illuminate its own target concepts, and, correspondingly, to the view that those concepts are a logic’s subject matter. MacColl is a unitarian about generic concepts such as implication and possibility, but is a pluralist about their respective subconcepts. He also holds to the ambiguity version of subconcept pluralism and of the diﬀerent- concept pluralism to which it generalizes. In none of these respects do MacColl’s pluralisms either require or take expression from any instrumentalism other than the weak and not quite aptly named thing I’ve been calling means-end instrument. And none of these pluralisms was in any very obvious way invented by MacColl. If RR-theses are to have a good showing, we’ll have to work harder and dig deeper.

~~The goose-gander problem~~

Quine’s quarrel with MacColl is—anachronistically—that since logical- term pluralism is out of the question (“They don’t know what they are talking about”), modal, many-valued, relevant, connexionist and probability logics aren’t logic; and that the right answer to Priest is “One”. Logical-term unitarianism gives rise to an interesting pair of questions. Does, as Quine seems to think, logical-term unitarianism imply unitarianism with regard to logic itself? That is, if the usual run of connectives classically interpreted are the sole logical terms, does it follow that the classical logic of propositions, and it only, is the one true propositional logic? The same question bites in the other direction. Is it possible for one to be a unitarian about the one true logic and yet a pluralist about the systems’ own logical terms? These questions give us what we might call the goose-gander problem; that is, if unitarianism is sauce for the goose of logical systems, how could it not also be the sauce for the gander of logical particles? It is the problem of determining whether the one true logic—were there to be such a thing— precludes or permits non-standard use of its own connectives. Let us say that there is a negative solution for the goose-gander problem if and only if both standard and non-standard uses of a system’s connectives does not expose the system itself to a systems-pluralism. A pair of well-known cases invoke a negative solution to this problem—Putnam on the relation between quantum logic and classical connectives, and Dummett on the relation between intuitionist logic and classical connectives. Seen Putnam’s way, although quantum logic is the one true logic, the classical distributive laws can be allowed to apply to macro-objects because of the insignificance of super- positions for them [Putnam 1968]. Similarly, Dummett claims that, although 218 John Woods intuitionism is the one true logic, there are specific purposes—e.g., matters of effective decidability—for which classical logic may properly be employed [Dummett 1978]. In each case, classical logic is said to be a legitimate part of logic, made so in part by the sublogic relation. 26 Then we have it that: SUBLOGIC PLURALISM: When the goose-gander problem has a negative solution with respect to a logic L and its connectives K1, ... K2 there exists a sublogic L∗ of L whose treatment of the Ki is non-standard by the lights of L. This would also be a good place to recognize the important idea of application pluralism. In the Putnam and Dummett examples, classical uses of the connectives are justified when they serve particular applications of the home logic—the description of meso-objects in the first case and the exploration of effective decidability in the second. By substituting “application” for “treatment” in the indented lines just above, we give this applicational pluralism official notice. Before quitting this section, it is well to note that there are conflicts which sublogic pluralism doesn’t resolve. Especially notable is Putnam’s and Dummett’s own rivalry about the one true logic, never mind their agreement about classical logic as their respective sublogics. Complicating the picture is the selection by Kripke and others of intuitionist logic as a sublogic of classical logic. 27 Taken in conjunction with Dummett, we would seem to have it that intuitionist and classical logics are both super and sublogics of one another. What this suggests is that sublogic pluralism is a conflict-dissolver (if at all) only in relation to some fixed point. The fixed point is the logic chosen for the role of superlogic. As we see, not only can the fixed point change, it can do so in ways that reverse the logic-sublogic relationship of a given arrangement. My reading of MacColl suggests a greater affinity to applicational pluralism than to the sublogic pluralisms of Putnam, Dummett, Kripke and the others, which would have been technically beyond his reach. In light of MacColl’s conviction that “all symbolic conventions may be altered when convenience requires it, in order to adapt them to new conditions or to new classes of problems” [MacColl 1906, 1], which affirms his applicational pluralism, he might have accepted the sublogic pluralisms exemplified by the cases under review, had he but known of them. But, again, interesting as it would be to speculate about the possibility, it is not a question on today’s order paper. So we will move on. This leaves the question of a MacCollian nativity. Is there reason to think that MacColl originated the idea that a logic’s legitimacy is a function of how it is applied? The answer is No. Applicational relevance has been with us

26. Similar arguments are available for the one and only status of classical logic, notwithstanding its provisions for stand-alone, purpose-specific non-classical—in fact, intuitionist—uses of the connectives. Beall and Restall are a case in point [Beall & Restall 2006, 32 & 72–73]. 27. See [Field 2009, 344] who attributes to B & R the attribution of it to Kripke. MacColl’s elusive pluralism 219 since logic’s founding. Aristotle’s syllogistic again is a case in point. In Topics and On Sophistical Refutations, Aristotle wants a theory of deduction that would serve as the theoretical core of a wholly general account of argument, an account that would help regiment the problematic distinction between good- looking arguments and good arguments. Syllogisms were purpose-built for this purpose. Their distinctive properties reflect it, whether the irredundancy constraint on syllogistic premisses or the non-circularity constraint on syllogistic conclusions. Aristotle also had a logic of immediate implication, the best- known parts of which are enshrined in the square of opposition. But immediate implications are no good for argument. For one thing, they are serial question- beggers. Aristotle was an applicational pluralist about logic—syllogisms are needed for argumentative contexts, and immediate implications, while they chart the interaction of quantification and negation, aren’t for Aristotle rules of argument. 28

~~The vernacular~~

A subconcept pluralism is often bound up with a certain philosophical approach to subconcept individuation. On this view, there are as many subconcepts of the concept C as there are in the vernacular different senses of the term “C” and its cognates. If the word ‘possible’ is n-wise ambiguous then possibility is a concept of which there are precisely n subconcepts. As we saw, B & R are generic-concept unitarians about implication and pluralists about its subconcepts. Moreover, since they hold that subconcept pluralism arises from the ambiguity of “case” in the generic definition of implication, their pluralism is of the ambiguity kind. What is not clear, to me at least, is whether the senses of “case” on which the pluralism of B & R turns are required (or assumed) to be ambiguities in the vernacular. If so, the ambiguities they attribute to “case”, in virtue of which “valid”, and “implies” and “consequence” in turn are ambiguous, are ambiguities present in English Today, as Paul Ziff once called it. The vernacular question is critical for the blindness problem. Let’s revisit M, M′ and M′′. Are there as many senses of ‘possible’ or of ‘implies’ as there are irredundant members of M or the vastly greater number in M′ and M′′? Are “possible” and ‘implies’ more than 50-wise ambiguous in English? One would hardly think so. Accordingly, if the ambiguity version of subconcept pluralism is true and if vernacularism is given its head, most of the logics in M, M′ and M′′ are blind. And, if this is so, the incompatibility problem is correspondingly solved, but only by default and not in any intellectually satisfying way. For if blind logics don’t say anything, it is trivial that they

28. Not to overlook that Frege’s and Russell’s own logics were purpose-built to implement a particular thesis from the philosophy of mathematics. These logics were the hand-maidens of logicism. 220 John Woods don’t say anything incompatible with each other. And, if this is the case, existential denial fails for blindness and succeeds for incompatibility only on a technicality. Given his own vernacularism, MacColl would have rejected most of present-day modal logic. There aren’t nearly enough different senses in English Today to service modal logic’s unseemly sprawl. MacColl’s vernacularism? Yes, let’s not forget that MacColl “tried to reflect the distinctions of natural language in his formal system, rather than the other way around, as Frege would”. This is nowhere more evident than MacColl’s dispute with Russell over the conditionality of ‘⊃’. AsR&R point out, MacColl’s central notion is that of the conditional. He not only acknowledges that this connective holds a privileged place in his logic, he also makes the conditional the center of his philosophy. [Rahman & Redmond 2008, 545] MacColl himself writes: For nearly thirty years I have been trying in vain to convince [Russell et. al.] that this assumed invariable equivalence between a conditional (or implication [SIC]) and a disjunctive is an error. [. . . ] Thus, Mr. Russell, arguing correctly from the customary convention of logicians, arrives at the strange conclusion that (among Englishmen) we may conclude from a man’s red hair that he is a doctor, or from his being a doctor that (whatever appearances may say to the contrary) his hair is red. [MacColl 1908, 152] MacColl’s position is that there is no sense of “if . . . then” in any natural language for which “If he has red hair then he is a doctor” is a true conditional sentence. Right or wrong, this puts him squarely in the vernacularist camp. 29 In our consideration of it up to now, conflict-resolution pluralism can claim the attraction of win-win settlement. But when it comes to the real nature of implication, there is nothing win-win about MacColl’s solution. It is a zero-sum resolution, with the strict-implication analysis the big winner and the material-“implication” analysis the loser hands down. So, on this question, MacColl’s resolution strategy is an application of what, in Paradox and Paraconsistency [Woods 2003], I call Surrender. Surrender, which is exemplified by reductio arguments and Aristotle’s and Locke’s ad hominem refutations, 30 is the psychological opposite of Reconciliation. It is a means not

29. Their further dispute concerns the necessity or otherwise of having in one’s object language a genuine conditional-expressing connective such that whenever ϕ implies ψ, there is a true object-language sentence whose conditionality is expressed by that connective with ϕ as antecedent and ψ as consequent. Since MacColl thought that ‘ ’ wasn’t such a connective, he also thought the one-to-one correspondence claim was moot.⊃ 30. For Aristotle, see On Sophistical Refutations, 22, 178b 17, 170a 13, 17–18, 20, 177b 33–34, 183a 22, 24; Topics ϕ 11, 161a 21; and Metaphysics. For Locke, see the chapter “Of reasoning” in [Locke 1961]. Note that ad hominem manoeuvres in Aristotle’s and Locke’s sense are not fallacies. MacColl’s elusive pluralism 221 of saving your opponent’s face, but rather of beating him into submission. Nothing pluralist here. 31 The vernacularity issue puts a good deal of pressure on the following question: If a term ‘C’ is n-wise ambiguous, where are those n-different senses to be found? Vernacularists are drawn to a predictable answer. If ‘C’ is n-wise ambiguous in L, then these various senses of ‘C’ will be present in the linguistic practice of competent users of L. So, if “implies” is ambiguous in English, all its senses will be located in English Today. It takes little effort to see that vernacularism is a spoke in the wheels of modern logic. True, some of logic’s most abiding targets are concepts anchored to meanings fixed by the usages of English Today. Even so, aren’t these meanings the products of language use, and isn’t language use subject to change? Isn’t what things mean in English subject to the linguistic conventions of English? And aren’t conventions solutions to coordination problems? Don’t they settle what, in uttering an expression, its utterer should be understood as meaning? If the meanings of English as we have it just now are too few to generate the concepts constituting the multiple subject matters required by our sprawling logics, couldn’t we hurry things along? Couldn’t we adopt new conventions? Couldn’t we make new meanings? Suppose that we could. Then we would have as many concepts for logic to pronounce upon as it is possible for us to make meanings for. And this would give us made-meaning pluralism.

~~Making meaning~~

Of its several forms, conventionalism is the only form of instrumentalism which is philosophically interesting in its own right and convincingly linked to a like pluralism. Whether it can be attributed to MacColl is another matter to which I shall return two sections hence. Before attending to it head-on, I want ﬁrst to say something about this business of hurrying things along, that is, making new meanings to suit one’s logical convenience. In so doing, it is helpful to mark a distinction between a system’s conceptual adequacy and its

31. It is important to note that both Reconciliation and Surrender are negotiation tools, and like all negotiation tools are consensual in nature. For Reconciliation to work, both parties to the conflict have to agree about what their respective positions are true (and false) of. Some relevantists do not agree that what relevant logic is false of is implication, and some strictists do not agree that what relevant logic is true of is inference. Similarly, for an Aristotelian or Lockean refutation to work, the refuted party has to agree that his position does genuinely conflict with other commitments he is not prepared to give up. There are situations in which such agreements are impossible to get. One person’s reduction of an absurd proposition can equally be another person’s sound demonstration of a surprising (even shocking) truth. This presents negotiators with what in [Woods 2003] I called Philosophy’s most difficult problem. 222 John Woods mathematical virtuosity, which, in turn, is presaged by an earlier distinction of Kant. I’ll deal with mathematical virtuosity in this section and the Kantian contrast in the section after. Mathematical virtuosity is tied up with what I once heard Joke Meheus call a “decent” logic. 32 One way in which to achieve this decency is to have a logic with a recursively specifiable formal grammar, a suitably rigorous proof theory, an appropriately set-theoretic semantics, and the demonstrable presence or absence of metatheoretical properties such as consistency, completeness, soundness, and decidability. 33 Also important is that the syntax of the logic should be able to regulate the flow of its proof procedures, and that its syntax and semantics also be structured to admit of metaproofs that employ the various forms of mathematical induction. Recurring to M, no one doubts that all its members are decent. 34 It would be hard to overestimate the importance of mathematical virtuosity in modern logic. There are people for whom decency is logic’s be-all and end- all. If that were so, then the right kind of multiplicity for logic would be as many or as few as meet the requirement of decency. A less triumphalist posture cuts that claim in half; that is, to say it makes of decency a necessary, but not sufficient, condition of logic’s tolerable size. For suppose it were otherwise. Then the force of eliminating the decency requirement would—let me put this without delicacy—leave it open that some of our logics could be just any old rigged-up formalism. Without prior constraint, a debauchery would ensue, in which Anything Goes in logic. 35 As it happens, not anything a woebegone scribbler scratches together counts as a system of logic, never mind the formal purity of his intentions. In days not long gone, some of the worst old duffers of the Soviet Academy were self-announced dialectic logicians. But it is not going too far to say that they weren’t in any sense relevant to our present concerns logicians at all. Suppose it were otherwise. Then there would be a bona fide notion of implication—for breezy ease of reference call it Stalinist implication—which these hacks got exactly right. 36 Generalizing, what we get when we suppress the decency requirement is the Jazz Age in logic, the no- holds-barred tumble of Anything Goes. If anything is certain about MacColl’s own pluralism, it is that it is not the Anything Goes variety. It is interesting to reflect on the extent to which MacColl himself may have been drawn to the decency requirement. Whatever their respective differences, decent logics are nice pieces of mathematics. What is distinctive

32. At the session on ‘Abduction’ at the Cognitive Science Society Conference in Chicago in July, 2004. 33. Others, of course, are less impressed by completeness than by categoricity. But the basic idea remains the same. 34. Not always with ease. Getting a semantics for S1 was particularly brutal. 35. With thanks to [Stove 1996]. See Cole Porter, “Anything Goes” (Copyright 1934 WB Music Corp). 36. Of course, as some wags have it, dialetheic implication is Stalinist implication. But that’s a light-heartedly intended in-joke about Priest’s politics, not his logic. MacColl’s elusive pluralism 223 about the newly emerging logics of MacColl’s time was not so much the range of their subject matters—modal, relevantist, connexionist, many-valued, and so on—but rather the expectation that the analyses of those subject matters would be given mathematically adequate formulations. This gives modern logic two masters to serve: the course of conceptual adequacy—of getting, say, the concept of implication right—and the cause of mathematical adequacy—of producing the system’s theorems decently. MacColl had reser- vations about the mathematization of logic, as witness the epigraph at the head of this paper. He was not much taken with reductions of logic to algebra, by the idea that logic had a mathematical subject matter. Although he himself was inﬂuenced by the algebraist methods of the day, he favoured a propositional rather than an algebraic approach, in the interests of “pure logic”. In the epigraph, MacColl counsels against warfare between the logicians (he means philosophically-minded logicians) and mathematicians about the form that a successful rapprochement might take. In “Symbolic logic” (1880) he gives a hint: Symbolic reasoning may be said to have pretty much the same relation to ordinary reasoning that machine-labour has to manual labour. [MacColl 1880, 45] MacColl goes on to say that the right kind of formalism enables any ordinary mind to obtain by simple mechanical processes results which would be beyond the reason of the strongest intellect if left entirely to its own resources. [MacColl 1880, 45] I take this to mean that a well-behaved logic is one whose procedures are mathematical in a methodological and abstract sense. This is the way in which “mechanical processes” are made possible. But what is neither necessary nor desirable in logic is that it be endowed with an expressly mathematical subject matter.

~~Analysis and synthesis~~

Vernacularism requires that the usages that generate the concepts which serve as the subject matter of a logic have a prior presence in natural language. There is a signiﬁcant tradition in philosophy which regards the vernacularist stance as a parochialism to no good end. This is where Kant enters the picture. Throughout his philosophical life Kant advanced a distinction between analysis and synthesis. 37 Analysis, says Kant, is the business of making our concepts clear. Synthesis is the business of making clear concepts. Analysis is the business of philosophers. Synthesis is the business of mathematicians. A

37. Early is Inquiry Concerning the Distinctness of Principles of Natural Philosophy and Morality, and Logic, (1704) [Kant 1974a]. Late is Logic (1800) [Kant 1974b]. 224 John Woods similar distinction is invoked, with attribution to Moore rather than Kant, by Russell in his Principles of Mathematics. It is the distinction between philosophical analysis and mathematical deﬁnition, which latter “is widely diﬀerent from that current among philosophers” [Russell 1903, 15]. Russell writes:

It is necessary to realize that definition, in mathematics, does not mean, as in philosophy, an analysis of the idea to be defined into constituent ideas. This notion, in any case, is only applicable to concepts, whereas in mathematics it is possible to define terms which are not concepts. [Russell 1903, 27] 38

Mathematics is the business of making clear concepts. What would we want clear concepts for if not for an interest in the things they instantiate and the things they make true. Making up concepts is a way of making things true. It is a heady idea—making things true. Theories, says Quine are free for the making up, echoing Eddington’s crack that they are put-up jobs. Is this really true? What is the source of this creative potency? And what, if any, are its limits?

The distinction between analysis and synthesis in logic also gives us the attractive means to re-engage the distinction between a logic’s mathematical adequacy and its conceptual adequacy, and to do so in a way that sheds light on logic’s contemporary pluralism and on the further consequences it holds for the vernacularity thesis. According to the synthesis part of the present distinction, it is possible for a logical system to generate new concepts. When the system is merely decent—that is, when it is merely mathematically adequate—the concepts it innovates are nevertheless bona ﬁde concepts of logic. If this is right, then the vernacularity thesis is wrong. It is wrong because mathematically adequate systems originate concepts that have no presence in English Today. And, mark well, this now provides that mathematically adequate logics meet the conceptual adequacy condition as a matter of course. For how could they not be faithful to the very concepts their theorems create?

38. There is a point on which Kant and Russell appear to be at odds. Kant thinks that it is possible to make concepts up. Russell thinks that it is possible to innovate definitions which, had they defined concepts, would have been made-up concepts. Russell is driven to this rather striking hesitation by having embraced the Moorean folly that since the intuitive logic of classes is inconsistent, there simply is no concept of class, leading Russell to say “I have failed to perceive any concept fulfilling the [post-paradox] conditions required for the notion of class” [Russell 1903, v–vi]. This is not a matter on which anything of deep importance for our purposes hangs. So I shall simply pass it by. Whether Kant is right in thinking that concepts are free for the making or Russell is right in thinking that there are mathematical definitions free for the making, the fact remains that for both men, these inventions are entirely legitimate in mathematics, that they are the source of genuinely principled descriptions, some of which are provably valid. MacColl’s elusive pluralism 225 Rainy-day pluralism

The present idea is that mathematical adequacy is a sufficient condition of synthetic content, hence guarantor of its possessor’s conceptual adequacy. No mathematically adequate logic is blind. Of course, if mathematical adequacy solves the blindness problem, it does not, just so, solve the application problem. If synthesis is given its head, mathematical adequacy gives conceptual adequacy, but it does not guarantee applicational adequacy. Consider an example. Dialetheic logic. About seven logicians in the wide world think—I mean, really think—that there are true contradictions. Yet true contradictions are the whole motivation for dialetheic logic. It is a logic (a family of logics) that caters for a highly selective set of true contradictions. Nearly everyone thinks that the very idea of a true contradiction is nonsense, that there is no conceivable sense of “true” and no conceivable sense of “contradiction” for which any contradiction is true. But this is wrong. Since approximately 1979, the year in which Priest’s LP début’d, dialetheism has been a mainstream research programme in logic. Its papers appear in the leading journals and its monographs appear with the best university presses. Why would this be so? The answer is that the dialetheic systems of the present day are decent logics. They are accomplished pieces of abstract mathematics. They have good generative grammars, rigorous and complete proof methods, mathematically lucid set-theoretic semantics, and they permit principled discernment of the presence or absence of the usual run of metalogical properties—consistency, soundness and so on. And withal it builds yet another concept of implication, and another conception of logical truth. The moral to draw from the dialetheic project parallels is that in logic, as in mathematics, mathematical versatility suffices for synthetic conceptual adequacy, without guaranteeing a concomitant applicational adequacy. Prior to Riemann there was no sense in which space was bent. Prior to dialetheic logic there was no sense in which contradictions are true. But for anyone who accepts the role of Kantian syntheses in logic, after dialetheism there is a concept of true contradictions and interesting attempts to find fruitful applications for it. 39 We have been assuming throughout that multiple systems of logic are to be tolerated to the extent to which they are both mathematically and conceptually adequate. Of late we have taken note of a difference in practice— if not always an expressly voiced one—between vernacularity and syntheticity interpretations of what conceptual adequacy requires. What we have not yet determined is whether a third condition should be imposed—the requirement of applicational adequacy in the here and now. If actual practice is anything to go on, the answer to this question is emphatically in the negative. Imagine

39. Most notably, a revival of intuitive set theory and intuitive semantics, contradictions and all. For a brief interim progress report see [Woods 2003, 169–174]. 226 John Woods that one of your students comes to you asking: “What is the road to a solid career in logic?” The answer, for good or ill, is: “Let your constructions be graced with mathematical versatility.” Bearing on this is a remark of Johan van Benthem’s: Now that we all understand the virtues of a model-theoretic semantics satisfying general Montagovian standards of rigor and clarity there is joy in playing around with virtually every speciﬁc detail of Montague’s original paradigm. The following [. . . ] illustrate various aspects of this new wave of free speculation. [Van Benthem 1979, 337] 40 On the face of it, this is disturbing—the triumph of homo ludens over homo sapiens. But that’s not the way it plays in logic’s present-day pluralism. Not only can playful logics produce new concepts whose existence owes nothing to their applicational fruitfulness, but—as the Riemann example reminds us— whose present want of applicability can be tomorrow’s applicational triumph. This gives rise to a further species of pluralism. Let us call it rainy-day pluralism. 41 Rainy-day pluralism is the pluralism of inventories of conceptually innovative unapplied logics awaiting the possibility of future employment. They are systems of logic put away for a rainy day, as the expression goes. They are the proceeds of logical investigation at its most abstractly experi- mental. 42 It is a pluralism that relieves logic of the necessity of capturing pre-ordained, pre-existing concepts, and gives to logic free hand at making new ones up, ready for application if such there be, but worthy objects of our attention all the same, as nice pieces of abstract mathematics. Modern logic is wedded to the idea of making things up. Not everything is fair game, but if your formal apparatus is decent, if your creative eﬀorts are crowned with the requisite mathematical virtuosity, you have a remarkably free hand.

~~Instrumentalism~~

~~We come at last to MacColl’s instrumentalism. It is an instrumentalism characterized by R & R as a kind of conventionalism. They say that:~~

40. Emphases added. 41. It is well to note that, for all its considerable latitude, rainy-day pluralism is not Anything Goes pluralism. Not anything goes in rainy-day pluralism. Rainy-day logics must meet the tough standards of mathematical virtuosity. Rainy-day logics have to be decent. 42. Of course, historical accuracy requires that we note that in its early days, modern dialetheic logic was never thought of as an applicationless plaything built for the shelf of logics in waiting. From the outset, dialetheic logicians had clear targets in mind. There were true contradictions awaiting them pre-theoretically. Their job, as they saw it, was to produce a logic that would accommodate that fact. However from the point of view of virtually everyone else, since there is no pre-existing datum for dialetheic logic, the only virtues it can have—and does in fact have so far—are the virtues of mathematical virtuosity. MacColl’s elusive pluralism 227

MacColl was probably the first to extend conventionalism and instrumentalism to logic. [Rahman & Redmond 2008, 534] MacColl himself also speaks of conventions in logic: [A]ll symbolic conventions may be altered when convenience requires it, in order to adapt them to new conditions or to new classes of problems. [MacColl 1906, 1] Both these ideas have generated large and important literatures. It would not be far wrong to say that instrumentalism and conventionalism are the dominant themes in 20th century philosophy of science, in forms ranging from theoretical terms and sentences, nominal definitions, theoretical models, ideal- izations, abstractions and fictions. These practices raise two obvious questions, one of which conventionalism purports to answer and the second of which is the draw for instrumentalism. The first question is, “How are these structures wrought?” The second question is, “What are they good for?” To the extent possible, I want not to entangle myself in their formidable complexities. I want to know whether MacColl is a conventionalist, and I want to know whether, if he is, this has played a role in his neglect. To make this as light a burden as possible, I want to try to find a minimal condition on conventionalism, and then to determine whether MacColl satisfies it. If he does, there is more work to do. If he doesn’t, our work is fortunately at an end. In this spirit, I will take it as given that any form of conventionalism which we could even consider ascribing to MacColl must include the Kantian principle of concept-making. Poincaré’s own conventionalism—of which MacColl would have been aware—is preceded by Kant’s doctrine of concept-making and would hardly have been possible without it [Poincaré 1905]. In so saying, I don’t want to make too much of Kant’s presence here. It is not in the least necessary to be a Kantian to accept that meanings can be made. For our purposes, Russell’s principle of mathematical definism would do just as well. THE CONVENTIONALIST MINIMUM: A theory is conventionalist only if some of its ineliminable findings are of the theorist’s making. COROLLARY: If a conventionalist theory’s subject-matter is fur- nished by its target concepts, at least some of these are of the theorist’s making, hence are internal to the theory. It is well to emphasize how little of conventionalism the conventionalist minimum captures. It offers us little more than a shot in the dark. If, perchance, MacColl rejects the principle, then he is no conventionalist, never mind the forms his pluralism might take. Acceptance is another thing altogether. It leaves the issue of MacColl’s conventionalism wide-open and unresolved. So which is it to be? Can we plausibly pin this conventionalist minimum on MacColl? Upon reflection, and on balance, I think that we cannot. There are three main considerations that should give us pause. One is MacColl’s vernacularism, his conviction that logic’s rightful conceptual targets have a prior presence in the vernacular, hence are external to the theory. Another is his 228 John Woods hesitation over the out-and-out mathematicization of logic, a hesitation which, whatever its details, would certainly have called into question the very idea of mathematical adequacy as guarantor of conceptual legitimacy. The third is unitarian obduracy about generic implication and conditionality. Since there is no sense—vernacular or otherwise—in which ‘⊃’ expresses a conditional, there is no sense—vernacular or otherwise—in which if ϕ ⊃ ψ then if ϕ then ψ. 43 Equally, since there is no sense of implies—vernacular or otherwise—in which, for example, arbitrary pairs of falsehoods imply one another. Not only was MacColl not the adapter of this conventionalism to his logic, it would have been an idea to invite his hostility.

~~Neglect~~

MacColl was plurally pluralist—different-concept, subconcept, ambiguity and vernacularist. Of none of these was he the inventor, and—with the exception of his vernacularism—none would have earned the scorn of his post- 1909 contemporaries, notwithstanding the various other things they disagreed about. Similarly, while made-meaning pluralism is grist for the mill of conventionalism, it is not a pluralism MacColl would have been happy with; and yet without it, it is difficult to see how we could get him to satisfy even our minimal conventionalism. What, then, explains MacColl’s neglect? Why did his contemporaries and correspondents stop writing about him? The first and almost last answer is that he died. The default position in logic, as in the rest of life, is post- mortem neglect. Beyond that, MacColl lacked an academic position and a cosmopolitan habitat. MacColl worked on logic in fits and starts. His writing is confused and his technical skills were limited. Maybe in large part it was just dumb luck (Why does everyone read Quine’s “Two dog- mas” and no one Morton White’s “Analytic-synthetic: an untenable dualism”? Why did everyone read Strawson’s “On referring” and hardly anyone Geach’s “Subject and predicate”?). I have heard it said that the principal reason for MacColl’s neglect was his outrageous prodigality with ontological commitment, his brazen subscription to existence-neutral quantification. Should we make something of this? Meinong was similarly minded. Not a logician, Meinong was a psychologi- cally trained philosopher drawn to the problem of intentionality. To that end, he contrived a theory of objects which was tailor-made for the kind of interest MacColl exhibited in pluralistic Universes of Discourse, within which he distin- guishes the class e of existent items and the class 0 of non-existents—objects such as centaurs and round squares [MacColl 1906, 1–4]. 44 It is interesting 43. And a jolly good thing, too. Even MacColl would allow that if (if ϕ then ψ) then ϕ ψ. But if we also put it that if ϕ ψ then if ϕ then ψ, we have the equivalence⊃ of ‘ ’ and “if . . . then”. Rather embarrassing⊃ that! 44. MacColl referred⊃ to 0 as the “null class”. However, by this he didn’t mean that 0 was empty, but only that its members were objects that didn’t exist. When MacColl’s elusive pluralism 229

that Russell in an appendix of Principles of Mathematics took after Meinong with a certain vigour. Although Russell didn’t like what Meinong had to say about these matters, an appendix in a major work can hardly be called neglect. Russell seems not to have danced a like attention on the similar views of MacColl. If this is neglect, it is not the existence-free character of MacColl’s logic that occasioned it. When it comes to objects, MacColl is Hamlet to Russell’s Horatio. “There are more things in heaven than are dreamed of in your logic, Russell.” MacColl’s existence-free logic is countable as yet another logic to throw into the mix, along with all the other existentially loaded ones, but not in ways that swell the ranks of systems-pluralism to any great extent. By far the greater part of MacColl’s impact on pluralism is in the size of logic’s ontology. But since this is not the pluralism that currently occupies us, I shall let it be. 45 I confess that I am bested by the neglect question. Most logicians come and go without creating the slightest occasion for neglect. Neglect implies prior recognition, and it is no small feat to have been noticed by the likes of Russell, Peirce and Schröder. If eventual neglect is the cost of such recognition, it is a price most of my friends would cheerfully pay. MacColl was in a very diﬀerent class. The heavy hitters of the day paid attention to him, although none was his champion in the way that Russell was of Frege. Then he died and was forgotten. That’s the way it goes.

~~Acknowledgments~~

For stimulating discussion at the MacColl Centenary or helpful conversation or correspondence afterwards, I warmly thank Stephen Read, Ivor Grattan-Guinness, Shahid Rahman, Amirouche Mokteﬁ, Andrew Irvine, Alirio Rosales and Jaakko Hintikka. My thanks, too, for helpful suggestions from two anonymous referees.

~~References~~

~~Apostoli, Peter & Brown, Bryson 1995 A solution to the completeness problem for weakly aggregative modal logic, Journal of Symbolic Logic, 60, 832–842.~~

it comes to fictions and the like, Frege would have their names denote the empty class 0. Whereas MacColl would have the referents of those names populate his 0. A different 0, needless to say. 45. I myself think that MacColl’s existence-neutral logic is one of his more impressive achievements, especially in its role as a logic of fiction. Recent work by Shahid Rahman and his colleagues reveals how robust the MacCollean scaffolding in bearing the weight of a fully developed free logic [Rahman & Redmond 2008]. 230 John Woods

Auman, Robert J. 1976 Agreeing to disagree, The Annals of Statistics, 4, 1236–1239. Beall, J. C. & Restall, Greg 2001 Defending logical pluralism, in J. Woods and B. Brown (eds.), Logical Consequence: Rival Approaches, Oxford: Hermes Science, 1–22. 2006 Logical Pluralism, New York: Oxford University Press. Brown, Bryson 2002 Approximate truth: a paraconsistent account, in Joke Meheus (ed.), Inconsistency in Science, Dordrecht: Kluwer, 81–103. Carnap, Rudolf 1937 The Logical Syntax of Language, London: Kegan Paul, Trench, Trubner and Co. (originally published in 1934). Dummett, Michael 1978 Truth and Other Enigmas, Cambridge, MA: Harvard University Press. Field, Hartry 2009 Pluralism in logic, The Review of Symbolic Logic, 2 (2), June, 342– 359. Follesdal, Dagfinn 2004 Referential Opacity and Modal Logic, London: Taylor and Francis (ﬁrst published in 1966). Gabbay, Dov M. 1994 What is a logical system?, in Dov M. Gabbay (ed.), What is a Logical System?, Oxford: Oxford University Press, 179–216. Grattan-Guinness, Ivor 1998 Are other logics possible? MacColl’s logic and some English reac- tions, 1905-1912, Nordic Journal of Philosophical Logic, 3 (1), 1–16. Haack, Susan 1978 Philosophy of Logics, Cambridge: Cambridge University Press. Hacking, Ian 1979 What is logic?, Journal of Philosophy, 76, 285–319. Harman, Gilbert 1972 Is modal logic logic?, Philosophia, 2, 75–84. 1986 Change in View, Cambridge, MA: MIT Press. MacColl’s elusive pluralism 231

Hughes George E. & Cresswell, Max 1968 An Introduction to Modal Logic, London: Methuen. 1996 A New Introduction to Modal Logic, London: Routledge. Kant, Immanuel 1974a Inquiry Concerning the Distinctness of Principles of Natural Philosophy and Morality, and Logic, Indianapolis: Bobbs-Merrill (originally published in 1764). 1974b Logic, translated and with an introduction by Robert S. Hartman and Wolfgang Schwartz, Indianapolis—New York: Bobbs-Merrill, 1974 (originally published in 1800). Locke, John 1961 An Essay Concerning Human Understanding, edited by John W. Yolton, 2 vols., London: Dent (originally published in 1690). MacColl, Hugh 1880 Symbolical reasoning, Mind, 5 (17), 45–60. 1906 Symbolic Logic and Its Applications, London: Longmans, Green, & Co. 1908 ‘If’ and ‘imply’, Mind, n. s., 17 (65), 151–152. Machery, Eduard 2009 Doing without Concepts, New York: Oxford University Press. Margolis, Eric & Laurence, Stephen (eds) 1996 Concepts: Core Readings, Cambridge, MA: MIT Press. Paglieri, Fabio & Woods, John 2009 Enthymematic parsimony, Synthese, SpringerLink date: 12 August, DOI: 10.1007/s11229-009-9652-3. Peacocke, Christopher 1976 What is a logical constant?, Journal of Philosophy, 72, 221–240. Poincaré, Henri 1905 Science and Hypothesis, London: Walter Scott. Priest, Graham 2001 Logic: One or many?, in J. Woods and B. Brown (eds.), Logical Consequence: Rival Approaches, Oxford: Hermes Science, 23–38. Putnam, Hilary 1968 Is logic empirical?, in R. Cohen and M. Wartofsky (eds.), Boston Studies in the Philosophy of Science, 5, Dordrecht: Reidel, 216–241. 232 John Woods

Quine, Willard Van Orman 1970 Philosophy of Logic, Englewood Cliffs, New Jersey: Prentice-Hall. 1992 Pursuit of Truth, Cambridge, MA: Harvard University Press. Rahman, Shahid & Redmond, Juan 2008 Hugh MacColl and the birth of logical pluralism, in Dov M. Gabbay & John Woods (eds.), British Logic in the Nineteenth Century, vol. 4 of the Handbook of the History of Logic, Amsterdam: North-Holland, 533–604. Russell, Bertrand 1903 The Principles of Mathematics, Cambridge: Cambridge University Press (2nd ed., 1937). Scriven, Michael 1976 Reasoning, New York: McGraw-Hill. Sher, Gila 2001 Is logic a theory of the obvious?, in J. Woods and B. Brown (eds.), Logical Consequence: Rival Approaches, Oxford: Hermes Science, 55–79. Stove, David 1996 Cole Porter and Karl Popper: The Jazz age in the philosophy of science, in Roger Kimball (ed.), Against the Idols of the Age: David Stove, New Brunswick, NJ: Transaction, 3–32. Tarski, Alfred 1956 On the concept of logical consequence, in A. Tarski, Logic, Semantics and Metamathematics, trans. by J. H. Woodger, Oxford: Oxford University Press, 409–420. Van Benthem, Johan 1979 What is dialectical logic?, Erkenntnis, 14 (3), 333–347. Woods, John 1968 Inference and Implication, Toronto: University of Toronto (mimeo). 1989 The relevance of relevant logic, in Jean Norman & Richard Sylvan (eds.), Directions in Relevant Logic, Dordrecht: Kluwer, 77-86 2001 Pluralism about logical consequence: resolving conflict in logical theory, in J. Woods and B. Brown (eds.), Logical Consequence: Rival Approaches, Oxford: Hermes Science, 39–54. 2003 Paradox and Paraconsistency: Conflict Resolution in the Abstract Sciences, Cambridge: Cambridge University Press. MacColl’s elusive pluralism 233

Woods, John & Brown, Bryson (eds.) 2001 Logical Consequence: Rival Approaches. Proceedings of the 1999 Conference of the Society for Exact Philosophy, Oxford: Hermes Science. Woods, John & Walton, Douglas 1972 On fallacies, Journal of Critical Analysis, 5, 103–111 (reprinted as chapter 1 of their Fallacies: Selected Papers 1972-1982, Dordrecht: Foris, 1989; 2nd ed., with a Forward by Dale Jacquette, London: College Publications, 2007).

~~On the growth and use of a symbolical language Hugh MacColl~~

[Editors’ note: This paper was communicated by the Rev. Robert Harley, F. R. S., and read at the Ordinary meeting of the Manchester Literary and Philosophical Society on 22 March 1881. The full paper, signed “Hugh M’Coll, Esq., BA.”, appeared first (without the abstract) in the Memoirs of the Society, series 3, 7, 1882, 225–248. The (English) abstract below appeared first in the Proceedings of the Society, 20 (8), session 1880-1881, 103. The French abstract below has been provided by the editors.] [Résumé : Cet article discute de façon générale les cas dans lesquels les symboles peuvent être utilisés de façon avantageuse dans le raisonnement logique et mathématique, et aspire, par un examen des symboles déjà existants, à déter- miner quelques règles ou principes, d’abord, quant à l’opportunité d’introduire de nouveaux symboles, puis, quant au types de symboles qui doivent être sélectionnés. Cet article contient aussi une brève histoire du développement graduel de la méthode symbolique propre à l’auteur, telle qu’elle a été présen- tée et expliquée par l’auteur dans ses articles publiés dans les Proceedings of the London Mathematical Society, le Educational Times, le Philosophical Magazine, et Mind.] Abstract: This paper discusses in a general way the cases in which symbols may be advantageously employed in logical and mathematical reasoning, and endeavours, from an examination of our existing stock of symbols, to deduce some rules or guiding principles, first, as to when new symbols should be introduced, and secondly, as to what kinds of symbols should be selected. It also contains a brief account of the gradual development of the author’s own symbolic method, as explained and set forth in his papers published in the Proceedings of the London Mathematical Society, the Educational Times, the Philosophical Magazine, and Mind.

In an article on “Symbolical Reasoning,” in a recent number of ‘Mind’ (no 17, Jan. 1880), I have described the relation between symbolical reasoning and ordinary verbal reasoning as analogous to that between machine labour and ordinary manual labour. To trace this analogy through all its various points of resemblance would take too long; but there is one point which deserves some notice, as it bears more especially upon the present subject.

~~Philosophia Scientiæ, 15 (1), 2011, 235–249. 236 Hugh MacColl~~

For what kinds of operations are machines usually invented? A little reflection will show that one common and prominent characteristic of such operations is sameness; we employ machines to perform operations which have to be frequently repeated, and repeated in the same unvarying manner. Sewing- machines, knitting-machines, reaping-machines, and, in fact, the great generality of machines, however widely they may differ in other respects, resemble each other in this. For what kinds of expressions and relations, mathematical or logical, do we usually invent symbols? We shall find, as before, that the common characteristic of such expressions and relations is sameness—that they are expressions and relations which have to be repeated frequently. When any complex expression or relation is perceived to have a tendency to recur again and again, we economize thought, time, and space if we denote this expression or relation by some simple, suggestive, and easily formed symbol which we may always recognize as doing duty for its more complex equivalent. The representive symbols thus invented combine afterwards among themselves into new expressions and relations of more or less complexity, and give birth, in their turn, when the necessity or convenience arises, to fresh representative symbols, whose abbreviating power bears, on an average, the same ratio to that of the symbols they displace, as the abbreviating power of the latter bears to that of their immediate progenitors. In strict conformity with this law of symbolical growth the science of mathematics has gradually attained its present wonderful power within the limits of its application; and in strict conformity with the same law, the science of logic, which is now evidently entering on quite a new phase of existence, will probably before long, and within much wider limits of application, surpass the achievements of mathematical science itself. Now it is clear that the power and progress of any symbolical language must depend very largely upon the judgment exercised, first, as to whether, in any proposed case, a new symbol is really required, or would on the whole be useful, and, secondly (supposing the need of a new symbol to be admitted), as to the kind of symbol that should be selected. With regard to the first point, we must remember that the introduction of a fresh symbol is always accompanied by the disadvantage that it adds a fresh item to the load which the memory has to carry, and it is only when its advantages more than outweigh this very serious drawback that it should be admitted as a permanent addition to the existing vocabulary. Can we discover any general principles or rules which should guide us in this important matter of admission or rejection? Let us examine a few of the symbols which we now possess, and see whether any such rules can be discovered. The ratio which the circumference of a circle has to its diameter, namely, 3.14159 &c., is one that occurs frequently, and for this reason mathematicians express it by a single arbitrary symbol π. The ratio which the diagonal of a square has to its side, namely, 1.41421 &c., is another ratio which also occurs On the growth and use of a symbolical language 237 frequently, and yet mathematicians do not express this by any single arbitrary symbol, nor would any mathematician think the introduction of such a symbol desirable. Why is this? The answer is obvious: the latter ratio may be expressed, without any fresh definition or explanation, by a very brief and simple combination of existing symbols, namely by the combination √2; while we know of no brief and easily formed combination of existing symbols, requiring no fresh definition, which would accurately and unambiguously express the former ratio. From these and other analogous examples we may safely assume as one guiding principle, that some conventional symbol of abbreviation should be used as a substitute for any expression that has a tendency to recur frequently, provided that no suitable combination of existing symbols (i.e. a combination short, simple, and requiring no fresh definition or explanation) can be found to replace it. The next point is, as a rule, more important and also less easily decided. It is this: —Granting the necessity for some new symbol of abbreviation, what kind of symbol should be selected? In the ease of the symbol π, to which we have already alluded, this question of suitable selection is, it is true, of secondary importance; almost any arbitrary symbol of easy formation would have done just as well; but this is an exception to the general rule. Consider the symbol an, which has been invented as an 2 abbreviation for the product of n equal factors, each equal to a; that is, a 3 for aa, a for aaa, and so on. If the first of these products, namely aa, were the only one that had a tendency to recur, we may be quite sure that mathematicians would remain satisfied with it in its original form, and would 2 never have accepted the innovation a as its equivalent. But since aaa, aaaa, &c., have also the same tendency of frequent recurrence, the appropriateness of the symbol selected is evident: the numerical index reminds us of the number of equal factors; and we are at once provided with a more effective notation for considering the properties and relations of all expressions that are products of equal factors, as, for instance, in the binomial theorem. Let us now examine the raison d’être of that remarkable class of symbols which were invented at a more advanced stage of the science (by whom I know not), and which give such a wonderful sweep and power to symbolical language generally, logical as well as mathematical; I refer to that class of symbols of which f x may be taken as a specimen. This symbol denotes any complex expression( ) whatever (mathematical or logical) that contains the simpler expression x, in any relation whatever as one of its constituents. What was the special need which this symbol was invented to supply? We have often to consider what an expression would become if one of its constituents were taken away and a fresh constituent put into its place, just as people sometimes speculate as to what would be the effect upon a ministerial policy if a certain member of the cabinet were to resign and a certain other person appointed in his place. If f x denote the expression of which x is ( ) 238 Hugh MacColl a constituent, then f a will denote the new expression which is formed by substituting a for x.( To) take a simple case, let f x denote the algebraical 6 6 ( ) expression 3 x x 5; then f 2 will denote 3 2 2 5, and will be equal to − + + 6 ( ) − + + 13; f 0 will denote 3 0 0 5, and will be equal to 7; f x 5 will denote 6 ( ) − + + 6 ( − ) 3 x 5 x 5 5, and will be equal to 8 x x; and so on. −( − ) + ( − ) + − + The last symbol f x 5 warns us of a danger to be carefully guarded against in the introduction( − of) fresh symbols, namely the danger of ambiguity. The meaning here attached to it might in certain cases be confounded with an older and commoner meaning; for the symbol f x 5 also denotes the product of the two factors f and x 5. How is this danger( − of) ambiguity to be guarded against? We might, it is true,− guard against it by adopting, instead of f, a totally new symbol of some unwonted shape; but this is a course to be avoided if possible. Strange-looking symbols somehow offend the eye; and we do not take to them kindly, even when they are of simple and easy formation. Provided we can avoid ambiguity, it is generally better to intrust an old symbol with new duties than to employ the services of a perfect stranger. In the case just considered, and in many analogous cases, the context will be quite sufficient to prevent us from confounding one meaning with another, just as in ordinary discourse we run no risk of confounding the meanings of the word air in the two statements—“He assumed an air of authority,” and “He resolved the air into its component gases.” In the special case of f x 5 no ambiguity exists when the letter f is used in no other sense throughout( − ) the investigation on which we happen to be engaged; and when it is used in another sense, all risk of confusion is obviated by simply employing instead of f, in the expressions f x , f x 5 , &c., some other symbolic letter, such as φ. ( ) ( − ) We may recapitulate the results so far arrived at thus: — 1. A new symbol, or an old symbol with a new meaning, to be accepted as a permanent addition to the existing stock, should represent all expression or relation that tends to occur frequently, and that cannot be logically expressed by any short combination of existing symbols. 2. Provided we can avoid ambiguity, it is better to employ an old symbol in a new sense than to invent a totally new symbol. 3. The symbol chosen should be of short and easy formation and lead to symbolic expressions of simple and symmetrical forms. The illustrations hitherto given have been borrowed from mathematics, because these are more familiar to the general reader than any that could have been taken from symbolical logic; yet the latter is in every respect the simpler science of the two,—simpler in the conceptions ∗ represented by symbols, simpler in the smallness of the number of symbols employed, and simpler in

∗. That the conceptions which underlie the very elements of symbolical mathematics are by no means easy to grasp will be admitted by any one who has attempted to explain to beginners the real logical meaning of the “Rule of Signs” in multiplication and division of ordinary algebra. This difficulty meets the tyro on the very threshold of the science. When he has advanced a few steps further, he is confronted with On the growth and use of a symbolical language 239 the mechanical operations that have to be performed. In claiming this advantage for logic over mathematics, I speak solely of that scheme of symbolical logic which I, rightly or wrongly, consider the simplest and most effective, namely the scheme which I have explained and illustrated in ‘Mind,’ in the ‘Proceedings’ of the London Mathematical Society, in the ‘Educational Times,’ and in the ‘Philosophical Magazine.’ According to this scheme the whole and sole duty of the logician is to investigate the relations in which statements (i.e. assertions and denials) stand towards each other. For all practical reasoning- purposes a statement may be defined as anything that conveys directly through a bodily sense (as the eye or ear) any information (true or false) to the mind. In this sense a nod or a shake of the head is a perfectly intelligible statement. The Union-Jack fluttering from the mast of a ship conveys as clear and definite information as the words “This is a British ship” shouted through the captain’s speaking-trumpet; and therefore the flag is as much a statement in the logical sense as the words; and, like the words, it may (as we know by experience) be a true or a false statement. Logic, then, being concerned with statements, the analogy of ordinary algebra suggests the propriety of denoting simple statements by single letters, and the relations in which statements stand to each other either by the relative positions of the statement-letters, or by separate and distinct symbols. Therefore a very important inquiry in laying the foundation of a practical symbolical calculus for solving logical problems is this: —What are the characteristics and relations which most frequently distinguish or connect the statements of an argument? Foremost among distinctions we shall find that of truth and falsehood. All intelligible statements may be divided into two great classes, the true and the false. Every statement must belong to one or other of these two classes, though we may not always know to which. If it were not for this element of uncertainty, reasoning would be purposeless, and logic would have no raison d’être. This uncertainty (sometimes real and sometimes only hypothetical) suggests the convenience of dividing the statements of any argument upon which we happen to be engaged into three distinct classes, the admittedly true, the admittedly false, and the doubtful. Borrowing a hint from mathematical probability, we may denote any statement belonging to the first class by the symbol 1, and any statement belonging to the second class by 0, while any doubtful statement (whether the doubt be real or hypothetical) may be denoted by any symbol we choose except these. Now, generally speaking, in the course of any consistent argument or investigation the boundaries of these three classes will be found to be gradually changing; the first two classes, the admittedly true and the admittedly false, though never encroaching upon each other’s ground, will both constantly encroach upon the ground of the third, the doubtful. a symbolical paradox which it has taxed the ingenuity of the subtlest mathematical intellects to explain, and of which no rational explanation whatever was given until within very recent times; I allude to the useful and important yet perplexing symbol √1. 240 Hugh MacColl

Statements may also, independently of their truth or falsehood, be divided into two distinct classes, namely assertions and denials. Every assertion either claims or has already obtained admission into the class denoted by the symbol 1; while its denial contests its right to this symbol, which it claims for itself, and seeks to brand it, as an impostor, with the symbol 0. As long as these two claimants belong to the class of doubtful statements, all that we can say about them is, that the one (either the assertion or its denial) must be true, and the other false. The denial of any assertion may be conveniently denoted by an accent, thus: —Let x denote the statement, “He is in England;” then x′ will denote “He is not in England.” The statements hitherto spoken of are simple or elementary statements— that is, statements represented each by a single letter, or a single letter and an accent. Any statement that requires more than one letter to express it may be called a complex statement. The principal relations by virtue of which simple statements combine into complex ones are three—namely, conjunction, disjunction, and implication, corresponding respectively to the three conjunc- tions and, or, if. The first relation is generally symbolized (like multiplication in ordinary algebra) by simple juxtaposition, and occasionally, though never necessarily, by the symbol ; the second (like addition in ordinary algebra) by the symbol ; and the third× by the symbol , as in the following examples: — + ∶ Let x denote the statement “He will go to Paris;” and let y denote the statement “I shall go to York.” Then x y denotes the compound statement “He will go to Paris and I shall go to York;” the symbol x y denotes the disjunctive statement “He will go to Paris or I shall go to York;”+ and x y denotes the implication “If he goes to Paris I shall go to York.” ∶ A compound statement, as a b c, claims the symbol 1 for every one of its factors; a disjunctive statement, as a b c, claims the symbol 1 for one at least of its terms; and the implicational+ statement+ a b claims the symbol 1 for the consequent b, provided the antecedent a is entitled∶ to it, but neither claims nor disclaims it for b if a is not entitled to it. Brackets are used when necessary to collect the different elements of a complex statement, and so prevent any uncertainty respecting to what complex statement any element or relational symbol belongs. Thus, the compound statement a b c , formed by the two factors a and b c, is a very different statement from( + the) disjunctive statement ab c, formed+ by the two terms ab + and c. So, again, is a b c ′, the denial of the whole implicational complex ( ∶ + ) statement a b c, a very different statement from a b c′ , in which the symbol of denial∶ + affects only the element c. ∶ + Reciprocal implications—that is, compound implications of the form a b b a —occur so frequently that a symbol of abbreviation is convenient.( ∶ Borrowing)( ∶ ) again from the existent mathematical stock, we may use † either ∷ †. To avoid the employment of brackets and repetition as much as possible, it will be convenient to use both, with this distinction, that the symbols ∶ and ∷ should be On the growth and use of a symbolical language 241 or =. Thus either the symbol a b or the symbol a = b may be taken as an abbreviation for the reciprocal implication∷ a b b a . ( ∶ )( ∶ ) The symbol f x has been already considered, and is employed in logic in the same sense( as) in mathematics; that is to say, it denotes any statement whatever that contains x as one of its constituents; but the symbol f ′ x , for which no logical meaning analogous to its mathematical one is likely( to) turn up, may be conveniently employed as an abbreviation for f x ′. { ( )} These are the only symbols that need be employed in the system of symbolical logic which I advocate, and they are amply sufficient not only for the complete solution of any logical problem that I have ever seen solved by any other method, but also for the complete solution of many problems which, I think, it would be difficult to solve by any other method with which I am at present acquainted. The rules which I have proposed for observance in introducing new symbols are, I believe, sound, and I have followed them myself to to the best of my ability. As the science advances, other symbols will, no doubt, become necessary; but they should be introduced slowly, and not till their utility is made clearly manifest.

My statement in ‘Mind,’ that though a b implies a′ b, it is not equivalent to it, has been called in question, my critics∶ maintaining+ that there is no real difference between the conditional statement “If a is true, b is true,” and the disjunctive statement “Either a is false or b is true.” Now, I admit at once that, in the ordinary language of life, disjunctive statements are often made which convey, and are intended to convey, a conditional meaning, and, further, that the example which I gave in illustration, namely “He will either discontinue his extravagance, or he will be ruined,” is one of them. Many statements, however, are made in common life which are tacitly understood to convey a stronger meaning than logically and literally belongs to them. Take, for instance, the well-known expression, “He will never set the Thames on fire.” In its literal sense this very harmless-sounding statement does not commit one to much; it may, with equal safety, be applied to the cleverest man living and to the most incapable idiot. What the practical reasoner would be concerned with in making use of any evidence conveyed to him in such terms would be the intended meaning of the speaker; and if his argument should be of such a nature as to necessitate the employment of symbols, the symbol for the coordinate (i.e. of equal reach in regard to the statements affected by them), but both subordinate (i.e. of inferior reach) to the symbol . Thus, a ∶ b + c ∷ d + e ∶ f ∷ g is an abbreviation for the complex statement = a ∶ b + c b + c ∷ d + e d + e ∶ f f ∷ g , ( )( )( )( ) while a ∶ b + c d + c ∶ f ∷ g is an abbreviation for = a ∶ b + c d + c ∶ f ∶∶ g ( ) = ( ) . 242 Hugh MacColl statement should denote its intended and not its literal meaning. The real question in dispute is this, does the conditional statement “If a is true b is true,” as I define and symbolize it, convey a meaning in any way different from the disjunctive statement “Either a is false or b is true,” as I define and symbolize it? My argument in ‘Mind’ was, that since the denial of the first, namely “a may be true without b being so,” conveys less information than “a is true and b is false,” which is the denial of the second, the conditional disjunctive statements of which these are the respective denials cannot be equivalent. As the non-equivalence of the denials, however, is much more evident than that of the affirmative statements, it will be well worth while to give, if possible, a more direct proof of the non-equivalence of the latter.

As it can easily be shown that a b is equivalent to 1 a′ b, the question ∶ ∶ + may be narrowed to this, is the implication 1 α, in which α denotes a′ b, equivalent to the simple affirmation α? It seems∶ to me that 1 α and α differ+ in pretty much the same way as the statement “It is well-known∶ that tin is heavier than zinc,” and the simpler affirmation “Tin is heavier than zinc;” that is to say, the former implies the latter, but is not implied in it. The statement 1 α, in addition to claiming the symbol 1 for itself, asserts that its protégé α ∶has fairly made good its right to it; whereas α only claims this symbol on its own account. The symbol 1 α ′, which is the denial of 1 α, may be ( ∶ ) ∶ read, “α is not necessarily true;” whereas α′, the denial of α, is much stronger, and asserts positively that α is false. It follows from the law of logic called contraposition that the denial of the weaker (or implied) statement is stronger than and implies the denial of the stronger (or implying) statement. The disjunctive statements of ordinary language may be divided into conditional disjunctives and unconditional (or pure) disjunctives—the former being those already referred to as conditional in meaning though disjunctive in form. Among these may be classed the famous disjunctive of Edward I., “By God, sir Earl, you shall either go or hang” (symbolically, g h), the meaning of which is evidently, “If you don’t go, I will have you hanged.”+ The king, by a not uncommon trick of speech, used the weaker statement in order to express a stronger meaning. The earl, in his reply, “By God, sir king, I shall neither go nor hang” (g′ h′), secures emphasis of a more direct kind by flatly denying even the king’s weak disjunctive, instead of the much stronger conditional statement which would more logically, though less emphatically, express the king’s real meaning. The denial of “If you won’t go, I will have you hanged,” would be the very mild assertion, “I may refuse to go without your having me hanged,” a mode of speech which would not at all have suited the temper of Earl Bigod. As an instance of a simple unconditional disjunctive, we may take the statement, “We shall either go to Brighton or Hastings this summer.” There appears to me to be a fundamental difference between the class of disjunctives of which this is a type, and the conditional class of disjunctives previously On the growth and use of a symbolical language 243 illustrated. At the same time it must be admitted that in common language, just as statements which are conditional in meaning are often expressed in a disjunctive form, so real disjunctives unconditional in meaning are often expressed in a conditional form. The last example, for instance, “We shall either go to Brighton or to Hastings this summer,” might, according to usage and without any perceptible difference of meaning, be expressed as “If we don’t go to Brighton this summer, we shall go to Hastings,” or in the same words with Brighton and Hastings interchanged. The fact, however, that conditionals and disjunctives are frequently confounded in ordinary untechnical language, is no reason why they should be so in formal or symbolical logic. Even if I have not succeeded in satisfactorily proving that a b and a′ b are not synonymous, it is safest, I think, to adopt my view in actual∶ practice.+ Let it be observed that the hypothesis of non-equivalence commits one to less, and therefore involves less risk to the inferred conclusions. My critics admit with me that a b a′ b is a correct formula; but they would also add the formula ( ∶ ) ∶ ( + ) a′ b a b , the validity of which I deny. If I am wrong, I am open to the charge+ ∶ of( seeking∶ ) to deprive logic of a new formula which might possibly prove useful, but whose utility has yet to be proved. If my opponents are wrong, they are open to the graver charge of seeking to introduce an erroneous formula, which not only can render no service in reasoning, but might even seriously endanger our conclusions. As this article is an attempt to explain and illustrate the laws which necessitate the growth, and the principles which determine the form, of a symbolical language, I hope it will not be considered either irrelevant or egotistical, if I give a brief account of the development of my own method. By “my own method” I mean simply the method which I discovered (including those features which it has in common with the prior methods of others), as well as those characteristics which are peculiar to itself. What these are I leave to others to decide. The question is certainly irrelevant to the expressed object of this article; and its discussion would only provoke the natural impatience of the reader. I only mention this at all in order to explain that I use the possessive pronoun my merely as a convenient abbreviation, and in a sense which cannot possibly give offence to any of my fellow workers. As I stated in my third paper in the ‘Proceedings’ of the London Mathematical Society, my method originated in a question in probability proposed in the ‘Educational Times’ for June 1871 ‡. The question (as I understood it) may be thus generalized : —“Given that the variables x, y, z are each taken at random between given limits, what is the chance that an assigned function of these variables, say φ x,y,z , will be real and positive?” When I began to solve the problem I found( that,) in addition to the particular event whose chance was required, I should have to consider the relations in ‡. The subject of probability was one which I had recently taken up at the request of Mr. J. C. Miller, the mathematical editor of the ‘Educational Times,’ who felt great interest in it himself, and strongly recommended it as “an unworked vein in which I should find many treasures.” 244 Hugh MacColl which this event stood towards several other events on which it more or less depended. It struck me that it would help the memory and facilitate the reasoning, if I registered the various events spoken of in regular numerical order in a table of reference. The event whose chance had to be found resolved itself into a concurrence of two distinct but not independent events 1 and 2; and I denoted the chance (or probability) of this concurrence by the symbol p 1 ⋅ 2 . The compound event 1 ⋅ 2 implied the occurrence of a third event 3, which( ) it was necessary also to take into account; I had therefore to replace p 1 ⋅ 2 by p 1 ⋅ 2 ⋅ 3 . But the consideration of 1 ⋅ 2 ⋅ 3 could not be separated( from) the consideration( ) of a fourth event 4, in conjunction with which it might happen, but not necessarily; and the probability of the concurrence 1 ⋅ 2 ⋅ 3 depended materially upon whether it happened in conjunction with this fourth event or without it. Denoting the non-occurrence of this fourth event by the symbol : 4, I thus had the equation p 1⋅2⋅3 = p 1⋅2⋅3⋅4 = p 1⋅2⋅3 ∶ 4 . Proceeding in this way, but in a somewhat( groping) and( tentative) manner,( I) finally resolved the problem into a form which brought it within the reach of the integral calculus; in other words, I had somehow determined the limits of integration, though I hardly knew how. I applied the same method successfully to two or three other problems in the ‘Educational Times,’ but without being able to make any material improvement in it. Whilst occupied with these researches, the editor of the ‘Educational Times’ sent me a very neat and simple geometrical solution by Mr. G. S. Carr of the very problem which had given me so much trouble. This so discouraged me (in the belief that I was only wasting my time) that I threw up the whole subject in disgust, and determined for the future to eschew all mathematics that did not fall within the very narrow limits of my requirements as a teacher. When six years afterwards, I broke my resolution and again took up the subject of probability, my mind naturally reverted to the old abandoned method; and it then struck me that, with all its defects, it had one important merit, namely independence of geometrical diagrams, and that, consequently, it would be well worth my while to apply myself patiently to the task of re- moving its defects and developing it, if possible, into something better. My first step was to drop the letter p (for probability), which I thought might, without ambiguity be left understood; so that, for instance, 1 ⋅ 2 ⋅ 3 should replace p 1 ⋅ 2 ⋅ 3 as an abbreviation for “the probability of the event 1 ⋅ 2 ⋅ 3.” My next( step was) to use letters instead of numbers, as A B C instead of 1 ⋅ 2 ⋅ 3, and an accent to denote non-occurrence, as ABC′ instead of 1 ⋅ 2 ∶ 3. But at this point a difficulty presented itself: how was A B C, the chance of the compound event A B C, to be distinguished from A B C, the product of the chances A, B, C? for the chance of the compound event would not generally be the product of the chances of the separate events. To guard against this risk of confusion I decided to use capitals, as A B C, when the chance of the whole compound event was meant, and small italic letters, a b c, when the product of the separate chances a, b, c was meant. Thus, though the chances A, B, C were separately equal to a, b, c, the symbol A B C would not (except in the On the growth and use of a symbolical language 245 case of independent events) be equivalent to the symbol a b c. The equivalence of A (B + C) and A B + A C, and of similar expressions, I discovered before I introduced letters instead of numbers. So far, I had made no real advance: the substitution of a literal for a numerical notation improved perhaps the appearance of the method; but it did not affect it practically. In applying the method to such questions as required the integral calculus it still remained a tentative method; I still groped my way towards conclusions in particular cases without the help of any general rules of procedure. At last I was struck by the fact that the events registered in my tables, and whose chances were denoted by the letters, were all of the form x > x1, x2 > x, x2 > x1, y > y1, y2 > y, y2 > y1, &c., and had all reference to the limits of the different variables. This suggested the idea of a partial return to the original numerical notation and classifying the events according to the variable spoken of. I denoted the event and also the chance of the event x > x1 by x1, the event x1 > x and its chance by x1′ , and so on for x2, x2′ , x3, x3′ , y1, y1′ , &c. This was a very important step so far as my method related to integration limits; and after this its development in this direction was comparatively rapid—too much so for me to remember very accurately its different stages. Still, I looked upon the method as essentially and inseparably connected with probability; and even when I had decided that it would be more convenient and less confusing to let my symbols denote logical statements rather than mathematical chances, I could not for some time turn to any account the independence of mathematics which I had thus secured for the method. The notion of the mutual exclusiveness of events (or statements) connected by the sign + clung to the method to a very late period; in fact, I was in the very act of writing my first article “On Symbolical Reasoning” for the ‘Educational Times,’ when the needlessness of this restriction occurred to me. I had written down my definitions of the equations A B C = 1 and ABC= 0 in the following words: — The equation A B C = 1 asserts that all three statements are true; the equation A B C = 0 asserts that all the three statements are not true, i.e. that at least one of the three is false; and I had to consider suitable definitions of the equations A + B+C= 1 andA+B+C= 0. It was quite evident that the equation A+B+C= 0, whether the statements A, B, C were mutually exclusive or not, must assert that all the three statements are false; and the very words used in the previous definitions of A B C = 1 and A BC= 0 suggested that, as a symmetrical complement of this, the equation A + B + C = 1 should assert that all the three statements are not false, i.e. that at least one of the three is true. The only question to decide was whether the rule of multiplication, (A + B) (C + D) = A C+ A D + B C + B D,would still hold good. A very little consideration showed that it would; so, though the method was correct, so far as it went, on either supposition, I judged it wiser to leave room for possible future development by adopting the wider rather than the narrower hypothesis for its basis. 246 Hugh MacColl

Finding myself thus, at the end of my investigation, on logical instead of mathematical ground, I naturally began to study the relation in which my method stood towards the ordinary logic, and especially towards the syllogism. The only book on logic that I possessed was Prof. Bain’s work; and to this I turned. The resemblance which my method bore to Boole’s, as therein described, of course struck me at once; but Boole’s treatment of the syllogism was more likely to put me on the wrong track than to help me. As my most elementary symbols denoted statements, not necessarily connected with quantity at all, I could not see how the syllogism, with its ever recurring all, some, none, could be brought within the reach of my method. The Cartesian system of analytical geometry at last supplied the desiderated hint as to the proper mode of procedure. In this system, as every mathematician knows, one single point is spoken of in every equation, but with the understanding that it is a representative point, and that the equational statement made respecting it is also true respecting every other point in the locus expressed by this equational statement. The symbol ∶, which I had already begun to use as an occasionally convenient abbreviation for the word “implies,” now became almost imperative. Syllogistic reasoning is strictly restricted to classification. The statement “All X is Y ” is equivalent to the conditional statement “If any thing belongs to the class X, it must also belong to the class Y .” Speaking then of something originally unclassed, if x denote the statement “It belongs to the class X,” and if y denote the statement “It belongs to the class Y ,” then the implicational statement x ∶ y (or x implies y) will be equivalent to the syllogistic statement “All X is Y .” It was evident after this that x ∶ y′ would be the proper symbolical expression for “No X is Y ;” but, strange to say, the discovery of the suitable symbolic expressions for “Some X is Y ” and “some X is not Y ” caused me no small trouble, even though I had previously more than once wondered under what circumstances the symbol x ∶ y ′ would be required. For a long time I ( ) did not recognize this x ∶ y ′ as the equivalent (in classi[fi]cation) of “Some ( ) X is not Y ,” and x ∶ y′ ′ as the equivalent of “Some X is Y .” In my second communication to( the Mathematical) Society I used the symbol ν ∶ xy to denote “Some X is Y ;” and it was only when I had read the very just objection made by one of the referees to my introduction of the arbitrary and possibly non-existent class V that it suddenly flashed upon me that the true symbolical expression for “Some X is Y ” should be x ∶ y′ ′, the denial of the implication ( ) x ∶ y′, and that the true symbolical expression for “Some X is not Y ” should be x ∶ y ′, the denial of x ∶ y. ( ) The next new symbol to be introduced into my symbolic system was the symbol xa, to express the chance of x being true on the assumption that a is true. The circumstances which suggested this symbol to me are curious and instructive. My first idea was to use the symbol xc to denote the chance of x being true, the suffix c being merely suggestive of the word chance and not denoting a statement. In fact, this was the notation which originally formed On the growth and use of a symbolical language 247 the basis of my fourth paper, “On the Calculus of Equivalent Statements,” when it was first communicated to the London Mathematical Society. While this paper was in the hands of the referees, I was occupied with a problem proposed to me by Mr. C. J. Monro, and involving among other things the consideration of a chance x z c, which I at first considered as equal to ( ) xc x ∶ z c, being under the idea that, since x ∶ z expressed the conditional ( ) statement “If x is true z is true,” x ∶ z c would be the proper symbol to express the chance that if x is true z(is true.) On reflexion I discovered that this, plausible as it sounded, would lead to inconsistency of notation. For, since x ∶ z is equivalent to z′ ∶ x′, consistency of notation required that x ∶ z c should ( ) denote the same chance as z′ ∶ x′ c; and, as I had interpreted the symbols, this would not be the case. The( chance) that z is true, on the assumption that x is true, is not generally equal to the chance that x is false on the assumption that z is false. I was thus forced to the conclusion that I had put a wrong interpretation on the symbols x ∶ z c and z′ ∶ x′ c, which must be equivalent, and that neither of them, therefore,( ) was( the proper) symbol for the chance which I wished to express. It became necessary, therefore, since there did not appear to be sufficient data for logically inferring a correct expression for this chance, to invent a new and arbitrary symbol for it; and then the important question presented itself as to what that symbol should be. It must, if possible, be brief and easily formed; it must be formed, at least partly, of the symbols x and z; and yet it must be some unambiguous combination of those symbols—that is to say, a combination which should convey no other meaning either by definition or by implication. One of several symbols that offered themselves as candidates for the important post to be filled, I at last selected the symbol zx as the one most likely to perform effectively the duties required of it.

The symbol zx being thus fairly installed, I was struck by the resemblance between it in some respects and the symbol zc. Both expressed the chance of the truth of z, though on generally different assumptions; and, what was more remarkable, some of the formulae which I had obtained involving the constant suffix c, as α + β c = αc + βc − αβ c, ( ) ( ) were also true when for c I substituted the variable suffix x. This suggested the propriety of considering c too as a statement, instead of a mere arbitrary abbreviation for “the chance of the truth of,” and it soon became evident what that statement must be. The constant suffix c, like the variable suffix x, must denote a statement taken for granted; but, unlike the variable x, it must denote a statement whose truth is taken for granted always—that is, throughout the whole of an investigation. In other words, the suffix c must be an exact equivalent for the logical symbol 1. This, however, necessitated other symbolical changes. As long as the suffix c did not denote a statement, I was at liberty to use this letter in conjunction with the letters a and b in other positions as a statement; so that cc (like ac and bc) would simply denote the chance of the truth of the statement c, and 248 Hugh MacColl its value might vary from 0 to 1; but with the new meaning of the suffix c, we should always have cc = 1. It thus became expedient to leave c at liberty to discharge other functions in company with its old comrades a and b, and to intrust the duty of denoting universally admitted statements to some letter whose services in other capacities could be more easily spared. I decided, after some hesitation, on the Greek letter ε, which is easily formed, pleasing to the eye, and not often wanted. It may be asked, why was I not satisfied with the symbol 1, which already denoted an admitted statement? My answer is, first, that I thought this numeral would not look well in frequent companionship with literal suffixes; and, next, that I thought it better to reserve it, in company with other numerals, for distinguishing statements of the same class or series, as a1, a2, a3 &c., which, though different statements, will generally be found to have some common factor or characteristic a. Having thus decided that ε should denote a statement of acknowledged truth, that xa should denote the chance of the truth of x on the assumption that a is true, and that therefore xε must simply denote the chance that x is true, with no assumption beyond the understood data of the problem, it soon became evident that this notation would express many of the laws of probability in neat and compact formulae, and also that it would contribute towards precision of reasoning from its constant reference, by means of its suffixes, to the assumptions on which any argument in probability rested. It is well known that of all mathematical subjects probability is the one in which mistakes are most apt to be made; and these mistakes are usually the result of correct reasoning based upon unperceived false assumptions. These assumptions, for the most part, would be readily seen to be false, if they were only expressed; a notation therefore that actually forces them on the attention must be considered as possessing one very important advantage in that fact alone. As this new scheme of probability-notation quite superseded that which formed the basis of the paper which had been already submitted to the referees of the Mathematical Society, these gentlemen naturally declined (on the scheme being communicated to them through the Honorary Secretary, Mr. Tucker) to pronounce any opinion either upon the original paper or on the proposed alterations, till the whole was recast and rewritten. When this was done, and the paper again submitted to them, they advised its publication. In my former paper in ‘Mind,’ “On Symbolical Reasoning,” I referred to the analogy between the relation connecting antecedent and consequent in logic and that connecting subject and predicate in grammar. Would it be presumptuous to suggest as a probable hypothesis that this analogy is more than a mere coincidence, and that it really points to an original identity? It does not seem unreasonable to suppose that in the very early stage of human speech, each separate word represented a complete statement and conveyed its own independent information. On this supposition, the growth of vocal language would proceed according to laws in some respects analogous to those which shape the development of a language of symbols. Our abstract nouns, On the growth and use of a symbolical language 249 for instance, seem to be nothing but abbreviations for original statements. Take the compact and well-known saying, “Unity is strength.” What is this but an abbreviation for the conditional statement, “If a company be united, they will be strong”? or, as it may be otherwise expressed, “If the statement symbolized by the abbreviation unity for ‘They are united’ be applicable to a company of persons, so will also the statement symbolized by the abbreviation strength for ‘They are strong.’ ” But here I must stop. Speculations as to the primaeval forms of human speech do not come fairly within the limits prescribed by the title of this article; and further discussion of the subject in this direction would therefore be irrelevant.

~~Adresses des auteurs~~

Francine F. Abeles Stein Haugom Olsen Department of Mathematics Dean, Faculty of Business, Languages, Kean University and Social Sciences 1000 Morris Avenue Østfold University College Union, NJ 07083 – USA 1757 Halden – Norway [email protected] [email protected]

Irving H. Anellis Shahid Rahman Peirce Edition U.F.R. de Philosophie Institute for American Thought Université Lille 3 (Sciences Humaines, Indiana University Lettres et Arts) Purdue University Indianapolis Pont du Bois – B.P. 60149 Indianapolis, IN 46202-5157 – USA 59653 Villeneuve d’Ascq – France [email protected] [email protected]

Michael Astroh Stephen Read Institut für Philosophie Department of Philosophy Baderstrasse 6-7 University of St Andrews D-17487 Greifswald – Germany Scotland, KY16 9AL – U. K. [email protected] [email protected] Fabien Schang Jean-Marie Chevalier LHSP Archives H. Poincaré 34, boulevard de l’hôpital UMR 7117 CNRS - Nancy-Université 75005 Paris – France Université Nancy 2 [email protected] 91, avenue de la Libération – BP 454 Ivor Grattan-Guinness F-54001 Nancy Cedex – France Emeritus Professor of the History of [email protected] Mathematics and Logic James J. Tattersall Middlesex University Business School Department of Mathematics and The Burroughs, Hendon Computer Science London NW4 4BT – England Providence College [email protected] Providence, RI 02918 –U.S.A. [email protected] Amirouche Mokteﬁ LHSP Archives H. Poincaré John Woods UMR 7117 CNRS - Nancy-Université Department of Philosophy Université Nancy 2 1866 Main Mall 91, avenue de la Libération - BP 454 University of British Columbia F-54001 Nancy Cedex – France Vancouver, B.C. – Canada V6T 1Z1 [email protected] [email protected]