Aristotle’s Syllogism and the Creation of Modern Logic: Between Tradition and Innovation, 1820s–1930s 9781350228849, 9781350228870, 9781350228856

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Aristotle’s Syllogism and the Creation of Modern Logic

Bloomsbury Studies in the Aristotelian Tradition General Editor: Marco Sgarbi, Università Ca’ Foscari, Italy Editorial Board: Klaus Corcilius (University of California, Berkeley, USA); Daniel Garber (Princeton University, USA); Oliver Leaman (University of Kentucky, USA); Anna Marmodoro (University of Oxford, UK); Craig Martin (Oakland University, USA); Carlo Natali (Università Ca’ Foscari, Italy); Riccardo Pozzo (Consiglio Nazionale delle Ricerche, Rome, Italy); Renée Raphael (University of California, Irvine, USA); Victor M. Salas (Sacred Heart Major Seminary, USA); Leen Spruit (Radboud University Nijmegen, The Netherlands). Aristotle’s influence throughout the history of philosophical thought has been immense and in recent years the study of Aristotelian philosophy has enjoyed a revival. However, Aristotelianism remains an incredibly polysemous concept, encapsulating many, often conflicting, definitions. Bloomsbury Studies in the Aristotelian Tradition responds to this need to define Aristotelianism and give rise to a clear characterization. Investigating the influence and reception of Aristotle’s thought from classical antiquity to contemporary philosophy from a wide range of perspectives, this series aims to reconstruct how philosophers have become acquainted with the tradition. The books in this series go beyond simply ascertaining that there are Aristotelian doctrines within the works of various thinkers in the history of philosophy, but seek to understand how they have received and elaborated Aristotle’s thought, developing concepts into ideas that have become independent of him. Bloomsbury Studies in the Aristotelian Tradition promotes new approaches to Aristotelian philosophy and its history. Giving special attention to the use of interdisciplinary methods and insights, books in this series will appeal to scholars working in the fields of philosophy, history and cultural studies. Available titles: A Political Philosophy of Conservatism, by Ferenc Hörcher Elijah Del Medigo and Paduan Aristotelianism, by Michael Engel Early Modern Aristotelianism and the Making of Philosophical Disciplines, by Danilo Facca Phantasia in Aristotle’s Ethics, by Jacob Leth Fink Pontano’s Virtues, by Matthias Roick The Aftermath of Syllogism, edited by Marco Sgarbi, Matteo Cosci The Reception of Aristotle’s Poetics in the Italian Renaissance and Beyond, by Bryan Brazeau The Scientific Counter-Revolution, by Michael John Gorman Virtue Ethics and Contemporary Aristotelianism, edited by Andrius Bielskis, Eleni Leontsini, Kelvin Knight

Aristotle’s Syllogism and the Creation of Modern Logic Between Tradition and Innovation, 1820s–1930s Edited by Lukas M. Verburgt and Matteo Cosci

BLOOMSBURY ACADEMIC Bloomsbury Publishing Plc 50 Bedford Square, London, WC1B 3DP, UK 1385 Broadway, New York, NY 10018, USA 29 Earlsfort Terrace, Dublin 2, Ireland BLOOMSBURY, BLOOMSBURY ACADEMIC and the Diana logo are trademarks of Bloomsbury Publishing Plc First published in Great Britain 2023 Copyright © Lukas M. Verburgt, Matteo Cosci and the Contributors 2023 Lukas M. Verburgt and Matteo Cosci have asserted their right under the Copyright, Designs and Patents Act, 1988, to be identified as Editors of this work. For legal purposes the Acknowledgements on p. xii constitute an extension of this copyright page. Cover image: John Venn, Symbolic Logic (1894 [1881]) All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage or retrieval system, without prior permission in writing from the publishers. Bloomsbury Publishing Plc does not have any control over, or responsibility for, any third-party websites referred to or in this book. All internet addresses given in this book were correct at the time of going to press. The author and publisher regret any inconvenience caused if addresses have changed or sites have ceased to exist, but can accept no responsibility for any such changes. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. ISBN: HB: 978-1-3502-2884-9 ePDF: 978-1-3502-2885-6 eBook: 978-1-3502-2886-3 Series: Bloomsbury Studies in the Aristotelian Tradition Typeset by Integra Software Services Pvt. Ltd. To find out more about our authors and books visit www.bloomsbury.com and sign up for our newsletters.

This book is dedicated to the memory of John Corcoran (1937–2021)

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Contents List of figures and tables List of contributors Acknowledgements Introduction: History of modern logic in a new key  Lukas M. Verburgt and Matteo Cosci 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Richard Whately’s revitalization of syllogistic logic  Calvin Jongsma Mill and the British tradition of inductive logic: The role of syllogism  Lukas M. Verburgt The Aristotelian roots of Bolzano’s logic  Mark Siebel George Boole and the ‘pure analysis’ of the syllogism  David E. Dunning Logic of relations by De Morgan and Peirce: A case study for the refinement of syllogism  Sun-Joo Shin Ernst Schröder’s algebra of logic and the ‘logic of the ancient’  Volker Peckhaus Brentano and Hillebrand on syllogism: Development and reception of the ‘idiogenetic’ theory  Matteo Cosci Hugh MacColl: Never twist the syllogism again  Jean-Marie C. Chevalier Frege’s relation to Aristotle and the emergence of modern logic  Erich H. Reck Christine Ladd-Franklin’s antilogism  Francine F. Abeles Syllogism and beyond in the Peano School  Paola Cantù Hilbert’s use of the syllogism  William Ewald The role of syllogistic logic in early set theory  José Ferreirós The fate of the syllogism in the Göttingen school  Curtis Franks

Index

viii ix xii

1 15 37 55 73 93 113 129 155 173 189 207 231 247 269 289

List of figures and tables Figures 4.1

George Boole’s development of a new mathematical treatment of the syllogism, opposite the version he published in 1847. MS/782 The Mathematical Analysis of Logic, 1847, page 37. ©The Royal Society 4.2 Boole’s mathematical expressions for possible syllogistic inferences. MS/782 The Mathematical Analysis of Logic, 1847, page 42. ©The Royal Society 14.1 ‘THINNING’ and ‘CUT’ rules 14.2 Example proof according to ‘THINNING’ and ‘CUT’ rules

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83 279 279

Tables 3.1 5.1 6.1 7.1 7.2

7.3

7.4 7.5 7.6

Bolzano’s reformulations of the categorical propositions Peirce’s multiplication table Schröder’s grouping of valid kinds of syllogism Brentano-Hillebrand: Simple conventions for logical notation are stipulated Brentano-Hillebrand: The traditional types of categorical propositions are understood in their existential form, and so are the respective judgements Brentano-Hillebrand: The applicable rules which govern the relations between categorical propositions in their existential form are established on the grounds of non-granted existential import Brentano-Hillebrand: An axiomatic starting point of the new system is set Brentano and Hillebrand’s Syllogistic Scheme Hillebrand’s categorical propositions as implying double judgements

58 107 121 134

134

134 135 135 136

List of contributors Francine F. Abeles is Professor Emerita at Kean University in Union, NJ, USA. For the ten years beforehand, she was distinguished professor of mathematics and computer science. She is co-editor of a volume in Birkhäuser’s Studies in Universal Logic book series titled Modern Logic 1850–1950. East and West (2016). It includes a paper she co-authored: ‘The Historical Sources of Tree Graphs and the Tree Method in the Work of Peirce and Gentzen’. The volume Research in History and Philosophy of Mathematics (2015) includes her paper ‘The Influence of Arthur Cayley and Alfred Kempe on Charles Peirce’s Diagrammatic Logic’. She has edited three volumes in the pamphlets of Lewis Carroll series for the LCSNA/University Press of Virginia (mathematical, political and logic pamphlets). She is the author of nearly one hundred papers and reviews on topics in geometry, number theory, voting theory, linear algebra, logic and their history. Paola Cantù is a senior researcher in philosophy at the Centre National de la Recherche Scientifique (CNRS) and co-director of the Philosophy and History of Science Axis at the Centre Gilles Gaston Granger (Aix-Marseille University). Co-editor of Logic and Pragmatism. Selected Writings of Giovanni Vailati (2009), Teorie dell’argomentazione. Un’introduzione storico-critica  alle logiche del dialogo  (2006) and of the transcriptions Kurt Gödel Maxims and Philosophical Remarks (vols. IX, X, XI, XII, XIV–XV), she is the author of Giuseppe Veronese e i fondamenti della geometria (1999) and  E qui casca l’asino.  Gli errori di  ragionamento nel dibattito pubblico  (2011).  She wrote numerous articles on the Peano School, on Bolzano, on structuralism, the philosophy of  mathematical practice and the applicability of mathematics. She is a member of the editorial board of History and Philosophy of Logic, Topoi and Annals of Mathematics and Philosophy. Jean-Marie Chevalier is Professor of Philosophy and the director of the Department of Philosophy at the University of Créteil (UPEC) in France. He is the author of  L'Empreinte du monde. Un essai sur les formes logiques et métaphysiques [The Imprint of the World. An Essay on Logical and Metaphysical Forms] (2013), Qu’est-ce que raisonner ? [What Is Reasoning?] (2016) and Peirce ou l'invention de l'épistémologie [Peirce or the Invention of Epistemology]  (2022). He is also the  co-editor of several books in the philosophy of knowledge. Chevalier is chiefly a Peirce scholar but has also published articles and chapters on the history and philosophy of logic, philosophy of knowledge, scientific reasoning and semiotics. Matteo Cosci is a research fellow at the University Ca’ Foscari Venice. His research work is centred on Aristotle, Galileo and the history of Aristotelianism. He is author of Verità e comparazione in Aristotele (2014), co-editor of The Aftermath of Syllogism (2018) and

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List of Contributors

principal investigator of the Marie Curie research project ‘The Ophiuchus Supernova: Post-Aristotelian Stargazing in the European Context (1604–1654)’. He is also assistant editor of the online journal Philosophical Readings. David E. Dunning is a historian of science, mathematics and computing in modern Europe and North America. His research explores the material and social dimensions of abstract knowledge. Much of his work  on the  history of logic focuses on the cultures around symbolic notation: his articles  include studies of the  pedagogy of Frege’s  Begriffsschrift, the political history of so-called Polish  notation and the connections between logical notation and early computing programming. He is also co-editing special issues on gender and domesticity in the history of mathematics and on the logical work of John Venn. He is currently a lecturer in the Integrated Studies Program at the  University of Pennsylvania and the 2022–2023 IEEE Life Member History Fellow. William Ewald is Professor of Law and Philosophy at the University of Pennsylvania, and the editor of  From Kant to Hilbert: A Source Book in the Foundations of Mathematics (1996) and co-editor of the Lectures of David Hilbert on Foundations of Mathematics and Natural Science (2000–). José Ferreirós  is Professor of Logic and Philosophy of Science at the Universidad de Sevilla and member of the Institute of Mathematics (IMUS). He is the author of Labyrinth of Thought (2007) on the development of set theory, and Mathematical Knowledge and the Interplay of Practices (2016), which presents a historical-cognitive approach to mathematics.  Ferreirós specializes in the history and philosophy of mathematics, having served as president of the Association for the Philosophy of Mathematical Practice, and has also done research on the physical sciences, history and philosophy of logic, and general philosophy of science. He has published books and articles on various topics concerned with the contributions of Gauss, Riemann, Dedekind, Cantor and Hilbert.  Curtis Franks is Associate Professor of Philosophy at the University of Notre Dame. He is the author of  The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited  (2009) and is currently the editor-in-chief of the  Notre Dame Journal of Formal Logic. Franks has published several essays on the history and philosophy of logic ranging from genealogical  accounts of central  notions in mathematical logic (completeness, inferential validity, deduction), to reconstructions of Talmudic derivations, to articulations of the variety of pragmatic and naturalistic conceptions of logic. Calvin Jongsma is Professor of Mathematics, emeritus, at Dordt University. His 1982 dissertation at the University of Toronto definitively established the initial publication date of Whately’s Elements of Logic and demonstrated how Whately revitalized British logic through a cogent and lively defence of syllogistic reasoning against its earlier critics in the context of a highly popular logic text. Jongsma is also the author of Introduction

List of Contributors

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to Discrete Mathematics via Logic and Proof (2019), but his research and writing have mostly focused on topics in the history and philosophy of mathematics and logic. Volker Peckhaus is Professor of Philosophy of Science and Technology at Paderborn University, Germany. He is the editor-in-chief of the journal  History of Philosophy of Logic  (since 2006, 2006–2010 together with John W. Dawson). He is author of Hilbertprogramm und Kritische Philosophie. Das Göttinger Modell interdisziplinärer Zusammenarbeit zwischen Mathematik und Philosophy  (1990) and  Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert (1997). He contributed to Heinz-Dieter Ebbinghaus, Ernst Zermelo. An Approach to His Life and Work  (2007, 2nd  ed., 2015), and published numerous papers and (co-) edited several books on the history of logic and the philosophy of mathematics. Erich Reck is Professor of Philosophy at the University of California, Riverside. He is the author of a series of articles on both the history of logic and the history of analytic philosophy, especially concerning Frege, Russell, Wittgenstein, Carnap and Gödel. He is also the editor of several collections of essays in these areas, including From Frege to Wittenstein (2002), Gottlob Frege: Critical Assessments (2005) and The Historical Turn in Analytic Philosophy (2013). Beyond that, he works on the philosophy of mathematics, the philosophy of science and nineteenth-/twentieth- century philosophy more generally. Sun-Joo Shin  is Professor of Philosophy in the Department of Philosophy at Yale University. She is the author of  The  Logical Status of Diagrams (1994)  and  The Iconic Logic of Peirce’s Graphs (2002). Her main interests are the logic of diagrams, visualization in mathematics, visual reasoning, Peirce’s logic and more. Mark Siebel is Professor of Philosophy at the University of Oldenburg, Germany. His research areas cover epistemology, philosophy of science, philosophy of language and mind, logic and the history of analytic philosophy. He is the author of  Der Begriff der Ableitbarkeit bei Bolzano [The Concept of Deducibility in Bolzano] (1996) and  Erinnerung, Wahrnehmung, Wissen  [Memory, Perception, Knowledge] (2000), editor of Kommunikatives Verstehen [Communicative Understanding] (2002) and coeditor of Bolzano and Analytic Philosophy (1997), Semantik und Ontologie [Semantics and Ontology] (2004) and Paradoxien [Paradoxes] (forthcoming). Lukas M. Verburgt is currently Gerda Henkel Stiftung Research Scholar and affiliated to the Netherlands Institute for Advance Study in the Humanities and Social Sciences (NIAS). He has held visiting research positions at Trinity College, Cambridge, the Department of History and Philosophy of Science, Cambridge, and the Max Planck Institute for the History of Science, Berlin. His research focuses on various themes in the history of science and philosophy from 1800 to the present. He is the author of John Venn: A Life in Logic (2022) and (co-)editor of several volumes, including A Prodigy of Universal Genius: Robert Leslie Ellis, 1817–1859 (2022) and the forthcoming Cambridge Companion to John Herschel.

Acknowledgements We would like to thank Marco Sgarbi for his willingness to include the book in the wonderful Bloomsbury Studies in the Aristotelian Tradition series; Jade Grogan and Suzie Nash for their support during the editing and production process; and Ursula Martin and Christopher Hollings for their help in obtaining the image permissions for Boole’s papers. We are also immensely grateful to all the authors who contributed a chapter to this collection, sharing the fruit of their research with us and the readers One of us (LMV) gratefully acknowledges grant support from the Gerda Henkel Stiftung (grant AZ 23/F/22).

Introduction History of modern logic in a new key Lukas M. Verburgt and Matteo Cosci

This book is dedicated to the memory of John Corcoran (1937–2021), who had agreed to contribute a chapter but sadly passed away before it was completed. We like to think of the aim of the book as doing for modern logic what Corcoran did for the work of George Boole, namely to make sense of and do justice to the idea that Aristotelian syllogistic logic contributed to its creation. More specifically, the chapters show that the period between the nineteenth and early twentieth century saw a parallel development of modern logicians reshaping syllogism and reflections on syllogism shaping modern logic. This might sound odd as it stands in striking contrast to the standard narrative about the history of modern logic, which says that its creation and development happened in spite of, or in direct opposition to, the old logic. W. V. Quine, for one, wrote that Aristotelian logic is to modern logic what the ‘arithmetic of primitive tribes’ is to modern mathematics: not even a scientific predecessor but a ‘prescientific fragment’.1 And there is no denying that, in terms of scope, power and analytic rigor, the Prior Analytics pales in comparison to Gottlob Frege’s Begriffschrift or A. N. Whitehead and Bertrand Russell’s Principia Mathematica. But using today’s standards to judge logic’s past is, of course, not a good starting point for a history of modern logic and for capturing its historicity. When modern logic is presented as non-Aristotelian, no wonder that the role of Aristotle’s syllogistic logic in its emergence disappears from view. For instance, one fact that has been neglected, and which Corcoran documented in great detail, is Boole’s agreement with everything Aristotle said, his disagreements concerning what Aristotle did not say.2 Boole’s position vis-à-vis Aristotle represents just one of a large number of different attitudes towards the syllogistic that modern logicians upheld. This book charts some of these attitudes and shows how their study gives rise to a more nuanced story about the creation of modern logic. Some parts of this story will be familiar. The nineteenth century was the century in which the mathematical revolution in logic achieved its breakthrough. It was also the century in which – after no less than two millennia – the hegemony of the syllogistic fell apart and Aristotle’s immense achievement ceased to be logic’s paradigm. It was replaced by what is today called

2

Aristotle’s Syllogism

classical or standard logic, consisting of propositional and first-order logic. All this is well known and well established.3 Other parts of the story that this book tells will be new. The collapse of the Aristotelian empire did not happen overnight. It was a lingering demise. Perhaps because the period between Kant – who in 1781 wrote that logic has not, will not and cannot improve upon Aristotle – and Frege – who in 1879 self-consciously moved logic far beyond Aristotle – is held to be an inactive time in the history of logic, this process of demise has hitherto received little attention. Far from being in slumber, however, the period was a highly complex one. No logician agreed with Kant’s notorious claim. At the same time, logicians had ‘little common ground except for their rejection of Kant’s conservatism’.4 There was an explosion of attempts to rethink logic. With this explosion came a fragmentation ‘in tone, in method, in aim, in fundamental principles’5 so extreme that logicians often did not even recognize each other’s work as logic. John Venn, in his contribution to the first issue of the journal Mind of 1876, wrote that it would not be going too far to say that the principal difficulty in the way of a student of Logic at the present day … consists not so much in the fact that the chief writers upon the subject contradict one another upon many points, for an opportunity of contradiction implies agreement up to a certain stage, as in the fact that over a large region they really hardly get fairly within reach of one another at all.6

Many historians of logic have ignored or neglected some of the most influential logicians during the period – such as Richard Whately, John Stuart Mill, Hermann Lotze and Christoph Sigwart, for example. (Not to speak of other perhaps less influential but equally or even more interesting figures like Hugh MacColl, Lewis Carroll and E. E. Constance Jones.) The diversity of approaches to logic seems too large and the number of logical traditions too big to make sense of the work of these and other logicians, though the case could be made that there has been a process of ‘Vergessenmachen’ at work in the historiography of modern logic – with pioneers of the new logic deliberately pushing older traditions into oblivion.7 The outcome is that historiography has been almost silent about everything there is to the nineteenth-century logic that does not ‘flow directly to the waters that created the Peircean-Fregean tsunami of mathematical logic’.8 For instance, British logic in the 1870s–90s remains a ‘very confusing intermediary period’,9 where the old syllogistic was still taught, the new Boolean logic not yet established, and Whitehead and Russell’s mathematical logic soon forthcoming. However, it is for much the same reason that the rise of modern logic is arguably still far from being fully understood. This can be seen from what has been called the ‘Quine-Putnam muddle’.10 Quine’s view was that traditional logic ended and modern logic began with Frege’s Begriffschrift, simply because this book contained the first system of propositional and predicate logic. Putnam, finding inspiration in David Hilbert and Wilhelm Ackermann’s 1938 textbook, protested and dated the beginning of modern logic to Boole’s Mathematical Analysis of Logic (1847) and Laws of Thought (1854). Unlike Quine, Putnam did justice to the two main schools, or origins, of modern logic. But by suggesting that both are part

Introduction

3

of one and the same development of ‘modern mathematical logic’, Putnam neglected the essential differences and tensions between the Boole-Schröder and Frege-Russell traditions. For example, as is well known, whereas Boole’s system was a calculus ratiocinator, Frege aimed for a lingua characteristica. And whereas Schröder reduced Frege’s system to a mere notational variant of Boole’s calculus, Frege refused to call this calculus logic at all. Another problem with the Quine-Putnam project of dating modern logic is that what could be called the ‘mathematical turn in logic’ was not a development beginning in the second half of the nineteenth century, let  alone an accomplishment of Boole or Frege alone. It can be traced back to the final quarter of the seventeenth century, notably in the prescient work of Leibniz. Also, neither Quine nor Putnam recognized the historical fact that it took until the 1880s for Boole’s work to make a mark and that Frege’s work went largely unnoticed until the 1910s. It makes the exclusive focus on Boole and Frege – at the expense of other, lesser-known or even forgotten, logicians, including female logicians – all the more surprising: as has long been the case in the history of analytic philosophy, the search in the history of modern logic has traditionally been for ‘founding fathers’.11 A related but arguably even more fundamental problem is that what Quine and Putnam both missed were the relations of what are now called the modern traditions to the older syllogistic tradition. Their search for a discrete event (or even an exact year) obscured the fact that the mathematical turn itself was part of the broader process of the lingering demise of the syllogistic, that is, of the gradual downfall of what for over two millennia had been logic’s paradigm. Modern logic is without a doubt a major achievement. And it is true that it came to replace the older syllogistic logic. But it is incorrect, historically and conceptually speaking, to suggest that traditional and modern logic are opposites and that logic simply started anew upon entering the modern period. First of all, in order to establish a break with tradition, the pioneers of modern logic had to engage with that very tradition. They had to show, for example, that their logic could do everything syllogistic logic could do, and more, while the reverse was not the case. Moreover, in the first decades of the twentieth century, when the syllogism had been left behind and largely reduced to a historical curiosity, logicians continued to critically and constructively engage with it. This, at least, is what some of the chapters in the present volume show. The book’s purpose is threefold. First, it examines the role of reflections on and engagements with Aristotelian syllogistic logic in the creation of modern logic, putting the focus on the longue durée from the 1820s to the 1930s. Second, it does so by tracing how this informed the debates over the nature, scope and proper method of logic and shaped the cross-pollination of the various logical traditions in this period. Third, it presents the multifarious engagements – whether constructive, critical or destructive – of logicians with syllogistic logic as a missing link in the historiography of logic. One of this link’s surprising aspects concerns the new (dis)continuities which it makes possible to uncover. For instance, it becomes clear that the process from which arose modern logic was set in motion by a revived interest in the details of traditional syllogistic logic, spurred almost single-handedly – at least in the Anglophone world – by Richard Whately. Furthermore, the logicians who initiated

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Aristotle’s Syllogism

the mathematical turn, like Boole and Augustus De Morgan, reformed and expanded rather than abandoned the Aristotelian heritage. Also, even in and after Frege, syllogistic logic continued to play a key role, albeit in a completely different way than before. Frege himself appealed to it by way of contrast with his own new logic. It influenced the early development of set theory in the work of Bernhard Riemann, Richard Dedekind and others. And Hilbert had novel things to say about it too, just as he had about Euclidean geometry. The nineteenth century was the century in which modern logic came into existence. The nineteenth century is also the century in which the hegemony of syllogistics fell apart. It has hitherto not been fully recognized, let alone fleshed out in depth, that these two fascinating epoch-making processes were interestingly and complexly related. The fact that modern logic replaced traditional logic, and that it was only when modern logic arrived on the scene that logic became a great discipline – to use Quine’s famous phrase – has given rise to the tendency to study the former largely at the expense of the latter. This, in turn, has made that their mutual relation as well as most of what went on in logic in the nineteenth century remained poorly understood. The chapters in this volume seek to redress the balance, presenting the creation of modern logic as a long-term development for which the syllogistic tradition, in different ways and for various reasons, was formative. Much more remains to be said about this and there is no pretension of exhaustiveness in terms of specific authors and topics or wider ramifications. We would like to think of the volume as a contribution to recent attempts at a full-blown history of ‘logic’ in the multifaceted ways in which it was understood in the period between the early nineteenth and beginning of the twentieth centuries.12 Such a history would have to go far beyond deductive logic in Europe. It would also have to include, for instance, inductive logic and developments in other cultures, such as in the Arabic world.13 It is our hope that the present volume offers starting points for such a fuller and more wide-ranging endeavour. What follows is a short introductory overview of central aspects of the fourteen, chronologically ordered, chapters in the book. The reader is referred to the chapters themselves for more details and further discussion of the subjects. *** The book opens with Calvin Jongsma’s account of the pioneering work of Richard Whately, whose entry on ‘Logic’ in the Encyclopaedia Metropolitana (1823) and especially Elements of Logic (1827) revived the study of traditional syllogistic logic – or, more generally, of formal logic – among English-language philosophers. Whately thus went against a long-established and widely shared preference for inductive, experimental and mathematical ways of reasoning. However, he was not completely alone in facing the influential criticism of Scottish philosophers like Thomas Reid, George Campbell and Dugald Stewart, as he was supported by his teacher Edward Copleston and aided by his student John Henry Newman. One of Whately’s main arguments was that some discoveries in natural philosophy are discoveries of things ‘implied in that which we already know’. Therefore, syllogistic has a role to play in the

Introduction

5

process of the advancement of knowledge. Moreover, Whately argued, even though not all sound reasoning is essentially syllogistic, any argument can easily be evaluated in terms of its soundness by expressing it in syllogistic form. The importance of Whately’s achievement for the revival of the study of logic was widely acknowledged by English logicians, even those – like Boole or Mill – who aimed to expand and transform logic in ways that were at odds with or went beyond Whately’s conception and treatment of logic. In the second chapter Lukas M. Verburgt explores the creative role of the syllogism in the British tradition of inductive logic by focusing on the work of John Stuart Mill and especially on its reception by one of his most prominent followers, John Venn. Both Mill and Venn – contra Whately – held that the conclusion of a syllogism cannot state anything more than what is already implied in the premises. Mill concluded that syllogisms, considered as inferences, are essentially question-begging insofar as they involve a petitio principii. From this there followed Mill’s famous view that all inference is inductive. Like Mill, Venn belonged to the inductivist school. But unlike Mill, Venn came to believe that syllogizing cannot at all be described as an inferential process. Although the premises, in the end, are bound to result from inductive generalization, the step of moving from the premises to the conclusion – especially in the case of complex mathematical reasoning – involves ‘mental labour’. In this way, Venn made it possible for the inductivist tradition to engage with recent developments in the mathematization of logic associated with De Morgan and Boole. Syllogistic could have developed into a much more comprehensive and extended field of logic earlier, considering that the Aristotelian definition of syllogism as ‘a discourse in which, certain things being stated, something … follows of necessity’ can be conceived in a much broader sense. As it was, for centuries no one questioned, for instance, whether the principle standing at the basis of traditional syllogistic – the famous dictum de omni et de nullo – is a tautological starting point. Moreover, no one took seriously the idea that the only thing which really matters in the definition of syllogism is ‘that the conclusion becomes true whenever the premises are true’. These and other considerations stand at the heart of Bernard Bolzano’s work, as shown in the third chapter by Mark Siebel. According to Bolzano’s prescient Theory of Science (1837), the proper object of syllogistic is neither mental occurrences nor linguistic signs but ‘sentences in themselves’ consisting of ‘ideas in themselves’. Much like Aristotle’s original definition of implication, Bolzano’s notion of deducibility is concerned with truth-preservation: the conclusion of an inference is deducible from the premises if they contain ideas in themselves whose substitution leads to a true conclusion if the premises are true. This notion does comprise not only formal validity but also material validity, i.e. a kind of validity being dependent on the meaning of non-logical expressions within the sentences involved in the inference. This is the stance from which Bolzano envisaged innumerable forms of logical inference beyond the limitations and constraints of traditional syllogistic. The fourth chapter concerns George Boole’s development and generalization of Aristotelian logic. Boole first used logic to argue for a new conception of mathematics, but in the process of treating logic mathematically he came to embrace novelty in logic itself with increasing confidence over time. His first step is found in The Mathematical

6

Aristotle’s Syllogism

Analysis of Logic (1847), where Boole realized that a system of equations could reveal the more fundamental structure of Aristotelian syllogisms. He subjected the traditional pattern to the algebra of his calculus or ‘pure analysis’, as expressed by a new notational symbolism. More specifically, the syllogism became a system of two equations in three variables and, more generally, the number of possible inferences was considerably augmented. For the second step, David E. Dunning focuses on an interleaved copy of The Mathematical Analysis of Logic, in which Boole returned to the topic soon after having published the book. In his handwritten notes, Boole included negative subjects, doubled the number of categorical propositions to eight and offered a more complete classification of syllogisms by mathematical derivation. Moreover, he identified three algebraically derived rules of inference, which were intended to replace the traditional ones. The third step is represented by a manuscript entitled Elementary Treatise on Logic not mathematical including philosophy of mathematical reasoning. As the title suggests, syllogistic was treated in a ‘non-mathematical’ way. According to Boole, the merit of traditional syllogistic forms decreased as a result of it. The fourth step corresponds to the publication of The Laws of Thought (1854), where Boole put forward a more critical understanding of Aristotelian syllogistic, whose limitations, incompleteness and arbitrariness were made explicit to the reader. As a last step, Boole planned a book for a non-specialist audience, entitled The Philosophy of Logic, which was never brought to completion. All things considered, it was the rewriting of Aristotelian logic in a mathematical form that made Boole realize how logic could encompass a wider range of possible inferences and take syllogistic to a new level. In the fifth chapter, Sun-Joo Shin analyses the transition from the dominant categorical syllogism to relational arguments as realized by Augustus De Morgan and Charles S. Peirce in their respective efforts to expand the domain of logic. De Morgan was an ardent defender of the efficacy of the syllogism and tried to extend its functionality beyond its traditional schematism, which he considered a restriction on its deductive potential. His major contribution, in this regard, was the development of a ‘logic of relations’ which widened the scope of syllogistic tout court. On his reading, the copula expressed every kind of linking relation between two terms and any logical relation could be considered a composition of two relations. Accordingly, a syllogism was defined as ‘a composition of two relations into one’. Later, Peirce developed an extended version of Boole’s algebra and restructured De Morgan’s relational arguments in such a way that they could be decomposed and analysed into more than one traditional syllogism. This approach made Peirce think of relations as compositions of relations which could be multiplied to handle complex chains of inferences, going much beyond the triadic pattern of syllogistic reasoning. Thus, De Morgan and Peirce shared the idea of interpreting syllogistic as a sort of inferential subset of relational reasoning – an idea that resulted in an unprecedented and highly versatile strengthening of deductive syllogistic reasoning. The sixth chapter discusses Ernst Schröder’s algebra of logic. The main focus is on his three-volume set Vorlesungen über die Algebra der Logik, completed between 1890 and 1905. Within the system presented there, Volker Peckhaus singles out in particular the treatment of the ‘Logic of the Ancient’, as Schröder called it. In the

Introduction

7

Vorlesungen, traditional syllogistic and early calculi of classes are replaced by a new propositional calculus, where the categorical forms are translated into algebraic expressions. The calculus is governed by two principles: the principle of propositional identity and the principle of transition, standing at the basis of the first syllogism of the system, namely the hypothetical ‘inference of substitution’. Schröder used Friedrich Ueberweg’s System der Logik (1857) as his starting point and Hermann Lotze’s Logik (1843) as his critical target. He constructed his system by further developing Boole’s elimination theorem, while also taking the work of De Morgan and of Joseph Gergonne into account. Like MacColl, Schröder advocated the requirement of existential import for categorical propositions and praised Christine Ladd-Franklin for her brilliant validation formula. On the basis of systematic combinatory calculations and elimination rules, Schröder was able to select and reorder fifteen valid forms of traditional syllogisms while recovering some of the invalid forms too. Despite his plan to develop a complete syllogistic, Schröder favoured the generality of his ‘algebraic logic’ and emphasized the pre-eminence of his elimination theorem. The next chapter presents Franz Brentano and Franz Hillebrand’s ‘idiogenetic theory’, a post-scholastic type of syllogistic theory involving acts of judging which were regarded as belonging as such to a special genus (idios genos) of psychical phenomena. The logical traits of the theory were first put forward by Brentano in his Psychologie vom empirischen Standpunkt (1874, first ed.) and then formally presented in Hillebrand’s Die neuen Theorien der kategorischen Schlüsse (1891). The most novel aspect of the theory was that all judgements were restated in existential form as singlemembered assertions, or rejections, whose subject and predicate could be simpliciter converted. The proposal provoked numerous reactions. Particularly the last part of Hillebrand’s system, namely the extension about ‘double judgments’ (existential and predicative judgments bound together), was criticized by Husserl and Meinong, among others. But it also received active support from Brentano’s student Anton Marty. In his chapter, Matteo Cosci recalls the Leibnizian antecedent that showed the character of supposition of the existential import holding in the traditional square of oppositions. That assumption was a matter of concern for Brentano, who may have been aware of its formulation (possibly via Leibniz’s Difficultates Quaedam Logicae) in the process of developing his own reform of syllogistic on new, intentionalistic grounds. Aside from its intrinsic merits and originality, Brentano and Hillebrand’s ‘idiogenetic theory’ had a considerable impact in the fields of descriptive psychology, analytic philosophy and early phenomenology towards the end of the century – not to mention its relevance for the great current in logic inaugurated by Kazimierz Twardowski, prominent student of Brentano and the standard-bearer of his reform in Poland at the beginning of the twentieth century. In the eighth chapter Jean-Marie Chevalier presents the surprisingly rich work that Hugh MacColl carried out on the syllogism at the turn of the twentieth century. MacColl first developed his own system of propositional logic, both working with and departing from Boole’s theories. He next explored the consequences of enriching propositional logic with operators, the basic one being ‘strict implication’, which added necessity to the simple material conditional. The operator was used to embrace all kinds of inferences such that, for instance, the relations between categorical propositions

8

Aristotle’s Syllogism

became equivalent to relations between conditional propositions. Accordingly, MacColl presented the syllogism as an inferential scheme whose nature is essentially hypothetical – or, more precisely, doubly hypothetical, as the hypothetical implication between premises and conclusion is reframed by MacColl as a second-order premise dependent on the general hypothesis which sustains the deduction as a whole. As a consequence, MacColl called into question the validity of all syllogisms expressed in the traditional form: the possibility of deducing something true had always been assumed without taking into account the underlying hypothetical character of the whole operation. For MacColl, every syllogism should be introduced by an ‘If ’ which is antecedent to, and dominant over, the ‘therefore’ between the premises and the conclusion. Only under this condition are what MacColl called ‘formal certainty’ and ‘syllogistic validity’ ensured. MacColl also solved the problem of the existential import in his own way. For, MacColl argued, once one identifies different senses of existence and distinguishes a universe of existent entities from a universe of non-existent entities, ‘[i]n pure logic the subject, being always a statement, must exist – that is, it must exist as a statement’. Accordingly, within MacColl’s system of propositional logic all statements denote something, even those concerning fictitious entities. His research also led him into the uncharted territory of non-classical, multidimensional logic. There he classified five different truth-values corresponding to five different types of syllogisms. Under his reading, syllogisms express relations of strict implication within the realm of possibility, namely a domain that is non-existent but also non-contradictory. In the ninth chapter Erich H. Reck reconsiders the role of Gottlob Frege’s Begriffsschrift (1879) in the transition from Aristotelian logic to modern logic. According to Frege, traditional ‘term logic’ did not come with a clear object-concept distinction and, more generally, was too attuned to ordinary language. For this reason, it was an inadequate tool for the analysis of the kind of mathematical concepts that Frege was interested in within his logicist project. Therefore, Frege broke with some of the basic tenets of the Aristotelian syllogistic. He rejected the Aristotelian predicative model based on the subject-predicate distinction, introduced the material conditional together with the modus ponens as the one sufficient kind of inference, reformulated the categorical proposition in a symbolic and more rigorous way, contrasted the idea according to which concepts are ‘sums of marks’ in favour of nested quantifications and, finally, distinguished multiple meanings of the copula on the basis of different notations. On closer inspection, however, while at odds with the Aristotle of the Prior Analytics, Frege’s commitments were rather similar to Aristotle’s in the Posterior Analytics. Put differently, Frege proved to be an Aristotelian in epistemology, if not in logic.14 He, perhaps unwittingly, inherited Aristotle’s definition-based approach, his kind‐crossing prohibition rule, the axiomatic-deductive model and the awareness that not everything can be proved on pain of circularity. Another similarity between Frege and Aristotle is the adoption of a first-person perspective in logic, for a logic oriented towards universal applicability. In this sense, it can be said that Frege’s trailblazing contribution to the emergence of modern logic went hand in hand with a pre-modern indebtedness to an Aristotelian model of science. One of the most long-standing open questions concerning the syllogism was how to identify syllogistic validity in a simple and effective way. This problem was as old

Introduction

9

as the syllogism itself and highly complex. ‘The Syllogism’s Final Solution’, as Susan Russinoff called it,15 was found by Christine Ladd-Franklin as late as 1883. This was the year in which Ladd’s test for the validity of any syllogism (or ‘antilogism’, as it was called) was published for the first time in a text titled ‘On the Algebra of Logic’, which appeared in a collection of articles by Peirce and his John Hopkins University students. Whereas Peirce had previously adopted two symmetrical types of copula (one for the propositions of existence and the other for the propositions of non-existence), LaddFranklin used a single type of quantified copula (positive for universal propositions always denoting non-existence and negative for particular propositions always denoting existence). The fundamental relation to be specified between two classes of – sets could now be expressed by the function of exclusion (symbolized by V) rather than 16 by the inclusion relation as debated by other logicians of the period. Through these innovations, Ladd was able to provide a general characterization of all valid syllogisms in one single and exact formula. After many centuries, Aristotle’s problem was solved by Ladd-Franklin’s simple and elegant validation rule, though it took time to be accepted and even more time to be formally proved. In the tenth chapter of the book, Francine F. Abeles suggests that the idea for the solution of the problem may have come to Ladd from an ‘eliminativist’ reading of Aristotle himself, in particular from a certain passage of De Interpretatione. What is certain is that both Evert W. Beth’s and Charles L. Dodgson’s (i.e. Lewis Carroll’s) subsequent developments in logic were indebted, at least to a certain extent, to this original result. Over the course of her life, in which she struggled for official recognition, Ladd-Franklin maintained that the most important form of reasoning is syllogistic reasoning and that scientific knowledge is nothing but a network of truths whose connections are conclusions of valid syllogisms. The eleventh chapter studies how the syllogism was conceived by Giuseppe Peano and his school (Burali-Forti, Padoa, Vailati, Pieri). Although Aristotle and scholastic sources were never explicitly mentioned, the Peano School’s study of syllogistic logic was not conducted with an obliterating approach. Indeed, syllogism was taken as the fundamental characterization of deduction as such. As soon as it was envisaged as a form of calculus of classes, syllogistic came to be regarded as a valuable part of mathematical logic, which in turn came to be seen as an extension of traditional syllogistic. Moreover, the introduction of a new mathematical symbolism in 1898 allowed the group to notice how the syllogism could operate not only as an axiom but also as a rule of logical identity and as a rule for the elimination of the common term. They realized that the syllogism could be understood as the property of transitivity holding between three terms and as the instantiation of logical transitivity itself. In addition, they realized that it could be used as a regulatory premise at the start of proofs and that the rule of inference known as modus ponens could be interpreted as a new kind of syllogism too. Peano’s group laid down additional inferential rules – or ‘various species of syllogism’, as they called them – which extended the traditional forms with reference both to classes and to propositions. Syllogistic moods and forms were ultimately reduced to three general types, dubbed ‘syllogism’, ‘singular syllogism’ and ‘sorites’. Such a taxonomy – as Paola Cantù explains – was the result of a collective effort to identify, simplify and, at the same time, generalize all the possible properties of deduction.

10

Aristotle’s Syllogism

In the twelfth chapter William Ewald presents the view of syllogistic which David Hilbert presented to his students as an integral part of his systematic metamathematical study of logic in his Göttingen lectures of 1917–18. On the model of Whitehead and Russell’s recent Principia Mathematica, and with the aid of his assistant Heinrich Behmann, Hilbert set up a logical system modelled on the requirements of consistency, independence and – most importantly – completeness. Using Russell’s axiom of reducibility as a guiding principle, the calculus of predicates was extended to a calculus of classes which could also deal with the traditional syllogistic. In this framework, Hilbert’s treatment of syllogistic was neither optional nor arbitrary, but rather key to proving the comprehensiveness and capacity of the overarching mathematical method. After all, the ideal of completeness that Hilbert was pursuing in his system was both systematic and historical, and the latter quality was regarded as no less important than the former. Therefore, syllogistic had to be included in the analysis with an eye to the historical completeness of the ‘Grundlagen der Logik’. Aristotle had to be encompassed or ‘nostrified’ – as Göttingen mathematicians used to say – within the mathematical system under development so as to achieve the general aim of ‘deepening the foundations’ of logic. In this sense, Hilbert’s reformulation of the Aristotelian syllogistic stands as an elegant and compelling example of the exhaustiveness and rigorous systematization of that foundational program. Early set theory emerged from innovations in mathematics and in the algebraicized logic of classes. Since the logic of classes mainly emerged from the analysis of syllogistic logic, it can be said that early set theory inherited a great deal from syllogistic logic. In fact, early set theory, as formulated by Riemann and Dedekind, basically followed this line of development. Traditional logic, after all, was an integral part of the syllabus of the German Gymnasium and classical logic textbooks such as those by Herbart and Drobisch were actively read and studied in the mid-nineteenth century. More specifically, early set theory was influenced by syllogistic logic through the consideration and adoption of some of its canonical elements, such as the primacy of concepts and their ‘extension’, the study of syllogistic figures by means of the analysis of inclusion relations, and the algebraicization of logic by a calculus of classes, as José Ferreirós explains in the thirteenth chapter of this book. Both Riemann’s introduction of the concept of ‘manifold’, which stands at the origin of the concept of ‘set’ (1854), and Dedekind’s development of the notion of ‘system’ (a synonym of ‘class’ or ‘manifold’), which was basic to his approach to number sets and to ‘ideals’ (1871), bear strong traces of syllogistic theory. The last chapter explores how the pattern of syllogistic inference was received, generalized and enhanced by the Göttingen school of logic and by Gerhard Gentzen in particular. In 1932 Gentzen put forward ‘a formal definition of provability’, as he called it, which brought to completion – and strongly resembled – the completeness proof that may be found in Aristotle’s Prior Analytics. Gentzen’s celebrated theorem was a rigorous formalization of a general notion of logical consequence – including the syllogism – which surpassed the traditional distinction between syntax and semantics by resorting to the so-called cut and thinning rules. Gentzen’s result was informed in Göttingen by Paul Hertz’s Aristotelianism, Hilbert’s inferentialist semantics and Jacques Herbrand’s notion of champs finis. By recasting the intuitive concept of synthetic

Introduction

11

consequence as an intrinsic feature of inference itself, Gentzen managed to find a way to reconcile the tension – first made explicit by Bolzano – between the requirement of exclusive analytical consequences and their logical ‘derivability’. Moreover, his elimination theorem included the syllogism rule (that Bolzano had tried to keep out) in a more comprehensive and synthetic inference rule so that in the end a single meta-syllogistic formula could stand in place of, or ‘eliminate’, all syllogistic forms. To conclude with the words of Curtis Franks, ‘the Göttingen logicians taught us that syllogism-free reasoning is significant because with it one can replicate the syllogism’s inferential scope. The syllogism is revealed to be, not deadwood or a redundancy of logistic theory, but the gold-standard of inference against which meaning-constitutive rules are measured. […] [T]he syllogism is able to capture, in concrete form, the whole abstract concept of logical consequence, so that its elimination confers on a logic the label of completeness. And this […] is just the role that Aristotle saw it playing from the beginning’.17 *** Taking the period between the 1820s and 1930s as its focus, this book looks at the central and sometimes surprising role of the syllogism in pioneering attempts to go ‘under, over and beyond’ Aristotle. One of its main overarching points is that these attempts played a formative part in the creation of (‘non-Aristotelian’) modern logic. The Aftermath of Syllogism – which appeared in this Bloomsbury series in 2019 – ended with Hegel. The present volume basically takes things from there. It tells a more complex and arguably even more surprising story than the one told in that book. Much remains to be said and many open questions remain to be answered. And, as already emphasized, we make no claim to exhaustiveness when it comes to relevant authors, themes and topics, let alone to the historical and theoretical ramifications of and connections between the various chapters. Our focus is squarely on the interplay between the old and new logic. We take it as a sign of fruitfulness that this focus offers so much food for thought, whether regarding dominant narratives and well-known themes in the history of logic and philosophy or the new and lesser-known authors and topics it sheds light on. Rather than ending this introduction with general remarks or conclusions, we would like to end with suggestions for further research on the topic, given in no particular order: – Aristotelianism in the nineteenth- and early twentieth-century history of logic, history of philosophy and history of science; – The work on philosophical and formal logic by female logicians such as E. E. Constance Jones, Sophie Bryant, and Augusta Klein; – The transition from syllogistic to modern logic on the level of education and university curricula; – Different kinds of influential, forgotten or otherwise minor works on syllogistic, including textbooks, new editions of older Latin tracts, didactic works, ‘grey literature’ (theses, dissertations, lecture notes, unpublished manuscripts), memory

12

– – – –

Aristotle’s Syllogism aids for remembering all the different syllogistic forms and moods, and scholarly and/or philological studies of the history of syllogistic, such as Karl von Prantl’s Geschichte der Logik im Abendlande (1855) and the monumental volumes by Heinrich Maier Die Syllogistik des Aristoteles (1896–1900); Attempts at the diagrammatic representation of syllogisms in relation to the literature on the history of visual reasoning; The syllogism as an object of study in fields beyond logic, such as cognitive psychology (Helmholtz, James, von Hartmann, Bain and Torrey Harris), medicine (Hector Donon) and epistemology (Ellingwood Abbot); The development of syllogistic logic outside Western Europe, including in the Polish, Soviet, Arabic, Chinese and Indian traditions; John Neville Keynes’s work on the non-categorical syllogistic and its place at the crossroads of syllogistic and modern logic.

To paraphrase Umberto Eco’s The Infinity of Lists, no list is ever complete – and the same holds for this one. All our suggestions merely serve as pointers towards a fuller understanding of the central topic of the present book.

Notes 1

Other sources of inspiration for this project have been James Van Evra, ‘The Development of Logic as Reflected in the Fate of the Syllogism, 1600–1900’, History and Philosophy of Logic 21, no. 2 (2000): 115–34, José Ferreiros, ‘The Road to Modern Logic – an Interpretation’, The Bulletin of Symbolic Logic 7, no. 4 (2001): 441–84, and Calvin Jongsma, Richard Whately and the Revival of Syllogistic Logic in Great Britain in the Early Nineteenth Century, PhD dissertation (advisors: Kenneth O. May and John Corcoran), University of Toronto, 1982. 2 See John Corcoran, ‘Aristotle’s Prior Analytics and Boole’s Laws of Thought’, History and Philosophy of Logic 24, no. 4 (2003): 261–88. 3 This part of the story of the emergence of modern logic is told particularly clearly in Ferreirós, ‘The Road to Modern Logic’. 4 Jeremy Heis, ‘Attempts to Rethink Logic’, in The Cambridge History of Philosophy in the Nineteenth Century (1790–1870), ed. Allen W. Wood and Songsuk Susan Hahn (Cambridge: Cambridge University Press, 2012), 95–132, on 96. 5 Robert Adamson, A Short History of Logic (London: W. Blackwood, [1882] 1911), 20. 6 John Venn, ‘Consistency and Real Inference’, Mind 1, no. 1 (1876): 43–52, on 43. 7 See, in this regard, Lorenz Demey, ‘Mathematization and Vergessenmachen in the Historiography of Logic’, History of Humanities 5, no. 1 (2020): 51–74. 8 Dov M. Gabbay and John Woods, ‘Preface’, in Handbook of the History of Logic. Volume 4: British Logic in the Nineteenth Century, ed. Dov M. Gabbay and John Woods (Amsterdam and Oxford: North-Holland, 2008), vii–xii, on viii. 9 Amirouche Moktefi, ‘Lewis Carroll’s Logic’, in Handbook of the History of Logic. Volume 4: British Logic in the Nineteenth Century, ed. Dov M. Gabbay and John Woods (Amsterdam and Oxford: North-Holland, 2008), 457–505, on 457. 10 See, for example, Wolgang Kienzler, ‘Three Types and Traditions of Logic: Syllogistic, Calculus and Predicate Logic’, in Philosophy of Logic and Mathematics, ed. Gabriele

Introduction

11

12 13

14 15 16 17

13

M. Mras, Paul Weingartner and Bernhard Ritter (Berlin and Boston: Walter de Gruyter, 2010), 133–52. For a forceful critique of the historiography of analytic philosophy see, for instance, Frederique Janssen-Lauret, ‘Grandmothers of Analytic Philosophy: The Formal and Philosophical Logic of Christine Ladd-Franklin and Constance Jones’, Minnesota Studies in Philosophy of Science, forthcoming. An important recent work, in this regard, is Sandra Lapointe (ed.), Logic from Kant to Russell. Laying the Foundations for Analytic Philosophy (London: Routledge, 2019). See, in this regard, for example Dov M. Gabbay, Stephan Hartmann and John Woods (eds.), Handbook of the History of Logic. Volume 10: Inductive Logic (Oxford and Amsterdam: North Holland, 2011), and Khaled El-Rouayheb, Relational Syllogisms and the History of Arabic Logic, 900–1900 (Leiden and Boston: Brill, 2010). See Willem R. de Jong and Arianna Betti, ‘The Classical Model of Science: A Millennia-Old Model of Scientific Rationality’, Synthese 174 (2010): 185–203. I. Susan Russinoff, ‘The Syllogism’s Final Solution’, The Bulletin of Symbolic Logic 5, no. 4 (1999): 451–69. See also Sara L. Uckelman, ‘What Problem Did Ladd-Franklin (Think She) Solve(d)?’, Notre Dame Journal of Formal Logic 62, no. 3 (2021): 527–52. See Amirouche Moktefi, ‘The Social Shaping of Modern Logic’, in Natural Arguments: A Tribute to John Woods, ed. Dov Gabbay, Lorenzo Magnani, Woosuk Park and AhtiVeikko Pietarinen (Rickmansworth: College Publications, 2019), 503–20. See Curtis Franks’s contribution in this book, 356–80, in particular on 379.

14

1

Richard Whately’s revitalization of syllogistic logic Calvin Jongsma

die Logik … seit dem Aristoteles keinen Schritt rückwärts hat tun dürfen … Merkwürdig ist noch an ihr, daß sie auch bis jetzt keinen Schritt vorwärts hat tun können, und also allem Ansehen nach geschlossen und vollendet zu sein scheint. – Kant’s ‘Preface’ to the Critique of Pure Reason (B-edition, 1787) Immanuel Kant’s epigram on Aristotle’s logic has an authoritative ring to it, but strictly speaking, it is false. Kant’s contemporaries, however, would likely have agreed that Aristotle’s core logic comprised a completed system of valid syllogistic forms. Half a century later, British logic put the lie to Kant’s appraisal. William Hamilton extended formal logic by quantifying the predicate; John Stuart Mill reinterpreted logic by incorporating inductive reasoning; Augustus De Morgan generalized syllogistic logic to include relations as well as properties; and George Boole used algebra to express propositions and deduce consequences from premises, opening up propositional logic in the process. Each of these trends – particularly those originating in De Morgan’s and Boole’s investigations – broke through the traditional bounds of Aristotelian logic. The baseline for this transformation was Richard Whately’s spirited defense of the syllogism mounted around 1825. By mid-century, British logicians uniformly acclaimed Whately as having restored Aristotelian logic, though they also recognized that this played no active role in further developments.1 Documenting Whately’s work as the catalyst in revitalizing syllogistic logic is the focus of this chapter. For background, we will first briefly catalogue the chief criticisms of logic still active in his day.2

1 Seventeenth-century challenges to Aristotelian logic: Syllogism’s competitors and critics Early modern mathematics and natural philosophy made little use of traditional forms of reasoning. Systematic emphasis on experimentation, induction and quantitative relationships left scientific practitioners less reflectively focused on the forms of

16

Aristotle’s Syllogism

deduction. And, as mathematics moved away from an axiomatic geometric basis, where syllogistic forms had never taken root, towards the analytic approach of algebra and calculus, deductive reasoning found less of a home there than earlier. Syllogistic reasoning was also attacked by advocates for these fields, and towards the end of the century traditional logic was challenged by an empiricist epistemology.    (i) In 1620 Francis Bacon opposed the syllogism from the perspective of natural science, proposing an inductive organon of discovery to compete with Aristotle’s logic.3 Syllogistic reasoning is of little value for acquiring knowledge of nature because it remains trapped in the realm of words and is prone to accept unverified principles from past authorities. (ii) In 1637 René Descartes challenged traditional logic from the vantage point of mathematics. To guide the mind in its search for truth, one must follow the method of analysis. Syllogistic reasoning is useful only for (artificially) organizing and conveying to others what is already known.4 (iii) In 1690 John Locke’s monumental Essay Concerning Human Understanding advocated relying on our native reasoning process, comparing sensation-based ideas with one another rather than constraining them by artificial syllogistic forms. In the well-known sarcasm of his Essay’s fourth edition (1700), Locke insists that ‘God has not been so sparing to Men to make them barely twolegged Creatures, and left it to Aristotle to make them Rational’.5

2 Eighteenth-century challenges to Aristotelian logic: Logic textbooks and logic critics Locke’s epistemic focus and his stigmatizing formal reasoning became central emphases in eighteenth-century logic.6 The popular logics by Watts (1725)7 and Duncan (1748)8 exhibit strong influences of Descartes and Locke: logic aims at rightly directing our mental faculties in the pursuit of truth. At mid-century, Bacon’s proposal for a logic of induction gradually came into its own, being embraced by Scottish common-sense philosophers, who also followed Bacon in opposing Aristotelian logic. (iv) Lord Kames’s Sketches of the History of Man (1774) accused syllogisms of superficially deducing conclusions that are better known than their premises: syllogistic conclusions are more contained in their premises than inferred from them.9 (v) Thomas Reid published ‘A Brief Account of Aristotle’s Logic’10 as a seventy-fourpage appendix to Kames’s book. His wide-ranging criticisms challenged both logic’s technical organization and its utility. Reid scorned reducing syllogisms to first-figure forms to show their validity and basing them upon Aristotle’s Dictum de Omni et Nullo, which Whately was later to promote as the ultimate foundation of reasoning. Reid also criticized logic’s utility. Syllogistic forms are rarely used, even in mathematics. And natural science advances by determining a solid foundation of first principles, and these are established by induction, not syllogisms.

Whately’s Revitalization of Syllogistic Logic

17

(vi) George Campbell’s Philosophy of Rhetoric (1776) reproached syllogisms as epistemically sterile, proceeding from things less known to ones better known.11 In fact, the syllogism is a petitio principii. In establishing the general premise by induction from particular cases, one first needs to demonstrate the conclusion – one such case. A syllogistic argument thus completes a vicious circle: ‘[T]here is always some radical defect in a syllogism which is not chargeable with this.’12 (vii) Dugald Stewart’s first volume of his Elements of the Philosophy of the Human Mind (1792) acknowledged the formal nature of valid arguments, even affirming the value of literal symbolism: conclusions are ‘most likely to be logically just, when the attention is confined solely to signs’ and not led astray by ‘casual associations’.13 In fact, he asserts that ‘every process of reasoning is perfectly analogous to an algebraical operation’.14 Stewart’s second volume (1814) backed away from the view that reasoning is grounded in a well-ordered, unambiguous language modelled on algebra,15 because reasoning involves the ‘silent habits of interpretation’ more than the formal application of rules.16 His assessment of traditional logic now relitigates and reinforces earlier criticisms. Like Reid, he believes that the project of demonstrating demonstrations is ill-conceived. He adds that this is circular, for the conclusive character of such demonstrations is established by putting them into syllogistic form, the very thing whose validity is at issue.17 Stewart also profiles Campbell’s petitio principii charge, declaring it ‘unanswerable’.18 Stewart likewise reaffirms the criticisms of Bacon, Locke, Reid and Kames regarding the inutility of syllogistic reasoning, as it proceeds in the opposite direction of what is needed for investigating nature, where observation, experiment and induction are required. A complete logic must include the study of induction as well as the syllogism.

3 Early nineteenth-century Oxford defence of Aristotelian logic Edward Copleston entered the arcane world of Aristotelian logic around 1797 when he discovered as an Oriel College tutor that he was expected to teach the subject to his students. This became more pressing after 1800 when Oxford began making examinations more rigorous.19 Not having studied the subject earlier, Copleston threw himself into the task, drawing ideas from various texts, especially Aldrich’s logic. Copleston’s outlook on logic was also shaped by two controversies. In The Examiner Examined, or Logic Vindicated (1809) he took apart Henry Kett’s inept potboiler Logic Made Easy, a loose synthesis of scholastic and modern logics20 aimed at capturing the Oxford market. This was a warm-up salvo for three pamphlets written shortly thereafter,21 in which Copleston ably defended Oxford’s classical education against repeated Scottish ‘calumnies’ fired anonymously by John Playfair in the Edinburgh Review and in response to a friendlier polemic by Henry Home Drummond, Lord Kames’s grandson.22 These writings plus extracts from Copleston’s commonplace book23 give insight into Copleston’s ideas on logic.

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Logic is an essential part of Oxford’s liberal arts curriculum, Copleston says, because it develops a disciplined mind and because it provides tools to test an argument’s validity and so ‘cut short wrangling’.24 Logic does not teach a method of inquiry, nor is it an engine of science. Consequently, the syllogism has not been superseded by Baconian induction, nor are they in competition.25 Replacing Aristotle’s syllogism with Bacon’s induction would be like discarding ‘the windmill, because the steam-engine has been invented, or … the mariner’s compass, since the introduction of gunpowder’;26 these things serve different purposes. Logic is not reason’s guide in the search for truth but is ‘an art of language’,27 regulating how words unpack what we know and convey our reasoning to others. Copleston wards off Campbell’s petitio criticism by asserting that syllogisms capture how the mind draws particular conclusions from general principles.28 Though he doesn’t elaborate, Copleston claims that syllogistic reasoning can prove new truths from inductively acquired premises.29 However, the ‘chief boast of Aristotelian logic’ is that it provides categorical forms into which any argument whatsoever may ultimately be resolved.30 These forms are justified by various rules but ultimately by Aristotle’s Dictum de Omni et Nullo, which is ‘the Nucleus of the whole system’.31 Copleston’s ideas regarding the structure and utility of Aristotelian logic, along with his stress on the importance of language for logic, was soon adopted as the core of Whately’s own understanding of Aristotle’s logic.

4 The genesis of Whately’s logic: Copleston, Newman and Whately Richard Whately matriculated at Oriel College in 1805, where Copleston became his tutor, colleague and friend. Whately was elected a fellow in 1811, tutoring students privately over the next decade. As a member of the Oriel community, Whately honed his ideas on various topics through robust conversations with others on rambling walks through the countryside rather than by bookish scholarship.32 Whately’s strength lay in clearly explicating the fundamentals of a subject,33 being little interested in pursuing them in depth. Whately’s first publication, Historic Doubts Relative to Napoleon Buonaparte (1819), was an adroit satirical adaptation of Hume’s scepticism towards Biblical accounts of miracles applied to testimony regarding (the still living) Napoleon. Employing arguments in defence of Christian beliefs was something Whately did repeatedly over his lifetime, believing it a key reason why students should be well versed in logic.34 Given his growing reputation, Whately was an obvious choice to contribute the article on ‘Logic’ to the inaugural edition of the Encyclopaedia Metropolitana. This was an English competitor to the Scottish Encyclopaedia Britannica, making Whately’s article a rival of theirs, which was essentially Duncan’s logic supplemented by Reid’s critique. Whately’s two-part article was published in the middle of 1823.35 While it drew upon materials Whately used for tutoring students in logic, these were substantially derived from Copleston.36 Whately’s article also benefitted from some editorial sculpting by his student John Henry Newman, who in the summer of 1822 was assigned the task

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of converting Whately’s manuscript ‘Analytical Dialogues’ into a discursive form.37 Whately then revised Newman’s draft, incorporated the more technical parts based on Aldrich’s logic and added a chapter on fallacies before submitting it to the Encyclopaedia Metropolitana. Whately’s ‘Logic’ drew immediate acclaim. It was deemed ‘at once profound and popular’, answering ‘the vituperative charges … [of] the wise men of the North’ and presenting the subject in a manner ‘both pleasing and familiar’.38 By November 1826 it also appeared in book form as Elements of Logic. Gaining a ready audience, a second edition quickly followed in 1827, with seven more appearing before 1850.39 In the end, Elements of Logic became the most widely used nineteenth-century logic textbook in Great Britain and the United States, greatly exceeding Whately’s expectations.40 Whately slightly revised his original article for the book edition, adding a twopage dedicatory letter of appreciation to Copleston plus a twenty-three-page ‘Preface’ defending the utility of logic and assessing the state of logic at Oxford. This was followed by material taken from his article: an ‘Introduction’ discussing the nature of logic and various attitudes towards logic through history; an ‘Analytical Outline of the Science’, which gave students a helpful synopsis of logic and a rationale for studying it; a ‘Synthetical Compendium’, which systematically presented the standard technical topics; a substantial chapter on ‘Fallacies’, which later thinkers judged masterful; and a concluding ‘Dissertation on the Province of Reasoning’, which explored induction and the epistemic import of logic. To discuss all of this, we will first focus on Whately’s view of the nature and scope of logic, then we will outline the contours of Whately’s technical treatment of logic, and finally we will examine Whately’s responses to the criticisms of Aristotelian logic.41

5 Whately’s view of the nature and scope of logic Whately opens his 1823 article by announcing that ‘[l]ogic … may be considered as the Science and also as the Art of Reasoning’.42 This statement sets out two significant theses. First, Whately insists that (Aristotelian) logic is a science and not merely an art.43 Neglecting the scientific character of logic forces one to focus too heavily on logic’s utility. This led earlier thinkers like Watts to expand logic’s domain beyond its natural boundary, making demands of it that could not be met. No one system can rightly direct the mind in all its mental operations on all subjects. Logic’s true aim is more modest, focusing on the reasoning process. That’s Whately’s second main point. Logic’s concern is with reasoning: ‘It investigates the principles on which argumentation is conducted, and furnishes rules to secure the mind from error in its deductions.’44 Logic determines those rules that govern the mind as it makes sound arguments.45 Logic is thus the theory (science) and practice (art) of reasoning.46 Advocating a leaner logic with reasoning as its chief focus is relatively new with Whately. Earlier logicians, particularly those following Descartes and Locke, held that logic should teach one how to arrive at clear and distinct ideas, classify and define things appropriately, and assess evidence for judgments and determine their truth,

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as well as evaluate arguments and avoid fallacious reasoning.47 For Whately, terms and propositions are considered only insofar as needed to understand the operation and use of sound reasoning.48 Of course, besides syllogistic technicalities, Whately highlights his defence of logic and his viewpoint on the proper nature and scope of logic; these are, in fact, what distinguishes Whately’s text from others. Whately adds an important qualifier to his original classification of logic as he begins his systematic exposition. All parts of logic, he says, require the medium of language – for expressing ideas, judgments and reasoning as terms, propositions and arguments.49 In 1826 he formulates this in stronger terms: ‘[L]ogic is entirely conversant about language.’50 He supplements this in 1834 by claiming that if any mental reasoning can occur without the use of language, this lies outside the province of logic.51 Whately’s ideas here are closely allied with Copleston’s, and Stewart placed a similar emphasis on language in 1792. This also seems not that different from Aldrich, though Whately begs to differ.52 For Aldrich, words are the necessary and convenient signs logic must use in place of what they signify,53 but this only makes language an instrument of logic, not its essential foundation. Whately elaborates his viewpoint on language by asserting that ‘the conclusiveness of [an argument] is manifest from the mere form of the expression, i.e. without considering the meaning of the Terms’.54 This is most clearly exhibited by using letters to stand for the terms. Here, too, Whately goes further than Aldrich, who, while acknowledging that formal consequences of premises are the proper domain of logic, still allows there to be two kinds of inferential consequences – material consequences, which depend on the meaning of the terms involved, and formal consequences, which depend instead on the argument’s logical form.55 Aldrich continues, however, by illustrating the latter sort, as does Whately, with an overly simple argument formulated symbolically using letters: ‘B is A, C is B, therefore C is A.’56 Whately supports using letters to exhibit the forms of propositions and syllogisms in contradistinction to critics like Reid.57 He notes that this practice is like using number symbols in arithmetic and letters in algebra58 to abstract the notion of quantity from the concrete contexts and specificity in which they occur. Symbolic representation of propositions makes one concentrate on the logical connection between premises and conclusion ‘without any risk of being misled by the truth or falsity of the conclusion’.59 Without the generality of symbolic representation, Whately says, no science can be developed, for either logic or mathematics. That an argument’s validity is strictly due to its logical form, independent of the meaning of the terms and propositions involved was widely acknowledged, at least by traditional logicians and its defenders, but also by Stewart, who endorsed this feature of conclusive arguments in 1792. So while logic’s formal character is a central tenet of Whately’s approach, it certainly was not new.60 What can be argued in this regard is that Whately effectively reclaims and underscores this time-honoured feature of reasoning. Emphasizing the value of language may have facilitated this: ‘the form of expression  … alone is regarded in Logic’61 is something Whately stresses in several contexts. Whately thus simplifies and purifies traditional logic by excising material (semantic) considerations and dropping certain philosophical topics, reaffirming logic’s formal core as its proper focus. However, his logic is far from formalistic in any

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modern sense;62 concrete everyday arguments on sundry subjects remain the grist for his logic mill, even though their meaning and their premises’ truth values are deemed irrelevant for determining their validity. Symbolically formulating an argument helps to filter out noise that would complicate assessing its validity. But it also reveals the logical form of an argument. Whately doesn’t explicitly tell the reader what this ‘form’ consists of, though he uses the word repeatedly. The form of an argument seemingly includes its premise-conclusion arrangement (its figured mood) and also the propositions’ quality and quantity. In the final analysis, logical form is determined by the connection holding between the propositions’ terms (their ‘agreement’ or ‘disagreement’) and whether these terms are to be taken universally or partially (their distributed status).

6 Whately’s organization and technical development of syllogistic logic Aristotle had defined syllogism broadly as discourse in which, some things being supposed, something else necessarily follows. Over time, syllogism had become more narrowly understood as an argument having two premises and a conclusion – categorical statements with two terms, linked by a common middle term. Whately calls Aristotle’s more general notion an ‘argument’ and reserves the term ‘syllogism’ for the latter.63 Nevertheless, Whately insists that a syllogism is ‘not a peculiar kind of argument, but only a peculiar form of expression’64 into which all arguments may be resolved when ‘stated at full length and in … regular form’.65 Cast into such a form, an argument’s validity can be definitively tested.66 Syllogistic forms also capture the deductive process taking place in our minds, regardless of what subject the propositions treat.67 Whately makes this outlook on syllogisms a basic part of his defence of logic. After examining several valid arguments, Whately claims that they can all be reduced to ‘the universal principle of Reasoning’, Aristotle’s Dictum de Omni et Nullo: ‘[W]hatever is predicated (i.e. affirmed or denied) universally, of any class of things, may be predicated in like manner  … of any thing comprehended in that class.’68 Furthermore, arguments that don’t match this Dictum are invalid.69 This summarizes Whately’s view of conclusive reasoning as presented in his ‘Analytical Outline’. Fleshing it out systematically is naturally more involved. A science, conceived in Aristotelian terms, must be based on intuited first principles arising from collective human experience of a circumscribed subject matter, with further truths being deduced from them. Both Whately and Aldrich follow this prescription in organizing and treating syllogistic reasoning,70 though neither one adheres to a strictly axiomatic approach. To facilitate testing, Whately gives two ‘axioms or canons by which [a syllogism’s] validity is to be proved’ in addition to Aristotle’s Dictum.71 One deals with the agreement of terms (for testing affirmative syllogisms), and the other treats disagreement of terms (for negative syllogisms).72 Whately claims that these sufficient conditions for validity are also necessary:73 if a form violates these canons, it is invalid.74 From these basic canons six rules for syllogisms are (loosely)

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argued75 and treated as if they, too, are (individually) necessary and (collectively) sufficient for validity. Two rules identify the overall structure of a syllogism, two tell how terms should be distributed, and two govern negative premises and conclusions.76 Two additional rules governing particular premises and conclusions are corollaries of the other rules. Whately next identifies sixty-four total moods, listing twelve that can’t be ruled out as invalid.77 He then introduces the four syllogistic figures and notes which figured moods must be rejected for violating rules about the distribution of terms. In the end, twenty-four syllogistic forms remain, six in each figure, though five are judged ‘useless’ because their conclusions are weaker than what is warranted. While Whately seems to believe that all these forms are valid because they violate none of the rules, he nevertheless grounds their validity in the Dictum de Omni et Nullo as well.78 First figure forms are valid because they satisfy Aristotle’s Dictum directly, while the validity of all other forms is confirmed by reducing them to first figure forms. In the end, Whately believes the validity of all syllogistic forms has been established by Aristotle’s Dictum, on which ‘all Reasoning ultimately depends’.79 Whately’s technical presentation of logic clearly follows Aldrich’s. While he prefaced his presentation with his own overview of the field, streamlined Aldrich’s canons and rules, and added his own striking examples and illustrations, in the end his system is still largely constructed from and confined to the logic that had been Oxford’s favourite for over a century. His canons still involve the indistinct idea of terms agreeing or disagreeing with one another, and his rules use the vague notion of distribution of terms. Aristotle’s Dictum holds a more central role for him, but Whately is not completely clear what sort of logical foundation it or the canons and rules provide for syllogistic reasoning. Are both sets of principles necessary and sufficient conditions for validity, or do they serve distinct purposes? And if Aristotle’s Dictum is the ultimate basis of all reasoning, how does this principle logically relate to the canons and rules; are both really needed for treating syllogisms? The well-deserved celebrity of Whately’s text is thus not due to any bold technical novelties or a firmer foundation for syllogistic forms. Its standing was partly due to Whately’s strong reputation as a controversialist, which comes through in the non-trivial examples he uses and also in his chapter on ‘Fallacies’. His classification of fallacious reasoning into logical and non-logical fallacies was greatly admired at the time80 and is still recognized as a seminal contribution to the history of this (amorphous) topic.81 But in the moment, as well as in the long run, this was also not the main reason for the popularity of Whately’s logic. Its success, as we will argue next, was due instead to his vigorous and able defence of logic against the criticisms that had been gradually accruing in number and intensity over the preceding two centuries.

7 Whately’s defence of Aristotelian logic against its detractors Whately takes up two main types of criticisms of logic – criticisms of technical matters (logic as a science) and criticisms of logic’s inutility (logic as an art). These are somewhat connected, but we’ll look at them separately.

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One structural criticism addressed by Whately is Reid’s and Stewart’s accusation that demonstrating demonstrations is absurd and unnecessary: syllogisms are conclusive all on their own. Whately replies that this is not the way to think about Aristotle’s Dictum. It doesn’t add certainty to a syllogism’s conclusion; it merely encapsulates the core process of all syllogistic reasoning.82 Now, while this may be true, Whately does use Aristotle’s Dictum as a deductive basis for proving the validity of syllogistic forms, and the canons and rules are also used for this purpose. So Whately’s arguments do purport to demonstrate the conclusive character of syllogisms; his response doesn’t adequately repel Stewart’s attack. Furthermore, Stewart’s more insightful criticism that some circularity is involved in this deductive procedure – since all sound reasoning is, according to Whately, essentially syllogistic – goes unanswered.83 A second criticism along this line is Campbell’s caustic charge that all syllogisms embody the fallacy of petitio principii – that one must already know the truth of the conclusion in order to know the truth of the universal premise. Whately also has difficulty answering this criticism, which is understandable since he believes syllogisms exemplify Aristotle’s Dictum, where the conclusion is a special instance of the universal premise. Whately’s initial comeback to this criticism, however, is to leverage his opponent’s position to his own advantage: given that all valid reasoning is captured by Aristotle’s Dictum, Campbell’s critique, if correct, would be an indictment against all deductive reasoning, and that certainly can’t be right. ‘[Campbell’s] objections … lie against the process of reasoning itself universally, and will, of course, apply to those very arguments which he is himself adducing.’84 In the sixth edition of his text (1836) Whately adds the charming analogy that this is like ‘the woodsman, who had mounted a tree, and was so earnestly employed in lopping the boughs, that he unconsciously cut off the bough on which he was standing’.85 Whately thus alleges that this circularity criticism must be wrong, but he doesn’t explain why it is wrong. In fact, Whately accepts Campbell’s criticism, in a sense – it recognizes that valid conclusions are virtually contained in their premises, which is the nature of all deductive reasoning.86 Whately does have the resources to defuse the petitio criticism more fully, but he fails to employ them when dealing directly with Campbell. In his chapter on fallacies, Whately classifies petitio principii as a nonlogical fallacy. Its problem is not with logic; no rules are violated. Rather, the problem is epistemic in nature: knowledge is not advanced by such arguments. In classic Aristotelian terms, a petitio argument is a valid deduction but doesn’t demonstrate the conclusion.87 Of course, with respect to the syllogism, if the universal premise can be established other than by an inductive survey of the included particulars, then its conclusion wouldn’t necessarily be known prior to the premise. Whately never objects to Campbell’s criticism precisely along these lines, but he does come close when defending the epistemic utility of logic. Whately’s defence of logic as the art of reasoning was broached already in his ‘Introduction’ but is elaborated more methodically in his final ‘Essay on the Province of Reasoning’. Here he takes on logic’s vocal critics who oppose induction to syllogism and who assert that syllogistic reasoning cannot produce new truths. Whately’s response to logic’s inductivist critics has two parts, both based on disambiguating induction as a process of investigation from induction as a type of

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argument,88 something emphasized earlier by his mentor Copleston.89 First, he readily acknowledges the value of inductive investigation for experientially establishing principles to be used as a deductive basis for any science. In this sense, there can be no genuine conflict between Bacon and Aristotle, between induction and syllogism:90 both are necessary, and both can flourish. Second, however, when induction is put into argument form, concluding a general principle on the basis of examining particular cases, this may be treated the way all reasoning is – as a syllogism. Whately, following Aldrich (and traditional logics generally), notes that an inductive argument can be considered an enthymeme whose missing premise is a generalizing warrant.91 The strength of the evidence for this induction premise determines the conclusion’s degree of certainty, but the inference itself proceeds the way all (syllogistic) arguments do. To reiterate, for Whately syllogism is not a type of reasoning but the form into which all arguments can be put. Here too then, there is no discord or competition between induction and deduction, only mutual support. As Whately pointedly notes, ‘[a] plough may be a much more ingenious and valuable instrument than a flail, but it can never be substituted for it’.92 Induction may prepare the ground for growing an epistemic crop, but syllogistic reasoning reaps the inferential harvest. As time went on, Whately fortified his defence of syllogistic logic over against detractors who would set up an opposing logic of one sort or another in its place.93 Whately still needs to confront the compelling criticism that syllogisms cannot advance our knowledge. This was a thorny issue for Whately. First, if a general premise is accepted on the basis of an inductive generalization, the argument’s conclusion might well have been part of the evidence used to support that premise. But regardless, proceeding from a general premise to a special case of that premise seems to be exactly what Aristotle’s Dictum demands. Acknowledging this, Whately maintains that deductive conclusions are always virtually contained in their premises. How then can syllogisms discover new truths, lesser known than their premises? Whately again attempts to clarify the issue by drawing some distinctions:94 what is meant by ‘discover’ and ‘new truth’? Some discoveries, he admits, are of ‘real matters of fact’ gained through observation or testimony, and these can’t be found simply through reasoning. Other discoveries, however, are of things ‘implied in that which we already know’.95 This happens in geometry, for instance, where one learns the truth of new propositions only through demonstrations. Such conclusions will be new to those who did not know they were contained in what was supposed, particularly if the subject matter is complex. This is like a person who owns a vein of buried metal on his property: it already belongs to the owner, but when it is dug up, its ownership is finally revealed.96 A result may be virtually contained in a syllogism’s premises, but until it is derived from them it may not be explicitly known to the reasoner. What is ‘new’ and what is ‘known’ is relative to each person, unlike what is logically contained or implied in the premises. An even more convincing argument for the epistemic utility of reasoning is given in a later edition. Suppose, Whately says, diggers unearth the remains of a horned animal. Not knowing that all horned animals are ruminants, they can’t conclude that it is the skeleton of a ruminant. On the other hand, a naturalist off-site knows this property of horned animals but is unaware that the discovered remains are of a horned

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animal. Only when the knowledge of the diggers and the naturalist are combined can they conclude that the skeleton is of a horned animal.97 Both premises must be known jointly in order for the conclusion to be known. The issue of logic’s utility arises in part because logical consequence has not yet been sufficiently distinguished from epistemic matters. Whately is moving towards a more formal understanding of logical inference, but what this means about argumentation is still not fully recognized. Notions of logical consequence, logically deduced conclusion and epistemic outcome are still inextricably bound together. Moreover, by accepting categorical propositions as the canonical form of propositions (interpreting them extensionally as containment assertions)98 and then adopting Aristotle’s Dictum as the basic principle underlying all valid reasoning (what is true of a class is true of any subclass), it’s understandable that one might be perplexed about the epistemic status of a conclusion logically contained in its premises. It’s difficult to see how deduction can arrive at anything truly new. How to think differently about all this would take some time, but Whately offered a helpful beginning analysis of how deductive reasoning might produce new knowledge.

8 Contemporaneous reactions to Whately’s logic: Bentham, Mill and Hamilton Whately’s logic attracted immediate attention from several quarters. Those who published commentary editions of Aldrich quickly took note of Whately’s ideas.99 Whately’s text also received three substantive reviews. The first critical analysis of Whately’s logic was an 1827 book on logic by George Bentham. Bentham concurred with Whately’s defence of logic against its British detractors,100 and, like Whately, he believed that taking logic as a guide to direct our reason in all areas of thought was too expansive. However, Bentham didn’t want to restrict logic to syllogistic reasoning. There is no real reason ‘why the deductive process should, in preference to the inductive, claim the sole right to the common denomination of Logic’.101 Bentham also thought limiting logic to reasoning was too narrow. Rational operations like classification (for terms) and agreement of terms (for propositions) are important in their own right. In analysing propositions, Bentham proposed quantifying both the subject and predicate to indicate their intended extent, making propositions a class equation. Unfortunately, Bentham’s ideas had little impact on further developments after only sixty books were sold, his publisher went bankrupt, and creditors seized the remaining copies for wastepaper.102 Mill’s anonymous 1828 review was the first widely read assessment of Whately’s logic. Mill deemed the critique composed by his acquaintance George Bentham hasty and rash.103 Mill gives a more thoughtful response to Whately’s logic, but those who know Mill only from his Logic written fifteen years later would never guess him as the author of this review.104 His mature ideas on logic took shape a few years later as he reflected further on aspects of Whately’s defence of logic and on the nature of inductive reasoning.

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In general, Mill appreciates Whately’s logic. He completely agrees that putting arguments into syllogistic form enables one to evaluate their conclusiveness with ease. Surprisingly, Mill largely consents to Whately’s restricting ordinary logic to deductive reasoning, a position he would later decisively abandon. An inductive logic would be a welcome supplement to Aristotelian logic, he says, but it will never supersede it: rules for induction ‘would not contribute, in the slightest degree, to the correctness of our reasoning. The syllogistic logic affords the only rules which can possibly be of any service to that end’.105 Mill even makes the astonishing claim that ‘all correct reasoning is syllogistic: and to reason by induction is … a misconception of the two words, as if the advice were, to observe by syllogism’.106 What Mill most admires is Whately’s defence of syllogistic reasoning ‘against the contemptuous sarcasms of some modern metaphysicians’.107 Mill finds that Whately’s ‘vindication of the utility of logic is conclusive: his explanation of its distinguishing character and peculiar objects, of the purposes to which it is and is not applicable, and the mode of its application, leave scarcely any thing to be desired’.108 Inductivists’ persistent opposition to the syllogism is simply misguided.109 Far from challenging Whately’s promotion of deductive reasoning, Mill would bolster it further. It’s clear from mathematics, where syllogistic reasoning is ubiquitous, that deduction actually produces new truths. The same is true of physical sciences, where conclusions are deduced from inductively ascertained principles. Regardless of subject matter, all reasoning proceeds in exactly the same way, as Whately emphasizes. That valid conclusions are virtually contained in their premises is no proof against the fact that reasoning brings them to light. Without argumentation ‘they might have remained for ever as completely unknown, as if they did not result from the knowledge we previously possessed’.110 Nevertheless, deduction’s being able to generate new truths remains a ‘paradox’ and a ‘mystery’ still to be philosophically resolved.111 This matter continues to vex Mill until he makes a careful study of Stewart’s ideas on the nature of deductive reasoning around 1830.112 His final inductive resolution of the problem gets enshrined in his 1843 Logic. All in all, Mill maintains that impediments to ‘the right appreciation of the importance of logic, Dr. Whately has for ever removed’.113 Five years later William Hamilton, while acknowledging Whately’s importance, holds that much more remains to be done on logic’s behalf. Hamilton’s erudite ‘Recent Publications on Logical Science’, focused mainly on Whately, established his reputation as a pre-eminent logician. Hamilton observes that when ‘logic seemed in Oxford on the eve of  … an academic grave, a new life was suddenly communicated to the expiring study’.114 Hamilton credits Whately’s Elements with this revival. Nevertheless, he also claims that it is ‘rarely, indeed, wise above Aldrich’,115 which ‘as a full course of instruction … is utterly contemptible’.116 After slamming the work he has just recognized as resuscitating logic at Oxford, Hamilton feels compelled to substantiate his assessment. With respect to logic’s subject matter, Hamilton finds Whately’s views contradictory since he says that logic’s focus is on ‘the process of reasoning’ but also that ‘logic is entirely conversant about language’.117 Hamilton takes issue with both characterizations because they don’t adequately treat logic as a formal science studying the ‘laws … of thought’. Logic, he says, studies ‘things

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in general’ insofar as they stand ‘under the general forms of thought imposed on them by the intellect’.118 Since thinking involves more than reasoning, Whately’s concept of logic is too confining.119 And, while language can be used to express thoughts, it’s the laws governing the forms of thought and their relations that are the object of logic.120 Hamilton admits being influenced here by Kant’s outlook on ‘the nature and province’ of logic: ‘Logic is a formal science … occupied solely about … the conditions of thought itself.’121 Whately lacks consistency in this regard. While he professes that logic is concerned only with formal inference, where an argument’s ‘conclusiveness is evident from the mere form of the expression’, something that can be clearly exhibited by using letters,122 Whately too often allows material and metaphysical considerations to intrude, thus contaminating his treatment of logic. The ‘most original and valuable portion’ of Whately’s logic, according to Hamilton, lies in its ‘insufficient correction of mistakes touching the nature of logic … lingering among the disciples of Locke’.123 Beyond this double-edged remark, Hamilton has little to say about Whately’s defence of logic. His concern is almost completely with logic as a science. Even what he has to say about induction is approached as a structural issue, as how to formulate it as an argument. He neither addresses Whately’s defence against logic’s inductivist critics, nor does he analyse whether the syllogism is inherently circular or whether deductive reasoning can produce new truths. The pointed criticisms of syllogistic logic by his compatriots Kames, Reid, Campbell and Stewart are all ignored in Hamilton’s review,124 as are Whately’s lively responses to them. In the end, then, Hamilton is so fixated on the perceived imperfections of Whately’s notion of logic vis-à-vis his own Kantian standpoint that he cannot bring himself to adequately recognize the merits of Whately’s logic. A grudging appreciation for Whately’s text having restored logic at Oxford seems to be all he can muster.

9 Historical assessment of Whately’s logic Whately’s text unleashed a flood of new works on logic in Britain during the subsequent quarter century.125 These often cite Whately on technical matters, but they primarily praise his dynamic defence of logic against its critics, an apology done with flair, confidence, and clarity in the context of a systematic presentation of logic’s fundamentals. Whately’s style of writing, his pedagogical overview of the field, his use of striking examples, his concrete illustration of how logic could be used to evaluate everyday arguments – not to mention writing in English instead of Latin – made his logic accessible and attractive to a very wide audience, not just Oxford students.126 Its value as a logic textbook must account for a good degree of its popularity. By refocusing attention on argumentation based ultimately on Aristotle’s Dictum his presentation was an improvement over texts such as Aldrich. In simplifying and accentuating certain features of syllogistic logic, Whately made logic’s formal features more pronounced than before, though as noted this is not a wholly new feature, nor is it what made his text truly stand out – Hamilton, for instance, judged it weak in this respect. Contrariwise, his logic text was found appealing on account of its relevance to concrete arguments that one might encounter in everyday controversies.

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In the final analysis, though, Whately’s logic became the leading traditional text of its time due to its vigorous defence of syllogistic reasoning. Arguing for logic’s utility, disarming critics who would replace syllogistic logic with inductive logic, explaining how deductive reasoning could generate new truths and extend our knowledge in mathematics and elsewhere – these are the things that made Whately’s logic the key impetus behind logic’s revival.127 This, along with his emphasis that logic should be considered a science studying the underlying logical structure of reasoning, is what made Whately’s work stand out. Whately’s unapologetic defence of logic gave it renewed scholarly respectability. Logic was now reckoned a proper field for study and investigation. So even though the system of logic Whately developed was (only) an updated version of traditional syllogistic logic, it opened up the possibility for bright people to explore logic further.128 While Elements of Logic didn’t itself plough any new ground, it cleared the field of the accumulated debris that had kept people from attempting to produce anything there, and it provided an opening for novel possibilities to be fruitfully cultivated. Whately’s logic is thus the pivotal work that revitalized British logic and allowed it in due course to fan out along multiple new paths.129

Notes 1

2 3 4 5 6

See Robert Blakey, Historical Sketch of Logic from the Earliest Times to the Present Day (London, Edinburgh, Glasgow, and Belfast: James Nichol et al., 1851); Samuel Neil, The Art of Reasoning: A Popular Exposition of the Principles of Logic, Inductive and Deductive, with an Introductory Outline of the History of Logic, and an Appendix on Recent Logical Developments, with Notes (London: Walton & Maberly, 1853); William Jennings, ‘Tendencies of Modern Logic’, Dublin Review 36, no. 72 (1854): 419–51; Augustus De Morgan, ‘Logic’, The English Cyclopaedia 5 (1860): 340–54; Alexander Campbell Fraser, ‘Province of Logic and Recent British Logicians’, North British Review 33, no. 66 (1860): 401–27; Samuel Neil, ‘Modern Logicians. The Right Hon. and Most Rev. Richard Whately, D.D.’, The British Controversialist 7 (1862): 1–12, 81–94; Alexander Campbell Fraser, Archbishop Whately and the Restoration of the Study of Logic (London and Cambridge: MacMillan and Co., 1864); Thomas M. Lindsay, ‘On Recent Logical Speculation in England’, in System of Logic and History of Logical Doctrines, ed. Friedrich Ueberweg, translated with notes and appendices by Thomas M. Lindsay (London: Longmans, Green and Co., 1871), 557–90; Anonymous, ‘Logic and Logical Studies in England’, The London Quarterly Review 38, no. 76 (1872): 301–47. An expanded version of this chapter, tracing the critical background to Whately’s logic in more detail, is located in Dordt University’s Digital Collections. Francis Bacon, The New Organon and Related Writings, ed., with intro. Fulton Anderson (Indianapolis: The Bobbs-Merrill Co., Inc., 1960), 23–6. See Rules II, IV, X and XIV in René Descartes, Regulae ad Directionem Ingenii (Amsterdam, 1628/1701) and Part II in Id., Discours de la méthode (Leyden, 1637). John Locke, An Essay Concerning Human Understanding (London: Thomas Basset, 1690; 17004) IV.XVII.4, 671. Buickerood’s classic article (‘The Natural History of the Understanding: Locke and the Rise of Facultative Logic in the Eighteenth Century’, History and Philosophy of

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Logic 6 (1985): 157–90) delineates Locke’s immense influence on eighteenth-century British logic. 7 Watts’ Logick was published almost fifty times by the mid-nineteenth century. 8 Duncan’s Elements of Logick was published in two forms more than twenty times before Whately’s logic text appeared. Additionally, the lengthy Encyclopaedia Britannica entry on ‘Logic’ (1771) is drawn from it. 9 Henry Home (Lord Kames), Sketches of the History of Man (Edinburgh, 1774), III.I ‘Appendix’, 166. 10 Reid’s work went through over a dozen printings in the next half century. Parts of it were also appended to the Encylopaedia Britannica’s article on ‘Logic’, beginning in the third 1797 edition. Reid’s ‘Account’ is reproduced in Hamilton’s annotated edition of Reid’s Works (1846), where Hamilton’s notes provide a running and clipped rebuttal to some of Reid’s criticisms. 11 George Campbell, The Philosophy of Rhetoric (London: W. Strahan, 1776), I.VI, 168–9. 12 Ibid., I.VI, 174. 13 Dugald Stewart, Elements of the Philosophy of the Human Mind. Volume I. Reprinted in volume II of The Works of Dugald Stewart. Edited by William Hamilton; completed, with a memoir, by John Veitch (Edinburgh: Thomas Constable, 1854–60), IV.2, 174. 14 Ibid., IV.2, 178ff. 15 Dugald Stewart, Elements of the Philosophy of the Human Mind. Volume II. Reprinted in volume III of The Works of Dugald Stewart (Edinburgh: Thomas Constable, 1854–60), II.2, 103–4n and III.I, 192. See also Note L, 385. 16 Ibid., II.2, 107. In 1827 (volume III of Stewart’s Elements of the Philosophy of the Human Mind; reprinted in volume IV of The Works of Dugald Stewart, I.III, 58–9). Stewart doubts whether developing a philosophical language analogous to algebra for formal reasoning is attainable. 17 Stewart, Elements of the Philosophy of the Human Mind, III.I, 191n. 18 Ibid., II.1, 74. 19 See Calvin Jongsma, ‘Richard Whately and the Revival of Syllogistic Logic in Great Britain in the Early Nineteenth Century’. PhD diss. (Toronto: Institute for the History and Philosophy of Science and Technology, University of Toronto, 1982), 2-III.1, 259–67 for a description of the state of logic at Oxford in the early nineteenth century. 20 Henry Kett, Logic Made Easy; Or, a Short View of the Aristotelic System of Reasoning (Oxford: University Press, 1809), ‘Advertisement’, iv. 21 Edward Copleston, A Reply to the Calumnies of the Edinburgh Review against Oxford. Containing an Account of Studies Pursued in that University (Oxford: printed for the author, 1810); Id., A Second Reply to the Edinburgh Review (Oxford: printed for the author, 1810); and Id., A Third Reply to the Edinburgh Review (Oxford: printed for the author, 1811). Copleston was awarded a D. D. by diploma for his defence of Oxford. 22 John Playfair, ‘La Place, Traité de Méchanique Céleste’, Edinburgh Review 11 (1808): 249–84. Payne R. Knight, John Playfair, and Sydney Smith, ‘Calumnies against Oxford’, Edinburgh Review 16 (1810): 158–87. John Playfair, ‘Woodhouse’s Trigonometry’, Edinburgh Review 17 (1810): 122–35; Henry Home Drummond, Observations, suggested by the Strictures of the Edinburgh Review upon Oxford; and by the two replies, containing some account of the late changes in that University (Edinburgh: J. Ballantyne, 1810).

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23 See Richard Whately, Remains of the Late Edward Copleston D.D., Bishop of Llandaff, with an introduction containing some reminiscences of his life by Richard Whately D.D. (London: John W. Parker, 1854). 24 Copleston, A Reply, 22. 25 Edward Copleston, The Examiner Examined, or Logic Vindicated (Oxford: C. Richards, 1809), 39. See also Id., Second Reply, 16, 20–5, and Id., A Third Reply, ‘Appendix’, 9. 26 Id., Second Reply, 34. 27 Whately, Remains of the Late Edward Copleston, ‘Logic’, 119. See also Copleston, A Third Reply, ‘Appendix’, 8–9. 28 Whately, Remains, ‘Syllogism’, 121. 29 Ibid., 122. 30 Copleston, The Examiner Examined, 39–40. This includes inductive arguments. Copleston’s analysis here differs from Aldrich’s and was taken over by Whately, who was thereafter credited with the change by later logicians. 31 Ibid., 42–3. 32 Elizabeth Jane Whately, Life and Correspondence of Richard Whately, D.D. Late Archbishop of Dublin. Third, revised edition (London: Longmans, Green and Co, 1875), 8–10. 33 In later years Whately wrote Easy Lessons books on several topics, including Easy Lessons on Reasoning, which first appeared as a weekly serial in 1843. 34 Richard Whately, The Elements of Logic. Comprising the Substance of the Article in the Encyclopaedia Metropolitana: With Additions, & c (London: Mawman, 1826), ‘Preface’, xxvii–xxviii. Theological arguments were occasionally used as illustrations in his logic text, supposedly causing it to be banned by the Roman Catholic establishment (E. J. Whately, Life and Correspondence [18753], 269–70). ‘Appendix III’, from the [18314] edition on, also contained an extended logical analysis of Paley’s argument for the divine origin of the Christian religion. 35 See Jongsma, ‘Richard Whately’, 523–4, note 39 for my definitive dating of Whately’s article. 36 Whately assiduously credits Copleston as the equivalent of a co-author (Whately, The Elements of Logic, ‘Dedication’, iii-iv; see also the ‘Preface’ of the 1840 edition, vii–viii, the ‘Preface’ of the 1844 edition, vii, and a July 7, 1845 letter from Whately to Copleston in E. J. Whately, Life and Correspondence [18753], 236–7). 37 See Raymie Eugene McKerrow, ‘Introduction’, in Elements of Logic, facsimile reprint of Whately’s 1827 edition (Delmar, New York: Scholars’ Facsimiles & Reprints, 1975), v–xiii and Raymie Eugene McKerrow, ‘Campbell and Whately on the Utility of Syllogistic Logic’, Western Speech Communication 40 (1976): 3–13. John Henry Newman, Letters and Diaries of John Henry Newman, vol. XV, ed. C. S. Dessain and V. F. Blehl (London: Nelson, 1964), 176–9 and Id., Letters and Diaries of John Henry Newman, vol. XXI, ed. C. S. Dessain and E. E. Kelly (London: Nelson, 1971), n. 72 give Newman’s account of preparing Whately’s article for publication. 38 Anonymous, ‘Review of Encyclopaedia Metropolitana’, British Critic 20 (1823): 303. 39 The National Union Catalogue lists over seventy-five English and American printings, and this doesn’t include reprints of the Encyclopaedia Metropolitana article or editions of Whately’s Easy Lessons on Reasoning. 40 Cf. Whately’s comments on the rising popularity of logic in the prefaces of his first, fourth and eighth editions of The Elements of Logic (18261, xxiv; 18314, xxxi; and 18448, xxxi).

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41 Our analysis will be based on Whately’s article ‘Logic’ in the Encyclopaedia Metropolitana, parts IX and X (July and September 1823). Citations will be from the 1872 reprint of the original 1823 article, supplemented by remarks on noteworthy changes introduced in later book editions. 42 Whately, ‘Logic’, in particular on the ‘Introductory Section’, 1. This remains the opening declaration in all later editions. 43 Very few writers at the time considered logic a science. Contrariwise, Watts and Duncan as well as British critics of Aristotelian logic take logic as an art. Aldrich labels it ‘an instrumental art’, a point Hill repeats in his 1821 commentary, even saying ‘Logic is … not a science’, (9) a claim Whately declared strange in 1826 (II.I.2, 56–7n). After Whately’s remark, Hill says instead: ‘Logic is both an art and a science. …[for] every art must have a science’ (Henry Aldrich, Artis Logicae Rudimenta, with illustrative Observations on each Section. Commentary by John Hill (Oxford: J. Parker, 18284), I.1.3, 10–11). 44 This is Whately’s second sentence throughout all editions (Whately, ‘Logic’, on the ‘Introduction’, 1). 45 Ibid., ‘Introduction’, 9. 46 Ibid., I.2, 24n. 47 Ibid., IV.8, 54–5. Whately here uses Copleston’s metaphor about foolishly expecting reading glasses to teach one how to read: no general rules can be devised for avoiding ambiguous terms or rejecting falsehoods. 48 In theory, that is. Whately still retains some matters that later thinkers considered superfluous. 49 Ibid., I.2, 23–4. 50 Whately, The Elements of Logic [1826], II.I.2, 56n. 51 Id., The Elements of Logic [1834], II.I.2, 60n. Whately makes this proviso after Hamilton’s 1833 review, which emphasized thought over symbolic expression. 52 Id., The Elements of Logic [1826], II.I.2, 56n. 53 Aldrich, Artis Logicae Rudimenta [18212] I.II, 5–6. 54 Whately, ‘Logic’, III.1, 37. Whately says ‘force’ instead of ‘form’ here and in all nine book editions (the Latin term for ‘force’ appears in Aldrich, Artis Logicae Rudimenta III.1 and III.2 as well: see John Hill in Aldrich, Artis Logicae, 109 and 113), but earlier he had said the ‘conclusiveness [of a valid argument] is evident from the mere form of [its] expression’, (‘Introduction’, 15), and he also changed ‘force’ to ‘form’ when characterizing a syllogism in his Easy Lessons on Reasoning. Reprinted from ‘The Saturday Magazine’ (London: J.W. Parker, 1843), I.IV.1, 24. 55 Cf. Aldrich, Artis Logicae Rudimenta [18212] III.I, in Hill’s commentary on 109–10 with Whately, ‘Logic’, III.1, 37. Hill notes, however, that all material arguments can be reduced to formal syllogisms, which represent the inferential process that necessarily takes place in the mind. 56 This ambiguous formulation is used as a (poor) short-hand representation of a syllogism. 57 Whately, ‘Logic’, in particular ‘Introduction’, 15. 58 Algebra was considered ‘universal arithmetic’ at the time. A more abstract symbolic version was proposed in George Peacock’s 1830 Algebra, but even then, algebra lacked the abstract formal character it acquired over the next century. 59 Whately, ‘Logic’, ‘Introduction’, 15. 60 James Van Evra, ‘Richard Whately and the Rise of Modern Logic’, History and Philosophy of Logic 5, no. 1 (1984): 1–18 emphasizes this feature of Whately’s logic

32

61 62

63 64 65 66 67 68 69 70

71 72

73 74

75 76 77

Aristotle’s Syllogism but overstates it, making Whately a harbinger of modern logic. While Whately advocates formulating an argument symbolically to test its validity (Whately 1823 [1872] ‘Introduction’, 14), he himself uses abstract symbolism only rarely and no more than Aldrich. In fact, Aldrich abstractly symbolizes all twenty-four valid syllogistic forms, while Whately presents only one from each figure for illustration purposes. Aldrich also uses symbolic forms to exhibit the reduction process, while Whately uses concrete arguments. Whately was too much a controversialist to engage in extensive formalism. Logic may be a formal science, but Whately still has its use very much in mind. Whately, ‘Logic’, II.2, 34; see also IV.8, 54 and V.2, 59. No formal syntax, semantics or deduction system can be found in Whately. Whately only identifies regular patterns of valid argumentation (syllogistic forms) and classifies some as more basic than others because of their connection to Aristotle’s Dictum and their ability to ‘perfect’ the rest. Whately, ‘Logic’, III.1, 37. Ibid., ‘Introduction’, 10. Ibid., III.1, 37. Ibid., ‘Introduction’, 5. Ibid., 8. Ibid., 12, 13. Ibid., 15–6. This conflicts with what Van Evra says (‘Richard Whately and the Rise of Modern Logic’, 10; repeated in Id., ‘Richard Whately and Logical Theory’ in Handbook of the History of Logic. Volume 4: British Logic in the Nineteenth Century, ed. Dov M. Gabbay and John Woods (Amsterdam: North Holland, 2008), 83), but Whately’s practice fits Aristotle’s classical notion of science quite well. Whately, ‘Logic’, III.2, 38. These vague ‘agreement’ canons are counterparts to mathematical axioms about equality: quantities equal to the same quantity are equal to each other, and (derivatively) if one quantity is equal to a second and a third is not, the first and third quantities are unequal. Aldrich posits six canons, treating terms as naming classes and considering various relations between them. Ibid., III.2, 38. Whately’s canons and rules function as a summary of known features about arguments, not as formal axioms that implicitly specify valid arguments in the way the axioms, say, of abstract group theory delimit its models (groups). Whately briefly entertains using a counterargument to show an argument to be invalid: find an argument in the same form that is patently invalid because the premises are true while the conclusion is false. However, while Whately uses this ploy often in arguing against others, he judges that using rules to test for validity ‘is evidently a safer and more compendious, as well as a more philosophical mode of proceeding’ (Whately, ‘Logic’, ‘Introduction’, 11). Corcoran notes that the method of counterargument is widely used by Aristotle and wonders whether in the final analysis it might be the only method for establishing invalidity (John Corcoran, ‘Argumentations and Logic’, Argumentation 3 (1989): 31). Whately, ‘Logic’, III.2, 38–9. The fuzzy notion of ‘agreement’ makes the argumentation less than rigorous. Aldrich is again less concise than Whately, giving twelve rules for rejecting invalid forms. Ibid., III.3, 40. Unlike Aldrich, who uses his rules to explain why all but twelve moods are invalid, Whately is content with ruling out only two specific moods as

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examples of the process. In 1827 (II.III.3, 95) Whately silently changes his count to eleven potentially valid moods, something Hill had corrected Aldrich on already in 1821 (III.5, 140) by considering when terms must be distributed in the conclusion. 78 This follows Aldrich, who explicitly uses the Aristotelian Dictum as a second method for demonstrating validity. Aristotle’s Dictum does not have the central prominence in Aldrich that it does in Whately, however; in Aldrich it is a theorem based on the canons, though he says it can also be considered a self-evident axiom (Hill, in Aldrich, Artis Logicae Rudimenta, III.6, 148). 79 Whately, ‘Logic’, III.4, 43. Of course, this is true only if forms satisfying Aristotle’s Dictum generate valid arguments (granted) and if all inferences involved in the reduction process are also justified by the Dictum, which is not true: Aristotle’s Dictum does not cover either conversion or the reductio manoeuvre. 80 Neil lauds it as being ‘almost beyond praise’ (Samuel Neil, ‘Modern Logicians. The Right Hon. and Most Rev. Richard Whately, D.D.’, The British Controversialist 7 (1862): 87). 81 See e.g. Hamblin’s classic 1970 treatise on this topic (Charles Leonard Hamblin, Fallacies (London: Methuen, 1970). 82 Whately, ‘Logic’, ‘Introduction’, 13–4. 83 So far as I know, Stewart, who died in 1828, never responded in print to Whately’s treatment of his criticisms. 84 Whately, ‘Logic’, ‘Introduction’, 13. Whately repeats the same sentiment in his concluding ‘Essay on the Province of Reasoning’ (ibid., 89). 85 Id., The Elements of Logic, [18366] I.4, 36. 86 Id., ‘Logic’, V.3, 61. 87 Unlike Aristotle, Whately never adequately distinguishes a deduction, which produces logical consequences from premises, from proof or demonstration, which produces (lesser-known) truths from (better-known) truths. Defending logic’s utility is naturally couched completely in epistemic terms. 88 Whately, ‘Logic’, ‘Essay’, 85–6. 89 Cf. Copleston, The Examiner Examined, 12 and 35–9. Whately’s discussion of induction occurs in the part of the text that Whately says Copleston contributed to the most (Whately, The Elements of Logic, ‘Dedication’, iv). 90 Whately (and Copleston before him) gives Bacon a pass on this, though Bacon clearly did oppose induction to syllogism at times in his writings. 91 Whately, following Copleston’s analysis, considers this inductive warrant as the major premise (Cf. Copleston, The Examiner Examined, 39). 92 Whately, Logic, ‘Essay.2’, 97. Striking illustrations like this are found throughout Whately’s writing. 93 In 1844, a year after Mill published his Logic, Whately mocks those who advocate replacing syllogistic logic with a ‘rational’ or ‘philosophical’ system of logic: that’s akin to seeking a ‘Universal Medicine’ while ignoring ‘the humble labors’ of pharmacists (‘Introduction.3’, 11). Ten years later he notes that logic’s opponents have had ‘a fair field … left open for them for a very long time, … so that they had full leisure for hatching the egg of their philosophical system, if there had been any vitality in it’ (Whately, Remains of the Late Edward Copleston, ‘Introduction’, 92). 94 Whately, ‘Logic’, ‘Essay.2’, 90. 95 Ibid., 90–1. 96 Ibid., 92. 97 Id., The Elements of Logic [18448], I.2, 26 and II.1, 241.

34 98 99

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Whately’s view of a proposition becomes more explicitly extensional as time passes. Hill’s fourth edition (1828), Huyshe’s first edition (1827), and Jackson’s edition (1836, based on Wesley’s mid-eighteenth-century translation) all treat Whately’s logic as authoritative. An anonymous English edition (1827) also refers to Whately (‘Review of Encyclopaedia Metropolitana’, British Critic XX (1823): 301–11). 100 George Bentham, Outline of a New System of Logic, with a Critical Examination of Dr Whately’s ‘Elements of Logic’ (London: Hunt and Clarke, 1827), I, 1. 101 Ibid. X, 175. 102 Benjamin Daydon Jackson, George Bentham. Facsimile Reprint of 1906 (New York: AMS Press Incorporated, 1976), 57. 103 Mill expressed these sentiments in a March 1828 letter, saying why he also had little interest in reviewing Bentham’s work (The Collected Works of John Stuart Mill, vol. XII, The Earlier Letters of John S. Mill 1812 – 1848. Part I (Online Library of Liberty, 2006), 36–7). 104 The older Mill evidently renounced his earlier views on logic. His 1828 review was omitted from a collection of his earlier writings, and it also goes unmentioned in his Autobiography (John M. Robson, ‘Textual Introduction’, in A System of Logic. Collected Works of John Stuart Mill, vol. XI (Toronto: University of Toronto Press, 1978), lxxx). 105 John Stuart Mill, ‘Whately’s Elements of Logic’, Westminster Review 9 (1828): 150. 106 Ibid. 107 Ibid., 137. 108 Ibid., 138. 109 Ibid., 147 and 169–70. 110 Ibid., 170. 111 Ibid. 112 Antis Loizides, ‘Introduction’, in Mill’s A System of Logic. Critical Appraisals, ed. Antis Loizides (London: Routledge, 2014), 4–5. 113 Mill, ‘Whately’s Elements of Logic’, 151. See Verburgt’s chapter in the present volume for Mill’s mature views on Whately. 114 William Hamilton, ‘Recent Publications on Logical Science’, Edinburgh Review 57 (1833): 199. 115 Ibid., 200. 116 Ibid., 198. 117 Ibid., 208. 118 Ibid., 209; see also 226. 119 Ibid., 207. 120 Hamilton doesn’t say much about logic’s relation to language here, only that he can’t believe Whately seriously wants to restrict our mental operations to what language permits. He says more about language as an instrument of thought in later writings. Cf. Hamilton’s remarks in Anonymous, Raymie Eugene McKerrowRaymie Eugene McKerrow Encyclopaedia Britannica. 7EDN Edition. vol. 13 (Edinburgh: A & C Black, c. 1837), 460–1. 121 Hamilton ‘Recent Publications on Logical Science’, 215. 122 Ibid., 217. 123 Ibid., 201. 124 In Reid’s Works (The Works of Thomas Reid., 2 vol., ed. by W. Hamilton. Edinburgh: James Thin, 1846–63) Hamilton himself takes exception to Reid’s criticisms of Aristotelian logic, but he fails to mention Whately’s responses.

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126

127

128 129

35

About a dozen nineteenth-century works on logic (other than a few Aldrich editions) were published prior to Whately. Between 1823 and 1850, besides the many reprintings of Whately’s book, roughly another dozen editions of Aldrich came out and almost four dozen other works on logic. See Augustus De Morgan, ‘Logic’, The English Cyclopaedia 5 (1860): 341, and Samuel Neil, ‘Modern Logicians. The Right Hon. and Most Rev. Richard Whately, D.D.’, The British Controversialist 7 (1862): 1–12, 81–94, 84 and 86. Whately’s logic also dominated the American market at the time. Peirce, who was later influenced mainly by De Morgan and Boole, was first introduced to logic in 1851 by studying Whately at the age of twelve (Max H. Fisch, ‘Introduction to Volume I’, in Writings of Charles S. Peirce. A Chronological Edition, vol. I: 1857–1866 (Bloomington: Indiana University Press, 1982), xvii). This is Mill’s early assessment; it is matched by what almost all later authors also conclude. Cf. Anonymous. ‘Logic and Logical Studies in England’, The London Quarterly Review 38, no. 76 (1872): 312ff, and James McCosh, The Scottish Philosophy, Biographical, Expository Critical, from Hutcheson to Hamilton (London: MacMillan, 1874), 292. See Alexander Campbell Fraser, Archbishop Whately and the Restoration of the Study of Logic (London and Cambridge: MacMillan and Co., 1864), 34. This chapter is dedicated to the memory of John Corcoran (1937–2021), who was my de facto PhD dissertation supervisor on the topic of Whately’s logic after the death of my advisor Kenneth O. May and who encouraged me to abridge and update my doctoral research to make it available in this form as well.

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2

Mill and the British tradition of inductive logic: The role of syllogism Lukas M. Verburgt

1 Introduction John Stuart Mill’s (1806–1873) A System of Logic was first published in 1843, going through a total of eight editions over the next thirty years. An epoch-making textbook, it provided an empiricist account of two types of reasoning: the syllogism (or ratiocination), treated in Book II, and induction, dealt with in Book III. It is for Book III rather than Book II that Mill was lauded by posterity, studied by his followers and enemies, and remembered today. The reasons for this are obvious enough. Mill belonged to the empiricist tradition which criticized the syllogism, preferring induction as the means for establishing knowledge. Moreover, Mill was and remained uninfluenced by the nineteenth-century developments in formal logic associated with William Hamilton, Augustus De Morgan and George Boole. Since he left the traditional elements of deductive logic – e.g. the subject-predicate form of the proposition, the restriction of deduction to syllogism – as he found them, thereby conceding deductive logic to Aristotle, Mill’s own work on deductive logic has been largely neglected. This chapter comes in two parts. The first part shows that philosophical reflection on deductive logic was crucial to the creation of the inductive logic on which Mill’s fame as a logician rests. More specifically, it argues that syllogism played a constructive rather than merely negative role in (the development of) Mill’s logical oeuvre. At the centre of this development stood Mill’s take on the ‘great paradox of the discovery of new truths by general reasoning’,1 on which he stumbled while reading Richard Whately’s Elements of Logic: the problem of how deductive inferences can be productive of new knowledge. Some of the key elements of this paradox – notably syllogistic petitio – are taken up to trace Mill’s changing views on deduction from his first publication on logic of 1828 to the System of Logic of 1843. During this period, Mill moved from an initial advocacy of the idea that syllogism involved inference to his revolutionary new theory of syllogism as interpretation. The second part of the chapter examines how Mill’s theory of syllogism was received, with ‘considerable reservations and modifications’,2 by one of Mill’s most dedicated followers, John Venn (1834–1923). What makes him an interesting case study – and a more interesting one,

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arguably, than any of Mill’s other followers, including Alexander Bain – is that, while remaining more or less committed to Mill’s outlook on logic, unlike Mill (and Bain), Venn took an active interest in the formal innovations which moved deductive logic beyond the syllogism. Venn’s Principles of Empirical or Inductive Logic (1889), one of the very last contributions to the principii tradition in logic, criticized Mill’s view of the relation of induction to syllogism on several accounts. However, Venn did so in order to accommodate algebraic logic, originated by George Boole and defended in his own Symbolic Logic (1881), within a broadly Millian framework. This may be seen as yet another indication of the constructive role that reflection on deductive logic played in the nineteenth-century British tradition of inductive logic.

2 Mill and deductive logic Any account of Mill’s thinking on logic must start with his early readings in logic.3 Mill’s logical upbringing commenced at the age of twelve when his father had him read Aristotle’s Organon, as well as several seventeenth-century scholastic logic texts, and Hobbes’s ‘Computatio sive Logica’ (1655). ‘Last year’, Mill wrote in 1819, aged fourteen, ‘I began to learn logic. I have read several Latin books of Logic: those of Smith, Brerewood, and Du Trieur, and part of Burgersdicius.’4 Mill, reflecting on the ‘logic of the schoolmen’ in his Autobiography, later observed that there was nothing in his education ‘to which I think myself more indebted for whatever capacity of thinking I have attained’.5 This positive judgement would be repeated in Book I of the System of Logic, for which Mill used the following motto, borrowed from Condorcet: ‘Scholasticism, which produced in logic […] a subtlety, a precision of ideas, the habit of which was unknown to the ancients, has contributed more than one can believe to the progress of good philosophy.’6 During a visit to France in 1820–1821, Mill read Robert Sanderson’s Logicae Artis Compendium (1615) and attended Joseph Gergonne’s (1771–1859) lectures on logic at the University of Montpellier. A few years later, around 1825–1826, Mill helped form the Society for Students of Mental Philosophy, which met twice a week in London to discuss a wide range of topics, including logic. The group started with Henry Aldrich’s Artis Logicae Compendium (1691), which, like Sanderson’s 1615 work, was a standard logic textbook at Oxford for almost two centuries. But, finding it too superficial, the group replaced it by Manuductio ad Logicam (1614) of the Jesuit Phillipe Du Trieu, which had been printed in Britain in 1662 and 1678. After finishing this, they took up Richard Whately’s Elements of Logic, first published in 1823 as two volumes in the Encylopaedia Metropolitana and then as a standalone volume in 1826.7 One of the most influential logic text of the nineteenth century, the Elements of Logic contained a spirited defence of the practical usefulness and theoretical importance of syllogistic logic against the attacks of the empiricists, notably Bacon and Locke, who had proposed its replacement by inductive logic. Whately’s re-conception of the nature and scope of deductive logic was so successful that his Elements of Logic would be widely credited with the revival of the study of logic in Britain. It was upon reading Whately, who had sparked ‘a wide range for metaphysical speculation’8 in him, that the young Mill

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conceived of the project of writing a book on logic. Whately’s book would, indeed, inspire Mill’s System of Logic and inform many of its fundamental viewpoints. More directly, Mill launched his new career as a (meta-)logician by reviewing it, thereby sparking his study of deductive logic.

2.1 Mill and Whately’s Elements of Logic Mill’s first publication on logic was the 15,000-word review of the Elements of Logic in the Westminster Review in early 1828, later praised by Bain as ‘a landmark not merely in the history of [Mill’s] own mind, but in the history of logic’.9 The fact is that the review, while taking issue with Whately on several points, was nothing more, but also nothing less, than an expression of Mill’s enthusiasm about Whately’s book. Mill believed Whately’s proof of the significance of syllogistic logic to be ‘conclusive’, leaving ‘scarcely anything to be desired’.10 Most importantly, Mill accepted most of Whately’s central – and at times far-fetched – arguments and viewpoints. Among other things, Mill agreed that logic is the art and science of reasoning, that reasoning is syllogistic and, consequently, that induction is not a form of reasoning: Syllogistic reasoning is not a kind of reasoning, for all correct reasoning is syllogistic: and to reason by induction is a recommendation which implies as thorough a misconception of the meaning of the two words, as if the advice were, to observe by syllogism.11

According to Whately and the young Mill, the rules of syllogism provide correct norms for all reasoning. The dispraise which usually fell to logic’s lot was not only ill-deserved but also foolish: Bacon, Locke and their empiricist followers had not put forward an alternative theory of valid inference, and even if they would have done so, they would have found that ‘there is no other way’12 in which the validity of an argument can be tried than by resolving an argument into a series of syllogism. It is to deductive logic that one turns for a description of the conditions of valid inference. This claim is today a commonplace. But it was made so by Whately and Mill, who would always stand behind it. Despite his praise and unquestionable acceptance of Whately’s outlook, Mill was deeply puzzled about one major topic, consisting of two distinct but often confounded parts: the petitio principii charge, which said that the syllogism is fallacious because the major premise presupposes the conclusion, and the complaint that, since its conclusion only makes explicit what is already included in the premises, the syllogism cannot lead to new knowledge. Whately’s strategy against the petitio principii charge was to make a distinction between physical and logical discoveries. Whereas the former establish something that was previously absolutely unknown, the latter merely ‘unfold the assertions wrapt up […] in those with which we set out’.13 Whately argued that the petitio principii charge rests entirely on the mistaken assumption that the aim of syllogistic reasoning is to make physical discoveries. This convinced Mill, who commented in his review that Whately had refuted the charge ‘triumphantly’14 and was fully correct in writing that syllogistic reasoning establishes unforeseen logical

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truths, as geometry proved. What did not convince Mill was Whately’s (lack of an) explanation of how new knowledge was discovered by syllogistic reasoning, that is, what it could possibly mean that geometry was ‘wrapt up’ in a few definitions and axioms. How, in Mill’s own words, to resolve the paradox ‘that mankind may correctly apprehend and fully assent to a general proposition, yet remain for ages ignorant of myriads of truths which are embodied in it?’15 Mill would wrestly profoundly with the problem of how syllogistic reasoning can be informative, how epistemic advance is produced by deductive inference, for a couple of years.

2.2 Mill and Dugald Stewart’s Elements of the Philosophy of the Human Mind After his 1828 review Mill continued to reflect on the nature of deductive inference, hoping to clear up the ‘mist’ that Whately had ‘left hanging over the subject’.16 The exact details of Mill’s train of thought are unknown, but, the mist appearing thicker than ever, he soon began to doubt whether a satisfactory explanation of how syllogistic reasoning can lead to new knowledge could be given in Whatelyian terms at all. It was in year 1830–1831, upon a third or fourth reading of the chapters on reasoning in the second volume of Dugald Stewart’s 1814 Elements of the Philosophy of the Human Mind that Mill obtained a clue of a better theory. What struck him was Stewart’s view on the use of axioms in deductive reasoning. Mill does not mention in his Autobiography which view he had in mind, but it was probably the following, found at the end of Section  1 (‘Of Mathematical Axioms’) of Chapter  1 (‘Of the Fundamental Laws of Human Belief ’): [T]he doctrine which I have been attempting to establish […], tends to identify [axioms] with the exercise of our reasoning powers; inasmuch as, in stead of comparing them with the data, on the accuracy of which that of our conclusion necessarily depends, it considers them as the vincula which give coherence to all the particular links of the chain.17

Stewart’s slightly vague point was twofold. First, geometrical axioms, rather than being the ground of geometry, are generalizations of ‘what, in particular instances, has been already acknowledged as true’.18 This reversed the priority of general statements and the particular statements deduced from them: inference does not consist in deriving theorems from axioms, but in establishing axioms, a process which, according to Stewart, occurs by reasoning from the particular to the general on the basis of experience. Secondly, once established, the truth contained in an axiom is ‘supposed or implied in all our reasonings’19 without the axiom itself adding anything to the reasonings themselves. Or, as Mill would summarize Stewart’s view on mathematical reasoning in the System of Logic, Stewart saw that axioms ‘are merely necessary assumptions, selfevident indeed, and the denial of which would annihilate all demonstration, but from which, as premises, nothing can be demonstrated’.20 What Stewart had said about axioms seemed to Mill to apply to all general propositions. From this realization – a rejection of Whately’s view that deduction can, by itself, lead to new knowledge – grew Mill’s radical and surprising theory of the

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syllogism, which he would put forward in Book II of the System of Logic, a sketch of which he had immediately put on paper in 1830–1831. It duly convinced him that he would be able to produce a book on logic ‘of some originality and value’;21 and so began the first phase (ca. 1830–1832) of Mill’s composition of a tract that would evolve into A System of Logic. Mill wrote the General Introduction, Book I (‘Of Names and Propositions’), Chapters 1–6 – from a ‘rough and imperfect draft’22 he had already made  – and Book II (‘Of Reasoning’), Chapters  1–3, entitled ‘Of Inference, or Reasoning, in General’, ‘Of Ratiocination, or Syllogism’ and ‘Of the Functions, and Logical Value of the Syllogism’.

2.3 Mill’s theory of the syllogism in the System of Logic The key ideas of Book II of Mill’s System of Logic steer a middle path between Whately and Stewart. On the one hand, Mill maintains Whately’s general definition of logic as the art and science of reasoning: the logician lays down the rules by which everyone must reason and analyses the mental process whereby everyone reasons. On the other hand, Mill follows Stewart’s views on the foundation and utility of deduction and on its relation to induction: syllogistic reasoning is based on the cogent induction of general premises, and the function of the syllogism is evaluative. Most importantly, in Chapter 3 of Book II Mill puts forward his own, new theory – a complete reversal of his 1828 position – of the logical value of syllogism: syllogistic reasoning is not inference at all but something that Mill calls ‘interpretation’. Book II of the System of Logic possesses ‘a very curious feature’:23 Chapter  3 presents the new theory of syllogism as a non-inferential process of interpretation, while Chapter 2 analyses syllogistic inference. One way to explain the anomaly is to see Chapter 2 as negative or destructive of the principle on which the traditional view of syllogism as leading to new knowledge was based (‘Dictum de omni et nullo’),24 and Chapter 3 as positive, or constructive of an alternative view of syllogism. For unlike Locke, say, Mill did not throw away syllogism altogether: if syllogism is not inference, then, Mill reasoned, it must be something else and even something important. Mill’s claim that syllogism is not inference is today seen as notorious and implausible. It relies on a set of outdated assumptions such as that all deduction is syllogistic, that a major premise is only a general name for certain particulars, that the relation between a set of premises and its consequences is that which holds between a major premise and its instances and, hence, that the truth of a major premise requires the truth of all its instances. Mill’s theory of syllogism is nonetheless worth studying, for considerations of intrinsic theoretical interest and historical significance. While these point to general issues – ‘What is it that makes deductive processes illuminating?’ – the focus will here be on two specific questions related to Mill: What is the problem that his theory sought to solve? And how did the theory solve this problem? In his 1828 review, Mill had been convinced that syllogism leads to new knowledge but was puzzled about how it could do so. By the time that the first edition of System of Logic appeared, in 1843, he had completely abandoned the idea that syllogism can advance knowledge, thereby dissolving the how. Interestingly, Mill’s motivation behind the claim that syllogism is not inference in his 1843 book

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had been entirely absent in his 1828 review: this was the central idea that ‘in every syllogism, considered as an argument to prove the conclusion, there is a petitio principii’25 or begging of the question. When we say,  All men are mortal,  Socrates is a man,  therefore  Socrates is mortal; it is unanswerably urged by the adversaries of the syllogistic theory, that the proposition, Socrates is mortal, is presupposed in the more general assumption, All men are mortal: that we cannot be assured of the mortality of all men, unless we are already certain of the mortality of every individual man: that if it be still doubtful whether Socrates […] be mortal or not, the same degree of uncertainty must hang over the assertion, All men are mortal: that the general principle, instead of being given as evidence of the particular case, cannot itself be taken for true without exception, until every shadow of doubt which could affect any case comprised with it, is dispelled by evidence aliundè; and then what remains for the syllogism to prove?26

Mill, believing the argument to be ‘irrefragable’, concluded that ‘no reasoning from generals to particulars can, as such, prove anything: since from a general principle we cannot infer any particular, but those which the principle itself assumes as known’.27 This allowed Mill to return to Whately with a vengeance: When you admitted the major premise, you asserted the conclusion; but, says Archbishop Whately, you asserted it by implication merely: this, however, can here only mean that you asserted it unconsciously; that you did not know you were asserting it; but, if so, the difficulty revives in this shape – Ought you to have known? Were you warranted in asserting the general proposition without having satisfied yourself of the truth of everything which it fairly includes? And if not, is not the syllogistic art prima facie what its assailants affirm it to be, a contrivance for catching you in a trap, and holding you fast in it?28

Mill believed there was only one possible solution: to simply deny that syllogism involves inference. Once it is established by inductive inference that ‘All men are mortal’, the conclusion ‘Socrates is mortal’ is not an inference drawn from but an ‘interpretation’ made according to the major premise, understood as a record or memorandum of the act of inductive generalization from particulars. This suggests that, while all inference is inductive, the syllogism is not entirely useless, as it has an important evaluative function. It offers a practical test of the validity of arguments, ‘by supplying forms of expression into which all reasonings may be translated if valid, and which, if they are invalid, will detect the hidden flaw’.29

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2.4 The role of petitio principii in Mill’s theory of syllogism Following the classical accounts of Mill’s theory of syllogism,30 more important than his idiosyncratic move of denying that syllogism is inference are his ideas about the (role of the) petitio charge. Mill saw it as a way of presenting the objection that syllogism does not advance knowledge. But the petitio problem is distinct from what is sometimes called the ‘hidden consequences problem’: the problem of ‘how someone can understand a set of premises without knowing’, of a given conclusion implied by them, ‘that it is actually implied by them’.31 Already in his 1828 review, where he did not touch on the petitio, Mill had wondered how it could be that ‘mankind may correctly apprehend and fully assent to a general proposition, yet remain for ages ignorant of myriads of truths which are embodied in it, and which, in fact, are but so many particular cases of that which, as a general truth, they have long known’.32 It was in the System of Logic that Mill ran this problem of hidden consequences together with the petitio problem, that of whether the syllogism is question begging. Here, Mill first observes in Chapter 3: It is universally allowed that a syllogism is vicious if there be anything more in the conclusion than was assumed in the premises. But this is, in fact, to say, that nothing ever was, or can be, proved by syllogism, which was not known, or assumed to be known, before.33

Mill then goes on to summarize the difficulty by saying that the petitio principii is present in every syllogism. Neither here nor anywhere else in the System of Logic does Mill explicitly recognize that this problem is not the same as the petitio problem; it is, however, ambiguously implied in the word ‘before’ in the passage just quoted.34 One way to read this word is that the conclusion has to be known before the premises can be known. If this were so, then every syllogism begs the question. Another way to read the word ‘before’ is that the conclusion is known as soon as the premises are known. If this is true, then every syllogistic deduction may be pointless but not question-begging, since the premises can be known without the conclusion first having to be known. This, arguably more accurate, reading points to Mill’s sceptical take on the hidden consequences problem: having found difficulty in making sense of how the consequences of premises can be hidden, he comes to doubt whether there can be hidden consequences at all, suggesting that syllogistic reasoning never does anything more than trivially reveal obvious consequences. Mill’s puzzlement even leads him to conclude that all valid arguments are obviously valid: when we understand the premises, we immediately know what their logical implications are. We will return to this sceptical reconstrual of the fact that the validity of deductive reasoning depends on the premises containing the conclusion in Section 3.2, where we discuss Venn’s critical reception of Mill’s petitio charge. Before doing so, it is important to take a step back and see the larger picture of Mill’s deductive logic, including his opposition to the innovations in formal logic found in the work of some of his contemporaries.

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2.5 Mill on formal logic Mill distinguished two logics in his oeuvre, the ‘Logic of Consistency’ and the ‘Logic of Truth’, which followers like Venn called ‘formal’ or ‘conceptualist’ logic and ‘material logic’, respectively.35 The formal logician is concerned with merely verbal propositions and apparent inferences: that is, with propositions (‘All fathers are parents’) which are true because the connotation of the subject (‘father’) includes that of the predicate (‘parent’), and with inferences in which the conclusion merely repeats (part of) what is contained in the premises. The material logician, on the other hand, deals with real inferences, which are ampliative in the sense that their conclusions say more than what is contained in the premises, consisting of real propositions. Unlike verbal propositions, real propositions are truth-apt: they are ‘susceptible of truth or falsity’, and not only of ‘conformity or disconformity to usage or convention’.36 Over the course of his career, Mill became more explicit about the relation between these two ‘logics’. Mill declares in the System of Logic that his logic is a logic of truth but emphasizes that it only deals with truth in one of two possible ways in which it can be known: it does not deal with truths which are known directly, that is, by intuition, but only with those which are known by inference from other truths, that is, which are inferred. Indeed, Mill wishes for logic to remain neutral about metaphysical disputes on the origin of human knowledge. A different line is taken in the Examination of Sir William Hamilton’s Philosophy (1865), where Mill insists that his logic also considers the original process by which knowledge is obtained. This position, already assumed throughout the early editions of System of Logic, is explicitly endorsed in the sixth edition of 1868. Its implication is twofold. First, material logic is also a philosophical theory which says that all inference is from particulars to particulars. Second, material logic includes formal logic as a special case, which plays a subordinate but indispensable role in the search for knowledge. Taken together, identifying inference with induction and deduction with syllogism, Mill assigns to formal logic the merely negative and practical task of maintaining consistency ‘between our general theorems from experience and our particular applications of it’.37 Given this outlook, it is no surprise that Mill did not have much to say about recent technical developments in formal logic, whether it was Hamilton’s quantification of the predicate, De Morgan’s logic of relations or Boole’s algebra of logic.38 Mill’s response to them is nonetheless telling and curious. It is well worth observing that, despite their fundamental disagreements about the nature and scope of logic, both camps, the material and the formal, had one thing in common: it was a long and deep reflection on the syllogism as defended by Whately that had led them to realize that logic is not syllogism and that there are valid non-syllogistic inferences. Their responses to this insight, however, were totally opposite. The formal logicians, finding that the syllogism was incapable of representing the truth-preserving structure of many valid arguments, sought to improve upon or even move deductive logic beyond it. Mill, who would always remain committed to the idea that syllogism embodied the form of all valid reasoning, concluded that all inference is inductive and that the proper function of the syllogism is merely evaluative. Mill’s response to what he called the ‘mania’39 of

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formal theorization among his contemporaries was twofold. First, considered as a contribution to the science of logic, he found the alleged expansions of syllogistic logic not representative of actual reasoning processes, writing that the new formal logic is ‘not merely superfluous, but erroneous: since the form in which it clothes propositions does not, like the ordinary form, express what is in the mind of the speaker when he enunciates the proposition’.40 This response is, indeed, curious, as Mill denied that the syllogism has a representative function.41 Second, considered as a contribution to the art of logic, Mill argued, the new forms proposed, such as Hamilton’s ‘Some B is all A’ or De Morgan’s ‘Most B’s are C’s/Most B’s are A’s/Thus, Some A’s are C’s’, did not improve the evaluation of reasoning: The sole purpose of any syllogistic forms is to afford an available test for the process of drawing inferences […]; and the ordinary forms of Syllogism effect this purpose completely. The new forms do not, in any appreciable degree, facilitate the process. […] The new forms have thus no practical advantage which can countervail the objection of their entire psychological irrelevancy; and the invention and acquisition of them have little value.42

Mill was more critical of Hamilton than he was of De Morgan. Both their innovations added to syllogistic theory, but, as purely ‘technical forms of reasoning’,43 they had no practical purpose. Unlike De Morgan’s statistical syllogism, Hamilton’s quantification of the predicate suffered from the additional problem of being at odds with how we actually reason: ‘not only we do not […] quantify the predicate in thought’, wrote Mill, ‘but we do not even quantify the subject. […] Even in an [sic] universal proposition, we do not think of the subject as an aggregate whole [“All A is B”], but as its several parts [“All A’s are B’s”]’.44 Mill’s second response to the formal ‘mania’ among his contemporaries is telling in the sense that it is arguably false. It seems problematic, if not impossible, to accept that the expansions of syllogism formalize valid inference forms that cannot be represented by the ordinary syllogism but to deny that these are helpful in testing possible inferences. Most importantly, to return to the topic introduced at the end of Section 2.4, their acceptance undermines Mill’s sceptical reconstrual of the problem of hidden consequences. This at least is the point that Venn seized upon in his attempt to reconcile algebraic logic with Mill’s material logic.

3 John Venn: Material Algebraic Logic The Cambridge logician John Venn (1834–1923) is mainly remembered as the inventor of the logical diagram that bears his name. During his active career as a logician, which started in the early 1860s and ended in the late 1880s, Venn was widely regarded as one of the most eminent British logicians. Like others working in this confusing period in the history of logic, such as William Stanley Jevons, Hugh MacColl and Lewis Carroll, Venn has been obscured by the contradictory influences

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he was balancing in his work. Within his oeuvre, consisting of three books, Logic of Chance (1866), Symbolic Logic (1881) and Principles of Inductive or Empirical Logic (1889), and about a dozen articles in Mind and other journals, Venn not merely sought to reconcile algebraic and material logic, itself a formidable task; he also defended algebraic logic against older formal logicians, such as William Spalding; waged a foundational debate with Jevons over the proper interpretation of Boole’s original system; protected deductive logic from the innovations of De Morgan, MacColl and Frege; and updated Mill’s material logic in the light of its faults and shortcomings, working alongside Herbert Spencer, Carveth Read and others.45 Venn was the most prominent and loyal follower of Mill’s work on logic. He recognized Mill as his intellectual mentor, and Mill’s material logic played a key role in his oeuvre, whether explicitly or implicitly. Perhaps the most helpful way of structuring Venn’s oeuvre is to see it as falling into three periods of engagement with Mill’s System of Logic: support (1860s), criticism (1870s) and disillusion (1880s). As Venn, looking back on his intellectual development in the late 1880s, noted, Mill’s all-pervasive ‘influence has subsequently generated the relation of criticism and divergence quite as much as that of acceptance’.46 The Logic of Chance, first published in 1866, aimed to reconceive probability theory in terms of material logic, criticizing the outlook on this field of the formal logicians Boole and De Morgan. During the years from 1876 to 1880 Venn published a steady stream of journal articles which were increasingly critical of Mill’s material logic.47 One of Venn’s main criticisms was that the idea that logic ultimately deals with objective facts, with what can be inferred from data obtained through experience and observation, is only an ideal. As things stood, the material viewpoint, which Venn still preferred to the formal or conceptual one advocated by Boole, De Morgan and Jevons, was conditioned ‘on every side by subjective or relative considerations’, that is, by ‘conventions and assumptions’, beginning with what it means for an object to exist.48 This quasi-conventionalist version of material logic, which Venn called empirical logic, would be worked out more fully in Principles. Here, he explicitly blamed Mill for ‘over-objectifying’ logic and wrote from the conviction that logic cannot be grounded on and will never lead to ‘ultimate objective certainty’, as Mill had supposed.49 This book, in which Venn put forward his mature position on logic, met with a largely negative reception; as the reviewer, James Sully, lamented in Nature, it reads as if Venn had just ‘sat down to write without a clear plan in mind’.50 Venn surely had a plan, but it was as poorly executed as it was original, not to say idiosyncratic; he wanted to cover all areas of logic, deduction and induction, from the position of empirical logic, understood as an ‘inductive system of logic’ in which logic is said to be grounded on objective and subjective assumptions.51 More specifically, Venn aimed to show that this new outlook made it possible to both a Boolean and a Millian, that is, to accommodate innovations in deductive logic while holding on to the view that all inference is at bottom inductive. This took two bold steps. First, Boole’s calculus of deductive reasoning had to be represented as the (only possible) extension of syllogistic logic and as a practical tool free from philosophical implications about human knowledge. Second, Mill’s theory of syllogism had to be reinterpreted in such a way that it allowed for hidden consequences, which could be revealed by symbolic deductive methods.

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The focus in the remainder of this chapter is on the second step.52 It will be pointed out that Venn’s modified acceptance of the Millian view on the relation of syllogism to induction essentially involved a compromise, albeit one with interesting implications, as is made clear in Section 4.

3.1 Venn’s new theory of syllogism Like other followers of Mill – such as Alexander Bain – Venn initially regarded Mill’s theory of syllogism as his ‘strongest claim to originality’ as a logician.53 Venn’s first paper in Mind, in its inaugural volume of 1876, accepted the theory as ‘the natural, simple, and almost necessary outcome’ of the material view of logic, and the petitio principii charge as a natural and key ingredient of the theory.54 As Venn summarized Mill’s position: Take, for instance, the proposition ‘All men are fallible;’ from this we obtain […] ‘Some infallibles are not men.’ Now regard these propositions as judgments […] and it can hardly be denied that one of them is an inference from the other. […] But penetrate to the facts to which these judgments refer, and we see at once that they are identical, or to speak more accurately, the one is a portion of the other. […] The conclusion, regarded as an objective fact, is the premises, or rather a portion of them. We are accordingly driven to carry our investigations a step further back, and we then perceive that the only step in the reasoning at which new facts were appealed to, instead of merely new judgments about them being made, was in the formation of the major premiss. […] Hence Mill’s view readily follows, viz.: that it is the major premiss which really contains the whole inference, the remaining part of the syllogism consisting merely in identification and interpretation of what had gone before.55

Venn returned to the topic about a decade later, in Chapter  15 (‘The Syllogism in Relation to Induction’) of Principles. After writing that Mill’s theory seems to him tenable, ‘with considerable reservations and modifications’,56 he proceeds as follows: he considers the possible grounds for accepting it and then goes on to criticize it. The most important ground for accepting Mill’s theory of syllogism is Mill’s material logic, as Venn had already pointed out in his 1876 article. However, since Venn now disagrees with this outlook, it has become a ground for rejecting the theory: Mill’s treatment of syllogism is put forward as the best illustration of his ‘over-objectifying’ of logic. Take the example of M, P and Q, and assume that M is a part of Q and P a part of M. Mill maintains, writes Venn, ‘that whoever asserts that “All M is Q”, must know, or, if not, ought to know, that “P is Q”, because the fact that P is Q is simply a part of the general fact that “All M is Q”’.57 According to Venn’s empirical logic, which says that logic is concerned with the operations of the human mind when drawing inferences about a world considered as objective, Mill neglects the fundamental distinction between ‘the objective facts and our subjective recognition of them’.58 It is perhaps true that, when facts are thought of ‘as they exist in nature and not as they come into a syllogism’,59 ‘P is Q’ is a part of the broader fact that ‘M is Q’. Importantly, however,

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from the perspective of empirical logic, it does not follow that the recognition of ‘P is Q’ is necessarily given in that of ‘M is Q’. That the man who has realized […] that All M is Q and that P is M, and who does not know yet that P is a Q, is standing, so to say, on an uncommonly narrow ledge, must be admitted; but the only question is, whether it can afford him a possible foothold. If it will […], then he can make use of his momentary position to climb higher. And for a justification of the syllogistic process this is all that is needed.60

This provides Venn with the starting point for a criticism of Mill’s theory of syllogism and an opening for his own, new theory of syllogism. Venn holds that it is possible to conceive of someone who has accepted a set of premises and who has not yet accepted the conclusion implied by them. More precisely, he not only recognizes that for the syllogism to involve a process of reasoning there must be hidden consequences but also insists that deductive arguments sometimes do involve hidden consequences. At this point, Venn refers to his work on algebraic logic, Symbolic Logic (1st ed. 1881; 2nd ed., 1894) to problematize Mill’s position. Having found difficulty in explaining how the consequences of premise can be hidden, Mill had questioned whether there are hidden consequences, only to conclude that syllogism does no more than reveal obvious consequences. According to Venn, the ‘apparent force’ of this sceptical argument is derived solely from ‘the extreme simplicity and familiarity’ of the examples that Mill considered and took as paradigmatic.61 Every one who has studied mathematics must have experienced a feeling of surprise at times in finding how far he has been carried on his way. He starts with a premise which it may take some trouble to distinguish from a pure identity, and finds that, starting from this, he may be imperceptibly led on by intuitively obvious advances into some profound and far-reaching algebraical formula. In the ordinary syllogism there is of course nothing corresponding to this, but if we were to select examples from the more complicated varieties offered under the Symbolic Treatment of Logic it would not be difficult to find instances which should approximate to those in mathematics.62

One of Venn’s favourite examples was the following, drawn from Jevons’s Elementary Lessons in Logic (1870): The members of a board are each of them either bondholders or shareholders, but not both, and the bondholders, as it happens, are all on the board. What conclusion can be drawn?63 Venn had used this example twice: once as a problem in syllogistic logic in the examination papers he had set at London University in the 1870s, and once as a problem in algebraic logic at Cambridge. At London, some five or six of a total of 200 students who had ‘tortured it to fragments with syllogism’, without having a clue ‘what to look for’, gave the correct answer, ‘No shareholders are bondholders’.64 At Cambridge, it was answered correctly by more than half of the class, putting the argument as follows: ‘There can be no bondholders who are shareholders, for if there were they must be either on the board, or off it. But they are not on it, by the first of the given statements; nor off it, by the second.’65

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For Venn, the point of the example, however simple, was twofold. First, it shows that more complex logical problems than those discussed by Mill can be solved easier and quicker with new, algebraic methods than with the old, syllogistic ones. Second, and more importantly, it shows that in such cases, let alone in cases involving three or four propositions and several terms, ‘no untrained acuteness would be capable of detecting at a glance’ what the solution is; this, in fact, involves a ‘mental act’, that is, a process of reasoning.66 Taken together, algebraic logic improves the syllogism but does not break away from it, yet the very fact that syllogism can be improved to better deal with complex arguments strongly suggests that deduction does more than trivially reveal obvious consequences. Venn’s new theory of syllogism, and its relation to induction, was essentially twosided, being partly logical and partly philosophical. On the one hand, contra Mill, it denies on logical grounds that the major premise contains the whole inference as, in setting out from a set of premises, the process of reaching the conclusion implied by them may involve mental labour and be illuminating. On the other hand, with Mill, it accepts that, philosophically speaking, any deductive reasoning is grounded on data derived from a long and complicated process of induction. Venn’s two-sided position offers a glimpse of the middle road that he sought between Boole and Mill, between formal and material logic: deductions can lead to logical insight, but not to physical knowledge; they are epistemically informative but not, so to speak, epistemically constitutive.

4 Mill, Venn and British logic in the nineteenth century By way of conclusion, this last section offers a brief comparison of Mill’s and Venn’s theories of syllogism and their views on the nature, scope and aim of logic, set against the background of the British tradition of inductive logic founded by Mill and continued by Venn and some of his commentators. Among the founders of contemporary analytical philosophy, notably G. E. Moore and Bertrand Russell, Venn was cast as one of the last surviving representatives of the outdated empiricist-utilitarianism of Mill. This is understandable, but it was also a mistake. For Venn found himself confronted with a problem situation much like theirs. Put roughly, how to combine an empiricist outlook on knowledge with formal logic? One major difference was the origin of their problem situations: that of Venn came from Mill and Boole and that of the early analytical philosophers from an opposition to the philosophical idealism of F. H. Bradley and J. M. E. McTaggart and developments in mathematical logic associated chiefly with Gottlob Frege. But perhaps the central difference was that in the 1870s–1880s the problem situation took on its sharpest form, and still had to be confronted, in the context of the syllogism. This points to the general take-home message of this chapter, namely that considerations about deductive logic helped shape the tradition of inductive logic. Given the centrality of syllogism, Venn’s dilemma may have looked outdated to Moore and Russell but, insofar as, after a spell of idealism, they returned to a native British empiricism, it very much applies to their twentieth-century work too.

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Both Mill and Venn struggled long and hard with the question whether syllogism involves inference in their attempts to come up with a material system of inductive logic. Neither of them sharply distinguished the petitio problem from the hidden consequences problem. But both showed awareness of the fact that, though the conclusion of a syllogism can contain no more than is implied in the premises, it can be illuminating to uncover what is implied in the premises. Because he did not maintain a firm grip on this insight, Mill slipped into thinking that, since in deducing conclusions we deduce what is already implied in the premises, syllogizing is question-begging and does not involve inference. Venn, for his part, reasoned that, since there are hidden consequences, as complex arguments show, deductive reasoning, and ipso facto the syllogism, involves inference and, hence, cannot be question-begging. By the time that Venn published Principles, which appeared some forty-five years after the first edition of System of Logic, he had abandoned his commitment to Mill’s material logic. Venn’s alternative, empirical logic amounted to a pragmatic compromise fashioned to incorporate formal logic – as he understood it, excluding, for instance, logic of relations – with a system of material logic: it essentially agreed with Mill’s philosophy – at the core of which arguably stood the idea that all knowledge is an induction from observed particulars – but insisted that for convenience’s sake it was better not to let its meta-logical theories restrict the practice of logic. Venn could write, for instance, that when we start from the premises, and ask no question about they were obtained, Mill’s theory of syllogism is wrong, but when we start from the individual facts from which the premises have been obtained, the theory is ‘the simplest and the best’.67 Hence, although ‘no general proposition can be a true ultimate starting point; […] we may nevertheless admit that such propositions are often the only starting point from which [we] actually did set out’.68 Venn’s mature work is interesting not only as standing at the apogee of the two logical traditions spurred by Whately’s Elements of Logic – formal, deductive logic and informal, inductive logic – but also as suggestive of the pivotal role that the syllogism played in their development. Most interesting, in this regard, is that, through its creative indecision on matters of foundations, Venn’s oeuvre reflects and highlights some of the underlying tensions of nineteenth-century logic. Or, put more strongly, it exemplifies the utter lack of consensus about what logic is, or should be, if it is not syllogistic. Take the relation between induction and deduction. On the one hand, inductive logicians sought to incorporate deduction within their work by presenting it as a noninferential process founded on inductive reasoning from particulars. On the other hand, deductive or formal logicians sought to assimilate induction by saying either that what the inductive logicians understood by induction did not belong to logic or that induction itself is a form of deductive inference. Another tension is that between logic as qualitative or quantitative and, relatedly, as an art or a science. Mill was often taken as arguing that logic is qualitative and was to be distinguished from mathematics as a quantitative science. Venn, though a self-proclaimed Millian, was notable for his work on how logical and mathematical methods are connected and how their analogy can help deductive logic move beyond the syllogism in terms of generality. For him, rather than proving Mill wrong, this showed that material logic needed to be freed from some of Mill’s philosophical assumptions. One of these was Mill’s psychologism: it was not

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the goal of logic, argued Venn, to settle debates on the ultimate sources of knowledge.69 Instead, logic ‘must be prepared to start at any moment from our present standing point, or at most from a step or two behind this, and to account for the connection of our beliefs over this limited range’.70 Another assumption was that of the standpoint of ordinary language. Mill had dismissed formal innovations not only because they did not improve deductive logic as a practical tool but also since they were, therefore, merely speculative exercises far removed from the logical difficulties of everyday life. Venn, for his part, believed otherwise. Since algebraic logic, among other things, greatly increases the power of deductive logic in the solution of complex problems, as compared to the syllogism at any rate, it is well worth pursuing it. Indeed, for this very reason, in generalizing the processes of syllogism, there is no need to be bound to the strictures of ordinary language. At the same time, Venn was still committed or remained tied to the position he questioned. Many pages in his Symbolic Logic are dedicated to showing that the algebraic generalizations can, in fact, be retranslated into syllogistic and popular forms of expression. Taken together, Venn’s work is best seen as a product of creative indecision. It embodies a struggle to combine opposites, ending up with watered-down versions of both. Another way of putting this is that Venn hit upon the limits of Whately’s dual heritage: a formal logic and an inductive logic that, despite all their irreconcilable differences, had one thing in common, namely that they both remained tied to the syllogism, critically and constructively.

Notes John Stuart Mill, Autobiography, vol. 1. of Collected Works of John Stuart Mill, ed. John M. Robson and Jack Stillinger (London: Routledge and Kegan Paul, and Toronto & Buffalo: University of Toronto Press, 1973), 1–290, at 189. 2 John Venn, The Principles of Empirical or Inductive Logic (London: Macmillan and Co., 1889), 372. 3 For Mill’s early education see Mill, Autobiography, Part I. The following account of Mill’s early and later thinking on logic draws chiefly on the following sources: Mill, Autobiography, Part IV; Steffen Ducheyne and John P. McCaskey, ‘The Sources of Mill’s Views of Ratiocination and Induction’, in Mill’s A System of Logic: Critical Appraisals, ed. Antis Loizides (New York and Abingdon: Routledge, 2014), 63–82, especially 65–70; David Godden, ‘Mill’s System of Logic’, in The Oxford Handbook of British Philosophy in the Nineteenth Century, ed. W. J. Mander (Oxford: Oxford University Press, 2014), 44–70; W. R. de Jong, The Semantics of John Stuart Mill (Dordrecht: D. Reidel Publishing Company, 1982), 18–44; Geoffrey Scarre, Logic and Reality in the Philosophy of John Stuart Mill (Dordrecht & Boston: Kluwer Academic Publishers, 1989), 15–37. 4 Mill quoted in Alexander Bain, John Stuart Mill. A Criticism: With Personal Recollections (London: Longmans, Green, and Co., 1882), 8. The books referred to are Samuel Smith, Aditus ad Logicam (1613); Edward Brerewood, Elementa Logicae (1614); and Franco Burgersdicius, Institutionem Logicarum Libri Duo (1626). 5 Mill, Autobiography, 23. 1

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See John Stuart Mill, A System of Logic Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation, Books I–III, vol. 7 of Collected Works of John Stuart Mill, ed. J. M. Robson, with an Introduction by R. F. McRae (London: Routledge & Kegan Paul, and Toronto & Buffalo: University of Toronto Press, 1974), 18. 7 See Calvin Jongsma’s chapter in the present volume for a discussion of Whately’s Elements of Logic. 8 Mill, Autobiography, 21. 9 Bain, John Stuart Mill, 36. 10 John Stuart Mill, ‘Review of Whately’s Elements of Logic’, in vol. 11 of Collected Works of John Stuart Mill, ed. J. M. Robson, with an Introduction by F. E. Sparshott (London: Routledge and Kegan Paul, and Toronto & Buffalo: University of Toronto Press, 1978), 3–35, at 3–4. 11 Mill, ‘Review of Whately’s Elements of Logic’, 15. 12 Ibid., 10. 13 Richard Whately, Elements of Logic, 2nd ed. (London: J. Mawman, 1827), 216. 14 Mill, ‘Review of Whately’s Elements of Logic’, 33. 15 Ibid., 34. 16 Mill, Autobiography, 188. 17 Dugald Stewart, Elements of the Philosophy of the Human Mind. vols. 3 (London & Edinburgh, 1814 [1792]). Vol. II: 49. 18 Stewart, Elements, vol. II, 29. 19 Ibid., 41. 20 Mill, System of Logic, 191. 21 Mill, Autobiography, 190. 22 Ibid., 190. 23 Scarre, Logic and Reality, 24. 24 ‘Whatever is true of a class, is true of everything included in the class.’ For a detailed discussion of Mill’s criticism of the Dictum de omni et nullo see Scarre, Logic and Reality, chapter 1. 25 Mill, System of Logic, 184. 26 Mill, System of Logic, 184. For a general discussion of the petitio principii fallacy see, for instance, John Woods and Douglas Walton, ‘Petitio Principii’, Synthese 31 (1975), 107–27. For its appearance in Mill’s System of Logic see David Botting, ‘Do Syllogisms Commit the Petitio Principii? The Role of Inference-Rules in Mill’s Logic of Truth’, History and Philosophy of Logic 35 (2014): 237–47. 27 Mill, System of Logic, 184. 28 Ibid., 185. 29 John Stuart Mill, An Examination of Sir William Hamilton’s Philosophy and of the Principal Philosophical Questions Discussed in His Writings, 2nd ed. (London: Longmans, Green, and Co., 1865), 439. 30 Reginald Jackson, An Examination of the Deductive Logic of John Stuart Mill (Oxford: Oxford University Press, 1941), John M. Skorupski, John Stuart Mill (London: Routledge, 1989), and Scarre, Logic and Reality. 31 Geoffrey Scarre, ‘Proof and Implication in Mill’s Philosophy of Logic’, History and Philosophy of Logic 5 (1984): 19–37, at 22. The analysis in this sub-section is based on Scarre’s account. 32 Mill, ‘Review of Whately’s Elements of Logic’, 34. 33 Mill, System of Logic, 183. 6

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34 See Scarre, ‘Proof and Implication’, 25–6. 35 See, especially, Mill, System of Logic, Book, Chapter 3, §9. During the 1870s–1880s, Mill’s logic of truth was variously called material logic, objective logic and matterof-fact logic by logicians frequently publishing in Mind such as John Venn, Carveth Read, Herbert Spencer and J. N. Keynes. 36 Mill, System of Logic, 109. 37 John Stuart Mill, ‘Grote’s Aristotle’, in vol. XI (‘Essays on Philosophy and the Classics’) of Collected Works of John Stuart Mill, ed. J. M. Robson, with an Introduction by F. E. Sparshott (London: Routledge & Kegan Paul, Toronto & Buffalo: University of Toronto Press, 1978), 473–510, at 479. 38 On De Morgan’s logic of relatives and Boole’s algebra of logic see, respectively, David Dunning’s and Sun-Joo Shin’s chapters in this volume. 39 See John Stuart Mill to John Elliot Cairnes, dated 5 December 1871, where Mill, referring to William Stanley Jevons’s work, speaks of the ‘mania for encumbering questions with useless complications, and with a notation implying the existence of greater precision in the data than the questions admit of. His speculations on Logic, like those of Boole and De Morgan, and some of those of Hamilton, are infected in an extraordinary degree with this vice’. Collected Works of John Stuart Mill. Volume XVII, ed. Francis E. Mineka and Dwight N. Lindley (London: Routledge and Kegan Paul, and Toronto & Buffalo: University of Toronto Press, 1972), 1862–3. A topic beyond the scope of the present chapter but well-worth exploring further, is that of the connection between Mill’s criticism of nineteenth-century developments in formal deductive logic and that of his arch-revival William Whewell, another innovator of inductive logic. 40 Mill, System of Logic, 173 (note). 41 For this point see David Godden, ‘Mill on Logic’, in A Companion to Mill, ed. Christopher Macleod and Dale E. Miller (Chichester: Wiley Blackwell, 2017), 175–91, at 181. 42 Mill, Examination of Hamilton’s Philosophy, 444–5. 43 Mill, System of Logic, 172. 44 Mill, Examination of Hamilton’s Philosophy, 437 (note). 45 On Venn’s work on logic see Lukas M. Verburgt, John Venn: A Life in Logic (Chicago and London: The University of Chicago Press, 2022). 46 John Venn, Principles, v. 47 It is a separate question whether Mill himself actually adhered to the material, matter-of-fact or objective logic ascribed to him by British logicians in the 1870s–1880s. For example, they rarely if ever took into account that Mill accepted intuition or consciousness as one of two sources of knowledge, instead focusing almost entirely on observation and experience. 48 John Venn, ‘The Difficulties of Material Logic’, Mind 4 (1879): 35–47, at 47 and 36. 49 Venn, Principles, 374 and vi. 50 James Sully, ‘Empirical Logic’, Nature 40, no. 1032 (1889): 337–9, at 339. 51 Venn, Principles, 19. Venn’s Principles contained many topics that would today not be recognized as forming part of logic but, instead, of philosophy of science and sociology. At the same time, Venn discussed several traditional logical topics in considerable detail, such as names and propositions, but did not include an account of the syllogism. 52 The first step is discussed in detail in Verburgt, John Venn, chapter 9. 53 John Venn, ‘Consistency and Real Inference’, Mind 1 (1876): 43–52, at 47.

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54 Ibid., 47–8. 55 Ibid., 48–9. 56 Venn, Principles, 372. 57 Ibid., 374. 58 Ibid. 59 Ibid. 60 Ibid., 376. 61 Ibid., 375. 62 Ibid., 377. 63 Rather than in the Elementary Lessons in Logic of 1870, as Venn mentioned on several occasions, the problem seems to have first appeared in Jevons’s 1874 The Principles of Science. Venn first made reference to this problem in print in John Venn, ‘Boole’s Logical System’, Mind 1 (1876): 479–91, on 487. It was subsequently discussed by several contributors to the ‘Mathematical Questions and Solutions’ in the Educational Times. 64 John Venn to William Stanley Jevons, 11 October 1874, JA6/2/360, John Rylands University Library, Manchester. 65 Venn, ‘Boole’s Logical System’, 485. 66 John Venn, Symbolic Logic (London: Macmillan and Co., 1881), x, and Venn, Principles, 376. 67 Venn, Principles, 378. 68 Ibid. 69 For a discussion of psychologism in Mill see, for instance, David Godden, ‘Psychologism in the Logic of John Stuart Mill: Mill on the Subject Matter and Foundations of Ratiocinative Logic’, History and Philosophy of Logic 26 (2005): 115–43. 70 Venn, Principles, 379.

3

The Aristotelian roots of Bolzano’s logic Mark Siebel

The Bohemian philosopher, theologian and mathematician Bernard Bolzano was born in 1781, when Kant’s Critique of Pure Reason was published, and he died in 1848, when Frege was born. These data blend in well with Bolzano’s philosophical growth because, in opposing Kant, he developed ideas showing a striking resemblance to Frege’s. However, since Bolzano resisted the Kantian zeitgeist, since he was dismissed from his professorship for political reasons and since many of his books were indexed as forbidden, his influence on nineteenth-century philosophy was regrettably petty. It took more than 100 years until Bolzano’s ideas awakened the attention they deserve. Bolzano’s logic is among the first topics to attract notice in the twentieth century. It is to be found in the first two volumes of his monumental Theory of Science (Wissenschaftslehre) from 1837. These volumes contain elaborated introductions both to basic logical concepts, such as ‘sentence in itself ’ and ‘deducibility’, and to a plethora of inference rules. In the contribution at hand, the Aristotelian roots of Bolzano’s logic are examined. Section 1 covers the objects of logic. In Bolzano’s view, logic is not concerned with sentences or judgements but with their potential contents, viz. so-called sentences in themselves. Since Bolzano takes the view that these entities invariably follow the subject-predicate pattern ‘A has b’ and are true only if there are A, he significantly deviates from the traditional understanding of the categorical propositions. In Section 2, Bolzano’s definition of the consequence relation, titled ‘deducibility’, is introduced. A conclusion is deducible from certain premises if they contain ideas that can be replaced by other ideas without thereby obtaining true variants of the premises and a false variant of the conclusion. Bolzano holds that his definition hides inside Aristotle’s definition of a valid syllogism. He is right insofar as, although these definitions differ in the details, they agree in defining implication via truthpreservation. Section 3 presents Bolzano’s treatment of syllogistic. Due to his specific interpretation of the categorical propositions, he rejects some of the relations in the square of opposition and some of the received mediate and immediate inferences. More importantly, however, syllogistic constitutes only a tiny part of Bolzano’s logic because he clearly sees its structural limitations.

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In Section 4, the results are summarized. Briefly, while Aristotle wins high praise from Bolzano for the fundamental notions behind syllogistic, he thinks poorly of their implementation into a system of inferential rules.

1 The objects of logic In his Logical Investigations, Husserl commends Bolzano as ‘one of the greatest logicians of all times’.1 The main reason for this accolade is that Bolzano fully cognizantly took up an anti-psychologistic position. Admittedly, he used ‘logic’ as a synonym for ‘theory of science’, and in this wide sense logic is dependent on psychology because it deals, inter alia, with ‘the conditions underlying the ability to know truths, especially for humans’.2 However, theory of science includes ‘pure logic’ or ‘theory of elements’.3 Logic in this narrow sense does not establish ‘the laws of thought’, i.e. the rules of ‘appearances in the mind of a thinking being’, but is ‘the theory of ideas, sentences, true sentences and inferences in themselves’.4 The crucial phrase here is ‘in themselves’. Sentences in themselves, or ‘objective sentences’, are not verbal expressions but can rather become the ‘sense’ of certain word compounds or the ‘material’, or ‘content’, of certain mental episodes.5 When Frieda utters the sentence ‘Wassermelonen sind süß’, and Harry judges that watermelons are sweet, the sentence in itself expressed by Frieda is identical with the one to which Harry agrees in thought. In contrast to Frieda’s utterance and Harry’s judgement, this sentence in itself lacks a position in space and time, does not stand in causal relationships and is independent of the existence of thinking beings or languages.6 Bolzano praises Aristotle’s definition of ‘statement-making sentence[s as] those in which there is truth or falsity’ as an early characterization of sentences in themselves.7 For just like Frege’s ‘thoughts’,8 sentences in themselves are conceived by Bolzano as the primary bearers of the unrelativized truth-values true and false. This means, first, that sentences and judgements have their truth-values in virtue of the truth-values of the sentences in themselves that are their contents. Second, what is expressed by, say, ‘I am hungry’ in each case does not possess relativized truth-values, such as true/false with respect to person S at time t. Third, there are neither truth-value gaps nor further truth-values. Every sentence in itself is either true or false.9 A sentence in itself consists of parts called ‘ideas in themselves’ or ‘objective ideas’. The sentence in itself expressed by ‘3 is a prime number’ can be decomposed into an idea of the number 3 and an idea of the property of being a prime number. Ideas in themselves are also neither linguistic symbols nor mental entities, but abstract objects that can turn into the contents of such things without existentially depending on them. In contrast to propositions, ideas are not true or false but ‘objectual’ if there is something that is represented by them, or ‘objectless’ if nothing falls under them.10 For the sake of simplicity, I will frequently use the shorter terms ‘propositions’ for sentences in themselves and ‘ideas’ for ideas in themselves. Moreover, I will refer to propositions and their components by putting the corresponding expressions in square brackets. Thus, [3 is a prime number] is the proposition expressed by ‘3 is a prime number’, and it contains the idea [prime number].

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There are two constraints on sentences in themselves exerting a significant influence on Bolzano’s understanding of the categorical propositions. First, Bolzano takes all propositions to be structured in the same way. Whatever their linguistic counterparts look like, the propositions expressed have the form ‘A has b’, or alternatively, ‘Whatever has a has b’. ‘A’ stands for the ‘subject-idea’, representing the object(s) the proposition is about; and ‘b’ stands for the ‘predicate-idea’, representing the property attributed to the object(s).11 To display the structure and constituents of a proposition as clearly as possible, one has to attune the given sentence to the canonical pattern ‘A has b’. For example, in affirmative universals ‘Every S is P’ the quantifier is regarded as not contributing an idea to the proposition expressed but as just pointing out that the extension of the subject-term consists of all S. The somewhat awkward paraphrase of ‘All men are mortal’ is thus ‘Man has mortality’.12 The second constraint is that a proposition [A has b] is true only if there exists at least one object represented by the subject-idea [A] and at least one property represented by the predicate-idea [b].13 Bolzano is aware of the fact that this proviso has drastic implications for Kantian analyticities, viz. propositions whose subject-idea contains their predicate-idea. For example, to verify ‘Every drake is male’, it need not only be shown that drakes are defined by being male but also that there exist drakes. In contrast to what Kant claims, such a proposition is therefore not a priori on Bolzano’s reading.14 In the Prior Analytics, Aristotle defines a proposition as ‘a statement affirming or denying something of something’.15 Bolzano concurs insofar as he also assumes subject-predicate structure, i.e. a constituent providing the objects the proposition is about and a constituent giving a property. However, Aristotle holds that the property can be related to the objects in two ways. It is attributed to them in ‘Every S is P’ and ‘Some S are P’, whereas it is denied of them in ‘No S is P’ and ‘Some S are not P’. Aristotle thus accepts two copulas, the affirmative ‘is’ (or ‘are’) and the negative ‘is not’ (or ‘are not’). Bolzano, on the other hand, discards a negative copula ‘has not’, or briefly ‘lacks’, but argues that ‘the concept of negation never belongs with the copula’.16 Along these lines, ‘Caius does not have wit’ does not express denial of an attribute, i.e. [Caius – lacks – wit], but ascription of a negative attribute, i.e. [Caius – has – lack of wit] or [Caius – has – the attribute non-wit].17 Bolzano’s refusal of a negative copula becomes apparent as well in his reading of the categorical propositions. As to particulars, he argues that they express ‘statements of objectuality’.18 While the affirmative ‘Some S are P’ states that the idea of an S that is P has objectuality, the negative ‘Some S are not P’ attributes objectuality to the idea of an S that is not P. Hence, their subject-ideas are meta-ideas insofar as they represent an idea, and their predicate-ideas represent the property of having an object. Since, if there is no S, the idea of an S that is P is as objectless as the idea of an S that is not P, both particulars are loaded with existential import. They are true only if ‘S’, i.e. their subject-term in the traditional sense, is not empty. Universal affirmatives can be used in two ways. On the one hand, ‘Every S is P’ can be equated with ‘S has P-ness’.19 On the other hand, ‘Every S is P’ may be used to convey the hypothetical proposition ‘If an object is S, then it is P’.20 Bolzano adds, however, that deploying a universal affirmative in the second sense is a ‘misuse’ violating the ‘proper sense’ of such sentences. Bolzano’s reason for this verdict is that ‘Every S is P’ in

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its proper sense has existential import because it implies ‘Some S is P’, which does not hold for the conditional.21 In modern Gricean terms, Bolzano could have said that, if ‘Every S is P’ is used in the non-committal way, there is an ‘implicature’ because there is a mismatch between the literal meaning of ‘Every S is P’ and what the speaker means by uttering this sentence.22 Negative universals can also be used in a committal and a non-committal way.23 ‘No S is P’ might express the denial of ‘Some S is P’, stating that the idea of an S that is P has no objectuality. According to this use, ‘No S is P’ lacks existential import but is rather true if there is no S. But ‘No S is P’ may also be taken to mean that all S have the property of being non-P.24 Then the proposition expressed is existentially loaded because it assigns a property to all objects of a certain kind. Here, Bolzano does not stigmatize the non-committal use as a misuse but even presents a geometrical example meeting with his approval.25 When it comes to the logical relations between propositions, however, Bolzano makes clear that he adopts the second reading by pointing out that ‘No S is P’ implies ‘Some S are non-P’.26 Table 3.1 summarizes Bolzano’s reformulations of the categorical propositions and their translation into predicate logic.27 Since Bolzano furnishes all propositions with existential import, he markedly deviates from the traditional reading to be found, e.g., in William of Ockham’s Summa Logicae.28 According to this reading, only affirmatives are existentially loaded, whereas negatives are considered true if ‘S’ is empty. By his treatment of negative particulars, Bolzano also derogates from Aristotle’s original formulation of them, which is ‘Not every S is P’ and hence does not entail the existence of an S.29 Morscher notes that ‘[w]hereas Aristotle’s theory of categorical syllogisms does not allow empty terms at all, Bolzano’s logic does so, but they cannot be the subject-ideas of true propositions’.30 This comment needs some qualification. Consider negative universals. Traditionally, they are rendered ‘No S is P’, with the idea expressed by ‘P’ being the predicate. As the example ‘No figure is a round square’ shows, Morscher is right in claiming that the predicate in this traditional sense can be empty even if the proposition is true according to Bolzano. In Bolzanese, however, a negative universal reads [S has non-P-ness], with the entire idea following [has], viz. [non-P-ness], being the predicate. If the predicate in this Bolzanian sense is empty, the proposition is false because there is no property attributed to the objects it deals with.31 In what follows, the context will make clear whether my remarks are about subject and predicate in the Bolzanian or the Aristotelian sense. Bolzano emphasizes at many places that logic explores neither mental occurrences (such as judgements) nor linguistic signs (such as sentences) but their potential Table 3.1  Bolzano’s reformulations of the categorical propositions Every S is P No S is P Some S are P Some S are not P

S has P-ness S has non-P-ness The idea of an S that is P has objectuality The idea of an S that is not P has objectuality

(∀x)(Sx → Px) & (∃x)(Sx) (∀x)(Sx → ¬Px) & (∃x)(Sx) (∃x)(Sx & Px) (∃x)(Sx & ¬Px)

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contents, namely propositions.32 In this spirit, he claims that ‘syllogistic should deal only with sentences in themselves, not with their verbal expressions’.33 What is more, Bolzano attributes to Aristotle, if not an express mention, but an implicit grasp of this point: How many theorems and inquiries that appear in Aristotle’s Organon  … have mere sentences and truths in themselves as their topic! What is the entire field of syllogistic other than a theory of certain relations that hold between sentences and truths in themselves!34

2 Deducibility Bolzano regards not only the items of logic as objective but also the logical relations between them. Let us consider Bolzano’s explication of the consequence relation he named deducibility.35 Deducibility is defined with the help of the method of variation, which is the imaginary substitution of ideas in a proposition (or an idea) by other ideas. By substituting them, we get variants of the original proposition (or idea) that may have a different truth-value (or extension). For example, replacing [3] by [6] in [3 is a prime number] leads to a false proposition, whereas additionally substituting [even number] for [prime number] results in the true variant [6 is an even number]. Variation has to be systematical: same ideas must be replaced by same ideas. [Green chairs are blue] is not a permissible variant of [Red roses are red]. Furthermore, the variants must be non-empty insofar as their subject-ideas have to represent at least one object.36 Substituting [the greatest prime number] for [3] in [3 is a prime number] is not allowed. Due to Bolzano’s requirement that a proposition is true only if its subjectidea is objectual, it would otherwise be too easy to obtain false variants of a proposition. The method of variation plays a key role in Bolzano’s logic because he defines a multitude of notions by it, such as logical truth, necessary truth, analyticity, compatibility and deducibility. The definition of deducibility reads as follows:37 The conclusion of an inference is deducible from the premises with respect to certain variable ideas if and only if every substitution of the variable ideas leading to true variants of the premises also leads to a true variant of the conclusion.

Along these lines, [Donald is male] is deducible from [Donald is a drake] with respect to [Donald]. Given that the above-mentioned requirements on variation are satisfied, replacing [Donald] by other ideas never results in a true variant of the premise and a false variant of the conclusion. In keeping with his anti-psychologistic stance, Bolzano lays great stress upon deducibility being ‘a relation that holds objectively, i.e. regardless of our faculties of representation and cognition’. Unfortunately, some logicians described it ‘as a relation between judgements’ and claimed that it ‘consists in the fact that the acceptance of one sentence brought about the acceptance of another’.38 This is misguided because the deducibility of a proposition from other propositions depends neither on what we think nor on what we say.

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Deducibility is a triadic relation because it does not only interrelate a set of premises and a conclusion but also involves a set of variable ideas. The latter contains the elements that are irrelevant for the transfer of truth from premises to conclusion. Due to its triadic nature, deducibility differs from the consequence relation underlying syllogistic. Syllogisms are meant to be formally valid. They are inferences in which the premises imply the conclusion by sheer logical form. To recognize that a syllogism is valid, it therefore suffices to know the meaning of the logical expressions contained in it, such as the quantifiers ‘all’ or ‘some’ and the negation ‘not’. By contrast, Bolzano is after a broader concept including material validity. This kind of validity depends on the meaning of non-logical expressions, such as ‘drake’ and ‘male’ in ‘Donald is a drake; therefore, he is male’. However, Bolzano also points to a narrower conception of deducibility that is formal. He states that there are ‘propositions that can be deduced from [a proposition] just in virtue of its form (i.e. that are deducible when all those parts in it that logicians do not count among its form are taken as variable)’.39 Let us call this relation formal deducibility. Its specific feature is that only logical ideas are held constant, in order that all non-logical ideas in the given propositions are free for substitution: The conclusion of an inference is formally deducible from the premises if and only if the conclusion is deducible from the premises with respect to their nonlogical ideas.

Hence, [Donald is male] is formally deducible from [Donald is a drake] and [All drakes are male] because replacing the non-logical ideas [Donald], [drake] and [male] leads to a true variant of the conclusion whenever the variants of the premises are true. Perhaps surprisingly, [Donald is male] is also formally deducible from the sole premise [Donald is a drake]. Since the idea [drake] is identical with [male duck], the proposition [Donald is a drake] is identical with [Donald is a male duck]. [Donald is male] is therefore deducible from [Donald is a drake] with respect to the non-logical ideas within these propositions, namely [Donald], [male] and [duck]. This might raise the suspicion that there are no genuine cases of material deducibility, i.e. cases in which material deducibility is not accompanied by formal deducibility. But consider [This is a plane triangle] and [This has an angular sum of 180 degrees]. The latter is deducible from the former regarding [this]. Since the definition of plane triangles does not mention the angular sum of 180 degrees, however, the idea [plane triangle] does not contain [angular sum of 180 degrees]. Hence, if all non-logical ideas in these propositions are free for substitution, there are true variants of the premise and false variants of the conclusion, e.g. [This is a square] and [This has five sides]. Put another way, [This has an angular sum of 180 degrees] is not formally deducible from [This is a plane triangle] because [Plane triangles have an angular sum of 180 degrees] is synthetic in the Kantian sense.40 There is a further feature of Bolzano’s definition yielding genuine material deducibility. Today’s logical systems rest on the criterion that an argument is valid only if it is absolutely necessary that its conclusion is true if its premises are true. Thus, the argument ‘Donald is a drake; therefore, he is male’ is valid because drakes are male by definition. On the other hand, the argument ‘Donald is a drake; therefore, he is at most thirty years old’ is invalid.

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For even if there are biological laws preventing a drake from getting older than thirty years, they do not provide the strong kind of necessity needed for validity. By contrast, Bolzano’s definition of deducibility does not contain any modal concept. It does not state that the variants of the conclusion must be true if the variants of the premises are true but merely says that there is de facto no substitution leading to true variants of the premises and a false variant of the conclusion. Thereby, Bolzano’s definition gives free rein to inferences going beyond absolute necessity. Let drakes become at most thirty years old for biological reasons. Then [Donald is at most thirty years old] is deducible from [Donald is a drake] with respect to [Donald], although the connection between being a drake and being at most thirty years old is not one of absolute necessity. Bolzano thus allows material inferences to be based on laws of nature. In Ryle’s words, such laws may function as ‘inference-tickets’ that entitle to pass over from a premise to a conclusion.41 Inferences of this type provide further examples of genuine material deducibility. While [Donald is at most thirty years old] is deducible from [Donald is a drake] with respect to [Donald], it is not deducible if all non-logical ideas are released for substitution. For it is easy, then, to construct a true variant of the premise and a false variant of the conclusion. Bolzano even goes beyond Ryle in not requiring any kind of necessity at all, whether absolute or nomological. Deducibility is characterized in such a way that the facts legitimizing an inference need not even be laws of nature. Let us assume that, for contingent reasons, the names of my favourite bands start with a ‘W’. Then [The name of this band starts with a ‘W’] is deducible from [This is one of M.S.’s favourite bands] with respect to [this]. Bolzano thereby opens the doors to material inferences resting on merely contingent facts. When it comes to former explications of the consequence relation, Bolzano cites Aristotle’s definition of a valid syllogism as ‘one of the best’:42 A [valid] syllogism is a discourse in which, certain things being stated, something other than what is stated follows of necessity from their being so.43

The reason for Bolzano’s praise is that he spots variation of ideas in the wording ‘follows of necessity’. This formulation, Bolzano argues, is meant to point out ‘that the conclusion becomes true whenever the premises are true’. Since ‘propositions none of whose parts change are not sometimes true and sometimes false’, Aristotle appears to have in mind that ‘whenever the exchange of certain ideas makes the premises true, the conclusion must also become true’.44 If we ascribe the temporal frequency model of modality to Aristotle,45 there is indeed a common thread with Bolzano’s account. This model identifies necessity with omnitemporality in order that it is necessary that p just in case it is always true that p. By applying this model to the necessity phrase in Aristotle’s characterization of valid syllogisms, the characterization reads as follows: A syllogism is valid if and only if, for every time t, if the syllogism’s premises are true at t, then its conclusion is true at t.

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Aristotle’s definition thus corresponds with Bolzano’s in identifying implication with truth-preservation. Both subject premises and conclusions to variation (of ideas in Bolzano and points in time in Aristotle) and demand that truth is transferred from premises to conclusions throughout. Moreover, since the omnitemporality defining necessity does not embrace possible worlds but only the actual one, Aristotle would not require necessary but only factual truth-preservation. Thereby, his definition also allows for material inferences based on contingent facts. Assume again that, for contingent reasons, the names of my favourite bands invariably start with a ‘W’. Then, the inference from ‘This is one of M.S.’s favourite bands’ to ‘The name of this bands starts with a “W”’ is valid according to Aristotle as well. Despite these common features, however, we should resist the temptation to identify the given definitions. First of all, in contrast to Bolzano’s sentences in themselves, the premises and conclusions of a valid syllogism in Aristotle’s sense must not be true or false simpliciter. As pointed out in Section 1, a sentence in itself is ‘either true or false, and remains that way always and everywhere’.46 Hence, any old true sentences in themselves would form a valid syllogism according to Aristotle’s definition because, for every time t, if one of them is true at t, the other one is also true at t. One could reply that Bolzano actually wanted to assign atemporal (not omnitemporal) truth-values to sentences in themselves. But then Aristotle’s definition is not even applicable to them. Additionally, Bolzano’s quantification over variants of propositions is much more general than Aristotle’s quantification over points in time. They converge if the variable ideas are ideas of points in time. Just as ‘Donald is male’ is true at t if ‘Donald is a drake’ is true at t, so replacing the temporal idea in [Donald is male at t] leads to a true variant if the variant of [Donald is a drake at t] is true. But Bolzano allows a lot more than variation of temporal ideas. Hence, although Aristotle’s definition can be counted as a precursor of Bolzano’s, the latter clearly extends the Aristotelian template.

3 Syllogistic Bolzano’s logic follows the traditional pattern by addressing first ideas, then propositions and finally inferences. The alternative term for this approach to logic, namely ‘theory of elements’, was also used, among many others, by Kant.47 Bolzano strongly disagrees with Kant, however, on the improvability of logic: [O]ne of Kant’s literary sins was that he attempted to deprive us of [the] wholesome faith [in the perfectibility of logic] through an assertion very welcome to indolence, namely, that logic is a science that has been complete and closed since the time of Aristotle.48

Bolzano clearly saw that there is not only the straitened number of inferential schemas mentioned in traditional logic but infinitely many distinct forms of inference.49 The aim of the part ‘Of Inferences’ in the Theory of Science is to introduce in a systematic manner the most important of them.50 Within this presentation consisting of 120 pages and a 50-page appendix on previous treatments, syllogistic constitutes only a tiny part. Since

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Bolzano proceeds by enumeration, not axiomatization, his logic is still remote from the revolutionary character of Frege’s logic. Nevertheless, by convincingly highlighting the narrow boundaries of syllogistic, Bolzano is one of the first philosophers who made clear that it is time to go beyond them. This is one of the rare places where Aristotle is criticized: ‘Aristotle began with such a broad definition of the word “syllogism” that one is astonished all the more that he could have subsequently restricted the concept of this kind of inference so severely.’51 Many of these restrictions concern the structure of inferences. It was assumed that a valid inference is either mediate or immediate, with mediate inferences containing one premise and immediate ones two premises. Furthermore, premises and conclusion are supposed to be categorical propositions. In immediate inferences, these propositions include exactly three variable terms. The minor and the major terms are the subject and the predicate of the conclusion, respectively, and they are contained in one premise each, whereas the middle term is part of both premises.52 Let us consider some of Bolzano’s counterexamples. The following inferences do not contain three variable terms but two or four:53 All A are B. All B are A. Therefore, the idea of A and the idea of B represent the same objects. All A are B. All C are D. Therefore, a whole including an A and a C includes a B and a D.

The second example also refutes the claim that there has to be a common middle term in the premises.54 Moreover, if one of the variable terms in the first example is considered the middle term, the remaining term is neither the minor nor the major term in the customary sense. For it is neither the subject nor the predicate of the conclusion but only a part of the subject. Bolzano assumes that such inferences have been overlooked because their conclusions do not fit the patterns for categorical propositions.55 Something similar holds for the famous canon that nothing follows from mere negatives. If ‘negatives’ does not only refer to propositions of the types ‘No S is P’ and ‘Some S are not P’ but also to ‘It is false that every (non-)S is (non-)P’, then the following example proves that this canon is wrong:56 It is false that every S is P. It is false that every non-S is P. Therefore, the idea of a P does not subsume every object.

For the sake of fairness, Bolzano adds that the canon in question is usually limited to categorical propositions. But he thinks that he has a counterexample even then:57 No M is P. No M is S. Therefore, some non-S are non-P.

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However, traditionalists need not be impressed by this inference because it is valid only if negative universals have existential import. These scholars can therefore reply that, although Bolzano assumes existential import, a negative universal is true according to their reading if nothing falls under the subject. Hence, if ‘M’ is empty and ‘S’ and ‘P’ express mutually exclusive and exhaustive ideas, such as ‘real’ and ‘unreal’, the premises are true while the conclusion is false. As to the canon that nothing follows from mere particulars, Bolzano admits that he does not have a counterexample when this canon is restricted to syllogisms in the traditional sense.58 He should have said the same about the canon that nothing follows from mere negatives. As an ‘inference not having the form of a syllogism at all’, Bolzano cites an example of conjunction introduction:59 Caius is learned. Titius is learned. Therefore, Caius and Titius are learned.

Bolzano does not mention the option, already considered in medieval times, to translate a singular affirmative ‘a is P’ into a universal affirmative ‘Every object that is identical with a is P’. With small adjustments, this trick can also be employed on the conclusion: Every object that is identical with Caius is learned. Every object that is identical with Titius is learned. Therefore, every object that is identical with Caius or Titius is learned.

However, although we thereby obtain categorical propositions, the constraints on variable terms are not met. The minor term, i.e. the subject of the conclusion (‘object that is identical with Caius or Titius’), is nowhere to be found in the premises, while the major term, i.e. the predicate of the conclusion (‘learned’), is included in both premises. Further intractable inferences arise from Bolzano’s rule of conditionalization. He explains that, if a conclusion Q is deducible from premises P1 to Pn, then any conditional ‘If Pm+1 … and Pn, then Q’ is deducible from the remaining premises P1 to Pm (m < n).60 From this rule, inferences like the following emerge: If P1, P2 and P3, then Q. Therefore, if P1 and P2, then (if P3, then Q).

It is hardly possible to squeeze an inference of this type into the bodice of syllogistic. As early as in the paragraph on deducibility, Bolzano emphasizes that a valid inference may contain more than two premises. One of his examples is the following ‘sorites’:61 Every A is B. Every B is C.

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Every C is D. Therefore, every A is D.

Although this inference displays that a general restriction to two premises is indeed wrong, Bolzano concedes that ‘we could  … arrive at the conclusion by a series of ordinary syllogisms’, which is twice Barbara in the given case. If such an adaption is always feasible, ‘the theory of non-syllogistic forms of inference would at least be dispensable’. For this reason, Bolzano presents further inferences that contain more than two premises but cannot be adjusted to syllogistic standards. Here is one of them:62 Either A or B. Either A or C. Either A or B or C. Therefore, A.

Bolzano might have overlooked that his rule of conditionalization allows to convert any inference with more than two premises into a chain of inferences with only two premises each. For example, an inference ‘P1, P2, P3; therefore, Q’ can be converted into ‘P1, P2; therefore, if P3, then Q’ and ‘P3; if P3, then Q; therefore, Q’. Hence, Bolzano’s examples can be attuned at least to the constraint that every inference is reducible to inferences including at most two premises. However, this is cold comfort for traditionalists because, such as in the case above, the resulting propositions often resist translation into categorical propositions or will not contain the required number of variable terms in the right position. Bolzano does not only object to the structural requirements and some canons of syllogistic but also finds fault with the dictum de omni et nullo, which was considered ‘the highest principle of all inferences’. The ‘omni part’ of this principle states that whatever holds of all A also holds of any individual A. Bolzano realizes that this is false if ‘all’ is understood ‘collectively’, viz. as referring to the whole of all A. There are many attributes, such as weight, that are not inherited from a whole to its parts. Rather, ‘all’ has to be understood ‘distributively’, i.e. as referring to each individual A. But then the first part of the dictum de omni et nullo boils down to the ‘empty, useless tautology’ that ‘[w]hatever holds of each A holds of each A’.63 Bolzano’s negative attitude towards the strictures of syllogistic does not prevent him from introducing the traditional inferences that are covered by his logic. In Section 1, it was pointed out that Bolzano deviates from tradition in furnishing all propositions with existential import. Since he thus allows for further false-makers for negatives, the syllogistic part of his logic contrasts with traditional syllogistic. First of all, the resulting square of opposition is a downsized version of the received square. In Bolzano, ‘Every S is P’ and ‘No S is P’ are still contraries. They cannot be true together, but it is possible that both are false.64 Likewise, the particulars are still subalternate to the corresponding universals. ‘Some S are P’ is entailed by ‘Every S is P’, and ‘Some S are not P’ is entailed by ‘No S is P’.65 However, ‘Every S is P’ and ‘Some S are not P’, as well as ‘No S is P’ and ‘Some S are P’, are not contradictories anymore but only contraries. For if there are no

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S, both elements of these pairs are wrong.66 Similarly, the particulars ‘Some S are P’ and ‘Some S are not P’ are not subcontraries but are both false if the subject-idea is empty.67 Bolzano is fully aware of these incongruities. The same holds for the deviations concerning immediate inferences. He registers simple conversion of ‘Some S are P’ and conversion by contraposition of ‘Some S are not P’, with the latter’s result ‘Some non-P are not non-S’ being reduced to ‘Some non-P are S’.68 However, both conversion by contraposition of ‘Every S is P’ and simple conversion of ‘No S is P’ are invalid in Bolzano’s reading. ‘Every S is P’ is true if the subject-idea is not empty and the predicateidea is ‘one of the ideas that have the largest extension (e.g. the idea of something in general)’. But ‘All non-P are non-S’ is wrong under these conditions because the subject ‘non-P’ ‘(not something, i.e. nothing) obviously does not have any extension at all’. Furthermore, ‘No S is P’ is true if ‘S’ has an object while ‘P’ is objectless. But then ‘No P is S’ is false in Bolzano’s reading because there are no P.69 As to mediate inferences, Bolzano lists nineteen traditional moods in §§227 and 235, omitting Barbari, Celaront, Camestros, Cesaro and Calemos. Elsewhere, he points out that Leibniz accepted twenty-four moods, ‘in that for every mood with a universal conclusion he added another with the same premises but a particular conclusion’.70 The five moods omitted by Bolzano are of this type. They are the weakened variants of Barbara, Celarent, Camestres, Cesare and Calemes because they contain as conclusions the subalternates ‘Some S are P’ or ‘Some S are not P’ instead of ‘Every S is P’ or ‘No S is P’, respectively. It thus appears that Bolzano’s reason for dropping them lies in this special feature.71 Apart from quoting the traditional moods in the standard formulation, Bolzano introduces them in his canonical form. In this way, Barbara reads as follows:72 Whatever has a has b. Whatever has c has a. Therefore, whatever has c has b.

Since all categorical propositions are existentially loaded in Bolzano, he has to reject Calemes, which is called Calentes by him:73 All P are M. No M is S. Therefore, no S is P.

The minor premise ‘No M is S’ is true if ‘S’, but not ‘M’, is objectless, whereas the conclusion ‘No S is P’ is false if ‘S’ is objectless. Along these lines, the true premises ‘All drakes are ducks’ and ‘No drake is a round square’ lead to the false conclusion ‘No round square is a drake’ because the latter states that round squares are non-drakes, thus implying that there are round squares. For the same reason, Calemos (or Camenop) is invalid in Bolzano’s syllogistic. Calemos is the weakened relative of Calemes because it contains the conclusion ‘Some S is not P’ instead of ‘No S is P’. Its invalidity is probably not mentioned by Bolzano because he ignores all of the five weakened moods.74

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4 Conclusion Bolzano’s attitude towards Aristotelian syllogistic is ambivalent. In Section  1, it was shown that he deviates from the traditional understanding of the categorical propositions not only in providing all of them with existential import. He also does not accept a negative copula because he takes all propositions to be affirmative. However, when it comes to the fundamental issue that the objects of logic are neither verbal expressions nor mental occurrences, but sentences in themselves, Bolzano attributes to Aristotle an, at least implicit, insight into this truth. In Section 2, a second agreement on a fundamental point became apparent. Bolzano lays great stress upon Aristotle’s definition of a valid syllogism as a precursor of his own definition of deducibility. Although there are differences in the details, they in fact agree in defining implication via truth-preservation. While Bolzano subjects premises and conclusions to variation of ideas, Aristotle varies points in time. At any rate, both require that true premises are always connected with true conclusions. Moreover, since both are satisfied with factual truth-preservation, they have to admit material inferences resting on contingencies. However, since Bolzano’s variation of ideas is much more comprehensive than Aristotle’s variation of points in time, Bolzano’s definition substantially extends the Aristotelian template. Section  3 focused on Bolzano’s dissatisfaction with the system of inference rules provided by syllogistic. The fact that he rejects some relations in the square of opposition and some classical inferences, such as Calentes and contraposition of universal affirmatives, is relatively innocuous because it results from his particular understanding of the categorical propositions. Much more significant are Bolzano’s objections to the severe structural confinements of syllogistic, some of its canons and the dictum de omni et nullo. In summary it can be stated that, while the fundamental notions behind Aristotelian syllogistic hold a lot of promise for Bolzano, he considers their concrete implementation into a system of inferential rules highly disappointing. For in the light of the infinite variety of valid inferences, it resulted in a straitjacket.

Notes 1 2

3

Edmund Husserl, Logische Untersuchungen. Band 1: Prolegomena zur reinen Logik, 2nd, rev. edn. (Halle: Niemeyer, 1913), 225; transl., Logical Investigations. Volume 1, transl. J. N. Findlay (London: Routledge & Kegan Paul, 1970). Bernard Bolzano, Wissenschaftslehre. Versuch einer ausführlichen und größtentheils neuen Darstellung der Logik mit steter Rücksicht auf deren bisherige Bearbeiter (Sulzbach: Seidel, 1837), vol. I, §6; §15, 59; repr. in Bernard Bolzano Gesamtausgabe 1, 11, 1 to 1, 14, 3 ed. J. Berg (Stuttgart: Frommann-Holzboog, 1985–2000); transl., Theory of Science, ed. and transl. P. Rusnock and R. George (New York: Oxford University Press, 2014). I refer to the Wissenschaftslehre by ‘WL’ plus number of volume, section and page. WL I, §15, 59; §16, 67.

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Ibid., §16, 62; §15, 59. Bernard Bolzano, ‘Einleitung zur Größenlehre’, in Bernard Bolzano Gesamtausgabe 2, A, 7, ed. J. Berg (Stuttgart: Frommann-Holzboog, 1975), 25–216, 47. I refer to this manuscript by ‘EG’. For sentences in themselves as ‘senses’ of linguistic signs cf. WL I, §28, 121; WL II, §148, 89; EG, 47. For sentences in themselves as ‘material’ of mental episodes cf. WL I, §19, 78; §22, 90; §34, 154; WL III, §291, 108. 6 Cf. WL I, §19, 77–8; §25, 112; WL II, §122, 4; EG, 47. 7 Aristotle, De Interpretatione, ed. Immanuel Bekker (Berlin: Georg Reimer, 1831), 17a; cf. WL I, §23, 93–4. 8 Gottlob Frege, ‘Der Gedanke. Eine logische Untersuchung’, Beiträge zur Philosophie des deutschen Idealismus 2 (1918): 58–77; repr. in Logische Untersuchungen, ed. G. Patzig (Göttingen: Vandenhoeck & Ruprecht, 1966, 3rd edn. 1986); transl., ‘Thoughts’, in Collected Papers on Mathematics, Logic and Philosophy, ed. B. McGuinness (Oxford: Basil Blackwell, 1984). 9 Cf. WL I, §24, 108; §25, 113; WL II, §125, 7; §147, 77–8. For a comprehensive comparison of Bolzano’s and Frege’s primary truth-bearers see Wolfgang Künne, ‘Propositions in Bolzano and Frege’, in Bolzano and Analytic Philosophy (Grazer Philosophische Studien 53), ed. W. Künne, M. Siebel and M. Textor (Amsterdam: Rodopi, 1997); repr. in W. Künne, Versuche über Bolzano: Essays on Bolzano (St. Augustin: Academia, 2008). 10 Cf. WL I, §48, 216–8; §49, 220; §54, 237–8; §66, 297; EG, 47–8, 51. 11 Cf. WL II, §127, 9. For a criticism of that hypothesis cf. Mark Textor, ‘Bolzano’s Sententialism’, in Künne, Siebel and Textor, Bolzano and Analytic Philosophy. 12 Cf. WL I, §57, 248–50. 13 Cf. WL II, §127, 16; §225, 399. The first part of this constraint can also be found in WL II, §130, 24; §196, 328. 14 Cf. WL III, §305, 178. For a similar problem raised by Kant’s acceptance of the inference from ‘Every S is P’ to ‘Some S are P’ cf. Mark Siebel, ‘“It falls somewhat short of logical precision”. Bolzano on Kant’s Definition of Analytic Judgements’, Grazer Philosophische Studien 82 (2011): 91–127, 94–5; ‘Kant on Infinite and Negative Judgements: Three Interpretations, Six Tests, No Clear Result’, Topoi 39 (2020): 699–713, 707–8. 15 Aristotle, Prior Analytics, ed. Immanuel Bekker (Berlin: Georg Reimer, 1831), 24a16–21. 16 WL II, §136, 45. 17 Cf. WL II, §136, 46–50. Compare Frege’s view that seemingly negative judgements are actually judgements in which a negative proposition is affirmed (‘Die Verneinung. Eine logische Untersuchung’, Beiträge zur Philosophie des deutschen Idealismus 1 (1919): 143–57; repr. in Logische Untersuchungen, ed. G. Patzig (Göttingen: Vandenhoeck & Ruprecht, 1966, 3rd edn. 1986); transl., ‘Negation’, in The Frege Reader, ed. M. Beaney (Oxford: Basil Blackwell, 1997)). Compare also the socalled infinite judgements to be found, inter alia, in Immanuel Kant, Kritik der reinen Vernunft, 2nd edn. (Riga: Hartknoch, 1787), B 95; transl., Critique of Pure Reason, ed. and transl. P. Guyer and A. W. Wood (Cambridge: Cambridge University Press, 1998); cf. Siebel, ‘Kant on Infinite and Negative Judgements’. 18 WL II, §137, 53. Cf. WL II, §194, 317; §225, 401, 404; §233, 431; §234, 435. 19 Cf. WL I, §57, 248–9. 20 In Bolzanese, such conditionals amount to ‘statements of deducibility’. Cf. WL II, §164, 198–200; §179, 224.

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21 WL II, §255, 404–5. 22 Cf. H. Paul Grice, ‘Logic and Conversation’, in Syntax and Semantics 3: Speech Acts, ed. P. Cole and J. L. Morgan (New York: Academic Press, 1975). 23 Cf. WL II, §138, 54–5. 24 Cf. WL I, §89, 426; WL II, §136, 46–7. 25 Cf. WL II, §138, 55. 26 Cf. ibid., §225, 401. 27 Cf. Edgar Morscher, ‘Bolzanos Syllogistik’, Philosophia Naturalis 24 (1987): 447–51, 448; Ettore Casari, Bolzano’s Logical System (Oxford: Oxford University Press, 2019), 66, 132–3. Casari adds Bolzanian translations of the non-committal uses of both universals. 28 Part I, ch. 72, par. 19. 29 Cf. Michael V. Wedin, ‘Negation and Quantification in Aristotle’, History and Philosophy of Logic 11 (1990): 131–50. 30 Edgar Morscher, ‘Bernard Bolzano’, in The Stanford Encyclopedia of Philosophy (Winter 2018 Edition), ed. E. Zalta, https://plato.stanford.edu/archives/win2018/ entries/Bolzano, sect. 3.4. 31 Cf. WL II, §255, 520, where it is pointed out that the Bolzanian subjects in the propositions expressed by ‘Every S is P’ and ‘Some S are P’ are ‘completely different’ because the former is [S] and the latter [the idea of an S that is P]. 32 Cf. WL I, §7, 23; §12, 47; §16, 61–4; WL II, §155, 128. For Bolzano’s theory of judgement cf. Mark Siebel, ‘Bolzanos Urteilslehre’, Archiv für Geschichte der Philosophie 86 (2004): 56–87; ‘Bolzano’s Theory of Judgment’, in The Act and Object of Judgment: Historical and Philosophical Perspectives, ed. B. Ball and C. Schuringa (London: Routledge, 2019). 33 WL II, §261, 532; cf. §265, 556–7. 34 WL I, §16, 63. 35 The most comprehensive study of deducibility is to be found in Mark Siebel, Der Begriff der Ableitbarkeit bei Bolzano (Sankt Augustin: Academia, 1996). For summaries cf. ‘Bolzano über Ableitbarkeit’, in Bernard Bolzanos geistiges Erbe für das 21. Jahrhundert, ed. E. Morscher (St. Augustin: Academia, 1999); ‘Bolzano’s Concept of Consequence’, The Monist 85 (2002): 580–99; ‘La notion bolzanienne de déductibilité’, Philosophiques 30 (2003): 171–89. Among other things, these publications contain a critical analysis of the claim that Bolzano’s definition is highly similar to Tarski’s definition of logical consequence. Cf. Alfred Tarski, ‘Über den Begriff der logischen Folgerung’, in Actes du Congrès International de Philosophie Scientifique 7 (Paris: Hermann & Cie, 1936), 1–11; transl., ‘On the Concept of Logical Consequence’, in Logic, Semantics, Metamathematics: Papers from 1923 to 1938, 2nd edn., ed. J. Corcoran, transl. J. H. Woodger (Indianapolis: Hacket, 1983). 36 Cf. WL II, §147, 80. 37 Cf. ibid., §155, 114. This is a simplified version. Bolzano defines deducibility for any number of conclusions, and he grants deducibility only if premises and conclusions are compatible, which means that there is a substitution of the variable ideas leading to true variants of all of them (see the title of §155; the theorems in subsections 4, 16, 19, 20, 22, 23 of §155; §248, 474–5). 38 WL II, §155, 128; cf. §268, 566. 39 WL I, §29, 141; cf. EG, 38. 40 Cf. Kant, Kritik der reinen Vernunft, B 744–5.

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41 Gilbert Ryle, The Concept of Mind (London: Hutchinson’s University Library, 1949), 122; cf. Stephen Toulmin, The Philosophy of Science (London: Hutchinson’s University Library, 1953), sect. 3.8. 42 WL II, §155, 128. Cf. also Bolzano’s eulogistic reference to Aristotle when he introduces the ‘relation of ground and consequence’, nowadays better known as ‘grounding’ (cf. WL II, §198, 341; WL IV, §525, 261–2). 43 Aristotle, Prior Analytics, 24b19–20. 44 WL II, §155, 129. 45 Cf. Oskar Becker, Untersuchungen über den Modalkalkül (Meisenheim am Glan: Anton Hain, 1952); Jaakko Hintikka, Time and Necessity. Studies in Aristotle’s Theory of Modality (Oxford: Clarendon Press, 1973). 46 WL II, §125, 7. 47 Immanuel Kant, Logik (Königsberg: Nicolovius, 1800); repr. in Gesammelte Schriften IX, ed. Königlich Preußische Akademie der Wissenschaften (Berlin, 1900–), 89; transl., Jäsche Logic, ed. and transl. J. M. Young, in Lectures on Logic (Cambridge: Cambridge University Press, 1992). 48 WL I, §9, 40; cf. Kant, Kritik der reinen Vernunft, B VIII; Logik, 20. 49 Cf. Bolzano’s early work Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (Prague: Caspar Widtmann, 1810; repr., Darmstadt: Wissenschaftliche Buchgesellschaft, 1974), §12, 63–7; transl., Contributions to a Better-Grounded Presentation of Mathematics, in The Mathematical Works of Bernard Bolzano, ed. and transl. S. Russ (Oxford: Oxford University Press, 2004). There, Bolzano argues that proofs giving the objective ground of a truth can be ultimately reduced to four basic inferences, namely Barbara and three non-syllogistic rules. 50 For a systematic reconstruction of this part cf. Casari, Bolzano’s Logical System, ch. 5. 51 WL II, §262, 534. 52 Cf. ibid., §262, 534; §265, 553–4. 53 Cf. ibid., §226, 406–7; §228, 416; §265, 554. 54 Cf. ibid., §228, 417. 55 Cf. ibid., §227, 415. 56 Cf. ibid., §232, 426–7. 57 Cf. ibid., §265, 559–60. 58 Cf. ibid., §238, 458. 59 Ibid., §155, 131. Cf. basic inference (b) in Bolzano, Beyträge, §12, 66. 60 Cf. ibid., §224, 396; §255, 520–1. 61 Cf. ibid., §155, 128, 130; §229, 419; §262, 541; WL IV, §683, 574. 62 WL II, §262, 541–2; cf. §255, 517; WL IV, §683, 575. In the German original, Bolzano erroneously offers as the third premise ‘Either B or C’. Rusnock and George have corrected this in their translation. 63 WL II, §263, 544. 64 Cf. ibid., §159, 159. 65 Cf. ibid., §155, 114; §225, 400–1, 405. 66 Cf. ibid., §159, 160; §230, 421–2; §234, 435–7; §257, 525–6. 67 Cf. ibid., §234, 436–7; §257, 526. 68 Cf. ibid., §233, 431. 69 Cf. ibid., §225, 401–2; §258, 526; §259, 527. 70 Ibid., §265, 558. 71 Šebestik argues that these moods are missing ‘because in Bolzanese several different forms of inferences merge’ (‘Bolzano’s Logic’, in The Stanford Encyclopedia of

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Philosophy (Fall 2017 Edition), ed. E. Zalta, https://plato.stanford.edu/archives/ fall2017/entries/bolzano-logic/, sect. 8.1). However, the missing moods and the non-weakened variants differ only in their conclusions, and translating the latter into Bolzano’s standard form ‘A has b’ does not result in identical formulations. 72 Cf. WL II, §227, 410, 412. 73 Cf. ibid., §227, 415. 74 For a list of resemblances and differences that takes into account not only Bolzano’s and Aristotle’s syllogistic but also the usual translations of categorical propositions into predicate logic cf. Morscher, ‘Bolzanos Syllogistik’: 449.

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George Boole and the ‘pure analysis’ of the syllogism David E. Dunning

1 Introduction In his 1847 Mathematical Analysis of Logic, the English schoolmaster and mathematician George Boole (1815–1864) translated traditional logic into mathematical notation and proceeded to elaborate upon the resulting algebraic reformulation.1 In doing so, he inspired a series of followers to develop related, mathematically oriented systems of logic. Boole is thus undeniably a pivotal figure in logic’s celebrated transformation from an ancient syllogistic tradition into a modern mathematical discipline. The value and ultimate impact of the particular pivot his work represents, however, have long been contested. Many commenters have seen Boole as an interesting but ultimately unproductive prelude to a story of modern logic that began in earnest only with Gottlob Frege in 1879.2 In 1977, James van Evra observed that ‘the exact nature of [Boole’s] importance remains elusive’.3 Van Evra sought to rehabilitate Boole in the face of common complaints that his use of mathematics was confusing and unsound.4 Also in Boole’s corner is Luis M. Laita, who proclaimed that ‘mathematical logic was an original creation of Boole’s genius’.5 Alongside invocations of Boole’s supposed role as father of modern logic, Boole’s leading biographer Desmond MacHale deemed him moreover the father of computer science.6 In an important shift towards interpreting Boole with respect to the tradition he was rooted in rather than developments he did not live to see, John Corcoran has emphasized the Aristotelian roots of Boole’s approach to logic. Corcoran ranked Boole’s sequel to the Mathematical Analysis, the Laws of Thought (1854) – alongside Aristotle’s Prior Analytics – as one of the two most important texts of pre-modern logic, crediting Boole with producing ‘the world’s first mathematical treatment of logic’ but also stressing that his system ‘does not fully merit being called a logic in the modern sense’.7 Weighing the various arguments for and against attributing the birth of modern logic to Boole, Volker Peckhaus has argued that modern logic’s lineages are irreducibly multiple; he concluded in well-reasoned exasperation, ‘We should therefore stop searching for the father of modern logic!’8 Once we have dispensed with the search for a solitary founder, the question of Boole’s significance remains. How exactly did Boole conceive of existing logical

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tradition, and how did he intervene? For all his novelty, Boole initially framed his logic around the analysis of traditional syllogistic forms, and he drew even more directly on established practices in the tradition of English symbolical algebra. Nor was he original in considering these enterprises together; as Marie-José Durand-Richard has shown, Boole engaged with long-running debates around the relationship between mathematics, logic and the use of abstract symbols.9 More generally, he contributed to an already vibrant English culture of symbolical algebra that saw its object of inquiry in abstract operations rather than in specifically quantitative values.10 Boole’s novelty was to construe the mental activities of logical inference as an example of precisely such operations, thereby rendering logical propositions fit for representation by the specific inscriptive techniques of mathematics. His basic project as a logician was to rewrite logic in the symbols of algebra and thereby subject it to rule-bound manipulation in the style of the science he called ‘pure analysis’. Much of the difficulty of evaluating Boole’s contribution comes down to the ambiguity of what exactly it means for ‘mathematical logic’ to be ‘mathematical’. One widespread usage reserves the label ‘mathematical logic’ for the tradition that takes logic as an epistemological and perhaps foundational resource for mainstream mathematics, in contrast to the ‘algebraic logic’ tradition to which Boole belonged.11 This perspective largely centres logicism, the doctrine that mathematics is derivable from a foundation of logical axioms alone, which motivated key figures in the ‘mathematical logic’ tradition such as Gottlob Frege and Bertrand Russell.12 This classification is convenient but can be misleading in that it makes little sense to claim that Boole did not do ‘mathematical logic’ because he failed to participate in a philosophical debate that took place after his death. Such anachronism is visible even in historically sensitive accounts, such as Sriram Nambiar’s insightful study of traditional logic’s influence on Boole, with its suggestion that Boole ‘is obscure on the relation between mathematics and logic’.13 Indeed if we assume that relationship is a matter of epistemological foundations, we will not find a developed position on the matter in Boole’s work. Boole, however, made logic mathematical in the most practical sense: he conducted logical deduction in existing mathematical symbolism. To construe ‘mathematical’ logic in this way might seem naively literal in contrast to the conceptual stakes of the logicist thesis. But while prevailing understandings of concepts can change over time, a borrowed inscriptive technique establishes a historical connection between two practices that holds independently of how those inscriptions were interpreted. Boole and those who followed him observably did the same kind of activity when they worked out a syllogistic conclusion as when they solved a system of algebraic equations (even if the rigor of that activity strikes today’s reader as dubious). And whatever one’s evaluation of the ultimate relevance of Boole’s logic to mathematics, it is clear that his nineteenth-century successors saw his logic as undoubtedly a kind of mathematical practice, as well as a novel and important system.14 At the turn of the twentieth century, mathematically inclined logicians still praised Boole as a pioneer in their field of study  – even those, like Giuseppe Peano, whom later logicians would elevate above Boole as having mathematized logic in a deeper and more productive way.15

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As Boole continued to work on his mathematical treatment of logic, he became increasingly confident that logic possessed more general foundations and wider applications than Aristotle had recognized. Thus, he deviated farther from traditional logic, but he did so in a register of additive progress rather than rupture. As John Corcoran has observed, ‘Boole never found fault with anything that Aristotle produced in logic  … Boole’s criticisms were all directed at what Aristotle did not produce’.16 Aristotle’s syllogism remained, for Boole, rightly admired even as he grew more confident that logic was due to advance beyond its historical origins; syllogism retained its place in Boole’s system even as the system expanded around it. Across successive attempts to present his logic, Boole modified his framing significantly while keeping the underlying practices stable. The central technique was to translate logical premises into algebraic symbolism, operate on them according to fixed laws and then translate the results of those operations back into ordinary language. In the Mathematical Analysis Boole had set out to demonstrate that logic belongs to mathematics by showing that these algebraic-logical practices work. He did not assume they would work more powerfully than the prevailing system of traditional logic, but in the course of writing the booklet he concluded this was the case, and he came to believe so with increasing confidence over the following years.

2 A novel analysis Boole’s first publication on logic was a work of tremendous originality; he insisted, however, that much of the system’s novelty had taken him by surprise. The Mathematical Analysis of Logic, published in 1847, was a slender, hastily composed demonstration that algebraic symbolism could be used to solve problems from traditional logic. Rather than setting out to advocate a radically new understanding of logic, Boole initially sought to defend a relatively new and still controversial conception of mathematics, which would be illustrated by a mathematical treatment of traditional logic. That this treatment shed new light on logic, he implied, was an unexpected benefit. The possibility of subjecting logic to mathematical analysis depended on how one defined mathematical analysis. The central principle of that science, according to Boole, was that ‘the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination’.17 He held that this idea – that the rule-bound manipulation of symbols possessed a validity independent of those symbols’ meaning – was the true essence of analysis. Analysts’ traditional preoccupation with functions of numbers was historically contingent: only by ‘accidental circumstances’ had it ‘happened in every known form of analysis, that the elements to be determined have been conceived as measurable by comparison with some fixed standard’.18 This historical accident put magnitudes and ratios in the foreground, but Boole insisted that ‘the history of pure Analysis is … too recent to permit us to set limits to the extent of its applications’.19 According to this view, the fact that analysis had developed in the context of quantitative investigations did not imply that such investigations were its only legitimate domain.

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By invoking the recent history of ‘pure Analysis’, Boole called attention to that term’s shifting meaning over the preceding two centuries. In the early seventeenth century, ‘analysis’ and the ‘analytical art’ were used more or less interchangeably with ‘algebra’ to denote, in the words of William Oughtred’s influential textbook, a technique ‘in which by taking the thing sought as knowne, we finde out that we seeke’.20 Gradually, however, analysis becomes associated with the techniques of differential and integral calculus. In the eighteenth century, the towering figure of Leonhard Euler (1707–1883) influentially positioned analysis as a major branch of mathematics consisting essentially in the study of functions.21 In keeping with Euler’s functionbased conception of analysis, Boole’s first logical work culminated in the treatment of what he called ‘elective functions’.22 But whereas Euler’s functions were necessarily numerical values dependent on other numerical variables, Boole’s elective functions involved symbols understood to have interpretations in the realm of logic. (He presented them as specifically logical in meaning, but by insisting that analysis was too new for its limits to be known, he indicated that logic was likely not the only possible extension.) The meaning of analysis had already expanded from the determination of unknown quantities to include the treatment of infinite and infinitely small quantities and ultimately the study of numerically valued functions as abstract objects. Boole saw no reason to assume that its proper scope might not be wider yet. As Durand-Richard has observed, this belief resembles opinions already articulated by the Cambridge mathematicians Robert Woodhouse (1773–1827) and George Peacock (1791–1858), though Boole did not cite them.23 Like other English algebraists in his milieu, Boole expected the domain of analysis to grow. By 1847 he had already earned acclaim for contributions to this capacious notion of mathematical analysis. In an 1844 paper ‘On a General method in Analysis’, awarded a gold medal in mathematics by the Royal Society, Boole sought to extend the methods of the so-called calculus of operations, which considered symbols according to laws of combination divorced from numerical meanings. Observing that certain methods therein were restricted to linear equations with constant coefficients, he proposed ‘to develope a method in analysis, which, while it operates with symbols apart from their subjects, and may thus be considered as a branch of the calculus of operations, is nevertheless free from [these] restrictions’.24 Soon, in logic, he found a specific non-quantitative domain into which the abstract calculus of operations might extend. ‘That to the existing forms of Analysis a quantitative interpretation is assigned’, he wrote in the Mathematical Analysis, ‘is the result of the circumstances by which those forms were determined, and is not to be construed into a universal condition of Analysis’.25 In traditional Aristotelian logic, Boole located a counterexample demonstrating the historical contingency of the numerical focus of existing analysis. In the course of treating traditional logic analytically, Boole claimed to have unintentionally stumbled upon a method of deduction more powerful than the Aristotelian syllogism. ‘The aim of these investigations was in the first instance confined to the expression of the received logic, and to the forms of the Aristotelian arrangement,’ he wrote, ‘but it soon became apparent that restrictions were thus introduced, which were purely arbitrary and had no foundation in the nature of

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things’.26 Having set out merely to do syllogistic logic in a new way, Boole became convinced that such a logic restricted the scope of its deductions unnecessarily. Thus, he adopted different attitudes towards different varieties of novelty. He confidently argued for the new conception of analysis as bearing no essential relation to number. With respect to logic, he enthusiastically pursued the notational and methodological novelty of expressing the received logic in a new way. But he relegated the doctrinal novelty of describing new forms of logical inference to the secondary status of an unintended finding. Boole’s care to avoid seeming to pursue novelty suggests that he worried his project would be construed as presumptuous. Assuming the posture of a reluctant innovator, he wrote that in the course of his work ‘it was found to be imperative to dismiss all regard for precedent and authority, and to interrogate the method itself for an expression of the just limits of its application’. Thus disowning the decision to dismiss precedent and authority with the passive voice, he reiterated that ‘there was no special effort to arrive at novel results’.27 This cautious tone softened the boldness of Boole’s daring to rewrite traditional logic. The Mathematical Analysis was indeed a rewriting in a literal sense. Boole claimed that by translating the problems of Aristotelian logic into mathematical symbolism, he could reliably find any possible conclusions: ‘The premises of a syllogism being expressed by equations, the elimination of a common symbol between them leads to a third equation which expresses the conclusion, this conclusion being always the most general possible.’28 Boole’s system was first and foremost a notation, a symbolic calculus consisting in rules of translation and manipulation. Making his case for the use of symbols led Boole to break from the traditional view that logic belonged to philosophy, reinforcing his position that logic was in fact a branch of mathematical analysis. Taking philosophy to be ‘the science of a real existence and the research of causes’, he concluded that ‘according to this view of the nature of Philosophy, Logic forms no part of it. On the principle of a true classification, we ought no longer to associate Logic and Metaphysics, but Logic and Mathematics’.29 He went on to elaborate that in the Mathematical Analysis his reader would find [l]ogic resting like Geometry upon axiomatic truths, and its theorems constructed upon that general doctrine of symbols, which constitutes the foundation of the recognised Analysis. In the Logic of Aristotle he will be led to view a collection of the formulæ of the science, expressed by another, but, (it is thought) less perfect scheme of symbols.30

Though Boole casually evoked the ‘recognised Analysis’ as if it were a stable entity, we have already seen that this term was steadily evolving. In one sense he returned to an older meaning, privileging symbolic method over the subject matter of numerical functions. But he also pushed analysis towards further evolution, allowing the values of symbols to encompass non-numerical logical relationships. By repurposing the notational techniques of analysis to represent traditional logic, he aimed first and foremost to change analysis, but he would not leave logic unchanged.

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3 Rewriting the syllogism Boole packaged the Mathematical Analysis as a kind of commentary on Aristotelian logic. Half of the booklet’s chapters began with concise, traditional statements of the material to be treated mathematically in the chapter. In each case, a few paragraphs, typographically distinguished from the rest of the chapter in the manner of an epigraph or abstract, introduced the relevant concepts. The first such text included a footnote explaining, ‘The above is taken, with little variation, from the Treatises of Aldrich and Whately,’ the reigning standard presentations of traditional logic at the time.31 Though Boole’s knowledge of Greek was strong, he chose these comparatively recent treatises as his primary representatives of logical tradition rather than engaging in detail with Aristotle’s writings directly.32 The body of each chapter then went on to treat that material in Boole’s algebraic symbolism. That symbolism revolved around lowercase variables like x, which ‘when operating upon any subject comprehending individuals or classes, shall be supposed to select from that subject all the Xs which it contains’.33 The universe of discourse was represented by 1, and the implied factor of 1 meant that a free-standing x selects all Xs in the universe. These ‘elective symbols’ could be combined according to arithmetic operations: xy selects all individuals that are both Xs and Ys, while x + y selects those that are Xs along with those that are Ys.34 The arrangement of beginning chapters with a traditional presentation before rewriting it algebraically recalled the format of a commentary on an ancient text, though Boole’s source text, so to speak, was a synthesis of traditional approaches rather than a reproduction of a specific existing text. Taking Aristotelian logic as an effectively unitary tradition, Boole overlooked the developments and variations that characterized the centurieslong story of the syllogism.35 But this simplification was a plausible approximation, and it served to frame the Mathematical Analysis as a translation and elaboration of an ancient tradition into algebraic form. One such traditional exposition opened the chapter that dealt most directly with Aristotle’s logic, titled ‘Of Syllogisms’. This introduction to the syllogism, with its terms, figures and moods, was followed by a body text that began by considering two propositions relating X to Y and Y to Z respectively. Boole observed that in his notation these yield two equations which both include y: If from two such equations we eliminate y, the result, if it do not vanish, will be an equation between x and z, and will be interpretable into a Proposition concerning the classes X and Z. And it will then constitute the third member, or Conclusion, of a Syllogism, of which the two given Propositions are the premises.36

A syllogism was thus recast as a system of two equations in three variables. Representing two arbitrary such equations by ay + b = 0 and a'y + b' = 0, Boole stated the elementary algebraic result that the elimination of y could be represented by ab' – a'b = 0. A few pages later Boole suggested rearranging these equations ‘so that y shall appear only as a factor of one member in the first equation, and only as a factor of the opposite member in the second equation’, yielding ay = – b and b' = – a'y. He then instructed the reader

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to ‘multiply the equations, omitting the y’.37 Omitting the y here does not correspond to a mathematical operation; rather Boole has rearranged the relevant factors such that the product of the left sides set equal to the product of the right sides (ignoring y in both cases) constitutes an algorithmic recipe for the expression ab' = a'b, a simple rearrangement of the desired general elimination result. This resulting equation could then be interpreted back into a prose proposition: the syllogism’s conclusion.38 The bulk of the chapter consisted in Boole using this method to derive example syllogisms (one of which is discussed in detail below). The chapter’s central argument was that this equational form represented a more fundamental structure of the reasoning in question than did the usual syllogistic form. Boole contended that ‘the essential structure of a Syllogism is, in some measure, arbitrary’.39 He advanced this claim on two levels: he highlighted the prominence of arrangements that he could uncontroversially deem conventional, and he argued that his method revealed certain forms of valid conclusion that the syllogistic theory arbitrarily excluded. First, to exhibit the conventional nature of syllogistic arrangements, he reflected on the generic pair of a major and a minor term as determined by the order in which the premises are given. He observed: [I]t is purely a matter of choice which of the two shall have precedence in the Conclusions. Logicians have settled this question in favour of the minor term, but it is clear, that this is a convention. … Convenience is perhaps in favour of the adopted arrangement, but it is to be remembered that it is merely an arrangement.40

That important features of the syllogistic form were matters of convention did not render them any less valid, but it did raise problems for the view that privileged syllogism over any alternative representation of inference. The syllogism’s arbitrary restrictions did not end there, according to Boole; more serious than arbitrary conventions of ordering was the exclusion of certain valid inferences altogether. ‘The Aristotelian canons’, he wrote, ‘beside restricting the order of the terms of a conclusion, limit their nature also – and this limitation is of more consequence than the former’.41 In contrast, Boole claimed, his system would impose no such limits. He promised to locate the traditional system within his own but saw little reason to use it: ‘We may by restricting the canon of interpretation confine our expressed results within the limits of scholastic logic; but this would only be to restrict ourselves to the use of a part of the conclusions to which our analysis entitles us.’42 The syllogism chapter’s task would be to determine the scope of unrestricted analysis of syllogistic premises. Boole’s expansion of syllogistic reasoning hinged on the fact that in Aristotelian syllogism, subjects are positive: we might conclude all or some Xs are or are not Ys; we cannot conclude something like ‘Some not-Xs are not Ys’ in which the subject is the negation of a subject X that appeared in a premise. ‘Yet there are cases in which such inferences may lawfully be drawn,’ Boole wrote, ‘and in unrestricted argument they are of frequent occurrence’.43 He proceeded to work through several sample syllogisms, several of them not included under the Aristotelian umbrella.

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The following example exhibits his method in general as well as its claim to go beyond Aristotle: The following, EE, Fig. 1, … is an example of a lawful case not determinable by the Aristotelian Rules. No Ys are Xs, xy = 0, 0 = xy No Zs are Ys, zy = 0, y = v(1 – z) 0 = v(1 – z)x ∴ Some not-Zs are not Xs.44

In keeping with the procedure described above, Boole translated the traditional premises in the first column into the mathematical equations of the second and then in the third column rearranged those equations into the forms ay = – b and b' = – a'y where (in keeping with another rule we need not discuss here) at least one of b or b' is nonzero. Multiplying these equations and omitting y, he arrived at 0 = v(1 – z)x. The factor v(1 – z) represents ‘some not-Zs’; multiplication by x selects the members of this class that are also Xs; this product being equal to 0 indicates that all of these particular not-Zs are in fact not Xs. Thus, Boole concluded, some not-Zs are not Xs. This inference, which Boole asserted was both valid and syllogistic, did not belong to traditional logic. Under the traditional classification, the premises are EE; that is, both are universal and negative. A syllogism’s figure specifies the relationship between the conclusion’s subject (known as the minor term), its predicate (the major term) and the term that appears in both premises but not in the conclusion (the middle term). A syllogism is said to be in the first figure when its major term is predicated of its middle term, and its middle term is predicated of its minor term. To label this example as a syllogism in the first figure is thus to assert that the not-Zs of the conclusion constitute ‘the same’ subject as the Zs, despite the change of quality – an identity that Aristotelian logic does not grant. Because these premises justify no conclusion about Zs in the affirmative, in the Aristotelian scheme they do not license a valid syllogism. But, Boole argued, they do justify a conclusion about not-Zs.45 Sitting in the middle of the short book, the syllogism chapter functioned as a hinge. The preceding chapters built directly to it, developing the techniques – all mathematical reworkings of traditional processes – on which it depended. The following chapters then looked beyond the syllogism, briefly considering an alternative interpretation of Boole’s algebraic logic in terms of hypotheticals, and finally developing a theory of arbitrary functions (x) of his elective symbols. In the Mathematical Analysis, Boole claimed to have gone beyond Aristotle’s syllogism, but only incrementally. He emphasized the novelty of his symbolic technique for handling the material of traditional logic in a new way. One advantage he claimed for his method was that it uncovered a bit of new ground, but its value did not lie exclusively or even primarily in performing a few new forms of inference. Having formulated this initial presentation of his method, however, Boole continued to tinker with his logic. He became increasingly convinced of its importance and diverged correspondingly farther from syllogistic tradition.

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4 Rewriting the rewriting Boole wrote the Mathematical Analysis ‘within a few weeks’ and seems to have grown dissatisfied with it almost immediately upon publication. By 1851 he described it as ‘a hasty and (for this reason) regretted publication’.46 Some of his earliest efforts to rethink the project are preserved in an interleaved copy of the text in which he responded to his own work on blank facing pages.47 On the leaves opposite the pages of the syllogism chapter, he worked out extensive new treatments that leaned more heavily on the algebraic manipulation of symbols. Opposite the fourth page of the syllogism chapter, Boole penned the heading ‘General Laws of Syllogism’. Over the next fourteen blank leaves he elaborated an approach purged of references to Aristotelian figures or traditional mnemonics. Instead, this version of the method operated entirely in algebraic forms. He defined the domain of inquiry as ‘combinations of premises … the separate propositions of which can be referred to any of the following forms at the same time that they involve in their expression a common symbol y​ ’.48 As basic forms he gave the classic four categorical propositions along with four variations obtainable by allowing negative subjects. He listed these eight propositions and translated each into his notation:49 All Zs are Xs No Zs are Xs Some Zs are Xs Some Zs are not-Xs All not Zs are Xs All not Zs are not-Xs Some not Zs are Xs Some not Zs are not-Xs

z = vx z = v(1 – x) vz = v'x vz = v'(1 – x) 1 – z = vx 1 – z = v(1 – x) v(1 – z) = v'x v(1 – z) = v'(1 – x)

As we have seen, in the Mathematical Analysis he argued for the inclusion of negative subjects along with the traditional Aristotelian forms and comfortably worked with them in his examples. In this annotation he incorporated them more systematically, producing up front a uniform list of basic propositions that assumed the inclusion of negative subjects without comment. Boole’s list of eight basic propositions showed a user how to translate any two premises of a possible syllogism into mathematical form. He proceeded to discuss various possible cases, operating throughout with reference to their new mathematical forms rather than to their traditional equivalents. As he waded into the technical weeds of the analysis, he created a striking visual contrast with his original published presentation (Figure 4.1). We see how on this page of the Mathematical Analysis, Boole organized numerous cases by pairs of vowels, the traditional labels for the four Aristotelian categorical propositions. Even when identifying conclusions not determinable by the Aristotelian rules, he remained within the Aristotelian notational framework: his mathematical notation entered only momentarily to illustrate the derivation of a conclusion framed by prose propositions, vowel labels and references to Aristotelian figures. On the blank leaf opposite, however, Boole departed from traditional classifications entirely, working

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Figure 4.1  George Boole’s development of a new mathematical treatment of the syllogism, opposite the version he published in 1847. MS/782 The Mathematical Analysis of Logic, 1847, page 37. ©The Royal Society

exclusively in his extensive algebraic derivations. Manipulating lengthy equations according to his symbols’ rules, he arrived at formulations he deemed ‘symmetrical + elegant’. In the contrast between many-termed mathematical expressions on the left and copious traditional vowel labels on the right, this spread neatly visualizes Boole’s transition from performing a mathematical exegesis of Aristotelian logic to presenting an autonomous mathematical theory of his own. After several such pages of dense calculations, Boole tidied up his results under the heading ‘Recapitulation’, offering six long algebraic equations (Figure  4.2). Dividing all premises into two cases (those with like or unlike middle terms), Boole gave expressions for x, vx and 1 – x (that is, all Xs, some Xs, and all not-Xs) in each case. From these algebraic expressions he extracted three rules of inferences that he stated in plain language. In the case of like middle terms, on the condition that (at least) one of them be universal, a conclusion can be reached by equating the extreme terms. In the case of unlike middle terms, if one extreme term was universal, Boole instructed the reader, ‘Change the quantity and quality of that extreme and equate the result to the other extreme.’ Or, if both middle terms were universal, he instructed, ‘Change the quantity and quality of either extreme and equate the result with the other extreme.’ He argued that ‘[a]ll the cases of Syllogism are … included in the three rules above given’.50 In Boole’s future writing on logic, these algebraically derived rules replaced the treatment in terms of elimination he had given in the Mathematical Analysis. As Boole continued to develop the mathematically oriented logic introduced in the Mathematical Analysis, he became increasingly committed to letting mathematics, rather than tradition, structure his presentation. But this commitment did prevent him from exploring the possibility of approaching logic in other ways for other audiences; he simply held the mathematical view to be fundamental. In a manuscript thought to date from approximately the same period, Boole began to draft an ‘Elementary Treatise on Logic not mathematical including philosophy of mathematical reasoning’.51 His

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Figure  4.2  Boole’s mathematical expressions for possible syllogistic inferences. MS/782 The Mathematical Analysis of Logic, 1847, page 42. ©The Royal Society

plan in this unfinished work was ‘to speak of Logic chiefly as an Art’, noting that the corresponding science of logic ‘is a branch of the larger science of Reasoning by Signs, another form of which is exhibited in ordinary mathematics’.52 Most of Boole’s writing on logic focused on this underlying science rather than the art of applying it. Given its

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mathematical nature, however, he recognized that some readers would benefit from a less technical presentation. ‘What I here mean to exhibit is so much of the rules and forms of Logic as can conveniently be exhibited without any exact inquiry into first principles,’ he explained.53 Even in the context of an accessible, non-mathematical treatise, Boole by now did not hesitate to show his disdain for traditional conventions: ‘I shall without scruple disregard those technical forms and overstep those limitations which have [been] imposed upon the Art of Logic by almost every writer from the days of Aristotle to the present time without at the same time possessing any real foundation in Nature.’54 As he inched towards a more iconoclastic view of logic, he could articulate his critique of tradition independently of that critique’s underlying mathematical perspective. This manuscript included a discussion of syllogistic inference that indeed disregarded traditional technical forms while also avoiding mathematical formulations and techniques. The closest he came to explicitly using mathematical practices here was to invoke the notion of equality. He explained, ‘I shall speak of classes or parts of classes as equal when the members of one are members of the other. In this sense of the word equal, as well as in its mathematical sense it is an evident axiom that Things which equal to the same thing are equal to each other.’55 Operating entirely in prose, frequently with concrete example premises, other times with traditional capital-letter variables for classes, Boole developed and stated the same schematic rules he had derived algebraically in the interleaved copy of the Mathematical Analysis; unlike that presentation, here he gave no indication of the rules’ mathematical origins.56 Boole still entertained the usefulness of a non-mathematical syllogistic technique, but he abandoned much of the Aristotelian framing in favour of practices based directly on his mathematical analysis. Archival materials from the years immediately following the publication of the Mathematical Analysis show that as Boole reflected on his system, he increasingly allowed it to stand on its own terms. In his interleaved copy of the Mathematical Analysis, he recorded a commentary on his own text – a text that was, as suggested above, itself a commentary on traditional logic. The rules of syllogistic inference he developed there emerged from complicated algebraic computation, though they were ultimately expressible in prose. Whether appearing in their native mathematical context or restated in a draft of an ostensibly non-mathematical treatise, these new rules decisively replaced traditional Aristotelian logic in Boole’s eyes.

5 Tradition reconsidered In 1854, Boole published An Investigation into the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities. Though he would continue to mull over questions of logic and its philosophy in the following years, he never completed a more mature presentation of his theory and Laws of Thought remains the classic presentation of his logic. As previous readers have observed, Aristotle’s footprint was much smaller in this work than in the Mathematical Analysis.57 To some extent this change was epiphenomenal to Boole’s inclusion of a wider range

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of topics. Most substantially, he devoted well over a hundred pages in Laws of Thought to explaining how his logic could be applied to the theory of probability. Both books included exactly one chapter directly discussing the Aristotelian syllogism, but Laws of Thought comprised nearly three times as many chapters. As Boole continued to study logic, his confidence in his system’s range of applicability grew, and it followed that the syllogism’s relative share decreased as he covered a wider range of questions. Beyond this effect of a widening perspective, however, Boole’s regard for Aristotle’s logic had decreased in absolute terms as well. The longer he worked with his own algebraic system, the more critically he viewed syllogistic logic. Aristotle’s diminished standing in the Laws of Thought was evident from the outset. In the first chapter Boole confronted the question of Aristotelian logic head on, acknowledging that some readers might believe Aristotle’s system encompassed all reasoning with sufficient generality. Again he adopted a posture of wishing to avoid controversy, but with a new degree of confidence in his approach and its claims to greater scope than Aristotle’s. ‘I have no desire to point out the defects of the common Logic,’ he wrote, ‘nor do I wish to refer to it any further than is necessary, in order to place in its true light the nature of the present treatise’.58 To this end, he asserted that ‘syllogism, conversion, &c., are not the ultimate processes of logic. It will be shown in this treatise that they are founded upon, and are resolvable into, ulterior and more simple processes which constitute the real elements of method in Logic’.59 Moreover, he maintained that even if this were not the case, ‘there would still exist the same necessity for a general method’.60 Thus, he largely set Aristotle aside for the next two hundred pages, developing his general method based on the rule-bound manipulation of algebraic symbols. The system of the Laws of Thought retained the basic rules of the Mathematical Analysis, though the presentation was much changed by Boole’s decreased reverence for Aristotle. In addition to providing clearer exposition in general, in the later work Boole presented his logic in a new, autonomous manner – now largely detached from syllogistic precedent. Only after extensively developing his algebraic system did Boole eventually return to the question of traditional forms. He noted that Aristotle’s system ‘occupies so important a place in academical education, that some account of its nature … seems to be called for in the present work’.61 He thus implied that he had no choice but to assess the relative merits of his system and Aristotle’s, and insisted he did so ‘in no narrow or harshly critical spirit’.62 As in the Mathematical Analysis, Boole projected discomfort at finding himself in the position of criticizing Aristotle. But after suggesting his reluctance, he proceeded to articulate a much stronger diagnosis of the shortcomings of traditional logic. The mathematical substance of the syllogism chapter closely followed what Boole had sketched in his interleaved copy of the Mathematical Analysis as described above. He presented systems of equations and generated the same schematic rules for several cases of syllogistic premises. He described his rules on their own terms without relying on traditional vowel labels or mnemonic verses. Ultimately, while still striking a respectful tone, by 1854 Boole had come to see only historical value in Aristotelian logic. As before, all the shortcomings Boole identified took the form of limitations. He never suggested the Aristotelian system contained errors, nor even that it necessarily fell short of perfection with respect to its aims;

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rather he argued that those aims involved settling for arbitrarily narrow inferential power. He concluded that ‘the scholastic logic … is not a science, but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest’.63 Aristotle’s logic consisted of scientific truths, but they were neither comprehensive nor fundamental. At the book’s outset Boole had indicated that logic would be scientific insofar as it could be made to ‘rest on observation’ and ‘determine the laws’ of the mind, and he defined ‘fundamental’ as ‘those laws and principles from which all other general truths of science may be deduced, and into which they may all be again resolved’.64 Thus, in Boole’s evaluation Aristotle had succeeded in describing real laws of thought, but not all such laws, and those he discovered were insufficient to deduce the rest. To Boole, the incompleteness of Aristotle’s logic undermined its value as a classification of inferences, and its non-fundamental status meant that future systems would do well to seek alternative foundations rather than building upon it. Boole allowed that traditional logic’s legacy was sufficiently great that it ‘cannot be altogether unworthy of attention’, but he signalled scepticism as to whether it merited close study in most cases: But whether the mnemonic forms, in which the particular rules of conversion and syllogism have been exhibited, possess any real utility – whether the very skill which they are supposed to impart might not, with greater advantage to the mental powers, be acquired by the unassisted efforts of a mind left to its own resources – are questions which it might still be not unprofitable to examine.65

Perhaps there was a chance that, upon examination, the particulars of the traditional syllogism would be found to offer something that an unassisted mind could not construct just as profitably on its own – but Boole would not be the one to examine this question.

6 Conclusion Boole lived only another decade after the Laws of Thought appeared in print. His death at the age of forty-nine in 1864 was unexpected, and he might reasonably have expected to have many years remaining to perfect his system.66 He frequently set out to reframe his logic, trying out new approaches to the conceptual presentation. As he described this process to his fellow mathematician and logician Augustus De Morgan (1806–1871) in 1859, ‘I have written at different times as much as would make two or three books but when returning to a subject I can seldom make much use of old materials. They have lost their freshness & I can only begin again de novo.’67 In particular, after publishing Laws of Thought he tried again to offer a non-mathematical version that would be accessible to a wider audience. He called this project ‘The Philosophy of Logic’ and wrote that whereas Laws of Thought had been intended for mathematically trained readers, this work would serve a general audience. ‘Mathematics will not appear except in the notes,’ he promised.68

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He believed non-mathematical accessibility was desirable, but he seems never to have doubted the soundness of the underlying mathematical system’s notation and rules. The stable core of Boole’s logic was a set of practices: translating premises into algebraic symbolism, solving equations and translating the solution back into ordinary language. He was never satisfied for long with any philosophical packaging: the symbolic practices were the sturdy foundation upon which he continually built and rebuilt philosophical structures. As Boole studied logic, he gradually but steadily came to see less and less value in the Aristotelian tradition he had initially taken to be a satisfactory theory of reasoning. But he never deemed traditional logic wrong or worthless, merely incomplete; it was time to progress beyond it, he believed, and the way to do so was to recognize logic as properly subject to mathematical analysis. At first he applied mathematical techniques somewhat cautiously, hedging his claims to novelty and allowing a traditional approach to guide his exposition in the Mathematical Analysis. A system of equations could serve as an illuminating gloss on Aristotelian syllogisms, thereby illustrating that mathematical analysis was not confined to the realm of magnitude. When Boole revisited this work after publication, he developed a more thoroughgoing algebraic approach, abandoning much of the traditional framework and devoting far less attention to syllogism in the first place. By rewriting Aristotelian logic in mathematical notation, he had convinced himself that logic warranted more thorough reimagining to encompass a wider range of allowable inferences and deploy a new set of syllogistic rules. Boole began his treatment of logic deeply rooted in tradition but moved decisively away from it over time. His trajectory shows that treating logic mathematically was not a stark alternative to syllogistic logic but rather a plausible and fertile development of it in the context of Boole’s symbolically inclined mathematical culture. Boole developed inscriptive techniques based on the well-established practices of mathematics, and then followed those techniques when they led beyond the scope of traditional logic. He understood these practices to be manifestly mathematical and thus to prove that mathematics was not restricted to the study of quantity. By performing the deductions of traditional logic with his symbolic techniques, he put forward a capacious understanding of mathematical analysis and a mathematical understanding of logic.

Notes 1 2

3

George Boole, The Mathematical Analysis of Logic, Being an Essays towards a Calculus of Deductive Reasoning (Cambridge: Macmillan, Barclay, & Macmillan, 1847). According to Jean van Heijenoort, for example, Boole ‘tried to copy mathematics too closely, and often artificially’ and his era ‘would not count as a great epoch’ in the history of logic. Jean van Heijenoort, ‘Preface’, in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, ed. Jean van Heijenoort (Cambridge: Harvard University Press, 1967), vi–viii, at vi. James W. van Evra, ‘A Reassessment of George Boole’s Theory of Logic’, Notre Dame Journal of Formal Logic 18, no. 3 (1977): 363–77, at 363.

88 4

5 6 7

8 9 10

11 12

13 14

Aristotle’s Syllogism The charge of mathematical obscurity appeared as early as W. Stanley Jevons, Pure Logic or the Logic of Quality apart from Quantity: With Remark’s on Boole’s System and on the Relation of Logic and Mathematics (London: Edward Stanford, 1864). Influential mid-twentieth-century histories of logic emphasized the ‘defects of rigour’ and ‘shortcomings’ in Boole’s system, William Kneale and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), 422; see also I. M. Bocheński, A History of Formal Logic, trans. Ivo Thomas (Notre Dame: University of Notre Dame Press, 1961). Luis M. Laita, ‘Influences on Boole’s logic: The Controversy between William Hamilton and Augustus De Morgan’, Annals of Science 36, no. 1 (1979): 45–65, at 65. Desmond MacHale, The Life and Work of George Boole: A Prelude to the Digital Age, new edn. (Cork: Cork University Press, 2014), 82. John Corcoran, ‘Aristotle’s Prior Analytics and Boole’s Laws of Thought’, History and Philosophy of Logic 24, no. 4 (2003): 261–88, at 261; George Boole, An Investigation into the Laws of Thought on which Are Founded the Mathematical Theories of Logic and Probabilities (New York: Dover Publications, Inc., 1958). Volker Peckhaus, ‘Was George Boole Really the “Father” of Modern Logic?’, in A Boole Anthology: Recent and Classical Studies in the Logic of George Boole, ed. James Gasser (Dordrecht: Kluwer Academic Publishers, 2000), 271–85, at 282. Marie-José Durand-Richard, ‘Logic versus Algebra: English Debates and Boole’s Mediation’, in Gasser, Boole Anthology, 139–66. On Boole’s place in his mathematical context, see Maria Panteki, ‘The Mathematical Background of George Boole’s Mathematical Analysis of Logic (1847)’, in Gasser, Boole Anthology, 167–212 and Durand-Richard, ‘Logic versus Algebra’. On English symbolical algebra more generally, see Elaine Koppelman, ‘The Calculus of Operations and the Rise of Abstract Algebra’, Archive for History of Exact Sciences 8, no. 3 (1971): 155–242; J. Richards, ‘The Art and the Science of British Algebra: A Study in the Perception of Mathematical Truth’, Historia Mathematica 7 (1980): 343–65; Helena M. Pycior, ‘George Peacock and the British Origins of Symbolical Algebra’, Historia Mathematica 8, no. 1 (1981): 23–45; Kevin Lambert, ‘A Natural History of Mathematics: George Peacock and the Making of English Algebra’, Isis 104, no. 2 (2013): 278–302. I. Grattan-Guinness, ‘Living Together and Living Apart. On the Interactions between Mathematics and Logics from the French Revolution to the First World War’, South African Journal of Philosophy 7 (1988): 73–82. The standard history of the complex mathematical developments around logicism and its alternatives is I. Grattan-Guinness, The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton: Princeton University Press, 2000). Sriram Nambiar, ‘The Influence of Aristotelian Logic on Boole’s Philosophy of Logic: The Reduction of Hypotheticals to Categoricals’, in Gasser, Boole Anthology, 217–39, 221. Ernst Schröder held up Boole’s calculus as a superior alternative to Frege’s system in his review of Gottlob Frege, Begriffsschrift, eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle: Louis Nebert, 1879) in the Zeitschrift für Mathematik und Physik 25. Historical Literary Section (1880): 81–94. Christine Ladd acknowledged ‘five algebras of logic, – those of Boole, Jevons, Schroder, McColl, and Peirce’, and deemed ‘the later ones … all modifications, more or less slight, of that of Boole’. Christine Ladd, ‘On the Algebra of Logic’, in Studies in Logic by the Members of the Johns Hopkins University (Boston: Little, Brown, and Company,

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1883), 17–71, at 17. Logicians who were less enthusiastic about Boole’s reliance on difficult mathematics nonetheless agreed that his logic was on the one hand highly mathematical and on the other hand very important. William Stanley Jevons developed an alternative notation that he claimed gave ‘the same formal results’ but ‘with self-evident force and meaning’ in contrast to Boole’s ‘dark and symbolic processes’. Jevons, Pure Logic, 75. John Venn, though less critical than Jevons, hoped that ‘the present accidental dependence of the Symbolic Logic upon Mathematics’ could be eliminated in the future, while conceding that ‘at present it is almost inevitable’ given the mathematical character of Boole’s work. John Venn, Symbolic Logic (London: Macmillan and Co., 1881), ix. 15 In an early foray into mathematical logic, Peano credited Leibniz with having ‘stated some analogies between the operations of algebra and those of logic’, but held that ‘only in this century, in the works of Boole, Schröder, and many others, were these relations studied, so that deductive logic has become … part of the calculus of operations’. Giuseppe Peano, ‘Principii di Logica Matematica’, Rivista di Matematica 1 (1891): 1–10, at 1 (my translation). For an influential statement of the classification that strongly distinguishes Peano’s tradition from Boole’s, see Jean van Heijenoort, ‘Logic as Calculus and Logic as Language’, Synthese 17, no. 3 (1967): 324–30. Also influential is the less value-laden and more nuanced, though still firm in its bifurcation, Grattan-Guinness, ‘Living Together and Living Apart’, especially the schematic outline at 78. But as Peckhaus has shown, ‘the modern distinction between certain currents or traditions in the history of logic played no significant role’ in the institutional establishment of symbolic logic as a discipline. Peckhaus, ‘Was George Boole Really the “Father” of Modern Logic?’ 281. 16 John Corcoran, ‘Aristotle’s Prior Analytics and Boole’s Laws of Thought’, History and Philosophy of Logic 24, no. 4 (2003): 261–88, at 265. 17 Boole, Mathematical Analysis of Logic, 3. 18 Ibid. 19 Ibid., 4. 20 William Oughtred, The Key of the Mathematicks New Forged and Filed (London: Printed by Tho. Harper, for Rich. Whitaker, 1647), 4. 21 Among Euler’s vast publications, particularly relevant is Leonhard Euler, Introductio in analysin infinitorum (1748), in Opera omnia, Series 1, vol. 8–9. See Giovanni Ferraro, ‘Euler and the Structure of Mathematics’, Historia Mathematica 50 (2020): 2–24. 22 Boole, Mathematical Analysis of Logic, 60–81. 23 Durand-Richard, ‘Logic versus Algebra’, 159. 24 George Boole, ‘On a General Method in Analysis’, Philosophical Transactions 134 (1844): 225–82, at 226. On Boole’s gold medal, see MacHale, Life and Work of George Boole, 69–71. 25 Boole, Mathematical Analysis of Logic, 4. On the connections between Boole’s earlier mathematical work and his turn to logic, see Luis M. Laita, ‘The Influence of Boole’s Search for a Universal Method in Analysis on the Creation of His Logic’, Annals of Science 34, no. 2 (1977): 163–76 and Panteki, ‘Mathematical Background’. 26 Boole, Mathematical Analysis of Logic, 7–8. 27 Ibid., 8. 28 Ibid. 29 Ibid., 12–13; emphasis in original. 30 Ibid., 13; emphasis in original.

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31 Ibid., 20 n. Aldrich and Whately’s views and legacies are discussed in Chapter 1 of this book, written by Calvin Jongsma. 32 On Boole’s talent for Greek, see MacHale, Life and Work of George Boole, 8–18. 33 Boole, Mathematical Analysis of Logic, 15. 34 One oft-criticized idiosyncrasy of Boole’s system is that x + y is only a valid expression when x and y are disjoint. As Nambiar has argued, however, the common assertion that Boole’s addition is exhaustive disjunction is incorrect, for it is an operation on classes rather than a propositional connective. Nambiar, ‘The Influence of Aristotelian Logic on Boole’s Philosophy of Logic’, 228–31. 35 See Marco Sgarbi and Matteo Cosci, (eds.), The Aftermath of Syllogism: Aristotelian Logical Argument from Avicenna to Hegel (London: Bloomsbury Academic, 2018). 36 Boole, Mathematical Analysis of Logic, 32. 37 Ibid., 34. 38 My aim here is to offer a charitable interpretation of Boole’s project as he understood it; I do not argue for the validity of these methods. For a detailed discussion of Boole’s shortcomings from a modern point of view, see John Corcoran and Susan Wood, ‘Boole’s Criteria for Validity and Invalidity’, Notre Dame Journal of Formal Logic 21, no. 4 (October 1980): 609–38. 39 Boole, Mathematical Analysis of Logic, 33. 40 Ibid., 33; emphasis in the original. 41 Ibid.; emphasis in the original. 42 Ibid., 34. 43 Ibid. 44 Ibid., 36. 45 For Boole’s conclusion here to be valid, one must also presuppose existence of terms; if it is allowed that there might be no Ys, this syllogism commits the existential fallacy and is invalid. 46 Boole, Laws of Thought, preface; George Boole, ‘On the Theory of Probabilities, and in Particular on Mitchell’s Problem of the Distribution of the Fixed Stars’, Philosophical Magazine series 4, no. 7 (1851): 521–30, at 525. 47 Notebook S7 Logic, Additional Box 3, George Boole Papers, MS 782, Library of the Royal Society, London, United Kingdom (hereafter ‘Interleaved MAL’). The notebook is marked ‘George Boole Lincoln Feb 7th 1848’ on the first blank page, probably the date of acquisition rather than completion. G. C. Smith, ‘Boole’s Annotations on “The Mathematical Analysis of Logic”’, History and Philosophy of Logic 4, no. 1 (1983): 27–39 describes this document and another interleaved copy, privately held, with fewer annotations, but essentially identical to the other copy in those that it does contain. 48 Boole, Interleaved MAL, opposite 34. 49 Ibid. 50 Ibid., opposite 44–6. 51 George Boole, ‘Elementary Treatise on Logic Not Mathematical Including Philosophy of Mathematical Reasoning’, in id., Selected Manuscripts on Logic and Its Philosophy, ed. Ivor Grattan-Guinness and Gérard Bornet (Basel: Birkhäuser, 1997), 13–41. For notes on dating the manuscript, see Boole, Selected Manuscripts, 208. 52 Boole, ‘Elementary Treatise on Logic Not Mathematical’, 13. 53 Ibid., 14. 54 Ibid. 55 Ibid., 25.

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56 Ibid., 27. 57 Ivor Grattan-Guinness, ‘George Boole, An investigation of the laws of thought on which are founded the mathematical theory of logic and probabilities (1854)’, in Landmark Writings in Western Mathematics 1640–1940, ed. I. Grattan-Guinness (Amsterdam: Elsevier Science, 2005), 470–9, at 474; Corcoran, ‘Aristotle’s Prior Analytics and Boole’s Laws of Thought’, 264. 58 Boole, Laws of Thought, 10. 59 Ibid., 10. 60 Ibid. 61 Ibid., 226. 62 Ibid. 63 Ibid., 241. 64 Ibid., 3; 5. 65 Ibid., 242. 66 There has been some controversy around Boole’s fatal illness and treatment; see Desmond MacHale and Yvonne Cohen, New Light on George Boole (Cork: Atrium, 2018), 331–42. 67 George Boole to Augustus De Morgan, 21 March 1859, in G. C. Smith, The Boole–De Morgan Correspondence 1842–1864 (Oxford: Clarendon Press, 1982), 77. 68 Boole, Selected Manuscripts, 117–53, at 120.

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Logic of relations by De Morgan and Peirce: A case study for the refinement of syllogism Sun-Joo Shin

1 Introduction – On the bridge The mystery of the status of mathematical truth led Kant to an even more mysterious category, the synthetic a priori. First, the source of mathematical truth does not lie in contingent aspects of the world, but a priori. However, there is a major difference in the way a prioricity is obtained between an a priori sentence ‘All bachelors are unmarried’ and ‘The sum of the interior angles of a triangle is 180 degrees.’ Linguistic meaning guarantees the a priori true status of ‘All bachelors are unmarried’, hence, its analyticity. The meaning of ‘triangle’ does not seem to contain ‘180 degrees’. The demarcation between logical truth and mathematical truth was drawn in Kant’s book, and the synthetic a priori category was born. Kant’s long journey to transcendental idealism started from there. Commentators would jump into the discussion and point out that the meaning or at least the extension of logic Kant adopted is different from modern logic. Kant’s logic, as he himself made it clear, is Aristotelian logic. Hence, Kant’s synthetic a priori category, some claim, originates from his narrow concept or extension of logic. When logic is extended, mathematical proofs turn out to be analytic as well. How is logic extended? This question invites the history of modern logic to step in, and the chapter aims to highlight one tiny, but interesting, aspect of this complicated history. The extension of logic is composed of two different, but closely related, dimensions – territory (what to cover) and notation (how to express). In some sense it sounds inevitable that a change in substance demands a change in form. However, there is plenty of room to take the matter the other way around: New notational systems, for example, quantifiers and variables, empower logic to venture out to a new territory, that is, to relations. Or, modern quantification logic could be understood as a leap from Aristotelian logic both in terms of what and how. Taking the birth of modern logic to stem from multiple sources, the chapter does not subscribe to any specific view and does not deal with the delicate interaction between substance and form, either. On the other hand, I do believe that one way to get to modern logic is through Aristotelian syllogism. What I mean by ‘through syllogism’ is a rather concrete and intimate process from syllogism to modern logic. Instead of

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discarding syllogism, some philosophers not only embraced syllogism but also took it as a guide to an adventure. Zeroing in on the spot where syllogism and modern logic meet, I picture two bridges. On one bridge I place Auguste De Morgan and Charles S. Peirce, and on the other George Boole and (again) Charles S. Peirce. Both bridges have the same theme – to generalize syllogism, but with different agenda. The De Morgan/ Peirce bridge is a story about the expansion of territory and the Boole/Peirce bridge1 is the expansion of notation. This chapter explores the De Morgan/Peirce bridge. The chapter is neither about modern logic itself nor about Peirce’s full-blown predicate logic. Focusing on the way how syllogism served as an important background for De Morgan and Peirce’s way to modern logic, I find their progress delicate and fascinating. Let’s start with a basic structure of syllogism with two premises and one conclusion. Importantly, there are slightly different versions which we will discuss in the next section. (1a) Each proposition (in a subject-predicate form) connects two terms by a copula. (1b) Each proposition states a relation (mainly, inclusion relation) between two classes (two terms denote). (2a) With one and the same term appearing in the two premises, the conclusion establishes a new connection between the other two terms. (2b) With one and the same term appearing in the two premises, the conclusion establishes a new relation between the other two terms. (3) Each term denotes a class of individuals.

Not contradicting a fundamental gap between syllogisms and a new logic towards which they were heading, I would like to show in the next two sections how De Morgan and Peirce struggled over the representation of relations within the syllogism framework. Their effort was constructive and conservative. De Morgan’s generalization of syllogism hinges on the version (1b). Embracing Boole’s formal method to handle syllogistic reasoning (which focuses on (2a)), it is Peirce who appreciates strength and weakness of these two different kinds of effort so that the project may go forward. The third section examines Peirce’s ways to cover relational logic under syllogism and composition of relations beyond syllogistic reasoning. In the conclusion, we will evaluate the importance of one of the bridges from syllogism to modern logic – the bridge where we placed De Morgan and Peirce.

2 Syllogism extended by De Morgan One of the many interesting and amusing lessons we could learn from studying history is to appreciate different responses to the same event or phenomenon. The traditional syllogism provides us with a fountain of these learning moments. This section starts

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with the consensus among philosophers and logicians since the eighteenth century. Mathematical reasoning seems to be beyond what the traditional syllogistic reasoning covers. Thomas Reid (1710–1796), Immanuel Kant (1724–1804) and Augustus De Morgan (1806–1871) all acknowledged this. As briefly mentioned at the beginning of the chapter, Kant’s metaphysic started with his clear distinction between categorical syllogistic and mathematical reasoning. Reid also made it clear that a copula-connected subject-predicate form would not be enough to cover much of our reasoning, especially mathematical reasoning, for example, ‘greater than’, ‘less than’, and even ‘is equal to’. De Morgan cites some of the following passages in his work.2 [I]t is to be observed, that the propositions of mathematics are not categorical propositions, consisting of one subject and one predicate.… We observed before, that this conversion, A is greater than B, therefore B is less than A, does not fall within the rules of conversion given by Aristotle or the Logicians; and we now add, that this simple reasoning, A is equal to B, and B to C, therefore A is equal to C, cannot be brought into any syllogism in figure and mode.3

According to Reid, a categorical sentence ‘All dogs are animal’ is of a simple subjectpredicate form, while relational sentences ‘A is equal to B’ and ‘A is greater than B’ are about relations between two terms. The relational aspect of mathematical statements did not catch Kant’s interest. Instead, Kant’s inquiry into the source of mathematical truth took us to a vast metaphysical realm – how non-analytic, hence, synthetic, truth could be non-empirically, hence, a priori, true. De Morgan did not deny the difference between categorical and relational statements. He most certainly did not follow Kant’s philosophical journey. Unlike Reid, De Morgan was an ardent defender of syllogism. Being perfectly aware of the shortcomings of syllogism, how could De Morgan stay faithful to syllogism and receive credit as a front runner of the logic of relations? That is the main story of this section. De Morgan’s extension to various relational arguments was quite ambitious. He wanted to keep the form of syllogism and at the same time extend the territory of syllogistic inference to relational arguments. From Reid’s point of view, this is an impossible project since relational reasoning cannot be carried out within a syllogistic form. Merrill correctly and interestingly labelled De Morgan’s spirit as ‘an intriguing combination of conservatism and novelty’.4 What Merrill meant by De Morgan’s ‘conservatism’ resides in his effort to preserve syllogism and syllogistic reasoning. Without objecting to this label, some might quickly add that De Morgan’s conservatism became an obstacle to his otherwise more successful work on the logic of relations.5 I believe the ‘conservatism’ label for De Morgan could be somewhat misleading since De Morgan’s novelty starts with the way he conserves the syllogism. As I will show below, he is not defending syllogism with its shortcomings. It would be an indefensible defence. He preserves syllogism by providing a new angle on the traditional syllogism. Thanks to this new angle, the traditional syllogism is a sub-case

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of relational reasoning! The meaning, hence, the extension of syllogistic reasoning being changed, De Morgan’s conservatism lets the traditional syllogism be subsumed under relational arguments. We will analyse the process of arriving at this new perspective. He takes several steps to present a new perspective where syllogism represents relational inference. First of all, I claim that the different routes Reid and De Morgan took are directly related to their difference in interpreting categorical sentences, hence, the traditional categorical syllogism. The sentence ‘All dogs are animal’, according to Reid, shares the form with a sentence ‘Susan is tall’. The word ‘to be’ is a copula which links a subject and a predicate. There is no relation in this form of subject-predicate sentence. Hence, according to the structure presented in the Introduction, Reid takes (1a) and (2a). He would not endorse (1b), period. Importantly, I will argue that De Morgan’s main contribution to preserve syllogism starts with his preference of (1b) over (1a). Some might say this move reflects De Morgan’s conservatism: ‘De Morgan approaches the logic of relation from a conservative point of view, considering the logic of relations to be but a generalization of the traditional syllogistic patterns.’6 Some might take this position of De Morgan in a more adventurous spirit: ‘De Morgan’s enquiry lay in the desire to obtain a more abstract view of syllogistic reasoning, and to provide a general rubric from which the Aristotelian forms, and many others, could be derived.’7 Pushing this direction further, I show that by taking (1b) instead of (1a) De Morgan successfully brought a ‘relation’ talk into the discourse on syllogism. De Morgan’s first step is to treat the copula ‘to be’ (which always appears in the traditional syllogism) as abstract, focusing on its formal characteristics, not its accidental meanings.8 ‘By an “abstract copula” of course is meant a formal mode of joining two terms which carries no meaning, and obeys no law except such as is barely necessary to make the forms of inference follow.’9 Joining two terms is the essential characteristic of the copula, according to De Morgan, and he called it ‘abstract copula’.10 By eliminating any specific content of a copula and focusing on a formal mode, De Morgan paved a way to generalization of relations between the classes that terms denote. This step of De Morgan supports my claim that De Morgan would take (1b) as basic structure of syllogism rather than (1a) in the above. The copula ‘to be’ in Aristotle’s categorical syllogism is reinterpreted by De Morgan. Let me illustrate the significance of re-interpretation of the copula through examples: (A) All dogs are animals. All animals are mortal. Therefore, all dogs are mortal.

Reid would not think the copula ‘are’ here is a relation at all, but just a linking device between a subject and a predicate, telling us that each dog has the property of being an animal. On the other hand, some may say that the copula ‘are’ is interpreted as being included. If so, inclusion is a relation between these two classes, say, the class of dogs

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is included in the class of animals. This subtle difference becomes more obvious in the case of the following syllogism: (B) Line AB is equal to line CD. Line CD is equal to line EF. Therefore, line AB is equal to EF.

As we discussed in the above, taking ‘equal to’ as a relation, Reid would say Argument (B) is a relational, non-syllogistic, argument unlike Argument (A). Syllogism cannot get us to the validity of (B), which is so common to mathematical reasoning. How about ‘greater than’ or ‘less than’? It is time to leave syllogism, Reid said. De Morgan’s position is more complicated. He wants to treat both ‘are’ in (A) and ‘is equal to’ in (B) as copulas, formally speaking. In his early writing, he says ‘the copula, or manner in which the two are joined together, which is generally the verb is, or is equal to’.11 However, in the case of De Morgan, I do not think it matters what we call these phrases, since both cases express relations. De Morgan takes both (A) and (B) as syllogism and draws our attention to different kinds of relations expressed in (A) and (B). The copula ‘are’ in (A) represents an inclusion relation, while the copula ‘is equal to’ in (B) expresses an identity relation. Hence, their formal properties are different: One is symmetric but the other is not. On the other hand, both are transitive. Similarly, we may embrace ‘is greater than’ and ‘is less than’ as copulas with their own formal properties. De Morgan is ready to include many other relational expressions in the same category. Merrill succinctly captures this move in the following passage: We have raised the possibility of adding ‘is greater than’ to the list of copulas. But why stop with it? It appears that any two-pace relational expression would work equally well. If, as we have suggested, De Morgan had already adopted a relational analysis of the proposition, asserting that a proposition consists of a subject term related in some way to a predicate term, then it seems that any relation will do.12

De Morgan’s symbolization of copula comes from the same spirit: ‘In the form of the proposition, the copula is made as abstract as the terms.’13 In order to block specific, hence, accidental, meanings of terms in syllogisms, we symbolize them: (C) All P are Q. All Q are R. Therefore, all P are R.

Similarly, we suggest symbolization for copulas as well:14 (D) All P … X … Q. All Q … X … R. Therefore, all P … X … R.

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Both arguments (A) and (B) fit in this format. One may go on to conclude that logical form (D) is behind the validity of arguments (A) and (B). However, we do not want to say the following argument with form (D) is valid: (E) Tom is a parent of John. John is a parent of Susan. Therefore, Tom is a parent of Susan.

How can we block the validity of argument (E) without withdrawing form (D)? Putting the matter somewhat backwards, one may say that the property of relation X determines whether form (D) is adopted. The relation ‘being a parent of ’, lacking certain property, does not fit in the form (D). From this point of view, whether a given argument fits in form (D) is determined by what kind of relation an argument is about. The relations – ‘are’ in (A), ‘is equal to’ in (B), ‘is greater than’ and ‘is less than’ – are all transitive. Hence, we say these relational arguments have the form (D). Even though the meanings of these relations are different from one another, something common is found in the process of abstraction: transitivity. I think this is what De Morgan meant by the following well-known statement: ‘Logic considers, not thought, but the form of thought, the law of action of its machinery.’15 I believe the way he connects properties of relation and a form of argument contributes to the extension of syllogistic reasoning. According to De Morgan, the following argument is valid in the same way as (A) and (B) are, having the form (D): (F) P is greater than Q. Q is greater than R. Therefore, P is greater than R.

Reid’s sharp division between syllogism (A) versus non-syllogistic relational arguments (B), (C) and (F) seems to lose its force. Henry L. Mansel’s strong criticism gets in the picture. By substituting X for ‘killed’ in form (D), he asks, ‘Is this knowledge [whether a copula has certain property or it fits in (D)] formal or material? Is it derived from the general laws of thinking, or from a special knowledge of the nature of the actions denoted by the several verbs?’16 According to Mansel, De Morgan mixed up the material and formal nature of arguments. We know certain relations are transitive, thanks to our material knowledge about them, the critic says. De Morgan is vehement in arguing for the complexity of the form-matter distinction.17 I would like to sum up his defence in the following way: Both form and matter could come in at different levels. At the first level, everything, including a copula in the traditional categorical argument, could be material, but when we step up and abstract, we find a form based on material information. This form provides us with a way to handle material knowledge. When we take a traditional syllogism, say (A), to be an argument of form (D), aren’t we relying on the meaning of the copula ‘are’? This is material knowledge at that level. However, after abstracting its transitivity and expressing it as form (D), we are carrying out a form of thought. Another important property of a relation is convertibility.18

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De Morgan’s following comments on Mansel’s logical/material distinction predicts his expansion of syllogistic reasoning: When the last named writer [Mansel] makes it ‘material and therefore extralogical’ that Alexander was the son of Philip because Philip was his father, he uses the word historically. The formal connexion of relation and correlation exists, though Aristotle did not recognize it as copular except when the relation is its own correlation, and then only in a limited case. Certainly the matter, in the case of father and son, supplies the knowledge of the correlative relation existing, but not the mode of using it in inference, when known to exist.19

The passage hints that we may bring in more kinds of form to syllogistic reasoning. Since we know the meanings of parent and child, that is, material knowledge, the following form of thought may be licensed: from P.. X.. Q to Q.. X-1.. P. Not denying a difference between material and formal, De Morgan draws our attention to subtle and important aspects of the matter at hand: Whether some relation fits in a given form is material knowledge, but after that judgement, formal reasoning carries out the rest of the job. Pushing this line of thought, De Morgan combines relations to produce another relation. Here is his well-known example for the composition of relations:20 (G) John can persuade Thomas. Thomas can command William. Therefore, John can control William.

Three related relations are presented in one syllogism – persuade, command and control. De Morgan suggests the following form for (G):21 (H) P … X … Q. Q … Y … R. Therefore, P … XY … R.

De Morgan not only justifies the validity of argument (G) based on form (H) but also points out that form (D) is a special case of form (H), when X = Y. The algebraic equation y = φx has the copula =, relatively to y and φx; but relatively to y and x the copula is = φ. This is precisely the distinction of ‘Johan can persuade Thomas’ and ‘John is {one who can persuade Thomas.}’ The distinction of y = φψz from y = φx, x = ψz is the formation of the composite copula = φψ.22

At this point his generalization moves up to another level. First, we have seen his generalization over copulas (or linking terms between subjects and predicates) to relations. Let me call it the generalization from copulas to relations.23 Under this generalization, linking terms are symbolized in (D); hence, De Morgan treats the traditional syllogism as a special case of syllogistic form (D). The second level of generalization is over relations themselves. Let me call it the generalization from

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relation to relations. Any relation can be considered as a composition of relations, since it can be the composition of itself. These two stages of generalization, along with his subtle formal and material distinction, are nicely summed up in the following passage: A little consideration will shew us that every inference which is anything more than pure symbolic representation of inference is due to the presence of something material: … Here are two purely formal propositions, in which P, Q, R, S, represent individual objects of thought, and – A – indicates a relation A: –

P – A – Q   R – B – S P stands in A– relation to Q and R in B– relation to S. What are we to infer? Now, rub R, and for it write Q. This is material: … And now we have P – A – Q, and Q – B – S. Can we infer anything? With the form of combination for relations in our thoughts, we may symbolize it, and say P – AB – S. Now make the relations material: let –A– and –B– each be ‘is:’ then we have a material inference P is Q, Q is S, therefore P is S.24

The form (C), which Reid took to be the only form of the syllogism, turned out to be a special case of generalized form (H), (C) being a special case of (D), and (D) being a special case of (H). Here are two steps of specification: (i) from (H) to (D), being X = Y, and (ii) from (D) to (C), being X = ‘is’. In 1859, De Morgan reached the following conclusion: When by the word syllogism we agree to mean a composition of two relations into one, we open the field in such manner that the invention of the middle term, and of the component relations which give the compound relation of the conclusion, is seen to constitute the act of mind which is always occurring in the efforts of the reasoning power.25

The Aristotelian categorical syllogism is a sub-area of this broadened form of syllogism. More power to syllogistic reasoning! The next layer of generalization is on the third item of our characterization of syllogism: (3) Each term denotes a class of individuals. Another of De Morgan’s wellknown examples is the following: (I)

Every man is an animal. Therefore, every head of a man is the head of an animal.

He himself presented this argument as a challenge to syllogistic reasoning. De Morgan’s expansion of kinds of copula does not seem to explain the validity of this argument. As far as his own solution (?) goes: A little consideration suggests as a necessary rule of inference, the right to substitute a larger term used particularly for a smaller one, however used, and a smaller, used in either way, for a larger used universally.26

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Since the class of men is included in the class of animals, we may substitute ‘man’ for ‘animal’ to get the conclusion. However, this is far from being satisfactory. Why is this different from the cases we have examined so far? This argument does not seem to fit in the traditional form of syllogism, not because it has a different kind of copula or does not have a copula, but for the following two reasons: (i) The conclusion is not about the relation between two simple classes, horses or animals but the relation between heads of horses and heads of animals. While simple classes are classes of individuals/objects, these new classes are classes of pairs of individuals/ objects. (ii) De Morgan did not have a formal tool to express cross-referencing in the conclusion.27 ‘The traditional doctrine of the categorical proposition, even as modified by De Morgan, does not contain the resources to exhibit the type of cross-referencing that is required.’28 I see this head-example as posing the possibility for a different kind of generalization than copula-generalization or composition-generalization, but a generalization in the kind of class a term denotes – from a class of individuals to a class of pairs of individuals. In hindsight we know De Morgan did not have enough tools to execute the expansion of logic as he wanted. At the same time we know his strong faith in syllogistic reasoning and his effort for generalization were the beginning of the logic of relations. De Morgan’s passion for syllogism is translated into a passion for formalization of relational reasoning, and Charles S. Peirce is one of the main figures who was more than ready to execute De Morgan’s blueprint.

3 Syllogism reconstructed by Peirce Although De Morgan and Peirce engaged in correspondence on these matters, how much they actually communicated with each other and how much Peirce’s logic of relations was influenced by De Morgan’s work – none of this is that clear.29 We know Peirce was perfectly aware of De Morgan’s battle with relational arguments: ‘[I]t may at least be confidently predicted that the logic of relatives, which he [De Morgan] was the first to investigate extensively, will eventually be recognized as a part of logic.’30 However, it is not clear whether Peirce started working on the logic of relations without knowing De Morgan’s work.31 This section will not get into any of these historical issues but explores how the syllogism is related to Peirce’s development of the logic of relations. Peirce was one of those who had an ambition to explore mathematical reasoning, facing the apparent discrepancy between relational and syllogistic reasoning. How did Peirce assess syllogistic and mathematical reasoning? Did he, like Kant and Reid, draw a clear line between syllogistic and relational reasoning so that he may dismiss the syllogism as unfit for the new logic of relations? Or was he another defender of syllogistic reasoning like De Morgan? Interestingly enough, the following passage provides us with Peirce’s view on the three different positions – Kant’s, Reid’s and De Morgan’s – which we briefly discussed at the beginning of the previous section: I will mention that logicians have found such extreme difficulty in reducing mathematical demonstrations to syllogistic form, that some have boldly

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pronounced it impossible, and on that impossibility have founded a peculiar philosophy of mathematics, and of space and time with which it deals. But a further study of syllogism has led to the discovery of these new forms, by which mathematical demonstrations can be reduced to syllogism, and thus not only is a false theory of mathematics and of space and time overthrown but a careful analysis of demonstrations by means of these forms has led to the discovery of the great principles of mathematics, and the essential nature of space and time upon which they are based.32

The main view Peirce is criticizing here is Kant’s philosophy of mathematics: Kant’s puzzle started with the failure of reduction of mathematical to syllogistic reasoning. Focusing on the a prioricity of mathematics, Kant located the a priori source of mathematics in our a priori framework of perception, space and time. Peirce claims that not being able to reduce mathematical to syllogistic forms itself was a failure which could be corrected. And he did so, according to this passage, thanks to ‘the discovery of these new forms’. Before we explore what Peirce meant by ‘new forms’, I would like to locate the story of this section in a bigger picture. The above passage was written in 1866. As if Peirce had the above passage in mind, he made the following confession in 1898: But to return to the state of my logical studies in 1867, various facts proved to me beyond a doubt that my scheme of formal logic was still incomplete. For one thing, I found it quite impossible to represent in syllogisms any course of reasoning in geometry or even any reasoning in algebra except in Boole’s logical algebra.33

There is no doubt that Boole’s algebraic approach is totally different from De Morgan’s approach.34 And Peirce’s final product was strongly influenced by the Boolean algebraic spirit. However, Peirce’s accepting Boolean algebra was neither the starting nor the ending point of Peirce’s project. He needed to extend Boole’s system. After all, Boole’s algebra is limited to monadic relations. Even for the Aristotelian syllogism, Boole’s expression was not enough to represent an existential statement easily. That part of Peirce’s journey is not the topic of this chapter. If I continue the visual metaphor suggested in the Introduction, the chapter is about the other bridge where De Morgan and Peirce stand, not the bridge of Boole and Peirce. As Peirce makes it abundantly clear in the above quotation, the Boolean algebraic method was crucial to representing mathematical reasoning. As seen in the previous section, De Morgan’s position towards syllogism and the logic of relations is quite clear-cut. These two areas are not exclusive of each other, and there is no conflict between them either. Generalizing syllogism, De Morgan attempts to take the traditional syllogism as a subcase of syllogism. Generalized syllogism embraces relational arguments, including the composition of relations. Although De Morgan’s specific effort could not get us all the way to the logic of relations as we know, his tireless work on syllogism reflects his faith in the extension of the territory of logic – the logic of relations. Extending the territory is common between De Morgan and Peirce, which is a strong motivation behind Peirce’s adoption of Boolean algebra into logic. The desire to cover mathematical or other

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ordinary relational reasoning within the scope of logic helps Peirce appreciate Boole’s notation and venture further to extend Boole’s algebra of logic to get to the modern logic. Hence, it is interesting to see Peirce’s work on relations before his writings on Boole in 1867.35 Scholars36 have agreed that Peirce’s Lowell Lectures 1866 show how Peirce worked on relations without Boole’s influence. Here is a well-cited example from those lectures:37 (J) Everyone loves him whom he treats kindly. James treats John kindly. Therefore, James loves John.

The main difference between this argument and the Aristotelian syllogism is that each sentence in (J) states a relation between two terms. In spite of its appearance in RuleCase-Result form, Peirce admits that ‘yet there is a peculiarity about it [argument (J)]’.38 The issue is the nature of word ‘he’ in the first premise, that is, how to handle coreference. This co-reference issue emerges since the sentence is about a relation, not just about a property. Peirce’s instinct is to keep argument (J) as a syllogism, and he backs up the instinct with some manipulations. The main reason why (J) looks different from argument (A) is that (J) does not exhibit form (C). According to Peirce’s terminology, there is no term in the premises which serves as subject of one proposition and as predicate of another. Having first then arranged the syllogism in three propositions and having accurately observed that there are but three terms, not apparently only but in meaning, the next step is to find that term which is subject of one proposition and predicate of another.39

Let’s call it a middle term. In form (C), term Q plays that role; hence it is a form of Aristotelian syllogism, according to Peirce. Argument (J) does not seem to have one term that plays this role. Following Peirce’s suggestion ‘you will find that it contains two propositions … There are therefore two Aristotelean syllogisms into which you can thus convert it’ (WP, 1.377), let me reconstruct (J) into two Aristotelian syllogisms in the following way:40 (J*) (1) Everyone who treats a person kindly loves that person. (2) James is a person who treats someone kindly. (3) Therefore, James loves the person whom he treats kindly. (4) The person whom James treats kindly is John. (5) Therefore, James loves John.

The (1)–(3) syllogism has middle term, ‘one who treats someone kindly’, and the (3)– (5) syllogism carries the middle term ‘those who James treats kindly’. Just like form (C), these middle terms disappear in the conclusion of each syllogism, by serving as a subject in one proposition and as a predicate in the other proposition. Note that this is

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a quite different way from De Morgan in taking up relational arguments: De Morgan expanded the extension of syllogism, making the traditional syllogism a special case of the general syllogism while Peirce restructured relational arguments so that they may be decomposed into more than one traditional syllogism. The difference between Peirce’s and De Morgan’s approaches becomes clearer with mathematical reasoning: (K) 3 × 4 = 12 Therefore, 4 × 3 = 12.

Citing commutative property of multiplication relation, De Morgan would take argument (K) as a valid syllogism. On the other hand, Peirce says:41 The rule here is ‘Whatever number results from the multiplication of one by another results also from the multiplication by that one of the other.’ The case is ‘12 is a number which results from the multiplication of 4 by 3.’ The result is ‘12 results from the multiplication by 4 of 3.’

At this stage in terms of preserving syllogism, Peirce’s approach was more conservative than De Morgan’s. Peirce sticks with the traditional syllogism and converts relational arguments into the Aristotelian syllogism. What I mean by ‘this stage’ has a double meaning: Time-wise, this is the stage where Peirce had not yet adopted Boole’s algebraic notation. Subject-wise, this is where De Morgan’s first level of generalization took place, that is, the generalization of copula, not a composition of relations. Hence, we should not be quite surprised to see Peirce’s (over)confidence in syllogism as quoted at the beginning of the section. At a superficial level, Peirce seems to go along with De Morgan’s method, but I do not think that impression is correct. Moreover, this hasty judgement of conformity does not do justice to either De Morgan or Peirce for the following reasons. De Morgan’s effort at generalization does not consist of moulding relational arguments into the traditional syllogism at all. On the contrary, he aims to locate both the traditional syllogism and relational arguments in a bigger territory of syllogism. Peirce’s effort shown so far is not the same as this style of expansion, but rather an attempt to tackle relational arguments within the traditional form of syllogism. Robert W. Burch’s following comments are consistent with this view: That Peirce’s logical analysis in his early works is accomplished by attending to relative terms rather than to verbs and verb phrases (as, of course, firstorder predicate logic does) might perhaps be explained by the fact that in his early  logical  works, Peirce is still tied to the idea that all propositions are basically  subject-predicate in form, no matter how complicated the subject or predicate.42

De Morgan’s way of generalization from copula to relation is not found in Peirce’e early work on syllogism. However, De Morgan’s next level of generalization, from a relation

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to relations, is revisited in Peirce’s well-known paper ‘Description of a Notation for the Logic of Relatives’. The paper starts with the following sentence: Relative terms usually receive some light treatment in works upon logic, but the only considerable investigation into the formal ways which govern them is contained in a valuable paper by Mr. De Morgan in the tenth volume of the Cambridge Philosophical Transactions.43

Later in the paper Peirce spells out De Morgan’s influence: I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, lw shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he appears not to have had multiplication in mind.44

I find this passage extremely interesting in that Peirce himself makes clear both the connection between his multiplication sign and De Morgan’s notation and their differences. In the syllogistic form (H) above, De Morgan introduces the notation XY in order to represent a new relation – the composition of two relations, X and Y. This is exactly what Peirce means when he says the notation is the same.45 Notation being in the same form, neither syntax nor semantics is the same. A syntactic difference is rather subtle: In the case of compound form XY, what De Morgan had in mind over X and Y is copula between subject and predicate or a relation between two terms in a proposition. When Peirce writes lw, l and w could be ‘a term [property], a correlate of it [binary relation], a correlated of that correlate [ternary relation], etc.’ (CP 3:72). I attribute this difference to the different framework where this notation was presented. De Morgan’s composition of relations was suggested in the process of extending syllogistic reasoning, while Peirce’s multiplication notation was totally independent of syllogistic reasoning. Semantically, as Peirce himself said, the idea of multiplication does not have a role in De Morgan’s composition of relations. Peirce’s notation has a computational operation of ‘product’, due to the influence of Benjamin Peirce’s linear associative algebra, as we will see shortly. Before we discuss new aspects of Peirce’s notation, I would like to point out some similarity between De Morgan’s and Peirce’s spirit which has not been pointed out in the literature. Peirce presented the multiplication between relations first and later moved to the multiplication between properties. In that process, interestingly enough, Peirce seems to take relations as a general case and properties as a special case: Thus far, we have considered the multiplication of relative terms [relations] only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms [unary predicates which denote properties] by considering them as relatives. Now the absolute term ‘man’ is really exactly equivalent to the relative term ‘man that is –,’ and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term.46

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Women (w) that are students (s) are denoted by ‘w, s.’47 Brandy offers this comment: We see here an interesting feature of Peirce’s 1870 paper. Peirce takes the multiplication of relative terms as primitive. When he subsequently defines multiplication for absolute terms, it is as a derived concept. Multiplication for absolute terms is explained as a more complex symbol – m, b – involving a comma as well as a simple juxtaposition.48

We could not help recalling De Morgan’s generalized copula and generalized relation of relations. The traditional syllogism is a special case of relational arguments, and one and the same relation is a special case of the composition of two relations, X and Y, when X = Y. I would like to say that De Morgan and Peirce coincided on how to expand the territory of logic. They treated monadic proposition as a subcase of relational proposition. In spite of similarity in notation and certain idea, their parting from each other shows up in the way they interpreted a composition of relations. De Morgan’s composition of relations in the form (H) was suggested in the context of syllogism, but not Peirce’s application of relations. Peirce’s composition of relations aims to be more expressive beyond syllogistic reasoning. A composition may be made out of any kinds of predicates. Roughly speaking, Peirce’s relative product of relations (eventually) works in the following way: Let w and p be unary predicates. Then, w, p = {x| w(x) ∧ p(x)}. Let l be binary and w be unary. Then, lw = {x| ∃y[l(x, y)∧w(y)]}. Let s and l be binary. Then, sl = {< x,z > | ∃y[s(x,y)∧l(y,z)]}.

De Morgan’s notation for composition of relations, XY, is generalized once for all! That does not mean Peirce followed in De Morgan’s footsteps on relations. On the contrary, by interpreting XY as multiplication and having classes in mind, ‘Peirce saw the analogy between the composition of linear transformations and the relative product of relations.’49 I think that is the major break Peirce took from the traditional syllogism. Brunning said about Peirce’s 1870 paper: De Morgan’s methodology is governed by the logic of syllogism while Peirce’s methodology is entirely algebraic. This algebraic model taken over from Boole is foreign to De Morgan’s methods.50

This is why Peirce’s 1870 paper ‘Description of a Notation for the Logic of Relatives’ (where algebraic representation of relations and relations of relations was presented) has been considered a milestone in the modern logic. Peirce’s own example illustrates how compositions of relations are computed in the manner of linear algebra style:51 Given mutually exclusive classes, say A and B, there are four kinds of relations between these two exclusive groups which he called elementary relatives. Out of these two mutually exclusive sets, there are four elementary relatives: A:A, A: B, B: A,

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and B: B. And A: B ‘denote[s] the elementary relative [binary relation] which multiplied into B gives A’ (CP 3:123). Let u be the body of teachers in a school and v be the body of pupils in a school. Let these two classes be mutually exclusive. Let c denote colleague, t teacher, p pupil, and s schoolmates, and all of them are elementary relatives. Then, c = u:u (colleague is the relation of teacher to teacher) t = u:v (teacher is the relation of teacher to pupil) p = v:u (pupil is the relation of pupil to teacher) s = v:v (schoolmate is the relation of pupil to pupil).

Peirce pushes this way of modelling relations as compositions of relations as well. Is a composition of two relations, say colleague and teacher, still a relation? Suppose John is a colleague of Susan, and Susan is a teacher of Mary. Then, what is the relation between John and Mary? It is a teacher relation. That is, John is a teacher of Mary. According to Peirce’s notation, colleague is expressed as u:u and teacher as u:v. The composed relation c:t is the relation (u:u) x (u:v), that is, u:v, which is the relation teacher. Peirce himself presents the following multiplication table (Table 5.1).52 Sixteen propositions expressed in the table are spelled out by Peirce. And, let me quote the first four, in the first row:53 The colleagues of the colleagues of any person are the person’s colleagues; The colleagues of the teachers of any person are that person’s teachers; There are no colleagues of any person’s pupils; There are no colleagues of any person’s schoolmates;

Even though Peirce might not have a notion of ordered pairs as we do, this way of modelling a product of relations formalizes composition of relations – how we obtain a new relation out of two relations. And its mechanics Peirce took from Benjamin Peirce’s linear associative algebra, as Brunning claimed.54 Together with a more generalized domain over predicates using the multiplication sign for composition, XY, the system cries out for quantification and, subsequently, variables. With that, it is time to enter the gate of the modern logic we know. Table 5.1  Peirce's multiplication table

c (u:u) t (u:v) p (v:u) s (v:v)

c (u:u)

t (u:v)

p (v:u)

s (v:v)

c(u:u) 0 p(v:u) 0

t(u:v) 0 s(v:v) 0

0 c(u:u) 0 p(v:u)

0 t(u:v) 0 s(v:v)

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4 Conclusion – and one more Question When we reach this point of the history, syllogism seems to remain as a legacy at best. The logic of relations is the essence of modern logic and it is easy to point out how the traditional syllogism lacks relational statements. That way to disconnect modern logic from Aristotelian syllogism is too easy and too convenient to be true, in my view. This disconnection is belied by De Morgan’s and Peirce’s specific ways of embracing relational arguments under syllogism. Their ideas and efforts are innovative and admirable enough for us to re-think the role of syllogism in the development of modern logic. Common and rather mistaken views about these two logicians are: De Morgan, unfortunately, had too much faith in syllogism to carry out relational logic. Further, Peirce followed De Morgan’s position at the beginning of his logic inquiry, and realizing syllogism would not deliver so, he jumped to Boolean algebra and made things work out. These superficial evaluations, I am afraid, miss an important matter in common between these two logicians: Acknowledging the necessity of inquiry into relational arguments, neither of them proclaimed their departure from syllogism. Neither of them dismissed syllogism. At the same time, their major difference in their attitudes towards syllogism is striking. De Morgan’s way of embracing relational arguments is to change the extension of syllogism. As a result, the traditional syllogism is one kind of syllogism and the relational argument is a more general kind of syllogism. Within that framework, De Morgan established formal properties of relations and reached out to the composition of relations as well. Peirce, being aware of De Morgan’s work and ambition, did not take this route but attempted to reconstruct relational arguments into more than one traditional syllogism. An algebraic approach and notation enabled Peirce to find a way to represent relations in terms of relations between two classes and this method was crucial for generalizing the representation of relations, including composition of relations. Was Boole’s algebraic method crucial for Peirce’s invention of modern logic? Definitely, the answer is ‘Yes’. Let me ask one more pointed question in this concluding section: Was Boole’s algebraic notation alone enough for Peirce to invent the logic of relations? I would like to answer ‘Extremely unlikely’. Tarski succinctly states: The title of creator of the theory of relations was reserved for C. S. Peirce. In several papers published between 1870 and 1882, he introduced and made precise all the fundamental concepts of the theory of relations and formulated and established fundamental laws.  … In particular, his investigations made it clear that a large part of the theory of relations can be presented as a calculus which is formally much like the calculus of classes developed by G. Boole and W.S. Jevons, but which greatly exceeds it in richness of expression and is therefore incomparably more interesting from the deductive point of view.55

While Peirce was pushing further the application of Boole’s and Jevons’s calculus of classes, he knew where he wanted to head. The seed of Peirce’s invention – much richer and more interesting than calculus of classes – was planted before he started writing on Boole.

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Peirce took the spirit of De Morgan’s enterprise – to extend the territory of logic to relations – to heart, but Peirce did not follow De Morgan’s approach itself. Both De Morgan and Peirce had the same destination to reach, but their road maps were different. Peirce was able to see the necessity of Boole’s algebra of logic for the logic of relations, thanks to De Morgan’s ambition for the extension of syllogism. That is, De Morgan’s work on syllogism was crucial for Peirce to take the algebraic method to be crucial for the extension of logic from the traditional syllogism to the modern logic of relations.

Notes 1 2

See David Dunning’s chapter in this volume. Augustus De Morgan, ‘On the Syllogism: II. On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula’, Transactions of the Cambridge Philosophical Society, vol. IX, (1850), 79–127 1850. in On the Syllogism and Other Logical Writings, ed. Peter Heath (London: Routledge & Kegan Paul, 1966), 67. 3 Thomas Reid, ‘A Brief Account of Aristotle’s Logic’ (1774), in Sketches of the History of Man, ed. Henry Home Kames, vol. II (Edinburgh: W. Creech, 1774), Ch. IV, Section V. 4 Daniel Merrill, Augustus De Morgan and the Logic of Relations (Dordrecht: Kluwer Academic Publishers, 1990), 26. 5 ‘De Morgan’s own outlook remaining essentially syllogistic, which … would liable to impede him, unfortunately, from definitively taking a radically novel point of view.’ Benjamin S. Hawkins Jr., ‘De Morgan, Victorian Syllogistic and Relational Logic’, Modern Logic 5, no. 2 (1995): 137. 6 Daniel Merrill, ‘De Morgan, Peirce and the Logic of Relations’, Transactions of the Charles S. Peirce Society 14, no. 4 (1978): 248. 7 Peter Heath, ‘Introduction’, in On the Syllogism and Other Logical Writings, ed. Peter Heath (London: Routledge & Kegan Paul, 1966), xxix. 8 Merrill’s monumental work on De Morgan goes into more details about the generalization of the copula, and these further details are not directly relevant to the discussion in the text. Instead, I aim to locate the main ideas of De Morgan’s generalizations in a bigger picture of the syllogism. 9 De Morgan, ‘On the Syllogism: II’, 51. 10 Ibid. 11 Augustus De Morgan, On the Study and the Difficulty of Mathematics. 1831. Reprinted (La Salle, Illinois: Open Court Publishing Company, 1943), 68. 12 Merrill, Augustus De Morgan and the Logic of Relations, 32. 13 Augustus De Morgan, Formal Logic, or, The Calculus of Inference, Necessary and Probable (London: Taylor and Walton, 1847) (reprinted Delhi Alpha Edition, 2020), iii. 14 De Morgan introduces lines and parentheses for symbolizing copulas in his ‘On the Syllogism II’. His symbolization shows up in ‘On the Syllogism III’. 15 De Morgan, ‘On the Syllogism III and on Logic in General’, Transactions of the Cambridge Philosophical Society, vol. X, 1858, 173–230. In On Syllogism and Other Logical Writings, ed. Peter Heath (London: Routledge & Kegan Paul Ltd., 1966). 16 Henry L. Mansel, ‘Recent Extensions of Formal Logic’, North British Review vol. 15 (1851): 107.

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17 Augustus De Morgan, ‘On the Syllogism: III; and on Logic in General’, 1858, in On the Syllogism and Other Logical Writings, ed. Peter Heath (London: Routledge & Kegan Paul, 1966), 74–116. 18 ‘More important than these is his recognition that the “is” of identity in the traditional syllogism performs its function only in virtue of possessing the more general properties of being transitive and convertible (or symmetrical), and hence that any other relations similarly endowed would serve equally well’ (Heath, ‘Introduction’, xxvii.) ‘[H]e wants to go on to a more general treatment of the subject. The canons of syllogistic reasoning are in effect a statement of the convertible and transitive character of the relation of identity’ (William Kneale and Martha Kneale, The Development of Logic, 1962. Reprinted. (Oxford: Clarendon Press, 1986), 427. 19 De Morgan, ‘On the Syllogism: II’, 68. 20 Ibid., 56. 21 Ibid., 57. He symbolizes relation X as a line, Y as a dotted line, and the compound relation XY as the line above the dotted line. 22 Ibid., 56. 23 This roughly corresponds to ‘the abstract copula’ in Merrill, Augustus De Morgan and the Logic of Relations, 49–59, and Merrill’s discussion is much more complicated than what is needed for my discussion here. 24 De Morgan, ‘On the Syllogism: III’, 81. 25 Augustus De Morgan, ‘On the Syllogism: IV; and on the Logic of Relations’, 1860, in On the Syllogism and Other Logical Writings, ed. Peter Heath (London: Routledge & Kegan Paul, 1966), 238. 26 De Morgan, ‘On the Syllogism: II’, 28. 27 ‘Look in standard introductory logic texts such as those by Quine, Kalish, and Montague, Copi or Suppes and you will find that this sentence [the conclusion of (I)] (or its equivalent) is schematized as: (y)[(∃x)(Fx.Hyx) → (∃x)(Gx.Hyx)]. … The correct schematization of the conclusion is: (x)(y)[(Fx.Hyx) → (Gx.Hyx)]’ (R. G. Wengert, ‘Schematizing De Morgan’s argument’, Notre Dame Journal of Formal Logic 15 (1974): 165–6.) 28 Daniel Merrill, ‘On De Morgan’s Argument’. Notre Dame Journal of Formal Logic XVIII, no. 1 (1977): 138. 29 ‘Perice’s own early developments in this area [arguments involving relations], considering required extensions of logic to account for relations, was independent of De Morgan’s discussion in his Formal Logic, and, as seems likely from the above argument, was also independent of De Morgan’s memoir on relations. Thus, we might conclude, that while it is not certain, it is likely that Peirce, around 1866, independently discovered (albeit in a rudimentary form) the logic of relations’ (Emily Michael, ‘Peirce’s Early Study of the Logic of Relations, 1865–1867’, Transactions of the Charles S. Peirce Society 10, no. 2 (1974): 73). ‘I believe we may conclude that Peirce’s early work on the logic of relations was largely independent of De Morgan’s’ (Daniel Merrill, ‘De Morgan, Peirce and the Logic of Relations’ (1978): 280.) ‘Although De Morgan introduced the notions of relative product, converse, inverse, involution (forward and backward), and negation in his paper ‘On the syllogism. IV (1860), Peirce’s discovery of the calculus of relatives was independent of De Morgan.’ (Geraldine Brandy, From Peirce to Skolem: A Neglected Chapter in the History of Logic (Amsterdam: Elsevier, 2000), 21). 30 Charles S. Peirce, ‘Augustus De Morgan’, Nation 1871. In WP, 2, 450.

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31 Michael, ‘Peirce’s Early Study of the Logic of Relations’ and Merrill, ‘De Morgan, Peirce and the Logic of Relations’. 32 Peirce, Lowell Lectures, 2, 1866, WP, 1.385–6. 33 Charles S. Peirce, The Collected Papers of Charles Sanders Peirce, vols. 3 & 4 ed. Charles Hartshorne, Paul Weiss, and Arthur Bruks (Cambridge: Harvard University Press, 1933), 4:4. 34 ‘[T]he methods of this paper have nothing in common with that of Professor Boole, whose mode of treating the forms of logic is most worthy the attention of all who can study that science mathematically, and is sure to occupy a prominent place in its ultimate system’ (De Morgan, ‘On the Syllogism: II’, 22). 35 Please note his celebrated paper ‘Description of a Notation for the Logic of Relatives’ was published in 1870. 36 Cf. Michael, ‘Peirce’s Early Study of the Logic of Relations, 1865–1867’ and Merrill, ‘De Morgan, Peirce and the Logic of Relations’. 37 Charles S. Peirce, ‘Lowell Lecture II’, Writings of Charles S. Peirce: A Chronological Edition, Volume 1: 1857–1866, WP, ed. the Peirce Edition Project, 1.382. 38 Ibid. 39 Ibid., 1.377. 40 My reconstruction is different from Michael’s: Every person who teats another kindly is one who loves that other. (Rule) James is a person who treats John kindly. (Case) Every person who treats John kindly is one who loves John. (Peirce’s principle; 1, 2) Hence, James is one who loves John. (Result by Barbara; 2, 3) (Michael, 70). 41 Ibid., 1.385. 42 Robert W. Burch, ‘Peirce on the Application of Relations to Relations’, in Studies in the Logic of Charles Sanders Peirce, ed. Nathan Houser, Don D. Roberts, and James Van Evra (Bloomington and Indianapolis: Indiana University Press, 1997), 210–1. 43 Peirce, CP, 3:45. 44 Ibid., 3.68. 45 ‘Possibly the idea of an algebra of relatives was suggested by De Morgan’s convention of writing lm as an abbreviation for l of an m of’ (Kneale and Kneale, The Development of Logic, 429). 46 Ibid., 3:73. 47 We may use Peirce’s commas in his representation of categorical statements in terms of Boole’s algebra: h, (1 − b) = 0 (Every horse is black.) and h, b = 0 (No horse is black.) 48 Brandy, From Peirce to Skolem, 35. 49 Ibid., 11. 50 Jacqueline Brunning, ‘C. S. Peirce’s Relative Product’, Modern Logic 2, no. 1 (1991): 36. 51 Refer to Peirce, CP 3:123–126 and Brunning, ‘C.S. Peirce’s Relative Product’, 42–4. 52 Ibid., 3:126. Slightly edited from his table. 53 Ibid. 54 Jacqueline Brunning. ‘C. S. Peirce’s relative product’, The Review of Modern Logic, vol. 2, no. 1 (1991), 33–49. 55 Alfred Tarski, ‘On the Calculus of Relations’. The Journal of Symbolic Logic 6, no. 3 (1941): 73.

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6

Ernst Schröder’s algebra of logic and the ‘logic of the ancient’ Volker Peckhaus

1 Introduction The emergence of modern formal logic in the middle of the nineteenth century coincided with the change of mathematics towards structural mathematics. In this context logic was taken over from philosophy by mathematicians and accommodated for solving foundational problems in mathematics. The German mathematician Ernst Schröder (1841–1902)1 was one of the most important representatives of the algebra of logic, a movement which originated in the work of George Boole2 but with precursors in early modern times.3 His logical work became the standard in mathematical logic at the beginning of the twentieth century as used for the emerging theories in the foundations of mathematics, at least until Alfred North Whitehead’s and Bertrand Russell’s Principia Mathematica, the first volume of which was published in 1910,4 attracted more interest of researchers in this field. In 1877 Schröder published the small pamphlet Der Operationskreis des Logikkalkuls,5 his first book on logic. There he presented a critical revision of George Boole’s logic of classes. He developed the idea of duality between logical addition and logical multiplication first introduced by William Stanley Jevons.6 As a professor of mathematics at the Darmstadt Polytechnic he began to lecture on logic. He gave his first course on ‘Logic on a mathematical basis’ in the summer semester of 1876. After having accepted a call to the Karlsruhe Polytechnic in the same year he continued his teaching on logic. From the winter semester 1883/84 until the winter semester of 1888/89 he gave courses on algebra of logic. With these courses he prepared his monumental Vorlesungen über die Algebra der Logik with more than 2,200 pages in three volumes and four parts between 1890 and 1905.7 Contemporaries regarded the first volume alone as having completed the algebra of logic.8 Among the topics treated were the calculi of domains, classes, propositions, including a full-fledged theory of quantification, and the logic of relatives (relations) in which ideas of Charles S. Peirce were elaborated. Nevertheless, the Vorlesungen remained unfinished. Schröder was not able to complete the second part of Volume 2 on the logic of propositions during his lifetime.

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It appeared only posthumously in 1905. The third volume deals with the algebra and logic of relatives, i.e. relational terms. The first part which appeared in 1895 treats the algebra of relatives only; the second part on the logic of relatives never appeared. The distinction between algebra and logic unveils the double character of Schröder’s theory ‘as an algebra on the one hand, and a stage of development of logic on the other, namely its formation as a logic of relations (and relational terms, “relatives”) as such’.9 Schröder considered himself an algebraist. It was only by chance that his life’s work is usually connected to logic. No doubt, he was most of his life concerned with logic, always regarding logic, however, as means to an end, and this was the idea to create a scientific universal language. In this program the algebra of logic was the main tool for providing a pasigraphy for all sciences, i.e. the structural (grammatical) side of a general script used for this purpose.10 Following Leibniz’s model of a characteristica universalis he demanded an adequate relation between sign and designated object or concept derived from a list of basic concepts as small as possible. The ideal example is the extensive building of the number system erected by using 10 ciphers only. The pasigraphy and the underlying logic were interpretations of a general, in its last step of development, ‘absolute’ algebra.11 Schröder writes: Such a designation, if applied to the whole field of objects of thought, will be, contrary to the signs constituting a word which are indifferent to the contents of the expressed ideas, a characteristic language of concepts, a ‘concept script’ [‘Begriffsschrift’], and contrary to the particular languages of the nations a general language of objects (pasigraphy). With this we arrive at the idea of a philosophical universal language.12

Schröder relates this idea to Descartes and Leibniz13 and repeats his criticism of Gottlob Frege’s Begriffsschrift which does not deserve its name because it is actually a logical judgement script, ‘although not appropriate’.14 The improvement of logic by presenting it in an algebraic form concerned above all methodological aspects. Logic could be reconstructed as a calculative discipline. These rules of the calculus were algebraic rules. The logical calculi were intended to serve as methodological tools for all deductions or generally for all inferences. As inference methods the calculi should prove their superiority over the traditional methods of inferences, whether they were direct or mediated, i.e. syllogisms. It has to be stressed that algebra of logic means algebra of logic. So the algebra of logic is no logic at all, but an algebra applied to logic. It provides, investigates and applies the structures of logic. Logic is an interpretation of an algebraic structure, the semantics consisting of the basic objects of this structure: domains, classes, propositions and relatives. For Schröder this results in the following research program: – Starting point is the given logic, i.e. the traditional logic as theory of concepts, judgements and inferences. – The structural investigation aims at providing a structural model for traditional logic. It allows to evaluate this logic in respect to coherence and exactness. – It allows to investigate the limits of the given logic.

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– It allows to overcome these limits and to extend the scope of the given logic by modifying the structural basis. Within this program it is possible to further develop the given system and to arrive at new logical systems.

2 Theory of judgements In his Operationskreis of 1877 Schröder explicitly writes that he refrained from applying his logical calculus to the old logic, calling such application as ‘inferior anyhow’, referring to the derivation of syllogisms. This would extend the size of the present publication (of 42 pp. in sum). He announces a special publication which will show that the topic has not yet dealt with in sufficient thoroughness.15 In the long introduction to the first volume of his seminal Vorlesungen über die Algebra der Logik of 189016 historical considerations are not topical. Aristotle is mentioned only once in passing.17 Aristotle is, however, prominent in Volume 218 devoted to the calculus of propositions intended to replace the traditional theory of judgements. This is prepared with a critical discussion of traditional syllogistic. The seventeenth lecture begins with the traditional theory of judgements. It is entitled ‘Traditional division of categorical judgement according to quality and quantity. Modified interpretation of universals in the exact logic and inadequacy of the former calculus for representing particular judgement’.19 His own theory of proposition extends the earlier calculi of domains and classes and precedes the calculus of relatives. Domains are the most elementary object of his algebra. Given a manifold of elements, any selection of elements in this manifold is called a domain of this manifold.20 Such domains can be interpreted as classes, i.e. kinds of elements or concepts in extensional interpretation. Schröder begins this volume of the Vorlesungen with the fifteenth lecture where he prepares the ‘transition to the calculus of propositions’.21 He introduces propositions intuitively as one-dimensional manifolds, i.e. linear manifolds. The ‘identical’ 1 is the manifold of points on a straight line, unlimited in both directions.22 a and b are some domains of points on this line, i.e. line segments on the straight line. He introduces the subsumption a  b with intuitive figures expressing that the line segment of a lies on or coincides with the line segment of b. The identical product ab is the line segment where the line segment of a overlaps with the line segment of b. The identical sum a + b adds the line segments of a to b. The negation aˌ is represented by the rays outside the line segment of a in both directions.23 The postponed subscript stroke is used as sign for negation. aˌ means not a. The linear model of propositions is then applied to domains of points in time. This is the tool for introducing truth and validity of propositions (Gültigkeitsdauer, duration of validity). The duration of validity lies between the poles 1 for always and 0 for never. The temporal aspect occurs in definitions like the definition of equality: ‘If a b and at the same time a b, then one can write a = b.’24 Finally, all theorems like Th. 6x) ab a are regarded as being always valid.25

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The advancement from the earlier calculi of domains and classes to the calculus of propositions is immediately given as soon as the letter symbols a, b, c,… are understood as propositions (assertions, judgements). Schröder writes: As soon as one calculates with these symbols […] they are interpreted as their durations of validity [Gültigkeitsdauer], i.e. that under some proposition athe time (more exactly: the domain, the totality of points in time) is understood during which the proposition is true, excluding all points of time during which it is not true.26

Furthermore, definitions, principles and theorems of the calculi of domains and classes continue to be valid. In the calculus of propositions, the initial propositions under investigation concern these domains and classes. They are called ‘primary’.27 Propositions about primary propositions are called secondary propositions. The only signs added in the calculus of propositions are sum ∑ and product ∏, i.e. quantifiers that are taken up from the Peirce school. ∏x expresses ‘that a proposition related to a domain x is valid or should be valid for every domain x’.28 ∑x expresses ‘that the proposition is not necessarily valid for each, but for a certain domain x or for several domains x (from our manifold 1) – short: for at least one x –, or, as a condition, e.g., has to be valid’.29 To show the complementary character of propositional logic, Schröder lists the theorems of the identical calculus developed in Vol. I of the Vorlesungen (§ 29) and adds theorems of quantified logic (§ 30). The foundational principles in the calculus of propositions are as follows (capital letters stand for propositions): Principle I of (propositional) identity30 A

A

and Principle II (transition) (A

B) (B

C)

(A

C)

Schröder comments on Principle II: This principle presents the first (and most important) ‘hypothetical’ syllogism and expresses the eligibility to derive from the premises, presupposed as simultaneously valid, i.e. the minor: If A is valid, then B is valid, and the major: If B is valid, then C is valid, the conclusion, ergo If A is valid, then C is valid. With other words: If B is determined by A and C by B, then C is determined by A as well. The inference from the inference from a condition is an inference from the condition, as well.31

Schröder calls this syllogism ‘inference of subsumption’ (Subsumtionsschluss), regarding it as a principle of steadiness and continuity in inferences and conclusions.32 For Schröder the calculus of propositions is merely an application of the calculus of

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classes. It is, however, characterized by a further axiomatic condition according to which each symbol representing a proposition can only have one of two meanings: 0, i.e. never (true), or 1,33 i.e. always (true). He formulates this ‘specific principle of calculus of propositions’34 as (A = 1) = A

A = 1 means that the proposition A is always valid, A = 0 means that proposition A is never valid.35 In § 33 Schröder introduces the traditional theory of judgements, i.e. quality and quantity of categorical judgements. He explains the traditional square of oppositions36 interpreting universal positive judgements (a) as universal negative judgements (e) as

A A

B or ABˌ = 0 and Bˌ or AB = 0.37

The problem arises to express the particular judgements i and o. Schröder discusses elective symbols as introduced by George Boole and William Stanley Jevons.38 They used, e.g. w as indefinite class symbol. According to Schröder they erroneously thought that the i judgement ‘Some A are B’ could be expressed by wA = wB. This equation is, however, always valid for w = 0 and w = AB, independently of some A being B or not. Schröder’s solution is to use the relation not equal ≠. The i judgement (‘some A are B’) can then be expressed by AB ≠ 0 and the o judgement (‘some A are not B’) by ABˌ ≠ 0. He illustrates the relations between the two domains A and B with circles, a geometrical representation taken from Joseph Gergonne, and not from Leonhard Euler.39 This leads him to the necessity of a quantification of the predicate. For a convenient algebraic representation Schröder introduces signs for identity = subordination superordination cut (Sekant)

With the help of these symbols Schröder expresses propositions with quantified predicates as follows: d = (A = B): All A are all B. f = (A B): All A are only some B. c = (A

B):

Only some A are all B.

g = (A

B):

Only some A are only some B.

AB = A = B ≠ 0

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3 Schröder on the syllogisms of the ancient In his twentieth lecture40 Schröder investigates syllogisms under the heading ‘The syllogisms of the ancient’. He focuses on ‘simple’ syllogisms, i.e. inferences from two premises. He distinguishes between ‘categorical syllogism’ using as terms classes (concepts) and in premises and conclusions categorical judgements, and ‘hypothetical syllogisms’ using propositions as terms and hypothetical judgements in premises and conclusions.41 On the following pages Schröder develops the traditional doctrine of syllogisms using Friedrich Ueberweg’s System der Logik und Geschichte der logischen Lehren as an appropriate textbook.42 He mentions the weakened forms of syllogisms introduced by Johannes Hospinianus and supported by Georg Wilhelm Leibniz. Schröder relates the traditional syllogistic to his theory as follows: From the standpoint of our theory we have to declare now a number of these modes as incorrect, among them in particular all ‘weakened’ forms, generally all those inferences where from two universal premises a particular judgement is inferred. In a closer view these will appear as enthymemes that elide an essential premise in silence – but as soon as it is explicitly formulated and added to the other premises, the inferences are based on three premises and they cease from being ‘simple’ syllogisms, resp. ‘syllogisms’ at all.43

For discussing the categorical syllogism he translates the standard forms into the language of his calculus of domains, adopting for time being an ad hoc symbolization for particularity: ‘Some a’ is expressed by a'. He translates   a   a'   a   a'

b with ‘All a are b.’ b with ‘Some a is b.’ bˌ with  ‘All a are not-b’ or ‘No a is b.’ bˌ  with ‘Some a are not-b.’

He uses ∴ for therefore, ergo.44 He prefers the use of this sign over the use of for two reasons. First, it helps to minimize brackets. He illustrates this with mode Barbara. He had already determined that the hypothetical syllogism Barbara is an interpretation of Principle II of the calculus of classes in the calculus of propositions. Mode Barbara would have to be symbolized with (a

b)(b

c)

(a

c).

In the symbolism used now the mode reads a

b, b

c∴a

c.

It is astonishing that Schröder stresses the equivalence of the two formulations, although in the first one sentence is expressed, whereas the second relates three

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sentences to each other. The second reason for using the ergo sign is that enthymemic reasoning symbolized with ∴ could be accepted whereas using would in these cases lead to false subsumptions in the calculus of propositions. Schröder gives then an overview of the traditional syllogisms and discusses the procedures of conversion to reduce the syllogistic modes of higher figures to the first figures. He criticizes the traditional doctrine in the following respects: The distinction between the different modes is highly ‘unessential’, rather governed by external aspects. He argues that a relation like ab ≠ 0 can be expressed with ‘some a are b’ or ‘some b are a’. Dependent on the chosen expression the syllogism, although being essentially the same, would be listed under another mode, even under another figure.45 In respect to the validity of the traditional twenty-four modes he again criticizes the implicit existence supposition. Therefore, he accepts only fifteen of the twentyfour modes as valid. The remaining nine invalid modes include four standard modes (Darapti, Felapton of the third figure, Bamalip, Fesapo of the fourth figure), and the five weak syllogisms (Barbari and Celaront of the first figure, Cesaro and Camestres of the second figure, and Calemos of the fourth figure). They are in fact enthymemes. If the existence supposition is added as necessary premise, the inference is drawn from three premises; i.e. they are no syllogisms at all. He illustrated the problem with an example using mode Darapti in the third figure, in his notation: b

a, b

c ∴ a'

c.

His example is from geometry:46 Minor: Major: Conclusion:

All equilateral right-angled triangles (b) are equilateral. All equilateral right-angled triangles (b) are right-angled (c). Some equilateral triangles (a') are right-angled.

This conclusion is true in spherical geometry, but not in plane geometry. For validating the conclusion it is necessary to assume that the concept of an equilateral right-angled triangle is not empty, i.e. b ≠ 0. As soon as this existence condition is added, we have an inference from three premises, i.e. no syllogism. Given these formalizations and eliminating the modes that can be derived by simple commutation of letters, e.g. ab = ba or cd = dc, there are only two types of valid syllogistic modes of the kinds represented by the modes Barbara and Darii (or Festino).47 In § 43 Schröder discusses ‘Miss Ladd’s calculative treatment of the 15 valid modes’. He refers to Christine Ladd(-Franklin)’s theory of the syllogism. Christine Ladd was member of Charles S. Peirce’s school at Johns Hopkins University.48 Her important paper ‘On the Algebra of Logic’ appeared in the volume Studies in Logic by Members of the Johns Hopkins University, edited by Charles S. Peirce.49 Schröder calls it a merit of the ‘brilliant young lady-mathematician’50 to have found a general expression for the fifteen valid modes. Ladd used a variation of the elimination theorem to represent all valid and particular syllogisms in a single formula.51 Schröder elaborated this concept

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using George Boole’s Elimination Theorem as a starting point. In his Laws of Thought Boole began his considerations on elimination52 with Proposition I: If f(x) = 0 be any logical equation involving the class symbol x, with or without other class symbols, then will the equation

f(1)f(0) = 0 be true, independently of the interpretation of x; and it will be the complete result of the elimination of x from the above equation. In other words, the elimination of x from any given equation, f(x) = 0, will be effected by successively changing in that equation x into 1, and x into 0, and multiplying the two resulting equations together. Similarly the complete result of the elimination of any class symbols, x, y, &c., from any equation of the form V = 0, will be obtained by completely expanding the first member of that equation in constituents of the given symbols, and multiplying together all the coefficients of those constituents, and equating the product to 0.53

Schröder’s modification of the resolution and elimination methods was closely connected to the methods of Charles S. Peirce of Hugh MacColl, granting the latter in several respects priority.54 Schröder’s simplified version of the Elimination Theorem is given as Th. 50+):55 The equation ax + bxˌ = 0 is equivalent to the pair of equations ab = 0 and x = buˌ + aˌu with u being an indefinite domain. The elimination theorem holds for the two first modes of the first and the second figure which do not contain particular judgements. In the given context Schröder used it in the formulation A0) (αβ + γβˌ = 0)

(αγ = 0).

From A0) theorem A) can be derived: A) (αβ = 0) (βˌγ = 0)

(αγ = 0).

The following forms are equivalent: A1) (αβ = 0) (βˌγ = 0) (αγ = 0), A2) (αβ = 0) (αγ ≠ 0) (βˌγ ≠ 0), A3) (βˌγ = 0) (αγ ≠ 0) (αβ ≠ 0).

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From A1), e.g. we get the mode Barbara by interpretation in the following way: For α = cˌ, β = b, and γ = a, we get (cˌb = 0) (bˌa = 0)

(cˌa = 0),

by commutative transformation of factors (abˌ = 0) (bcˌ = 0)

(acˌ = 0).

This is according to Th. 38x) equivalent with b) (b

(a

c)

(a

c),

i.e. the mode Barbara. Schröder executes these interpretations for all fifteen valid syllogisms and groups the resulting forms sometimes using conversion in the premises. This results in eight different kinds of forms which can be ordered in two groups dependent of being derived from A1) or from A2) viz. A3). Schröder gives the following table (Table 6.1):56 Schröder evaluates this result in the following way: I entertain the belief that it will be impossible to handle this in a more beautiful way as it is founded by Miss Ladd – as we have tried to illustrate previously: It seems to be unthinkable to reliably compress the syllogisms into less than one formula and to excel the transparency and simplicity of formula A).57

In § 44 Schröder shows how to deal with the ‘incorrect syllogisms of the ancient’ in exact logic. They can be justified by going beyond the syllogistic rules. In Darapti, Felapton and Fesapo an existential proposition (Existenzialurteil) for concepts in the premises has to be added. With b ≠ 0 the modified Darapti, e.g., can be expressed as (b

a)(b

c)(b ≠ 0)

or (aˌb = 0)(bcˌ = 0)(b ≠ 0)

(ac

0)

(ac ≠ 0).

Table 6.1  Schröder's grouping of valid kinds of syllogism First group

Cesare Barbara Celarent Second group

Calemes Camestres

Bocardo Darii Festino Disamis Baroco Datisi Ferio Dimatis Ferison Fresison

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The case of Bamalip is different,58 because it is possible to deduce a valid conclusion from the premises. The premises of Barbara and Bamalip are similar. Barbara would run (b

a)

(c

b)

(c

a).

Bamalip would, however, lead to a particular conclusion a' c or ac ≠ 0. The inference from c a to ac ≠ 0 (from ‘all c are a’ to ‘some a are c’) is only valid for non-empty c. The modified version of Bamalip is therefore (c

b)(b

a)(c ≠ 0)

(ac ≠ 0).

On the following pages Schröder discusses the five weak modi Barbari, Celaront, Cesaro, Camestros and Calemes. Schröder maintains that exact logic shows that the direct inferences by subalternation from, e.g., ‘All a are c’ to ‘Some a are c’ presuppose the existence of a.59 Subalternation in the classical form is therefore not valid in exact logic due to the necessary existence presupposition. In respect to inferences by conversion Schröder comes to the result, that ‘of the conversions of traditional logic only the conversio pura is valid in pure logic’.60 For Schröder, the inference from ‘Some A are B’ to ‘Some B are A’ (conversio simplex, simple conversion) and the conversion by contraposition from ‘No A is B’ to ‘No B is A’ or from ‘All A are not-B’ to ‘All B are not-A’ are ‘pure conversions’. The conversion by limitation (conversio per accidens) is not valid, however, because it uses subalternation rejected earlier. Schröder writes that the conversion by limitation is only justified in cases where Zero is not added, i.e. where it is not allowed to speak about the nothing. On this basis a consequent logic has not yet be formulated but ‘such a logic would have to lack a good deal of simplicity and quality – which inhere exact logic’.61 In this connection Schröder refers to Hermann Lotze’s example of double conversion: The judgement ‘All pugs are dogs’ can be converted per accidens to ‘Some dogs are pugs’, an inference that could be accepted in exact logic if demanding that there are pugs at all. The second conversion (simplex) from ‘Some dogs are pugs’ to ‘Some pugs are dogs’ is valid, but nevertheless problematic; because we do not arrive at the initial sentence, the conversion leads to ‘removing part of the truth’, an effect which Lotze had called an ‘indecorousness’ (Unschicklichkeit). In order to deal with this problem, Lotze had suggested to bind the quantifier closely to the term. The first conversion of ‘All pugs are some dogs’ would give ‘Some dogs are all pugs’; with the second conversion we get the initial formulation ‘All pugs are some dogs’. Lotze concludes, however: ‘It is not worthwhile to improve these fruitless formulas.’62 Schröder was not that critical. Such effects should be accepted since logical deductions are rarely simple transformations of the premises. In almost all deductions some information from the premises is omitted in the conclusion. In the mode Barbara, e.g. we lose all our knowledge about the middle term. Schröder gives another example from arithmetic: from the premise a = b + 5 we can infer the relation a > b, but information from the premise is lost, as well. These ‘irritations’ can be solved by sharpening the use of the term ‘some’. Taking up an example by Augustus De Morgan Schröder writes that a proposition like ‘Some horses can be distinguished from their equestrians by their shape’ would not be accepted in everyday language because of the connotation we have that, if this sentence

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is true, some horses cannot be distinguished from their equestrians by their shape. But following De Morgan,63 ‘some’ in logic means ‘one or more or maybe all’. It is therefore not acceptable to assume that once we say ‘some are’ that the rest is not.

4 Complete syllogistic In § 48 Schröder executes in an exemplary way the idea of a complete syllogistic on the basis ‘simple syllogisms’ in the widest possible form leading to a complete syllogistic. His starting points are De Morgan’s ‘eight standard forms of expression, with reference to the order XY’, in De Morgan’s symbolization:64

{

Aˌ or X)Y Every X is Y

Oˌ or X:Y Some Xs are not Ys

{

A' or Y)X Every Y is X

{

Eˌ or X.Y No X is Y Iˌ or XY Some Xs are Ys

{

E' or x.y Everything is either X or Y I' or xy Some things are neither Xs nor Ys

O' or Y:X Some Ys are not Xs

Schröder adopts these propositions in a modified version, calling them ‘De Morgan’s eight propositions’.65 They consist in four ‘primitive’ extensional relations with their negations: XVI0. a = {AB = 0}, c = {ABˌ= 0}, b = {AˌB = 0}, l = {AˌBˌ= 0}; Negations: aˌ = {A ≠ 0}, cˌ = {ABˌ≠ 0}, bˌ= {AˌB ≠ 0}, lˌ = {AˌBˌ≠ 0}.66 He then determines the task of a general syllogistic as to eliminate the ‘middle term’ from each of the 32767 judgments about A and B in connection with each of the 32767 judgments about B and C and search for the conclusion which possibly flow from such two premises in respect to A and C, namely as far as these yield a valid (independent of B) inference, at all.67

Thus, this investigation has to cover 327672 = 1073676289 cases. They are, however, significantly reducible, as Schröder calculates.68 In § 48 he partially implements this task, discussing the syllogistic systems of Augustus De Morgan69 and Joseph Gergonne.70 The comparison shows that a syllogistic for Gergonne’s forms of judgements is much more complicated than De Morgan’s. Both contain traditional syllogistic. None of the two syllogistic systems shows advances over the other in respect to exactness. The requirement of perfect accuracy made both syllogistic systems consider modes of inferences with conclusions not belonging to the sphere of judgements the premises were taken from. These kinds of modes of inferences were ignored by the traditional syllogistic. Schröder concludes: This circumstance lets the value of a syllogistic as such step back against the value of the method with the help of which the conclusion (as resultant) can be gained, similar to the much more general elimination problems, which can be conceived in the logic of the identical calculus of propositions.71

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5 Conclusion Ernst Schröder’s algebra of logic was created as a tool for dealing with the structural side of logic understood in a broad sense and embracing traditional logic as well. The rules of classical Aristotelian logic, in particular the theories of judgements and inferences, can be translated into the algebraic language, and with this be opened for the calculative method. Schröder was not very stringent in his symbolism. For reading convenience he used in parts an ad hoc symbolism like in the case of particularity. This might be the reason for the intriguing fact that he did not suggest a first-order interpretation of the categorical judgements as Frege did in his Begriffsschrift more than ten years earlier. This is the more astonishing as he had the necessary tools at his hands with his theory of quantification. Christine Ladd’s survey of the different approaches in the algebraic tradition to the propositional forms of traditional logic shows that Schröder was not alone.72 The translation of the syllogistic modes into the algebraic language forces to add existential suppositions where required, transforming the respective modes into inferences from three premises. So they lose their state as syllogistic modes. Without the additional premise, however, they would be invalid. Even the big program of a complete syllogistic would not change the limited character of syllogistic limited to inferences from two premises with three terms by eliminating the middle term. So syllogistic is only part of an algebraic logic, where inferences are made by calculation. The mode Barbara remains important (transitivity), but the main tool is the more general elimination theorem (normal forms).

Notes 1

2

3

On Schröder’s logic and its role in the emergence of modern mathematical logic see Volker Peckhaus, Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert (Berlin: AkademieVerlag, 1997), chapter 6, and Volker Peckhaus, ‘Schröder’s Logic’, in Handbook of the History of Logic, vol. 3 ed. Dov M. Gabbay and John Woods: The Rise of Modern Logic: From Leibniz to Frege (Amsterdam: Elsevier, 2004), 557–609. George Boole, The Mathematical Analysis of Logic. Being an Essay towards a Calculus of Deductive Reasoning (Cambridge: Macmillan, Barclay, and Macmillan/London: George Bell, 1847); George Boole, An Investigation of the Laws of Thought, on which Are Founded the Mathematical Theories of Logic and Probabilities (London: Walton Maberly, 1854). Cf., e.g. Thodore Hailperin, ‘Algebraical Logic 1685–1900’, in Handbook of the History of Logic, Vol. 3: The Rise of Modern Logic: From Leibniz to Frege, ed. Dov M. Gabbay and John Woods (Amsterdam: Elsevier, 2004), 323–88; Victor Sánchez Valencia, ‘Algebra of Logic’, in Handbook of the History of Logic, ed. Dov M. Gabbay and John Woods, (Amsterdam: Elsevier), 389–544, on Schröder 477–87; Volker Peckhaus, ‘The Mathematical Origins of Nineteenth-Century Algebra of Logic’, in The Development of Modern Logic, ed. Leila Haaparanta (Oxford: Oxford University Press, 2009), 159–95.

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Alfred North Whitehead and Bertrand Russell, Principia Mathematica, vol. 3 (Cambridge: Cambridge University Press, 1910–13). 5 Ernst Schröder, Der Operationskreis des Logikkalkuls (Leipzig: Teubner, 1977). 6 William Stanley Jevons, Pure Logic or the Logic of Quality apart from Quantity with Remarks on Boole’s System and the Relation of Logic and Mathematics (London: E. Stanford, 1864). 7 Ernst Schröder, Vorlesungen über die Algebra der Logik (exakte Logik), vols. 1 (Leipzig: B.G. Teubner, 1890); Id., Vorlesungen über die Algebra der Logik (exakte Logik), vol. 2, Pt. 1 (Leipzig: B.G. Teubner, 1891); Id., Vorlesungen über die Algebra der Logik (exakte Logik), Vol. 3: Algebra und Logik der Relative, Pt. 1 (Leipzig: B.G. Teubner, 1895); Id., Vorlesungen über die Algebra der Logik (exakte Logik), vol. 2, Pt. 2, ed. Karl Eugen Müller (Leipzig: B.G. Teubner, 1905). 8 Cf., e.g. Alexander Wernicke, ‘Review of Schröder, Vorlesungen, Vol. 1’, Deutsche Litteraturzeitung 12 (1891), cols: 196–7. 9 Schröder, Vorlesungen, Vol. 3.1, 1. 10 Ernst Schröder, ‘Über Pasigraphie, ihren gegenwärtigen Stand und die pasigraphische Bewegung in Italien’, in Verhandlungen des Ersten Internationalen Mathematiker-Kongresses in Zürich vom 9. bis 11. August 1897, ed. Ferdinand Rudio (Leipzig: Teubner, 1898): 147–62. English: Id., ‘On Pasigraphy. Its Present State and the Pasigraphic Movement in Italy’, The Monist 9, no. 1 (1898): 44–62, (1899) Corrigenda, 320. Cf. Volker Peckhaus, ‘Ernst Schröder on Pasigraphy’, Revue d’histoire des sciences 67 (2014): 207–30. 11 For the concept of an absolute algebra cf. Ernst Schröder, Über die formalen Elemente der absoluten Algebra (Stuttgart: Schweizerbart’sche Buchdruckerei, 1874; attachment to the programme of the Pro- und Real-Gymnasiums in Baden-Baden for 1873/74). Cf. Volker Peckhaus, ‘Wozu Algebra der Logik? Ernst Schröders Suche nach einer universalen Theorie der Verknüpfungen’, Modern Logic 4 (1994): 357–81; Davide Bondoni, ‘Structural Features in Ernst Schröder’s Work. Part I’, Logic and Logical Philosophy 20 (2011): 327–59; Davide Bondoni, ‘Structural Features in Ernst Schröder’s Work. Part II’, Logic and Logical Philosophy 21 (2012); 271–315. 12 Schröder, Vorlesungen, Vol. 1, 93. 13 Ibid., 94–5. 14 Ibid., 95, footnote, with reference to Schröder’s 1880 review of Gottlob Frege, Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Halle: Louis Nebert, 1879); Ernst Schröder, Review of Frege, Begriffsschrift, Zeitschrift für Mathematik und Physik, Hist.-literarische Abt 25 (1880): 81–94. 15 Schröder, Operationskreis, V. 16 Schröder, Vorlesungen, Vol. 1, 1–125. 17 Ibid., 92. 18 Schröder, Vorlesungen, Vol. 2.1. 19 Ibid., 85. 20 Schröder, Vorlesungen, Vol. 1, 157. 21 Ibid., Vol. 2.1, § 28, 1–24. 22 Ibid., 1. 23 Ibid., 3. 24 Ibid., 5. 25 Ibid., 6. 26 Ibid., 17. 27 Ibid., 25. 4

126 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

Aristotle’s Syllogism

Ibid., 26. Ibid., 27. Ibid., 49. Ibid., 50–1. Ibid., 51. Schröder marks the 1 with a dot on top to indicate that it is not the numerical 1. Ibid., 52. Ibid., 63. Ibid., 86–7. Ibid., 88. Ibid., 91. Ibid., 96–7. Ibid., 217–55. Ibid., 217. Friedrich Ueberweg, System der Logik und Geschichte der logischen Lehren, 5th ed. Jürgen Bona Meyer (Bonn: Adolph Marcus, 1857, 1882). 43 Schröder, Vorlesungen, Vol. 2.1, 220. 44 Ibid., 221. 45 Ibid., 225. 46 Ibid., 228. 47 Ibid., 233. 48 On Christine Ladd-Franklin see Francine Abeles’ contribution in this volume. 49 Christine Ladd, ‘On the Algebra of Logic’, in Studies in Logic. By Members of the Johns Hopkins University (Boston: Little, Brown, and Company, 1883), 17–71. 50 Schröder, Vorlesungen, Vol. 2.1, 228, n.: ‘According to a mail communication by a notable scholar and researcher at the University of Cincinnati, an expression which I gladly make to my own, because of his reliability know to me.’ 51 Ladd, ‘On the Algebra of Logic’, 40. 52 Boole, Laws of Thought, Ch. VII. 53 Ibid., 101, emphasis in the original. 54 Cf. in particular Schröder, Vorlesungen, Vol. 1, 573–92, on MacColl especially 589–92. In Vol. 2.1 of the Vorlesungen he discusses extensively MacColl’s methods in the calculus of propositions. MacColl’s priority is given in expressions like: ‘the Boole-McCol[!]l[sic!]-(Peirce)-Schröderian restriction of propositional values to 0 and 1’ (Schröder, Vorlesungen, Vol. 2.1, 482). Cf. the series of papers Hugh MacColl, ‘The Calculus of Equivalent Statements and Integration Limits,’ Proceedings of the London Mathematical Society 9 (1877/78): 9–20; Hugh MacColl, ‘The Calculus of Equivalent Statements (second paper), ’Proceedings of the London Mathematical Society 9 (1877/78): 177–86; Hugh MacColl, ‘The Calculus of Equivalent Statements (third paper),’ Proceedings of the London Mathematical Society 10 (1878/79): 16–28. On MacColl’s impact on the development of the algebra of logic, cf. Volker Peckhaus, ‘Hugh MacColl and the German Algebra of Logic,’ Nordic Journal of Philosophical Logic 3 (1998): 17–34; Irving H. Anellis, ‘MacColl’s Influence on Peirce and Schröder,’ Philosophia Scientiae 15 (2011): 97–128. 55 Schröder, Vorlesungen, Vol. 1, 447. 56 Ibid., Vol. 2.1, 232. 57 Ibid., 234. 58 Ibid., 241–2. 59 Ibid., 243–4.

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60 Ibid., 247, emphasis in the original. 61 Ibid., 244. 62 Hermann Lotze, Logik. Drei Bücher vom Denken, vom Untersuchen und vom Erkennen (Leipzig: S. Hirzel, 1874), 105. 63 Augustus De Morgan, Formal Logic: or, The Calculus of Inference, Necessary and Probable (London: Taylor and Walton, 1847), 56, n. 64 Ibid., 62. On De Morgan’s syllogistic theory cf. Anna Sophie Heinemann, Quantifikation des Prädikats und numerisch definiter Syllogismus (Münster: mentis, 2015). 65 Schröder, Vorlesungen, Vol. 2.1, 139. 66 Ibid., 136. 67 Ibid., 175. 68 Ibid. 69 De Morgan, Formal Logic, Ch. VI. 70 Joseph Gergonne, ‘Essai de dialectique rationelle’, Annales de mathématiques pure et appliques 7 (1816–1817): 189–228. 71 Schröder, Vorlesungen, Vol. 2.1, 370. 72 Ladd, ‘On the Algebra of Logic’, 24.

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Brentano and Hillebrand on syllogism: Development and reception of the ‘idiogenetic’ theory Matteo Cosci

According to the memoir of one of his many distinguished students, Brentano’s greatest merit as a thinker was extreme consistency and farsightedness in viewing lines of thought from top to bottom – premises and conclusions alike – that is to say, thinking vertically.  … His strength lay precisely in the deductive part of the method, in the conception of the most general points of view and the derivation of all their respective consequences for the interpretations of phenomena.1

Another ‘Franz’, namely Franz Hillebrand, also proved himself to be capable of thought-climbing ‘along the vertical line’, by standing on the shoulders of his teacher Brentano. For it was Hillebrand who brought to formal completion the ambitious project of logical reform that his supervisor had developed in teaching but only outlined in print and that otherwise might never have become accessible to public scrutiny. The aim of this chapter is to present their common ‘Denken in der vertikalen Line’, and particularly their proposed reform of syllogistic logic, which began and was developed in the last forty years of the nineteenth century in Vienna, when this city was the capital of the rising Austro-Hungarian Empire and one of the leading cultural centres in Europe.2 In the first part of the chapter (Sections 1–2) the main tenets of Brentano’s proposed reform of logic will be presented, together with an account of the first reactions they elicited. In the second part (Sections 3–4) Hillebrand’s formal derivation of Brentano’s syllogistic system will be schematized, followed by an account of the ‘second wave’ of reception. Finally (Sections 5–6), a possible source of influence will be discussed and a brief overview of some later developments within the ‘third wave’ of reception will be given by way of a conclusion.

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1 Brentano’s ‘teaser’ for a new logical system (1874) A proposal for a new reform of the syllogistic was emphatically presented in the first edition of Brentano’s Psychologie vom empirischen Standpunkt, more precisely in book II, chapter vii, paragraph 15.3 For all intents and purposes, those four pages were very much a ‘teaser’ for readers interested in logic and its psychological foundations. Brentano’s announcement was skilfully introduced there in strong opposition to the centuriesold school logic and presented as something revolutionary, ‘simpler, clearer, and more exact’ than the traditional syllogistic system.4 The proposal had been developed in lectures delivered over the course of almost fourteen years, and the results – Brentano announced – were to be published in a book after the completion of the Psychologie. Brentano characterized his forthcoming work on logic as a ‘complete overthrow, and at the same time, a reconstruction [of syllogistic] … built upon new bases’.5 The only aspect of Brentano’s proposal for a wholesale reform of syllogistic that readers could anticipate was that it would be a direct consequence of his new, underlying theory of judgement, described in the preceding pages. The starting point of the alleged reform was that all categorical propositions and judgements had to be restated in existential form and thus regarded as single-membered assertions or rejections. The categorical proposition ‘All men are mortal’, for example, was taken to mean that one accepts the fact that ‘there is no man who is immortal, i.e. not mortal’. However, in the Psychologie the process of deduction was not laid out on the basis of such an insight, nor was the alleged novelty of the ‘new’ syllogistic exposed in full in those pages. Only the most startling results of the anticipated reform were briefly outlined, leaving ‘to a future revision … the task of verifying and developing this in detail’.6 Brentano’s conclusions can be grouped into three sets of advancements: (A) new rules for the assessment of the validity of syllogisms; (B) the replacement of the old school-logic rules in syllogistic; and (C) the revision of the traditional rules for the conversion of categorical propositions. In particular, Brentano anticipated the following points: (A) Revision of the traditional rules for the conversion of categorical propositions: 1. Every categorical proposition can be simpliciter converted (ab = ba). 2. (So-called) universal affirmative propositions are no longer convertible into particular affirmative propositions. 3. Many of the traditionally accepted conversions between categorical propositions (e.g. conversion by subalternation) are no longer valid. (B)  Replacement of the old school-logic rules in syllogistic: 1. Four syllogistic forms traditionally regarded as valid, no longer are. 2. On the other hand, a high number of new valid forms are obtained. 3. Overall, the rules of substitution and derivation of traditional syllogistic are maintained.

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(C) Introduction of new rules for assessing the validity of syllogisms: 1. ‘Every categorical syllogism includes four terms, two of which are opposed to each other, and the other two appear twice.’7 2. ‘If the conclusion is negative, then each of the premises has in common with it its quality and one of its terms.’8 3. ‘If the conclusion is affirmative, then the one premise has the same quality and an identical term, and the other has the opposite quality and an opposite term’.9 As we shall see, the ‘teaser effect’ chosen for the presentation of the alleged logical reform did not elicit the kind of enthusiastic reactions that Brentano was expecting – and this, despite the fact that especially one of the above-mentioned points sounded very surprising and challenging to almost every student of logic of that time. This was the statement (listed as A2) according to which universal affirmative propositions were no longer convertible into particular affirmative propositions.10 It was a point that had been discussed before, for instance by Augustus De Morgan,11 but which now was placed at the centre of the new logical reform and considered for the first time within the framework of descriptive psychology.

2 The first wave of reception ‘If these doctrines can be made out, obviously all logicians from Aristotle downwards have been sheer impostors, but the probability is great that they cannot be made out, and that the views as to the nature of conception and judgement from which they have been derived are erroneous. We shall look for Prof. Brentano with the most lively curiosity.’ With these intimidating words, Robert Flint (University of St. Andrews) hailed Brentano’s anticipated logical reform in the first issue of the newly founded journal Mind, challenging the proposal even before its actual exposition.12 Flint’s remark was followed by a communication signed by Prof. Jan P. N. Land (Leiden University) which appeared in the scientific correspondence section of the same issue. Land specifically tackled the logical section of Brentano’s Psychologie. The so-called innovations put forward by Brentano were debunked as merely surreptitious changes made to long-established terminology, which in Land’s view just expanded upon some earlier ideas developed by Herbart, Drobisch and Mill.13 Surprisingly, Land managed to anticipate and put in writing almost the entire syllogistic system (Jevons’s adopted notation included) on the mere grounds of the scattered allusions in Brentano’s proposal. The main point of contention, however, was that universal propositions were said to have no existential import. For Land, existential import was always – and always had to be – understood and presupposed, while for Brentano it was an assumption that should never be taken for granted. But again, for Land changing categorical propositions into existential ones, as Brentano had proposed, was a huge mistake because for him relevant parts of the implied meaning were lost in Brentano’s conversion.14 Land reiterated this point in an extended version of his communication that appeared in

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a Dutch journal a couple of weeks later: ‘[I]n translating categorical universals into existential negatives, part of the meaning is dropt [sic] by the way, and precisely that part on which the condemned logical operations depend.’15 He countered Brentano’s point because he thought that ‘in a categorical proposition the subject is necessarily presupposed to exist in some way or another’.16 Likewise, Christoph von Sigwart, professor of philosophy at Tübingen, criticized the idea at the core of Brentano’s syllogistic, namely that the basic structure of judgements is a single-membered form always implying the acknowledgement (or rejection) of the existence of the ‘matter’ of representation.17 Brentano discussed the first critical note by Land with his student Anton Marty. In a letter dated 15 April 1876, Land’s attempt to disprove the propositional reducibility of categorical propositions to existential ones was taken as a sign that even for him the issue at stake was not just a play on words.18 In addition, Brentano thought to have scored a point that Land had implicitly conceded: if existential import was not presupposed, then his rejection of some syllogistic forms was indeed correct. He exemplifies the result to Marty by means of a syllogism in Darapti: ‘All devils have been damned by God’; ‘All devils are spirits’; ∴ ‘Some spirits have been damned by God’.

Brentano explains that ‘[n]o one, who does not believe in Hell [i.e. no one, who does not give assent to the existence of at least one spirit that is a devil] can assert the conclusion: “some spirits are damned by God”’.19 For if there are no devils, then none of them is a spirit and none of them has been damned by God, therefore nobody could deduce from those premisses that some spirits have been damned by God, contrary to what the conclusion asserts. Brentano would later praise Land for his understanding and give credit to him for having been ‘the only one of my critics who has understood what Windelband [sc. Wilhelm] has called my “mysterious” suggestions for reforming elementary logic … and derive[d] correctly their necessary connections from this principle’, i.e. the principle of not granting existential import to universal categorical propositions.20 More generally, the main issue in this debate was to try to assess the alleged novelty, if any, of the syllogistic scheme presented. More precisely, the issue at stake was twofold: while Brentano considered his work to be novel with respect to traditional syllogistic (as represented, for instance, by the Scholastic scheme of his own teacher Franz Jakob Clemens21), his critics were mainly wondering whether it was a real improvement given simultaneous advancements in the field. John Venn had noticed some similarities with remarks that he himself and the early George Boole had made regarding the semantic of categorical propositions. For instance, Venn recalled Boole’s rendering of ‘Some X is Y’ as ‘xy = v’,22 whereby a particular affirmative proposition was read as implying the exclusion of the value 0 or ‘none’. Exactly as Brentano had independently done, Venn noted: What we thus lay the stress on is the existential character of such propositions. The expression xv = v may be read off in words as ‘xy is something’, i.e. is not nothing; or more colloquially, ‘There is xy’, or ‘x and y are sometimes found together’. The

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last two of these forms … would, I think, be naturally accepted as equivalents of the logical particular affirmative. … In quite recent times the same arrangement substantially has been adopted by Professor F. Brentano (Psychologie vom empirischen Standpunkt, 1874), but he does not extend it beyond the four familiar propositions. He announces it as a startling novelty which is to spread dismay among orthodox logicians. … he springs to the conclusion that the new mode of notation is to supersede altogether the traditional one; instead of being as I should say, an alternative method, not necessarily hostile to the old one, but far more suitable for the treatment of complicated problems and broad generalizations.23

Because of the similarities with Brentano’s formulation, this reading later came to be referred to as ‘the Brentano-Venn interpretation’,24 although Brentano developed it from the point of view of descriptive psychology, while Venn adopted a purely logical perspective. In any case, the Cambridge logician John Neville Keynes would regard Brentano’s logic as ‘practically identical’ to that of Venn (with whom he fundamentally agreed), despite its having been ‘put forward in a more paradoxical form’.25 Venn, however, was not the only British logician who converged towards Brentano’s existential logical reductionism. Interestingly, some exponents of neo-Hegelian British idealism, such as Francis Herbert Bradley and Bernard Bosanquet, found in Brentano’s theory of judgement a modern reading of the ‘reality of rationality/rationality of reality’ postulate. For, as Bradley wrote with reference to Brentano, ‘wherever we predicate, we predicate about something which exists beyond the judgement, and which (of whatever kind it may be) is real, either inside our heads or outside them. And in this way we must say that “is” never can stand for anything but “exists”.’26 Bosanquet, for his part, pushed Bradley’s theory even further, anticipating through a different (Hegelian) path a conclusion that Brentano was to reach in the last phase of his thought, namely that by asserting or rejecting categorical judgements, one always implies the existence, or non-existence, of their content matter.27

3 Hillebrand’s formal presentation of Brentano’s syllogistic system (1891) Franz Hillebrand provides a formal presentation of Brentano’s projected syllogistic in his Die neuen Theorien der kategorischen Schlüsse (The New Theories of Categorical Inferences).28 This book was probably inspired by Alexius Meinong, who was a former student of Brentano’s himself. Indeed, the year before Meinong had published a recusant logical handbook, written with the assistance of Alois Höfler.29 From those pages Meinong and Höfler opposed Brentano’s theory, claiming that logical judgements are not just about existence but about relations. They also emphasized the impossibility of any legitimate deduction with reference to imaginary, fictitious or non-actuallyexisting entities as a serious limit of Brentano’s alleged logical reform. As a response, Hillebrand laid out Brentano’s system with the idea of clarifying the newly proposed syllogistic and defending it also against these charges.

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To this purpose, Hillebrand’s book first presents the so-called idiogenetic theory of judgement advanced by his teacher in opposition to several competing theories and then introduces the ‘thetic’ judgement as the most basic form of judgement that a subject can accept or reject. Hillebrand labels the theory as ‘idiogenetic’ because, according to Brentano, the act of judging belongs to a non-reducible and special genus of psychical phenomena (from the Greek ‘ídion génos’),30 while the most fundamental forms of judgements were called ‘thetic’ (thetische Urtheile) as they just ‘put forward’ the being of what they state (‘títhemi’ meaning ‘to put’ or ‘to lay’). Beyond nomenclature issues, however, the author proceeds to establish the applicable rules of the system and to outline the actual deduction of valid syllogisms on their basis. For the sake of brevity, the whole process of construction of the new syllogistic system can be explained in six steps, which will here be summarized in six tables (Tables 7.1–7.6). Table 7.1  Brentano-Hillebrand: Simple conventions for logical notation are stipulated Symbols + – S (capital letter) s (non-capital letter)

Meanings (there) is, i.e. is accepted in so far as it is (there) is not, i.e. is rejected in so far as it is not positive content or ‘matter’ of judgement opposite (by contradiction) content or ‘matter’ of judgement

Table 7.2  Brentano-Hillebrand: The traditional types of categorical propositions are understood in their existential form, and so are the respective judgements Types of categorical proposition univ. affirmative univ. negative partic. Affirmative partic. negative

A E I O

‘All S are P’ ‘No S is P’ ‘Some S is P’ ‘Some S is not-P’

Symbols

Stand for

Sp – SP – SP + Sp +

= = = =

Translated meaning ‘S which is not-P is not’ ‘there is no S which is P’ ‘there is an S which is P’ ‘there is an S which is not-P’

Table 7.3  Brentano-Hillebrand: The applicable rules which govern the relations between categorical propositions in their existential form are established on the grounds of nongranted existential import Table of inferential conversions (1) the conversio simplex (SP = PS) is allowed for all the four types of categorical propositions (2) the conversio per accidens is allowed from A-types to E-types, but not from E-types to I-types (3) inference by subalternation is never allowed (4) opposition by contradiction remains effective as usual (5) inference by contrariety is not allowed (6) inference by sub-contrariety is not allowed (7) aequipollence is reduced to propositional identity (8) two further inferences (i.e. ‘mereological laws’) are specified – See point (b) in the following table

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Table 7.4  Brentano-Hillebrand: An axiomatic starting point of the new system is set (a) non-contradiction and excluded middle hold as general principles (both a priori and empirically valid) (b) the affirmation of a matter of judgement includes the affirmation of each of its parts, and the negation of a matter of judgment is included in the negation of its parts31 (c) immediate inferences, or ‘proto-syllogisms’, α and β (below) are posited at the start of the process of deduction:



α). AB – Ab – β). AB – A + also in the form: A + Ab – ___ ___ ___ Ab AB + A –

Following the rules of substitution and given the axioms established before (in Table 7.3), and in accordance with the permitted inferential conversions (of Table 7.4), twenty-four forms of valid syllogism are deduced. In particular, syllogisms 1–16 are derived from the starting inference α (from its two equivalent forms), while syllogisms 17–24 are derived from inference β. Table 7.5  Brentano and Hillebrand’s Syllogistic Scheme. 01. MP – SM + ___ Sp +

02. Mp – SM + ___ SP +

03. MP – sM + ___ sp +

04. Mp – sM + ___ sP +

05. MP + SM – ___ sP +

06. MP + sM – ___ SP +

07. MP + SM – ___ sp +

08. Mp + sM – ___ Sp +

09. mP– Sm + ___ Sp +

10. mp – Sm + ___ SP +

11. mP – sm + ___ sp +

12. mp – sm + ___ sP +

13. mP + Sm – ___ sP +

14. mP + sM – ___ SP +

15. mp + Sm – ___ sp +

16. mp + sm – ___ Sp +

17. MP – Sm – ___ SP –

18. mP – SM – ___ SP –

19. Mp – Sm – ___ Sp –

20. mp – SM – ___ Sp –

21. MP – sm – ___ sP –

22. mP – sM – ___ sP –

23. Mp – sm – ___ sp –

24. mp – SM – ___ sp –

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Table 7.6  Hillebrand’s categorical propositions as implying double judgements Types of categorical proposition univ. affirmative

A

‘All S are P’

univ. negative

E

‘No S is P’

partic. Affirmative

I

‘Some S is P’

partic. Negative

O

‘Some S is not-P’

Symbols

Stand for

S Sp – S SP – S SP + S Sp +

= = = =

Translated meaning ‘There are some S’ and ‘S which is not-P is not’ ‘There are some S’ and ‘there is no S which is P’ ‘There are some S’ and ‘there is an S which is P’ ‘There are some S’ and ‘there is an S which is not-P’

Overall, twenty-four was a larger number of forms than that commonly accepted by traditional school-logic. It was a result Brentano and Hillebrand took pride in. Moreover, many of the traditional forms were also included within the new system.32 Four forms that were traditionally accepted (namely Darapti [AAI, 3rd figure], Felapton [EAO, 3rd figure], Bamalip [AAI, 4th figure] and Fesapo [AAO, 4th figure]) turn out to be invalid because of non-granted existential import and, for this reason, are excluded from the system. The three criteria (C1–3) anticipated by Brentano in his Psychologie are here fulfilled (Table 7.5): the quaternio terminorum and the expected qualifications of the syllogistic inferences are conditions that are met in each of the twenty-four forms. Retroactively, these rules can work as a consistency test for checking the validity of all the syllogisms within the system. Hillebrand adds one last section to the book (chapter 5) with the aim of widening the field of application of the logical theory he has outlined to complex, or ‘double’, judgements. This addition is also developed in consideration of the early criticisms received by Brentano, but some hint in this direction was already present in the logical section of the Psychologie.33 According to the extension, categorical propositions can be read as logically equivalent to their simple forms and the acknowledgement of the existence of at least one item of what they claim to be such-and-such, as in the last table (Table 7.6). In this way, the inferences by subalternation which were considered illicit can be restored, but so can those forms of traditional syllogisms which were previously considered invalid (Darapti, Felapton, Bamalip and Fesapo). The conclusion of Hillebrand’s book would eventually prove critical, as it soon became a speculative battlefield of contention for those students of Brentano who defended and developed it,  such as Marty,34 as well as for those who attacked it, such as – most notably – Edmund Husserl.

4 The second wave of reception Upon its publication, Hillebrand’s book was immediately recognized as the ‘Logic’ that Brentano had announced as forthcoming seventeen years earlier. So, as expected, it renewed the interest in the anticipated reform and consequently led to a second wave of reception of Brentano’s work across European and U.S. universities.

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One year after its publication, Alexius Meinong (by this time professor at the University of Graz) basically took Hillebrand’s book as a personal attack against him and his review was to all intents and purposes a negative one.35 There he demystifies Brentano’s quaternio teminorum as a mere linguistic trick, rejects the refusal of subalternation as completely arbitrary, and assesses Brentano’s whole reform as unnatural and uselessly cumbersome.36 On the opposite side of the spectrum of this reception a French reviewer, Charles Andler, presented Brentano and Hillebrand as the vanguard of the new trend in logic. Their syllogistic, Andler wrote, was a fruitful endeavour and one that had arrived at a time of general reconstruction of the discipline of logic, following a long period of anarchy in the field.37 Miss E. E. Constance Jones, who had been studying with Neville Keynes, dismissed the alleged novelty of the quaternio terminorum and the rule according to which ‘ex mere affirmative nihil sequitur’. She concluded that, on close examination, the former innovation is merely a terminological redundancy, while the latter is basically equivalent to the two traditional rules for particular premisses taken together, namely that ‘from negative premisses no conclusion follow’ and that ‘from two particular premisses no conclusion follows’. The affirmative propositions of the new proposal, in her view, were basically those that in traditional logic were considered particular ones. Moreover, Constance Jones understood that Hillebrand had tried to replace the traditional dictum de omni et de nullo principle with his ‘new’ bundle of α and β inferences, with the addition of the principle of the Excluded-Middle (as in Table 7.4 above). But the resulting starting point, she argued, ‘seems to be closely akin to the rejected dictum’.38 She did concede, however, that ‘what is really novel of this syllogistic scheme is the exclusion of any syllogism with universal premisses and particular conclusions’.39 She later returned to this issue in greater detail in a paper published in The Proceedings of the Aristotelian Society. Here, she considered the whole scheme ‘possible’, albeit ‘extremely remote from ordinary thought’.40 Her American colleague, Christine Ladd-Franklin, stripped Hillebrand’s system of the only actual novelty that Constance Jones had granted it, namely the exclusion from the system of any syllogism with universal premisses and particular conclusions. Instead, Ladd-Franklin attributed this reduction to the work of the Scottish logician Hugh McColl.41 Furthermore, she reclaimed for herself the idea of a proper alternative to the dictum de omni et de nullo principle, one that was based on a better and more general use of the principle of non-contradiction and the Excluded-Middle. The same goes for a special axiom of ‘under-statement’, as she called it, that more or less resembled Hillebrand’s ‘mereological principle’.42 Finally, Wilhelm Enoch, a humanities teacher at the Diedenhofen gymnasium, published a long article on Brentano’s reform of logic based on Hillebrand’s formal presentation.43 Like Meinong before him, Enoch attacked the pupil in order to attack the teacher standing behind him. The main charge against Brentano’s logical theory is that it entails a form of psychologism or subjectivism. For all judgements in their existential forms are said to be intentionally accepted or rejected by some subject, so that any deduction depending on them will basically result in some kind of selfreferential solipsism. Also from a formal point of view, Enoch holds, Brentanos’s thetic judgements do not make for a fruitful syllogistic, since they just posit something as

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‘existent’ in the consciousness of the one who judges without establishing any relation with the external world. Moreover, according to Enoch the introduction of negation (p as non-P) tacitly restores the quantification that Brentano and Hillebrand wanted to avoid by resorting to simple judgements. The bottom line for Enoch is that the twenty-four ‘new’ forms do not prove to be an adequate substitution for the old ones. Finally, the conclusive introduction of double judgements into the system strikes him as a last capitulation or at least as a mere stratagem to allow for the entrance through the backdoor of those predicative judgements that had already been thrown out of the window.44 Many of these issues were taken up by the early Husserl. The Logical Investigations can be considered his formal answer to Brentano’s logic.45 It is well known that Husserl formulated his articulate response to Brentano’s theory of judgement in his fifth logical consideration, whose critical remarks dated back at least to his 1896 lecture on logic.46 However, much of what Husserl wrote against the Dutch psychologist Gerardus Heymans47 in the sixth prolegomenon to the Investigations can also be understood as an indirect attack on Brentano and Hillebrand’s logic. Indeed, Heymans is openly criticized there for misidentifying the logical laws of syllogistic with the psychological laws of thought. It was a mistake that in Husserl’s eyes Brentano had committed as well, as he also made clear in a letter directly addressed to his former teacher: ‘The modus Barbara like any logical or mathematical axiom contains not the least bit of physical or psychological “nature”’ (dated 27th March 1905).48 According to Husserl, Brentano had an ambiguous concept of judgemental matter, since he conflated the object-matter of judgements in the sense of their material content with the subjectmatter of judgements in the sense of their referentiality or ‘aboutness’. Consequently, the acknowledgement of something and the content acknowledged could not be properly distinguished, either in the act of psychological representation or in any deductive process which might follow from it.49 In Husserl’s understanding, therefore, Hillebrand’s syllogistic was biased because of this undue conflation of objective truth with subjective correctness. In this sense, Brentano’s idea of judgement was accused of losing sight of ‘things in themselves’, whose reference for Husserl is always intentional and immediate, even in the case of hypothetical judgements.50 Finally, the Doppelurteil theory seemed to him to be lacking an explanation for the grounding of the unity of double judgements in single acts of intentional acceptance or rejection. Furthermore, double judgements were intended to describe a twofold mental process – acknowledgement and predication – that for the phenomenologist finds no correspondence in actual acts of judging.51 Also the use of negation was deemed as inconsistent since it was seen indiscriminately applied to the matter of the judgement and to the judgement in its entirety.52 In conclusion, for Husserl the condition of acknowledgement or acceptance (Anerkennen) was redundant and the attempt to reduce all categorical judgements to existential ones was misleading. It goes without saying that from his point of view the attempt to build a syllogistic on such premises could not find any grounding on either the logical or psychological level. It was precisely the attempt to develop counterarguments to these criticisms that eventually led Brentano and some of his students to a new phase in their thinking.

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5 A Leibzinian basis for Brentano’s logic? Brentano never referred to Leibniz in his published work on logic (and neither did Hillebrand, for that matter), but he often mentioned him in his lectures. Here, we find many references to Leibniz’s pursuit of a universal logical language (or ‘characteristica universalis’) as an objective to be taken up and implemented. Brentano almost certainly became acquainted with this ambitious project through the lecture ‘On Leibniz’s Sketch of a Universal Characteristics’ delivered by his teacher Trendelenburg at the Berlin Academy of Science in 1856.53 Leibniz’s logical vision appears both in Brentano’s Würzburg teaching notes on Logik from 1878 to 1879 and in the Vienna teaching notes on Elementare Logik und die in ihr nötingen Reformen from 1884 to 1885.54 In his notes from the latter course, Hillebrand mentioned that Brentano had regarded the development of a universal logical language in the style of Leibniz as something that should have been accomplished.55 In other words, Leibniz’s efforts towards an improved and universally understandable ‘art of inferring’ struck Brentano as a project that deserved to be continued and eventually brought to completion, and this despite Trendelenburg’s respectful scepticism about such a programme.56 The name of Leibniz (or Leibnitz) also crops up several times in Brentano’s Würzburg notes for the course on Logik – Deduktive und Induktive Logik from the winter semester of 1869–1870.57 Here Brentano not only refers to Leibniz’s main works, such as the Monadology and the Theodicy, but also mentions a couple of Leibniz’s letters to Samuel Clarke, as well as his long essay-letter to Gabriel Wagner The Use of the Art of Reason or Logic from 1696, which contained references to Leibniz’s own advancements in syllogistic.58 These passages may have sparked Brentano’s interest in non-standard takes on Scholastic syllogistic while he was lecturing on that topic too. Indeed, another passage from Brentano’s teaching notes for his Logik lectures offers a double crossreference to Leibniz’s advancements in logic in the form of side notes: It is commonly thought that all logic is finished and done. By thinking in this way, one does too great a honor to logic and belittles one’s own. So much is lacking. It is also wrongly said that there is no mistake in logical teachings as they are usually presented. [⇐ cf. Leibnitz]. We shall see. [⇐ Passages from Leibnitz].59

Interestingly, there are some similarities between Brentano’s and Leibniz’s development of an ‘alternative’ syllogistic in terms of their peculiar concern for the existential import of categorical propositions. Many have written that Brentano could not have been familiar with Leibniz’s achievements, because they were contained in a work – the Generales Inquistiones de Analysi Notionum et Veritatum – that was retrieved only as late as 1901, by Louis Couturat, many years after Brentano had published his own results.60 Nonetheless, Leibniz had also re-stated his conclusions in another short work, namely the Difficultates Quaedam Logicae, which was a text actually available to Brentano and Hillebrand, as it could be found in all the previous editions of Leibniz’s collected writings, such as those by Raspe, Erdman and Gerhard (available from 1765, 1840 and 1890, respectively).

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In the first part of that essay, Leibniz focuses on some of the problems that stand at the heart of Brentano’s later logical agenda. First, Leibniz notes that a singular proposition is implicitly understood in every universal and particular proposition. Secondly, Leibniz finds that associated subjects and predicates are logically equivalent and thus reducible to single terms of judgement (‘Omne A est B, id est AB aequivalet ipsi A’). Next, Leibniz presents his ‘old’ reading of categorical propositions, similar to the scheme he developed in his Generales Inquisitiones, in existential terms: ‘Every A is B’, i.e. ‘AB and A are equivalent’, or ‘A non-B is a non-entity (A non B est non-ens)’ ‘Some A is not B’, i.e. ‘AB and A are not equivalent’, or ‘A not-B is an entity (A non B est ens)’. ‘No A is B’, i.e. ‘AB is a non-entity (AB est non ens)’ ‘Some A is B’, i.e. ‘AB is an entity (AB est ens)’.

Leibniz realized that affirmative particular propositions imply the subsistence of their object while the corresponding universal propositions do not necessarily imply it. For this reason he came to doubt the legitimacy of the conversion per accidens and by subalternation from the latter to the former. In order to preserve that possibility, existential import had to be implied in the case of affirmative universal propositions and consequently in all the other quantified propositions (‘in omnibus tamen tacite assumitur terminum ingredientem esse Ens’, he wrote). In this passage, the reduction that was long thought to be found exclusively in his Generales Inquisitiones reappeared in a form in which the existential import was implied as ‘suppositus’. Supposition aside, the scheme of categorical propositions translated in existential form is the point of departure of Brentano’s reform of syllogistic. It is possible that the previously noted cross-reference to Leibniz concerned this passage, since, as already noted, as early as 1869 Leibniz’s work on logic was on Brentano’s teaching agenda. Of course, Brentano presented his results in a very different order. From the acceptance of the thetic form of judgements, he arrived at the revision of the conversions of categorical propositions, which in turn led to a renewed syllogistic, but one may possibly suggest that Brentano came to his conclusions the other way round, namely by proceeding from the reading of Leibniz to the recontextualization of the issue of existential import in the field of descriptive psychology. It could also be suggested that Brentano knew of the ‘existential import’ problem only indirectly, in the form in which it was discussed for instance by Johann Friedrich Herbart, Moritz Wilhelm Drobisch or Friedrich Ueberweg.61 Mill, one of his main and explicit sources, also held that analytic propositions do not involve the existence of their referents.62 Among his points of references, Hillebrand also acknowledged Trendelenburg, Lotze and Lange, and all three expressed doubts about the legitimacy of usual conversions of categorical propositions.63 Trendelenburg maintained that the whole doctrine of conversion standing at the basis of traditional syllogistic was doubtful and in some cases even incorrect.64 Lotze wrote that, in general, in ‘the universal affirmative judgement “all S are P” […] there is a part of P left which has nothing to do with S, and only impure conversion can take place into the particular judgment

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“some P are S”’.65 Expressing the same concern, Lange also referred to another work of Leibniz (namely De stilo philosophico Nizolii, which was accessible to Brentano and Hillebrand too), where it was made clear that a universal attribution should always be understood as a ‘totum distributivum’ and never as a ‘totum collectivum’ with respect to its constitutive class of individuals.66 Leibniz and Brentano followed more or less the same path. However, from this point onwards they parted ways. While for Leibniz the assumption of existential import had to be hypothetically supposed, for Brentano it was never granted and thus always had to be specified.67 For this reason, it has been said that Leibniz’s logic basically only worked on an intensional level, whereas Brentano’s worked on both an intentional and an extensional level.68 In actual fact, however, one step of the argument in the Difficultates seems to hinge on the extensional level too, in a passage which resonates with Brentano and Hillebrand’s concern about what is actually being understood in a universal affirmative judgement: ‘Every laugher is a laugher, every laugher is a man, ergo some man is a laugher’. On the present interpretation that means, ‘All laughers are among laughers, all laughers are among men, ergo some men are among laughers’. But what if no man actually laughs? I assert that the proposition ‘All laughers are among men’, i.e. ‘All laughers are men’ is also false, because for it to be true, the proposition for ‘Some laughers are among men’, or, ‘Some laughers are men’, will also be true. But that proposition is false, if no man laughs.69

As we have seen, when Leibniz provides a demonstration of subalternation and conversio per accidens, he feels compelled to specify that ‘A est Ens (ex hypotesi)’ by the introduction of an additional premise, which claims the presence of ‘A’ as given.70 In other words, Leibniz needed to specify the existential import in those propositions, which did not exhibit that feature in and of themselves in order to save the scheme of conversions standing at the basis of the traditional syllogistic. Therefore, he ended up attributing existential import to universal affirmative propositions as something that is always implicitly understood with them. As already noted, ‘this seems clear confirmation of the view that for Leibniz the universal affirmative proposition does not have existential import’.71 From the above-mentioned passage, it may indeed seem that Leibniz’s logic of categorical conversions and Brentano’s are even more alike than previously thought. This remarkable convergence did not go unnoticed among logicians and was highlighted both in print and in private writings. Jevons, for one, drew attention to the advancements proposed by Leibniz in his Difficultates, and in particular his ‘reduction of the universal affirmative proposition to the form of an equation, which is the key to an improved view of logic’.72 Venn noted this too, while also acknowledging that Brentano, alongside Boole, had ‘substantially’ adopted the Leibzinian pattern of propositional conversion.73 Finally, mathematician Giovanni Vailati,74 in a letter to Brentano, wrote about the similarities between the latter’s development of logic and Leibniz’s. Hence, we know that Brentano was at least informed about these similarities and that they were publicly recognized at the beginning of the century.75 In all three cases, the Difficultates Quaedam Logicae was explicitly mentioned, and in the last

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two it was directly associated with Brentano’s Psychologie. Brentano, for that matter, never retracted his syllogistic but limited himself to saying that ‘Leibniz was far from misunderstanding the meaning of syllogism as an instrument for expanding human knowledge!’,76 for ‘Aristotle agrees with Leibniz in the conviction that the syllogism is useful’ and indeed ‘yield[s] a true increase of knowledge’.77 It is interesting at this point to reread the actual source on the basis of which Oskar Kraus (Brentanian student via Anton Marty) excluded the possibility that his teacher may have derived his theory from Leibniz, rather than developed it on his own.78 For, the source he referred to (Willy Freytag, professor of philosophy at the University of Zürich) indeed instilled the suspicion of some possible borrowings: [Louis] Couturat wrote a short but important note about the general theory of judgment by Leibniz [Couturat 1901, 350, n. 4]. Leibniz’s theory seems to have first appeared in the aforementioned Inquisitiones Generales and to have found a sort of epilogue in his Difficultates logicae. The latter work had already been published by Erdmann (1840, 101–104) and Gerhardt (VII, 1890, 211–217), I must say, so it is possible that the existential theory of judgment as it is presented therein was not without influence on the corresponding theories of our time. In all events, McColl should not be considered a rediscoverer of the theory with his 1878 contribution [as Couturat maintained], but perhaps it is rather Brentano that should be considered the rediscoverer. As a matter of fact, Couturat does not seem to be aware of Brentano’s Psychology from 1874. And, as far as we know, it may be the case that Leibniz too was building his own theory on earlier attempts in his works. It should be remarked, however, that Leibniz’s theory is exactly like Brentano’s, or at least part of it.79

Brentano’s possible source of influence should not come as a surprise. Leibniz’s logic had a huge and wide impact during the nineteenth century, especially in Frenchand German-speaking countries. This influence spread more or less in parallel to the rediscovery and study of his logical writings.80 Brentano, like most of his contemporaries, was trained in neo-Scholastic logic.81 For many nineteenth-century logicians Leibniz increasingly came to represent the most natural alternative to what was perceived to be the straitjacket of a worn-out and stifling Aristotelian tradition. Perhaps, then, Brentano’s name should be added to the long list of logicians who developed their own contributions in the light of Leibniz’s early innovations.

6 Later reworkings and the third wave of reception In 1889 Brentano published an essay review of Franz Miklosich’s book on subjectless propositions.82 He appreciated that book as much as its first edition of 1865 because he saw Miklosich’s research as a sort of independent, linguistic counterpart to his own psychological endeavour. In consonance with the Slovene philologist’s work, Brentano remarked that all kinds of unitary judgement (i.e. the categorical, the hypothetical and the disjunctive) could be more fundamentally expressed in the form of impersonal, existential propositions. The latter – he added – can in turn be coupled to form double ‘judgments in which something is first accepted as existing and in which something

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else is then either affirmed or denied of the first thing’.83 On that occasion, he also recalled with dismay the effort that he put into all his logical work and the lack of understanding that had accompanied its reception: ‘I took great pains to put the view in a clear light and to show that the previous theories were untenable, but up to now what I said does not seem to have had much effect.’84 In the same year, Brentano returned to the topic of syllogism, which he came to generalize as any form of deductive reasoning. He sided with Descartes and Locke, in addition to Leibniz, in defending the view that syllogism and deductive reasoning in general are ampliative, i.e. can expand our knowledge. This went against the dominant view held by a long list of philosophical opponents whom he challenged: Plato, Bacon (partly), Hume, Kant, Bain, Lange, Mill, Helmoltz, Sigwart and Poincaré, among others. Brentano firmly believed that ‘it was possible to reason syllogistically long before the discovery of the rules of syllogism … since men were able to reason correctly for thousands of years without having reflected upon the principles of valid reason and even without knowing anything about them’.85 In a conference that he gave at the Philosophical Society of Vienna in 1890, he argued that the rules of inference, including syllogistic laws, are neither entirely empirical nor entirely a priori.86 As he explained in the essay presenting his theory of knowledge (‘Nieder mit den Vorurteilen!’ from 1903), syllogistic laws are cases of application of the principle of non-contradiction, but the principle itself is neither entirely empirical nor entirely a priori.87 For him syllogisms are ultimately grounded in evident judgements which are evident because they describe inner perceptions.88 Regardless the received criticism of psychologism, Brentano was persuaded that the kind of intrinsic causality which ‘sets in motion’ syllogistic conclusiveness and the subject’s acknowledgement of its truth was almost a sort of efficient causality, as if it were physically expressed within the mental act of reasoning.89 In 1911 Brentano republished the second volume of his Psychologie (under the title of Von der Klassifikation der Psychischen Phänomene), to which he added several relevant appendices. Remarkably, in Appendix IX he radicalized his view of the existential import of categorical propositions. Not only do judgements in the form of universal affirmative propositions not entail the existence of the object they concern but, on Brentano’s new point of view, they actually entail the non-existence of their object for the reasoner. Brentano realized that statements can thus remain in principle true even if the intended class of items results in an empty set.90 He outlined the consequence of this radicalization for hypothetical/disjunctive inferences,91 while leaving the work of its syllogistic extension in the light of the temporal mode and the apodictic character of judgements up to his students. Finally, in his 1916 essay against Poincaré’s theory of knowledge Brentano put forward the idea that a syllogism is a sort of complete arithmetic induction. More specifically, Brentano connected the syllogism to the proof by induction of the recurrence relation: if a property applies to 1 and n +1 in a set (where ‘n’ is any whole number), then it applies to each whole number in that sequence.92 At the end of his career, Brentano’s interest shifted from syllogistic and quantified deduction to mathematics and a more general ‘science of quantity’, with the goal of extending the latter to the former by adopting the existential judgement as a unit of measurement.93 By the beginning of the new century the deductive method developed by Brentano gained credit in the fields of cognitive psychology and early phenomenology, as well as

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in the emerging tradition of Austrian analytical philosophy.94 At the beginning of the century, while Frege’s doctrines were not yet widespread, Brentano and Hillebrand’s syllogistic was heatedly discussed and criticized.95 The legacy of Brentano’s logic was defended at first by his student Anton Marty96 and then by Marty’s students Oskar Kraus and Alfred Kastil.97 The latter even completed a book, posthumously published, where he described at length the mathematical implications of Brentano’s and Hillebrand’s syllogistic.98 Another illustrious student of Brentano, Kazimierz Twardowski, defended the Brentanian theory of judgement against the rival logistic tendency. He even started writing a book (which was never brought to completion) intended to enhance what he considered to be the most neglected aspect of Brentano’s philosophy, namely his logic. Twardowski not only found it original and interesting per se but also relevant from a historical and systematic point of view. His thought, in turn, brought Brentano’s logic to the attention of two generations of Polish students, as represented by Bandrowski, Kotarbiński, Ajdukiewicz, Leśniewski, Czeżowski and Słupecki. Łukasiewicz’s ‘resurrection of Aristotelian syllogistic’, as it has been called, ‘owes much to Brentano’s example in showing that modernized methods can be brought to bear on traditional forms of inference without compromising logical rigor’.99 In Italy the mathematician Giovanni Vailati, a member of Peano’s school, together with his collaborator Mario Calderoni, credited Brentano with the idea that asserting a universal affirmative proposition amounts to denying the existence of something (a ‘limitation of expectation’, in the words of Ernst Mach).100 They also recognized that in this respect Brentano was anticipated by Leibniz and followed by logicians belonging to Boole’s school.101 Vailati planned to write a monograph on Brentano-Hillebrand’s syllogistic, although that project would never see the light. Brentano was informed of this project by a common friend, Amato Pojaro, and recommended Vailati to take into account the ‘unreasonable criticisms’ of Land and Enoch.102 In those years, the Italian journal La Cultura Filosofica published some contributions on Brentano’s logic, too.103 There, Giovanni Calò critically contested Brentano’s theory of judgement104 and found an error (repeated twice) in the syllogistic derivation presented by Hillebrand (namely the inference of ‘MP –’ from ‘SMP –’ in the execution of the proof procedure) and showed that it infringed the stipulated ‘mereological rule’.105 A little while later, Guido Rossi analysed the whole of Brentano-Hillebrand-Marty’s logic in a series of four articles in the same journal, which were eventually collected in a single, largely neglected volume.106 He accepted their theory of judgement but wholeheartedly rejected the syllogistic based on that theory, which he considered defective and linguistically misleading. Despite the fact that Brentano’s logic has been deemed ‘much too inelegant and cumbersome by modern standards’,107 scholars have acknowledged that such a ‘historical curiosity’108 paved the way for Peano’s introduction of the existential quantifier,109 for Frege’s rejection of indefinite terms,110 for George Frederick Stout’s acceptance of the psychological aspect of the subject-predicate relation111 and for Russell’s development of his multiple relation theory of judgement112 and his theory of definite descriptions (the British mathematician was ‘substantially in agreement’ with Brentano on the subject of syllogism).113 From the mid-twentieth century onwards, analytic philosophers began to rework Brentano and Hillebrand’s syllogistic in the symbolic-mathematical terms of the functional calculus, reaching a level of formalization that swiftly surpassed the original

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proposal in terms of precision and axiomaticity. Despite the on-point caveat about the risk of extrinsicity and anachronism expressed by Kurt R. Fisher and Leon R. Miller,114 the process was able to take place through the progressive work of a few very committed scholars, such as Alonzo Church, Arthur Prior (also with reference to the syllogistic systems developed by Anders Wedberg and John C. Shepherdson), Otto Bird, Peter Geach, Burnham Terrel, Roderick M. Chisholm, Lynn Pasquerella, Peter M. Simons and Roberto Poli. This transition basically occurred through three main factors: (1) amendments in the form of ‘Łukasiewicz-ish corrections’ for the axiomatixation of a ‘Brentano-style’ syllogistic; (2) de-ontologization of the existential import requirement and recovery of the fundamental character of consciousness’s intentionality which originally informed Brentano’s theory of judgement; (3) the pursuit of consistency with Brentano’s post-correspondentist ‘criterium’ of truth, which had been made available and published in the meantime.115 As has been acknowledged, ‘the elimination of the universal import from universal propositions and the consequent repudiation of all rules of inferences that require it for their validity’ was ‘a clear break with the traditional views in logic’,116 so it may be argued that Brentano and Hillebrand’s ‘vertical thinking’ ended up being the first complete syllogistic system alternative to the centuries-old and unchallenged one of Aristotle and his Scholastic epigones.117

Notes Carl Stumpf, ‘Erinnerungen an Franz Brentano’, in Franz Brentano. Zur Kenntnis seines Lebens und seiner Lehre. Mit Beiträgen von Carl Stumpf und Edmund Husserl, ed. Oskar Kraus (München: Beck, 1919), tr. in Linda McAlister (ed.), The Philosophy of Brentano (London: Duckworth, 1976), 146–7/44. Emphasis added. 2 See Barry Smith, Austrian Philosophy. The Legacy of Franz Brentano (ChicagoLaSalle: Open Court, 1995), 7–14; Robert Weldon Whalen, ‘A Surprising Pre-History of Postmodernism: Franz Brentano, “Fin de Siècle” Vienna, and Contemporary European Thought’, Modern Austrian Literature 38, no. 3/4 (2005): 1–11. 3 Franz Brentano, Psychologie vom empirischen Standpunkt (Leipzig: Duncker & Humblot, 18741) Engl. tr. of the 2nd ed., Psychology from an Empirical Standpoint, intr. by P. Simons (London: Routledge, 1995), 230–4/302–5. 4 Ibid. 5 Ibid. 6 Ibid. 7 Ivi, 302–3/231. 8 Ibid. 9 Ibid. 10 See William Kneale and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 19621), 55–7. Even if Brentano is not mentioned by name there, the square of opposition contrasted with the Aristotelian one is his own, notation excluded. See also, ibid., 411, n.1, where he is tellingly mentioned in the section on Boole. 11 Augustus De Morgan, Formal Logic (London: Walton and Maberly, 1847), 110–11: ‘On looking into any writer on logic, we shall see that existence is claimed for the significations of all the names. Never, in the statement of a proposition, do we find 1

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any room left for the alternative, suppose there should be no such things. Existence as objects, or existence as ideas, is tacitly claimed for the terms of every syllogism.’ 12 Robert Flint, ‘IX. – Critical Notices’, Mind 1, no. 1 (1876): 116–22, on 122. 13 See Elizabeth Valentine, ‘British sources in Brentano’s Psychology from an empirical standpoint (1874), with special reference to John Stuart Mill’, Brentano Studien 15 (2017): 291–328. 14 Jan P. N. Land, ‘Brentano’s Logical Innovations’, Mind 1, no. 2 (1876): 289–92. See John Passmore, A Hundred Years of Philosophy (London: Duckworth, 1917), 180, n. 2. 15 Land, ‘On a Supposed Improvement in Formal Logic’, Abhandlungen der königlichen Niederländischen Akademie der Wissenschaften, 2nd s., pt. 5, 1876, 358–72, on 367. 16 Ibid., emphasis added. 17 See Christoph von Sigwart, Die Impersonalien, eine logische Untersuchung (Freiburg i.B.: Akademische Verlagsbuchhandlung, 1888), 50–8 and ff.; Id. Logik, vol. 1, Freiburg i.B.: Akademische Verlagsbuchhandlung, 2nd ed., 1889–1893; Engl. tr. Logic, vol. 1, The Judgment, Concept, and Inference (London: Sonnenschein & New York: Macmillan, 1895, 72, n. 1). He maintained that categorical universal propositions are reducible, not to their existential form, but to hypothetical conditions. 18 The letter has been partially reported in Brentano, Psychology, 232 n.27. 19 Ibid. 20 Brentano, Vom Ursprung sittlicher Erkenntnis (Leipzig: Duncker and Humblot, 19891); Eng. tr. on the 3rd rev. ed., The Origin of Our Knowledge of Right and Wrong (Oxon, NY: Routledge, 2009), 38. 21 Cf. Ms.FrSchr41 and Ms.Fr.Schr42. See Robin D. Rollinger, Concept and Judgment in Brentano’s Logic Lectures, Analysis and Materials (Leiden: Brill, 2021), 25–36. 22 See George Boole, The Mathematical Analysis of Logic. Being an Essay towards a Calculus of Deductive Reasoning (Cambridge: Macmillan, Barclay and Macmillan; London: Bell, 1847), 21: ‘To express the Proposition “Some Xs are Ys”. If some Xs are Ys there are some terms common to the classes X and Y. Let those terms constitute a separate class V, to which there shall correspond a separate elective symbol v, then v = xy. And as v includes all terms common to the classes X and Y, we can indifferently interpret it as Some Xs, or Some Ys’. Cf. Stanley N. Burris, Annotated Version of Boole’s Algebra of Logic 1847 (Waterloo: Pure Mathematics, 2019), viii–x. 23 John Venn, Symbolic Logic (London: Macmillan, 1881), 165 and n.1. 24 Edward Buckner, ‘Brentano-Venn thesis’, http://www.logicmuseum.com/wiki/ Brentano-Venn_thesis (2015). Accessed 29 August 2021. See also Arthur Norman Prior, The Doctrine of Propositions and Terms, ed. P. T. Geach and A. J. P. Kenny (London: Duckworth, 1976); Id., Formal Logic, rev. ed. (Oxford: Clarendon Press, 19622), 170; Joseph S. Wu, ‘The Problem of Existential Import’, Notre Dame Journal of Formal Logic 10, no. 4 (1969): 415–24, at 416. 25 John Neville Keynes, Studies and Exercises in Formal Logic (London: Macmillan, 1884), 128. See Abraham Wolf, The Existential Import of Categorical Predication. Studies in Logic (Cambridge: Cambridge University Press, 1905), 133–4 and 143, n. 4. 26 Francis Herbert Bradley, The Principles of Logic (London: Kegan - Trench, 1883), 42. 27 Bernard Bosanquet, Logic, or The Morphology of Knowledge, vol. 1 (Oxford: Clarendon Press, 1888); rev. ed. 1911. Cf. T. Case, ‘Logic’, in Encyclopedia Britannica, vol. XVI, 11th ed. (Cambridge and New York: Cambridge University Press, 1911), 879–918, at 888: ‘This reconstruction, which merges subject and predicate in one expression, in order to combine it with the verb of existence, is repeated in similar

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proposals of recent English logicians. … Bradley says that SP is real attributes SP, directly or indirectly, to the ultimate reality, and agrees with Brentano that is never stands for anything but exists; while Bosanquet, who follows Bradley, goes so far as to define a categorical judgment as that which affirms the existence of its subject, or, in other words, asserts a fact.’ 28 Franz Hillebrand, Die neuen Theorien der kategorischen Schlusse: eine logische Untersuchung (Wien: Holder, 1891). Engl. transl, now in Rollinger, Concept and Judgment in Brentano’s Logic Lectures, 282–373. 29 Alois Höfler and Alexius Meinong, Philosophische Propädeutik, Erstel Theil: Logik (Vienna: F. Tempsky and G. Freitag, 1890), 90, 103 ff. 30 See Artur Rojszczak, From the Act of Judging to the Sentence: The Problem of Truth Bearers from Bolzano to Tarski, ed. Jan Woleński (Springer: Dordrecht, 2005), 36–8; Wayne M. Martin, ‘Fichte Logical Legacy: Thetic Judgement from Wissenschaftslehre to Brentano’, in Fichte and the Phenomenological Tradition, ed. V. L. Waibel, D. Breazeale and T. Rockmore (Berlin and New York: De Gruyter, 2010), 379–406. 31 In the following I will refer to these rules as ‘mereological laws’. 32 The traditional forms actually correspond to some of the ‘new’ ones, as Land correctly understood before their formal display. In particular, n. 1 corresponds to Ferio, Festino, Ferison, Fresison; n. 2 correspond to Darii, or Datisi; n. 6 correspond to Bocardo; n. 9 correspond to Baroco; n. 17 correspond to Celarent; Cesare; n. 18 correspond to Camestres, Calemes. 33 Brentano, Psychologie. ‘We believe […] that the categorical proposition corresponds to a judgement which can be expressed just as well in the existential form and that the truly affirmative categorical propositions contain within them the affirmation of the subject’, 284. See also his Vom Ursprung sittlicher Erkenntnis, 57. 34 Anton Marty, ‘Über subjectlose Sätze und das Verhältnis der Grammatik zur Logik und Psychologie’, series of seven articles published in vol. 8, 18, and 19 of the journal Vierteljahrsschrift für wissenschaftliche Philosophie between 1884 and 1895; then republished in Id., Gesammelte Schriften, II.1, ed. J. Eisenmeier, A. Kastil, and O. Kraus (Halle: Max Niemeyer, 1901–20), 1–307. See in partic. art. VI and VII, with Robin D. Rollinger, ‘Brentano’s Logic and Marty’s Early Philosophy of Language’, Brentano Studien 12 (2006): 77–98. Consider also the implicit endorsement of the idiogenetic theory by Julius F. W. E. Bergmann, Die Grundprobleme der Logik (Berlin: Mittler, 1882) 1895 2nd new ed., 78–81). In fact, Bergmann and Brentano were criticized on the same grounds by Husserl. 35 Robin D. Rollinger, ‘Meinong and Brentano’, Meinong Studies 1 (2005): 159–98, at 167: ‘In 1891 Brentano’s reform of logic was presented by one of his students, Franz Hillebrand, The New Theories of Categorical Inferences, a small book which Meinong himself reviewed. Since this book was no doubt written as a reaction to the logic textbook of Höfler and Meinong, and particularly since Hillebrand wrote his logic book under the supervision of Brentano, Meinong’s review of it was an indirect confrontation with his mentor. […] Meinong’s review of Hillebrand’s was by no means a positive one, particularly regarding the theory of judgment, which was central in Hillebrand’s exposition of Brentanian logic.’ 36 Alexius Meinong, ‘Rezension von: Franz Hillebrand, Die neuen Theorien der kategorischen Schlüsse’, Göttingische gelehrte Anzeigen 155, no. 1, first part (1892): 443–66. 37 Charles Andler, ‘[Avis sur] Die neuen Theorien der Kategorischen Schluesse. Eine logische Untersuchung by Franz Hillebrand’, Revue Philosophique de la France et de l’Étranger 33 (1892): 343–5.

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38 Emily Elizabeth Constance Jones, ‘Review of Die neuen Theorien der Kategorischen Schlüsse by F. Hillebrand’, Mind 1, no. 2 (1892): 276–81, at 280. 39 Ibid., 280–1. 40 Ead., ‘Import of Categorical Propositions’, Proceedings of the Aristotelian Society 2, no. 3 (1893–4): 35–45, at 39; and Ead., ‘Import of Categorical Propositions’, Mind 2, no. 6 (1893): 219–22. Cf. also on these lines George Hayward Joyce, Principles of Logic (London: Longmans, 1908): ‘Such a theory carries its own refutation with it. It is manifest to any one who reflects, that as a matter of fact we do not think in these forms’. 41 Louis Couturat, La Logique de Leibniz. D’apres des documents inedits (Paris: Alcan, 1901), 350, n. 4 (who probably had in mind H. MacColl, La logique symbolique et ses applications (Paris: A. Colin, 1901), 17–19 rather than his previous papers), versus Kraus, Franz Brentano, 108/22, n.1. See also Kraus’ editorial note to Brentano, Psychologie, 232 and Alfred Kastil, Die Philosophie Franz Brentanos: eine Einführung in seine Lehre (Bern: A. Francke, 1951), 206. This latter reads: ‘There is a not uninteresting historical fact. Thirty years after the basic ideas of this derivation had been published for the first time, a previously unknown work by Leibniz, namely the Generales Inquisitiones de Analysi Notionum et Veritatum from 1686, came to light thanks to the French mathematician Couturat, where the transferability of all categorical propositions into existential ones were presented in exactly the same way. Leibniz himself had added the words “hic egregie progressus sum” to his record. He would have appreciated the successor’s achievement [i.e. Bretano’s] better than our logisticians, who carelessly neglected it’. My translation. See the next section for an alternative reading of this point. 42 Christine Ladd-Franklin, ‘Dr. Hillebrand’s Syllogistic Scheme’, Mind 1, no. 4 (1892): 527–30. See also the chapter on Christine Ladd-Franklin by Francine Abeles in this book. 43 See Denis Fisette and Guillaume Frechétte (eds.), À l’école de Brentano: de Würzbourg à Vienne (Paris: Vrin, 2007), 35, n.1. 44 Walter Enoch, ‘Franz Brentanos Reform der Logik’, Philosophische Monatshefte 29 (1893): 433–58. 45 Cf. Dermot Moran, Introduction to Edmund Husserl, The Shorter Logical Investigations, with a preface by Michael Dummett (London and New York: Routledge, 2001), xxxiv–xxxiii: ‘Husserl suggested that his Logical Investigations was originally inspired by Brentano’s attempts to reform traditional logic. As he put it in his “Phenomenological Psychology” lectures of 1925: “[…] the Logical Investigations are fully influenced by Brentano’s suggestions” […] In lecture courses Husserl had attended, Brentano had proposed a reform of traditional Aristotelian syllogistic logic […]. Despite its promise, Husserl recognised that Brentano’s project was destined to fail, since it lacked a proper clarification of the nature of meaning in general. Only a complete clarification of the “essential phenomenological relations between expression and meaning, or between meaning-intention and meaning-fulfilment” could steer the proper course between grammatical analysis and meaning analysis (Bedeutungsanalyse), Husserl claimed (Logical Investigations, I, p. 173, tr. Findley of Husserliana XIX/1, p. 19)’. 46 Edmund Husserl, Logik Vorlesung, 1896, ed. Elisabeth Schumann, Husserliana Materialenbände I (Dordrecht: Kluwer, 2001). 47 Gerardus Heymans, Die Gesetze und Elemente des wissenschaftlichen Denkens: Ein Lehrbuch der Erkenntnistheorie in Grundzügen, vol. 2 (1st ed., Leiden: S. C. Van Doesburgh - Leipzig: Otto Harrassowitz, 1890–1894).

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48 E. Husserl, Briefwechsel, ed. by K. Schuhmann, with E. Schuhmann (Dordrecht: Kluwer, 1994), 34. See also Wolfgang Huemer, ‘Husserl’s Critique of Psychologism and his Relation to the Brentano School’, in Phenomenology and Analysis: Essays on Central European Philosophy, eds. Arkadiusz Chrudzimski and Wolfgang Huemer (Frankfurt: ontos, 2004), 199–214, on 208. 49 See Risto Tiihonen, Theory of Judgement in Edmund Husserl’s ‘Logical Investigations’, MA Thesis, University of Helsinki, 2020, 46, n. 115 (See also nn. 200; 201; 271). 50 See Richard Cobb-Stevens, ‘Husserl’s Theory of Judgment: A Critique of Brentano and Frege’, in Husserl’s Logical Investigations Reconsidered, ed. Denis Fisette (Dordrecht: Springer, 2003), 151–61. 51 See Carlo Ierna, ‘Husserl’s Critique of Double Judgments’, in Meaning and Language: Phenomenological Perspectives, ed. F. Mattens (Dordrecht: Springer, 2008), 49–73. 52 Ivi, 61–2, and Andrea Altobrando, ‘La negazione: dal rifiuto al contrasto. Brentano e Husserl sul giudizio negativo’, Verifiche 47, no. 2 (2017): 139–77. 53 Friedrich Adolf Trendelenburg, ‘Über Leibnizens Entwurf einer allgemeinen Charakteristik’, in Philosophische Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin. Aus dem Jahr 1856 (Berlin: Commission Dümmler, 1857), 36–69. 54 Ms.Harvard.EL72, 105 and Ms.Leuven.Y2, 36, respectively. See C. Ierna, ‘Brentano and Mathematics’, Revue Roumaine de Philosophie 55, no. 1 (2011): 149–67, at 159, and Mauro Antonelli, Seiendes, Bewusstsein, Intentionalität im Frühwerk von Franz Brentano, (Freiburg-München: Alber, 2001), 291–320. 55 Hillebrand note from 11 November 1884. Tr. Rollinger, Concept and Judgment, 227. 56 See Volker Peckhaus, ‘Language and Logic in German Post-Hegelian Philosophy’, Baltic International Yearbook of Cognition, Logic and Communication 4 (2008): 1–17, at 6–8. 57 Brentano, Ms.Harvard.EL.80. Logik. 58 Ibid., 12.973 [10], online edition by R. Rollinger, http://gams.uni-graz.at/context:bag/ nachlass (Accessed 29 August 2021). 59 EL80–12.974–5, my translation from the online edition: ‘Gewöhnlich denkt man sich die Logik wie fertig. Man tut ihr hier zu große Ehre an, wie man dort ihre Ehre schmälerte. Viel ist mangelhaft. Mit Unrecht sagt man auch, dass kein Irrtum in den Lehren steckt, wie sie gewöhnlich vorgetragen werden ⇐ vgl. Leibnitz]. Wir werden sehen. ⇐ Stellen aus Leibniz.]’. Hillebrand reported this passage in his notes: ‘As for as the view that logic is an already perfected discipline is concerned, Leibniz already opposed it by saying “There si nothing more imperfect than logic. The art of inferring from probable reasons is not yet developed”.’ Note from 8 November 1884. Tr. Rollinger, Concept and Judgment, 223. 60 See, for instance, Gustav Bergmann, Realism: A Critique of Brentano and Meinong (Madison: University of Wisconsin Press, 1967), 298–9. 61 Johann Friedrich Herbart, Lehrbuch zur Einleitung in die Philosophie (Königsberg: Unzer, 18131; 18374), §§53–9; Moritz Wilhelm Drobisch, Neue Darstellung der Logik nach ihren einfachsten Verhältnissen (Leipzig: Voss, 18361; 18875), §§55–6; Friedrich Ueberweg, System der Logik und Geschichte der logischen Lehren (Bonn: A. Marcus, 18471; 18683), § 85. See Wolf, The Existential Import, 143 and n. 4; 133–134 and David W. Sullivan, ‘Frege on Existential Propositions’, Grazer Philosophische Studien 41, no. 1 (1991): 127–49, at 131–8 for Brentano’s praise of Herbart. 62 John Stuart Mill, A System of Logic, Ratiocinative and Inductive, vol. 1 (London: Longmand et al., 18431; 18728), 103–10. Cf. Wilhelm Jordan, Die Zweideutigkeit des Copula bei Stuart Mill (Stuttgart: Kleeblatt, 1870).

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63 Hillebrand, Die neuen Theorien, 56; 60. 64 Trendelenburg, Logische Untersuchungen, vol. 2 (Leipzig: Hirzel, 18401; 1870), 335. 65 Rudolf Hermann Lotze, Logik, Drei Bücher vom Denken, vom Untersuchen und vom Erkennen, vol. 1 (Leipzig: Hirzel, 18431; 1874), 104–5. 66 F. A. Lange, Logische Studien. Ein Beitrag zur Neubegründung der formalen Logik und der Erkenntnisstheorie (Iserlohn: Baedeker, 18771 (1894)), 72–3. Cf. Nikolay Milkov, ‘Hermann Lotze and Franz Brentano’, Philosophical Readings 10, no. 2 (2018): 115–22. 67 Leibniz himself become eventually aware of this condition too. See Wolfgang Lenzen, ‘Leibniz’s Logic’, in Handbook of the History of Logic, ed. D. M. Gabbay and J. Woods vol. 3 (Amsterdam et al.: Elsevier, 2004), 1–83, at 79–80. 68 Stefano Besoli and Gabriele Franci, ‘Zur Rückführbarkeit der kategorischen Aussagen auf Existentialsätze. Eine Auseinandersetzung Brentanos mit Leibniz’, in Leibniz. Werk und Wirkung. IV. Internationaler Leibniz-Kongress. Vorträge (Hannover: Leibniz-Gesellschafte, 1983), 31–41, on 34; Peter M. Simons, ‘Brentano’s reform of Logic’, Topoi 6 (1987): 25–38, at 27. 69 Gottfried Wilhelm von Leibniz, Difficultates Quaedam Logicae, in God. Guil. Leibnitii opera philosophica quae extant Latina Gallica Germanica omnia, Pars Prior, ed. Johann Eduard Erdmann (Berlin: Eichler, 1839–40), 103–4. See George H. Radcliffe Parkinson (ed.), Leibniz: Logical Papers. A Selection (Oxford: Clarendon Press, 1966), 120. 70 See Prior, The Doctrine of Propositions, 117. 71 G. H. R. Parkinson, Logic and Reality in Leibniz’s Metaphysics (Oxford: Clarendon Press, 1965), 22. Cfr. W. Lenzen, ‘Leibniz on Ens and Existence’, in Existence and Explanation. Essays presented in Honor of K. Lambert, ed. W. Spohn, B. C. van Fraassen, and B. Skyrms (Dordrecht: Springer, 1991), 59–75. 72 ‘There are one or two other brief tracts in which Leibnitz anticipates the modern views of logic. Thus in the eighteenth tract in Erdmann’s edition (p. 92), called Fundamenta Calculi Ratiocinatoris, he says: “Inter ea quorum unum alteri substituti potest, salvis calculi legibus, dicetur esse aequipollentiam.” There is evidence, also, that he had arrived at the quantification of the predicate, and that he fully understood the reduction of the universal affirmative proposition to the form of an equation, which is the key to an improved view of logic. Thus, in the tract entitled Difficultates Quaedam Logicae, he says: “Omne A est B; id est aequivalent AB et A, seu A non B est non-ens.” (ed. Erdman, p. 102).’ William Stanley Jevons, The Principles of Science. A Treatise on Logic and Scientific Method (London: Macmillan, 19741, 1883), xviii. See V. Peckhaus, ‘The Reception of Leibniz’s Logic in 19th Century German Philosophy’, in New Essays on Leibniz Reception in Science and Philosophy of Science 1800–2000, ed. R. Krömer and Y. Chin-Drian (Basel: Birkhäuser, 2012), 13–24, at 21–22. 73 ‘It deserves notice that Boole did adopt this form [for the particular affirmative proposition] (xy = v) in his earlier work (Math. Analysis of Logic [1847], p. 21) but afterwards rejected it in favour of vx = vy. But in neither context can I find any discussion of the real difficulty which arises when we are called upon to decide the limits of indefiniteness to be assigned to v. It may be remarked that Leibnitz with his usual penetration, had observed that particular propositions could be thus expressed. He says (Difficultates logicae; Erd. p. 102) that he had formerly adopted the following scheme: “Omne A est B … seu A non B est non-ens; Quoddam A non est B … seu A non B est ens; Nullum A est B, erit AB est non-ens; Quoddam A est B, erit AB est ens”. So far as it goes this exactly coincides with the arrangement adopted above. But he seems to have rejected it to some extent afterwards, owing to the consequent difficulties about the

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conversion of propositions. In quite recent times the same arrangement substantially has been adopted by Professor F. Brentano.’ Venn, Symbolic Logic, 165, n. 1. 74 Cfr. Giovanni Vailati, ‘Sul carattere del contributo apportato da Leibniz allo sviluppo della Logica Formale’, Rivista di filosofia e scienze affini anno VII, 1, no. XII (1905): 5–6; repr. in Id., Scritti (Leipzig: Barth; Firenze: Seeber, 1911), 619–24. 75 ‘I have finally been able, in the last few days, to take a look at the book by Venn (Symbolic Logic, Macmillan, 1880 [sic]) which I mentioned to you a while ago. He cites (p. 165) precisely the same theory of the syllogism espoused by you in your Psychologie vom empirischen Standpunkt. He observes that it is similar to the one Leibniz espouses in the essay Difficultates Quaedam Logicae (Ermann edition, p. 102), and to that of Boole (Mathematical Analysis of Logic, 1847). He says of you that ‘like some other symbolists he (Brentano) springs to the conclusion that the new mode of notation is to supersede altogether the old one, instead of being, as I should say, an alternative method not necessarily hostile to the old one, but far more suitable for the treatment of complicated problems and broad generalizations’. Vailati to Brentano, 24 July 1900. (Reported in Roderick M. Chisholm and Michael Corrado, ‘The Brentano-Vailati Correspondence’, Topoi 1 (1982): 30–43, at 16). 76 Brentano, ‘Nieder mit den Vorurteilen! Ein Mahnwort an die Gegenwart, im Geiste von Bacon und Descartes von allem blinden Apriori sich loszusagen’ [1903], in Id., Versuch über die Erkenntnis, hrsg. A. Kastil (Leipzig: Meiner, 1925); hrsg. F. MayerHillebrand (Hamburg: Meiner Verlag, 1970), 55. My translation. 77 Id., Aristoteles und seine Weltanschauung, intr. by Rolf George (Hamburg: Meiner, 19111; 19772), 41. 78 Kraus, Franz Brentano, 108/22, n. 1. 79 Willy Freytag, ‘Bemerkungen zu Leibnitzens Erkenntnistheorie im Anschluß an Couturats Werk ‘La logique de Leibnitz d’après des documents inédits (Paris 1901)’, Archiv für die gesamte Psychologie 33 (1915): 135–51, at 140, my emphasis and translation. 80 See V. Peckhaus, Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert (Berlin: AkademieVerlag, 1997); Id., ‘19th Century Logic between Philosophy and Mathematics’, Bulletin of Symbolic Logic 5 (1999): 433–50. 81 See Diter Münch, ‘Franz Brentano et la réception catholique d’Aristote au XIXe siècle’, in Aristote au XIXe siècle (Villeneuve d’Ascq: Presses universitaires du Septentrion, 2005), 230–48; Rollinger, Concept and Judgment, 25–36. 82 Brentano reviewed the second edition of that book (Subjektlose Sätz, Vienna: Braumüller, 1883) and published his sympathetic considerations as an appendix (‘Miklosich über subjecktlose Sätze’) in the first edition of his Vom Ursprung sittlicher Erkenntnis (Leipzig: Dunker & Humblot, 1889), 111–22. 83 Ivi, 120 n. Engl. tr. by Cecil Hague, The Origin of the Knowledge of Right and Wrong, 1st ed. (Westminster: A. Constable & Co., Ltd., 1902), 114 n. 84 The Origin of Our Knowledge of Right and Wrong, 2nd English transl. by Roderick Chisholm and Elizabeth Schneewind (London: Routledge, 1969), on the 2nd ed. by Oskar Kraus (Hamburg: Meiner 1921), 70. 85 Ivi 81, n.3/53. 86 Id., Ms.Harvard.E. L.70, ‘Moderne Irrtümer über die Erkenntnis der Gesetze des Schliessens’, [1889], in Studien zur Österreichischen Philosophie 44, ed. Denis Fisette (2013): 513–24; Id., Die Lehre vom richtigen Urteil, ed. F. Mayer-Hillebrand (Bern: Francke Verlag, 1956), 227–37. 87 Brentano, ‘Nieder mit den Vorurteilen!’, 53.

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88 Ivi, 27–8. See Stephan Körner, ‘On Brentano’s Objections to Kant’s Theory of Knowledge’, Topoi 6 (1987): 11–17, on 12–13. 89 Ivi, 33–5. 90 See Lucie Gilson, Méthode et métaphysique selon Franz Brentano (Paris: Vrin, 1955), 74–83; Leo Rauch, ‘Brentano’s Psychology and the Problem of Existential Import’, Philosophical Studies 17 (1968): 121–31. 91 Brentano, Psychologie, 156–8/299–301: ‘For example, the proposition, “If all A is B, then some C is not D,” can be converted into the existential proposition, “There is no non-being of A non-B without the being of C non-D.” If one adds to this the proposition, “The non-being of A non-B is,” then “the being of C non-D” follows by modus ponens, or if one adds the proposition, “The being of C non-D is not,” by modus tollens it follows that “The non-being of A non-B is not”. Substituting the letter α for the term “non-being of A non-B,” and β for the term “being of C non-D,” the arguments take this simple form: “There is no α without β. α exists. Therefore, β exists.”

“There is no α without β. β does not exist. Therefore α does not exist.”’

92 Brentano ‘Zu Poincarés Erkenntnislehre’ [1916], in Id., Versuch über die Erkenntnis, 207–36. 93 See Ierna, ‘Brentano and Mathematics’, and Id., ‘Brentano as a Logicist’, Studien zur Österreichischen Philosophie 49, Special Issue: The Philosophy of Brentano, ed. M. Antonelli and Th. Binder (Leiden: Brill, 2021), 301–11. 94 Kevin Mulligan, ‘Exactness, description and variation: How Austrian analytic philosophy was done’, in Von Bolzano zu Wittgenstein, ed. J. C. Nyíri (Vienna: Hölder, 1986), 86–97, at 89: ‘Deductive arguments, in particular the formulation and examination of the consequences of a given thesis, explicit consideration of possible objections, a concern to be consistent and definitions are an integral part of the Brentanist contributions to exact philosophy. … The pervasiveness of clearly formulated deductive argument in the work of Brentano’s heirs can be traced back, like so much else, to a rule enunciated in the latter’s lectures: after a domain of phenomena has been described, the results must be deductively turned to a good account (deductiv verwertet), since only in this way can the descriptive psychologist bring to light questions that would otherwise pass unnoticed’. 95 Case, ‘Logic’. 96 See Robin D. Rollinger, Philosophy of Language and other matters in the work of Anton Marty (Amsterdam and New York: Rodopi, 2010), 49–75. 97 See Thomas Binder, Franz Brentano und sein philosophischer Nachlass (Berlin, Boston: De Gruyter, 2019), 256–64. 98 Kastil, Die Philosophie Franz Brentanos: eine Einführung in seine Lehre, in partic. 201–9. 99 Peter Simons, ‘Judging Correctly. Brentano and the Reform of Elementary Logic’, in The Cambridge Companion to Brentano, ed. Dale Jacquette (Cambridge: Cambridge University Press, 2004), 45–65, at 63–4. Cf. Wioletta A. Miskiewicz, ‘La critique du psychologisme et la métaphysique retrouvée. Sur les idées philosophiques du jeune Łukasiewicz’, Philosophia Scientiæ 15, no. 2 (2011): 21–53. 100 For the historical context, see Antonio Santucci, ‘Franz Brentano e i pragmatisti italiani’, and Francesca Modenato, ‘Conoscere e volere. L’incontro di Vailati e Calderoni con Brentano’, both in Brentano in Italia. Una filosofia rigorosa contro

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positivismo e attualismo, ed. Albertazzi Liliana and Roberto Poli (Milano: Guerini, 1993), 21–46 and 47–66, respectively. 101 Vailati, Scritti [1909], 926. 102 Chisholm and Corrado, ‘The Brentano-Vailati Correspondence’, 8. 103 See Roberto Giannetti, ‘La presenza di Franz Brentano in Italia agli inizi del Novecento’, Rivista di Filosofia Neo-Scolastica 69, no. 1 (1977): 86–102, on 101–2, and Mauro Antonelli, Federico Boccaccini, Franz Brentano. Mente, coscienza, realtà (Roma: Carocci, 2021), 168–75. 104 Giovanni Calò, ‘Concezione tetica e concezione sintetica del giudizio’, La cultura filosofica 2 (1908): 337–68, at 360: ‘existential judgments are rather contained in predicative judgments, because one can obtain the former from the latter and not the other way around’, my translation. 105 Ivi, 364–5. 106 Guido Rossi, Giudizio e raziocinio. Studi sulla logica dei brentaniani (Milano: Sodalitas, 1926). This book consists in a collection of revised articles originally published in 1916–17. Brentano might have been aware of the original articles by the way of Giovanni Vailati. 107 Gabriel Falkenberg, ‘Review of P. Simons’ Philosophy and Logic in Central Europe from Bolzano to Tarski’, Erkenntnis 41, no. 2 (1994): 275–9. 108 Ibid. 109 George Boger, ‘Existential Import and an Unnecessary Restriction on Predicate Logics’, History and Philosophy of Logic 39, no. 2 (2017): 109–34. 110 Else Margarete Barth, The Logic of the Articles in Traditional Philosophy. A Contribution to the Study of Conceptual Structures (Dordrect-Boston: Reidel, 1974), 405–9. 111 After the exposition of Brentano’s theory of judgement in the first volume of his Analytic Psychology, Stout maintained likewise that ‘the subject-predicate relation […] is a purely psychological category and not a logical category,’ in Analytic Psychology, ed. George Frederick Stout (London: Sonnenschein, 1896), vol. II, 213–14 and vol. I, 145–8. Cf. Maria van der Schaar, G.F. Stout and the Psychological Origins of Analytic Philosophy (Houndmills, Basingstoke - New York: Palgrave Macmillan, 2013), 145–8. 112 Mark Textor, ‘Acquaintance, Presentation and Judgement: From Brentano to Russell and Back Again’, Inquiry (2021): 1–27. 113 Prior, The Doctrine of Propositions, 116. 114 Kurt Rudolf Fisher and Leon R. Miller, ‘Notes on Terrel’s Brentano’s Logical Innovations’, Midwest Studies in Philosophy 1, no. 1 (1976): 95–7. 115 Brentano’s relevant studies on truth dated back in 1889, although they were written down in their final form in 1915, published by Oskar Kraus in 1930 and made available in English translation only in 1966, so Burnham Terrel could took them into account within the framework of his ‘Brentano’s style’ syllogistic system in 1976. 116 Burnham Terrel, ‘Quantification and Brentano’s Logic’, Grazer philosophische Studien 5 (1978): 45–65, at 46. 117 See Jean-Claude Gens, ‘La doctrine du jugement correct dans la philosophie de F. Brentano’, Revue de Métaphysique et de Morale 3 (1996): 370–1.

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Hugh MacColl: Never twist the syllogism again Jean-Marie C. Chevalier

1 Introduction Christine Ladd-Franklin thought it ‘unfortunate that Mr. MacColl’s valuable work in Logic should have met with almost complete neglect in England’.1 Hugh MacColl still deserves a better place in the history of logic than he currently has, for his achievements are not few. He is famous for having developed a modern propositional logic a few years prior to the publication of Frege’s Begriffsschrift, claiming that logic should not be a calculus of classes but of propositions. Russell called him ‘the first to have found symbolic logic on propositions and implication’.2 Another of his contemporaries acknowledged that ‘in making the proposition the unit of thought, he has brought the symbolic logic into line with the general theory of modern logic today’.3 He is also known for debating with Russell the problem of the existential import of propositions and for their controversy on ‘if ’ and ‘imply’. MacColl challenged the use of material implication, used the notion of propositional function, and opened the way to non-classical and modal logics and to logical pluralism.4 It has been claimed by specialists that ‘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’.5 Ladd-Franklin too considered it one of MacColl’s greatest accomplishments to have defined universal affirmative propositions in terms of implication, writing that ‘[t]he logic of the non-symmetrical affirmative copula, “all a is b,” was first worked out by Mr. MacColl’.6 Nevertheless, MacColl’s goal was not to produce a new theory of syllogism, which he considered one of the many fields of application of his new symbolic logic: ‘No part of logic has received so much attention and given rise to so much discussion as the syllogisms of Aristotle. This is why I have selected them as illustrations of the application of my symbolical method.’7 But there may be more to it: his insistence on the subject may also be ‘justified by the fact that syllogistics is configured as a complete and structured deductive system, capable of providing a term of comparison for the deductive possibilities of a logical system’.8 He considered that the best illustration of the advantages of his symbolic logic was that he was not obliged, as so many logicians of time were, ‘to twist and torture simple sentences in order to adapt them for syllogistic reasoning’.9

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Starting from MacColl’s treatment of the syllogism (Section  3), this chapter intends to show how his theory of implication and of hypothetical propositions was embodied (Section 4), what his conception of ‘formal certainty’ in syllogistic reasoning was (Section 5), how this came with a new conception of pure logic as propositional logic (Section 6), and finally how he could provide a solution to the problem of the presupposition of existence in syllogistic propositions (Section 7).

2 Biographical sketch The Scottish mathematician and logician Hugh MacColl (1837–1909) lived in the northern French city of Boulogne-sur-Mer from 1865 until his death in 1909. His logical activity is usually divided into two periods, separated by an interval of literary creation. He referred to it as a long pause in his mental life.10 In the late 1870s and early 1880s, MacColl first expressed his strong preference for propositional calculus rather than the term/class approach, widespread in his day. It led him to controversies with prominent logicians such as John Venn and William Stanley Jevons, in particular.11 Here is an interesting remark on the reasons why he felt compelled to develop a new system of logic. Working in the framework of propositional logic, MacColl could not see how to deal with quantification: 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.12

Thus, the problem of how to deal with the syllogism in a Boolean context was a strong motivation for developing a new system of notation and finally a new logic. MacColl claimed that his calculus was the only suitable notation for the new, symbolic logic, understood as an abstract structure, not just as a calculus, but as a system of general rules with a plurality of fields of application. Such a notation allegedly turned logic into a practical science, a useful instrument of research. MacColl’s main papers, in this regard, were a series titled ‘The Calculus of Equivalent Statements’ in which he showed that what he thought to be a purely mathematical method could be applied to logical questions. ‘This I thought would be my final contribution to logic or mathematics,’ he later remarked.13 Although he never stopped contributing mathematical puzzles to The Educational Times, after this first, active part of his life as a logician, he did not work on logic

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for some twelve years, a long-lasting abstention devoted to writing several novels and other literary works. It was not until 1896 that he was led to resume his logical studies – sparked by his reading of Lewis Carroll’s Symbolic Logic. Following this stimulus, MacColl published his major logical ideas on classical and non-classical logic, mainly in a new series called ‘Symbolic Reasoning’14 where he classified statements into true, false, certain, impossible and variable. Again he came into conflict with some of the leading logicians of his time, notably Bertrand Russell, this time on the issues of modality, implication and existence.15 MacColl himself observed a strong contrast between his first period and his new developments in logic: [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.16

In other words, MacColl claimed that if his first system was accidentally analogous to Boole’s logic, his second was deliberately free from every Boolean influence.17 As compared with other available logical systems, the outward ‘resemblances of mere form hide important differences in matter, method, and limits of application’.18

3 From the calculus of equivalent statements to symbolic reasoning MacColl’s approach to the syllogism developed over a long period of time, mainly because it was marked by a break of more than a decade. This section attempts to identify the main features of his changes of opinion with respect to the syllogism. MacColl’s first essay, entitled ‘On the Calculus of Equivalent Statements’ and published in 1877 in the Proceedings of the London Mathematical Society, was an entirely mathematical paper. The second essay more clearly belonged to logic. In it, MacColl explained that the first paper was originally submitted with a passage containing an application of equivalent statements to the theory of the syllogism, which was deleted because of an objection from the referee. The referee had ‘very justly objected’ that MacColl had followed Boole’s inconvenient method of introducing an additional term v in order to express ‘Some X is Y’ as ‘v belongs to some class V common both to the class X and the class Y’.19 In the second article, MacColl proposed an improvement of the method. He simply transcribed the nineteen valid syllogisms of the four figures into three propositions, where x, y and z are the terms, the apostrophe the negation, ‘:’ the implication and ‘÷’ the negation of the implication. This is the main innovation, so that ‘A÷B’ means (A:B), i.e. ‘A does not imply B’, which MacColl translated in modal terms: ‘[T]he truth of B is not a necessary consequence of the truth of A; in other words, it asserts that the statement A is consistent with B’, but it makes no assertion as

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to whether A is consistent with B or not’.20 This simple notational transcription allowed MacColl to observe that all possible valid modes of syllogism come under one of the four following implications, if necessary by means of a substitution of equivalents: (1) (a:b)(b:c):(a:c) (2) (a:b)(b:c):(a÷c’) (3) (a:b)(a÷c):(b÷c) (4) (a:b)(a:c):(b÷c’). The first one is an axiom, and the other three can be deduced from it. Considering these four functions, and noticing that f(a,b,c) = f(c’, b’, a’), MacColl wrote that the nineteen valid modes can be reduced to eight syllogisms. Conversely, ‘several valid syllogisms which are not found among the nineteen might be introduced’21 by adding missing implications. The fifth paper in the series ‘Calculus of Equivalent Statements’, read in 1896 before the London Mathematical Society, is the first of the new period. It introduces radically different notations, based on indices and exponents. The subscripts refer to an implication, while the superscripts refer to membership of a class. MacColl concludes that the syllogisms in which a, b, c denote classes ‘remain valid when we change indices into subscripts, provided we interpret the class symbols as now denoting not classes but singular representative statements’.22 In other words, the approach in terms of inclusion and the approach in terms of implication are equivalent if we interpret the class letters as names of propositions. For, in the formula (A:B)(B:C):(A:C) the sign ‘:’ means the same thing everywhere, but if we replace it by a symbol of inclusion it is impossible to say that the premises are a class contained in the conclusion. ‘It is just the other way around; if the word contained is to be used at all in this case, it is the conclusion that is contained in the premisses.’23 This is an important turning point in the passage from extensional logic to intentional logic and to a propositional approach to the syllogism. The ‘Symbolic Reasoning’ series continues along this new path. The fourth part of the series focuses on the syllogism within the framework of a ‘pure logic’ which can only be ‘the logic of statements or propositions’24 since it is the simplest and most general part of it. MacColl reaffirms and reformulates the conclusions he had reached in the 1870s: all syllogisms can be expressed as a function applied to three propositions (the premises and the conclusion of the syllogism), taken in a certain order, and with or without a negation exponent. The system is evidently more satisfactory because MacColl has managed to get rid of the symbol of non-implication ‘÷’, keeping only the implication, and because by adding among the propositions η, which designates the impossible (hence the contradiction), a single function is enough to express all the syllogisms. Where in 1877 MacColl resorted to four functions φ, f, ψ, χ depending on the place of ÷, in 1902 he manages to reduce everything to a single function, provided the symbols of conjunction and contradiction are included in its domain. The formula φ(x, y, z) is thus the general formula of the syllogism, ‘of which all validated syllogisms are but particular cases’.25 Compared to the class approach, this approach ‘from the point of view of pure or abstract logic’ is ‘a far simpler, more symmetrical, and more general way of treating the syllogism’.26 An example will make the interest and the evolution of MacColl’s notation and ideas about syllogism appear clearer. Let us focus on Darapti. This is the first valid mode of the third figure of the syllogism, which traditionally takes the following form: ‘All A is B, all A is C, therefore some B is C.’ In its ‘corrected’, implicational form, it says this: ‘If every A is B, and every A is C, then some B is C.’ In 1877, MacColl would

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call it ‘χ = (α:β)(α:γ):(β÷γ’)’, namely (α implies β and α implies γ) implies that β does not imply not-γ. Hence, Darapti is of the form χ(x, y, z). In 1897, he would write that Darapti may be expressed as αβαγ:βoγ’. At the time, he came to believe that Darapti is defective and lacks a premise in order to justify the conclusion. It was one of his guiding questions to find out the weakest premise that must be added to the two given premises in each of the four defective syllogisms to make them valid too. He formally proved that the premise is no other than αoη, that is, that α is not empty.27 Importantly, MacColl later held that it was a mistake he had committed, much like the other Booleans, because he lost track of the fact that the three propositions must all three have the same subject, an individual taken at random from the universe of discourse.28 As we will see, this connects with the matter of existential import. In 1902, MacColl wrote that Darapti takes the form φ(B, AC, η), where φ(x, y, z)=(x:y)(y:z):(x:z). Darapti thus has the following form: (B:AC)(AC:η):(B:η), which can be proven from the fundamental form (B:C)(B:A):(A:C’)’.29 Even more generally, Darapti (like Felapton, Fesapo and Bramantip, the four syllogisms traditionally considered as defective) has the form AB:C’. Darapti was seen as a controversial syllogism, since the Boolean logicians supposed it to fail when A is non-existent and B and C are mutually exclusive (as becomes abundantly clear in a Venn diagram). But this reasoning is actually false, as MacColl proved,30 for in this case (x:y)=η1, (y:z)=η2 and (x:z)=η3, so that (η1η2:η3), which is completely consistent. ‘In general logic’ Darapti is not a formal certainty because traditional logicians would say that it violates the canonical rule that ‘the middle term must be distributed at least once in the premisses’.31 But MacColl managed to prove that, presented in its correct form, the syllogism is actually valid. For instance, the syllogism ‘If no centaurs are really existent, and no fairies are really existent; then some things (or thing) that are not centaurs are not fairies’ is valid, though ‘it violates the traditional canon that no conclusion can be drawn from negative premisses’.32

4 Syllogism as implication MacColl’s greatest contribution is to have interpreted the syllogism intentionally rather than extensionally, i.e. no longer in terms of class inclusions, but as a system of implications. However, the debate over propositional logic did not focus on the inference/inclusion opposition as much as on an inference/implication contrast.33 MacColl’s basic operator is the strict implication34 and not the material implication: ‘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.”’35 This choice for strict implication, which adds the modality of necessity to the material conditional, makes MacColl a precursor of C. I. Lewis.36 The choice itself is justified by the idea that a:b is not equivalent to (not-a or b) but to ‘it is necessary that (not-a or b)’.37 It is logically equivalent to ¬◊ (a&¬b). The implication in question occurs not only between propositions when a syllogism is made but also within each and every proposition. For the relation between subject and predicate in every verbal statement ‘is strikingly analogous to

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that connecting the terms antecedent and consequent in any implication a:b. Take for example the statement “Man is mortal”. Let a denote the statement “He is a man”, and let b denote the statement “He is mortal”. Then the implication a:b is an exact equivalent for the statement “Man is mortal”.’38 MacColl’s analysis here is not far removed from the Russellian approach in terms of propositional functions. After all, ‘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.”’39 Then, x:y is equivalent to the syllogistic statement ‘All X is Y’ with x denoting the statement that something originally unclassed belongs to X and y that the same thing belongs to Y. The formula of the syllogism, therefore, is not AB:C but (A:B)(B:C):(A:C) and commands: ‘from two implicational premisses A:B and B:C draw implicational conclusion A:C.’ According to MacColl it refers to the difference between concrete inductive reasoning in the first case and abstract, formal, deductive reasoning in the other. It is evident that the latter is not only more difficult, but also that it is on a higher and totally different plane. In the former, the two premisses and the conclusion are all three elementary statements, while the whole reasoning constitutes a simple implication. In the latter the two premisses and the conclusion are all three implications, while the whole reasoning is an implication of the second order, the premisses A and В the former are precepts supplied directly by the senses; the premisses A:B and B:C of the latter are hypothetical concepts of the mind – concepts which may be true or false (as may also the conclusion), without in the least invalidating the formula.40

That the syllogism requires a hypothetical formulation proves that it is properly human: only the human language has reached the required degree of propositional complexity. Venn objected to the use of the word ‘implication’, which he believed should not be applied to hypotheticals but should refer to a known connection between the terms. The ‘if ’ relation is not as strict.41 Venn’s point may seem right, since MacColl was keen on making a difference between implication and the hypothetical structure of the syllogism. However relevant, this objection is about the appellation rather than MacColl’s logical treatment in itself. Furthermore, MacColl tried to clarify the various senses of implication in his notation. In the 1880s, he used the symbol A:B to mean both implication between terms and between propositions, but from 1897 on, formal implication from proposition to proposition became symbolized by the exponential notation AB.

5 Syllogistic Validity ‘Syllogistic Validity’ is the title of the penultimate paper by MacColl in the ‘Symbolic Reasoning’ series. It mainly concerns the criterion of distributed and undistributed terms in syllogisms upon which ‘the validity tests of the traditional logic turn mainly’,

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as MacColl claimed.42 His conception of the relationship between truth and validity is original and also rather puzzling. MacColl put forward a radical critique of the syllogism, bluntly claiming that all syllogisms are false. ‘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.’43 The fact that almost no logician since Aristotle noticed this problem did not prevent him from asserting his point with the utmost vigour. As it was, however, the syllogism had already been the object of many criticisms, which are legitimate when addressed to its extensional form: ‘In the therefore-form (A∴B) the Syllogism is open to the attacks made upon it by Mill, Bradley and others. In the “If” or implicational form (A:B) its validity is unassailable.’44 The problem, MacColl argued, is that the traditional syllogism asserts the truth of its premises and conclusion. But not only can these propositions be false, it is also a mistake to believe that the syllogistic form depends on the assertion of their truth: ‘not one syllogism out of the whole nineteen is valid in its traditional form PQ∴R, as in this form it asserts without warrant that the two premisses P and Q are both true’.45 If a syllogism is, as it should be, true in virtue of its form, then it is neither in virtue of the fact that ‘All Greeks are mortal’ nor that ‘All Greeks are men’. These statements have nothing to do with the structure of the syllogism. If it were the case, then the syllogism would be invalid when at least one of the asserted propositions is false, which is obviously not the case. Let us take the argument of the form: All A’s are B’s All B’s are C’s Therefore all A’s are C’s.

It could very well be that the conclusion is false, for example if some A’s are not B’s. The conclusion does not follow from the form of the argument presented. Therefore, the formulation of the syllogism is incorrect. MacColl solved this difficulty on the basis of a ‘system of equivalent propositions including disjunctions and conditionals, which reflects the natural semantics of traditional hypothetical forms’.46 As already seen, MacColl wrote the propositions in a hypothetical form: If all A’s are B’s and all B’s are C’s, then all A’s are C’s. This is legitimate since there is no distinction between categorical and hypothetical propositions. Likewise, Peirce first generalized Boole’s attempt to reformulate Aristotle’s categorical syllogisms as chains of propositions whose fundamental connective is implication.47 In an entry of his ‘Logic Notebook’ for 14 November 1865, Peirce wrote 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’.48 Peirce virtually, though not yet actually, equated the copula of predication with implication, thereby opening the way to the representation of the syllogism ‘All S are M, All M are P, Therefore all S are P’ as (S ⊃ M)&(M ⊃ P) ⊃ (S ⊃ P).49 For MacColl this transition resulted in a difference between reasoning that is formally valid but only hypothetical and reasoning that asserts the truth of each of its

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propositions. This difference is sometimes formulated as valid versus sound, a sound reasoning being a valid reasoning whose premises and conclusion are true. MacColl emphasized that in practice this is not a minor but a crucial or even vital difference, as the following example reveals: Suppose a General, whose mind, during his past University days, had been overimbued with the traditional logic, were in wartime to say, in speaking of an untried and possibly innocent prisoner, ‘He is a Spy, therefore he must be shot’, and that this order were carried out to the letter. Could he afterwards exculpate himself by saying that it was all an unfortunate mistake, due to the deplorable ignorance of his subordinates; that if these had, like him, received […] a logical education, they would have known at once that what he really meant was ‘if he is a spy, he must be shot’.50

Commenting on the form ‘All A’s are B, All B’s are C, Therefore All A’s are C,’ MacColl wrote that the syllogism, or any other argument, thus worded is not a formal certainty, it is false, whatever the conclusion may be; and it is also false when the conclusion is false, whatever the premisses may be. Barbara should be worded as follows: ‘If all A is B, and all В is C, then all A is 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.51

The passage is not as plain as it may look at first sight. It seems that MacColl insists on the formal validity of the syllogism, that is, on the position and distribution of the letters standing for subjects and predicates. But this is not exactly what MacColl meant. His precise explanation is that a statement has formal certainty ‘when it follows necessarily from our formally stated conventions as to the meanings of the words or symbols which express it’.52 ‘Formally certain’ obviously refers to the arbitrary definitions of terms, not to the form of reasoning.53 It deals with the relation of meaning between subject and predicate, that is, it concerns the analytical or synthetic nature of propositions. In this sense, ‘All A is B’ is a formal certainty if it is an analytic proposition, or if the class of A’s is included in the class of B’s. It thus implies that the premises must be true. Such a claim seems to directly contradict MacColl’s own theory which says that since a syllogism is not composed of categorical propositions the truth of its premises is of no concern. The claim was not a slip of the pen, however. MacColl repeatedly claimed that the premises of a valid syllogism are necessarily true. He ‘emphatically dissent[ed]’ from the many logicians who thought that the validity of an argument does not depend on the truth and falsehood of its premises. ‘The premisses and the conclusion are, in my opinion, the most important factors of an argument, and if either of these be false […] the argument should not be considered valid.’54 But did not the very same MacColl previously claim that Barbara, and any other valid syllogism, ‘is true whether premisses or conclusion be true or false, and must, therefore, be classed amongst the formal certainties’?55

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A solution to this apparent paradox could be the following. MacColl meant that in its ‘therefore’ form, a syllogism is not a formal certainty, because if its propositions were false, the syllogism is not valid. That is why the validity of reasonings depends on the truth of their premises and conclusions: the validity is conditional. Some propositions are uncertain but not false or possible but not true: these variable propositions, which stand ‘between 0 and 1’, must be taken into account, while they are not formally certain. This is the role of modal logic. However, in its ‘if-then’ form, a syllogism is a formal certainty. Indeed, MacColl also wrote that a formal certainty ‘is that which is valid unconditionally’.56 This is the case in the inferential form, since an empty set does not imply a false premise: while ‘All men are immortal’ is traditionally considered false, for MacColl the proposition is false only in a universe which radically excludes this possibility. This position is due to his particular analysis of the truth of propositions in modal terms. Unlike ‘A therefore B’, or the inference that makes B known from A, the implication ‘A implies B’ only means that ‘the affirmation of A coupled with the denial of B constitutes an impossibility’.57 This modal approach should next be set in a more general, propositional framework.

6 Propositional logic MacColl’s revolutionary approach to the syllogism rests on a conception of logic centred on propositional notation with the inference relation standing at its heart, as opposed to the equational notation inherited from Boole. As MacColl recalled: ‘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.’58 Whereas ‘other logicians generally divide logic into two parts: the logic of class inclusion and the logic of propositions’, MacColl’s logic is ‘one simple homogeneous system which comprises […] all the valid formulae of their two divisions’.59 In MacColl’s view, logic is not about classes. The letters represent neither terms and classes nor properties, as they did for Boole, as well as for Jevons, Venn or Lewis Carroll in his wake. The letters are ‘temporary symbols’ which stand for ‘statements’.60 Propositions, or what could be called propositional functions, have ‘the immense advantage of being independent of the accidental conventions of language’.61 Indeed, the logical structure of propositions, with a subject and a predicate, does not necessarily correspond to the grammatical or linguistic structure (e.g. the grammatical difference between active and passive voice does not reflect anything in logic). Propositional logic was for MacColl at the very least a matter of convenience: ‘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’.62 Why? As we have seen, the treatment of concrete cases shows that the traditional approach is awkward and forces one ‘to twist and torture simple sentences in order to adapt them for syllogistic reasoning’.63 Singular propositions, which refer to a

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one-member class, remain a puzzle in Boole’s logic. Lewis Carroll, for one, has no other means but to translate ‘John is not well’ into ‘All Johns are men who are not well’, which is due in particular to his failure to distinguish a single thing and a one-member class and to conceive a null class properly.64 Furthermore, a logic centred on propositions is not only a matter of convenience. It also echoes the very structure of human thought. ‘The complete statement or proposition is the real unit of all reasoning.’65 We do not reason with things but with propositions. Propositional logic is what MacColl called ‘Pure Logic’. It is based on an intentional approach, as opposed to an extensional ‘Applied Logic’ approach. MacColl considered pure logic as a second-order logic since it is able to compose propositions of propositions of propositions etc., which are still propositions, while the same cannot be done with classes. AB is a statement, and ABC, which is short for (AB)C, is a statement, asserting that the statement AB belongs to the class of statements denoted by C. But it cannot be interpreted in terms of classes: ‘In Applied Logic A and В may both denote concrete things or classes but farther we cannot go.’66 As a consequence of the composition of propositions, a three-dimensional logic is required. Since propositions are implications, reasonings are implications of implications so 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’.67 At this more complex stage, the third degree is that of the ‘variable’, that is, of ‘statements whose truth or falsehood may be considered an open question’.68 An example of a variable statement is ‘x = 4’, which may be true or not according to the given hypotheses, that is, depending on the context.69 This explains MacColl’s insistency on the temporary character of symbols. It forms a basis for MacColl’s endeavours towards non-classical logics. This kind of three-dimensional approach is also applicable to the treatment of the syllogism. More precisely, MacColl distinguished at least five dimensions or values: certain, impossible, variable, true and false, while accepting that this is only one logic among other possible conventions. In a syllogism, the aim is to determine whether a statement is true, false, certain, impossible or variable on the basis of certain initial data, which constitute the premises. The ‘variable’ itself has several values: a proposition can for example be true in a particular case, but potentially false without contradicting the initial hypotheses or on the contrary be false in a particular case but possible as a general law.70 Thus, in a syllogism ‘the question to be decided may be not merely to decide whether A is true or false, but whether A follows necessarily from, or is inconsistent with, our definitions or admitted and unquestioned data’.71 The answer is a proposition of a higher order than a simple proposition. For this reason, MacColl decided to note the values assigned to the propositions by means of superscripts: in his symbolism, Aε means that A necessarily follows from the data, Aη that A is inconsistent with the data. Generally speaking, the proposition Ax is of higher degree than A, and the proposition Axyz is a third-degree proposition relatively to the ‘rootstatement’ A. MacColl sees it as an expression of a revision of the judgement: (Ax)y is a revision of the statement Ax. This results in a potentially infinite number of degrees of judgement.

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7 Existential import One of the great difficulties in the reconstruction of the syllogism in modern logic results from the status of the null-class and the problem of existential import. According to Radford, ‘[r]egarding the existential import of propositions and the null-class, what MacColl has to say in “Symbolic Reasoning VI” and “Reply” is both clearer and more correct than what Russell has to say about both of these topics in his “The Existential Import of Propositions” and about existential import in “On Denoting”’.72 Aristotelian medieval syllogistics made an assumption of existence, which is necessary for the relations in the logical square to hold. It seemed natural that ‘Some men are Greeks’ refers to existing individuals and can be paraphrased as ‘Some men exist who are Greek’. On the other hand, from an extensional point of view, and in particular in Boole’s algebra of logic, the null or empty class, being a set containing no element, is included in all other classes, which invalidates most of the relations of the traditional logical square. More specifically, as Ladd-Franklin pointed out: ‘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.’73 MacColl made the same comment as early as in 1880. Thus, to use an example from Jevons, ‘The sea-serpent is not found in the water’ and ‘The sea-serpent is not found out of the water’ are not mutually contradictory if sea serpents do not exist. It is even the only correct conclusion which can be drawn from that pair of propositions. Or more generally, ‘All A’s are B’ and ‘No A’s are B’ are not contraries when A = 0. For an empty class, the only subsisting relation in the square of oppositions is contradiction: a universal proposition can be contradicted only by a particular proposition, and a particular proposition contradicted only by a universal proposition, as emphasized by MacColl as well as Peirce and Venn, who was the first to draw attention to this problem. MacColl began his analysis by stating that the question whether the objects denoted by terms exist ‘does not appear to me to belong to the province of pure logic, which should treat of the relations connecting different classes of propositions, and not of the relations connecting the words of which a proposition is built up’.74 It is rather a matter of grammar. Existence depends on the meaning one agrees to give to the word. In contrast, ‘[i]n pure logic the subject, being always a statement, must exist – that is, it must exist as a statement’.75 MacColl’s view was that it is necessary to distinguish, within the universe of discourse, a universe of real existences and a universe of non-existing entities, either merely unreal like unicorns or contradictory like round squares. This is the position that is generally attributed to him and which he put forward in the following way: 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

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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.76

For reasons of symmetry, the symbolic universe or universe of discourse includes not only the syllogistic classes but also their mutually exclusive, complementary classes (for instance X and what MacColl writes ‘X’’), no matter whether or not these really exist in the world. In other words, all statements are about objects belonging either to the actual universe or to a world that is only possible (non-existent but not contradictory) or to a self-contradictory world. The relation between actual objects can be expressed in extensional terms, like a class inclusion. The relation between merely possible objects, on the other hand, corresponds to the syllogism: it is a relation of strict implication, which says that it is necessary that if A then B. The traditional syllogistic proposition (‘All A’s are B’) is thus not interpreted as a universally quantified proposition but as stating that the proposition affirming the implication between A and В is certain. It is a rule. As for the impossible, the notation must distinguish the impossible Aη (which contradicts something given or a definition) from the meaningless Ao (which does not contradict anything at all). The null-class, which includes all non-existent and contradictory objects, is therefore far from being empty: the nullclass exists and has members. It is not contained in any class, contrary to what the Booleans thought. All propositions denote, even those about the current king of France, so that the truth-value of ‘The current king of France is bald’ depends on the context: the subject has what MacColl called a ‘symbolic existence’, and its ‘real existence’ is given by additional pieces of information (according to whether we place ourselves in a fairy tale, a historical time, the genealogy of royalists etc.). MacColl considered the possibility of saying true things about non-existent entities, mythical worlds, invented stories, etc., paving the way to a promising theory of fiction. For Russell, on the other hand, the aforementioned proposition is simply false, as famously argued in ‘On Denoting’.77 The significance of MacColl’s position is that it offers a more flexible and sensitive treatment of fictions. Its shortcomings are twofold: on the one hand, those of Meinong’s theory (e.g. ontological inflation and the generation of contradictions), and, on the other hand, its reduction of all meaning to denotation.

8 Conclusion MacColl himself did not at first regard syllogism as the most important field of application of his calculus of equivalent statements. His symbolic system of logic was intended to ‘clear away with the greatest ease a complete jungle of difficulties which

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had vexatiously arrested the progress of mathematical science’, especially the problem of the limits of multiple integrals.78 It was also supposed ‘to discover the general laws which regulate the various phenomena of the material universe’.79 However, the inclusion of the syllogism into modern logic, which Boolean logic had not managed to do properly, is arguably one of its most beautiful achievements. The Aristotelian syllogism is completely transformed by the propositional approach. Previously a territory of predicate logic, it becomes the object of a unified treatment which puts implication at the centre, provided there is no essential difference between a categorical and a hypothetical proposition. ‘This translation of the syllogism as an implication was a crucial step in establishing propositional logic in its modern form.’80 Moreover, pace Russell, MacColl chose the strict implication, which involves a multivalent conception of truth-values. This opens the way to a fully formal treatment of syllogistics within the framework of a study of propositional functions and of a logic of relations which pushes the boundaries of traditional, extensional logic.

Notes Christine Ladd-Franklin, ‘Dr. Hillebrand’s Syllogistic Scheme’, Mind 1, no. 4 (1892): 527–30, at 527. 2 Bertrand Russell, ‘Review of H. MacColl’s Symbolic Logic and Its Applications’, Mind 15 (1906): 255–60, at 255. 3 John Grier Hibben, ‘Review of Symbolic Logic and Its Applications by Hugh MacColl and of The Development of Symbolic Logic by A. T. Shearman’, The Philosophical Review 16, no. 2 (1907): 190–4, at 190. 4 Nicholas Rescher, Many-Valued Logic (New York: McGraw-Hill, 1969), 4, considered MacColl one ‘founding father’ of the polyvalent logics with C. S. Peirce and Nikolai A. Vasiliev. 5 Irving Anellis, ‘MacColl’s influences on Peirce and Schröder’, Philosophia Scientiae 15, no. 1, Special issue: Hugh MacColl after One Hundred Years, eds. Amirouche Moktefi and Stephen Read, (2011): 97–128, at 99. 6 Christine Ladd-Franklin, ‘On Some Characteristics of Symbolic Logic’, The American Journal of Psychology 2 (1889): 543–67, at 562. 7 MacColl, ‘Symbolical Reasoning’, Mind 5, no. 17 (1880): 45–60, at 58. 8 Fania Cavaliere, ‘L’opera di Hugh MacColl alle origini delle logiche non-classiche’, Modern Logic 6, no. 4 (1996): 373–402, at 393. 9 MacColl, ‘Review of Lewis Carroll’s Symbolic Logic. Part I: Elementary (London: Macmillan, 1896)’, The Athenaeum 3599 (1896): 520–1, at 520. 10 MacColl’s first wife died in 1884, which could provide another reason for his retirement. 11 See Lukas M. Verburgt, ‘The Venn-MacColl Dispute in Nature’, History and Philosophy of Logic 41, no. 3 (2020): 244–51. 12 MacColl, ‘On the Growth and Use of a Symbolical Language’, Memoirs of the Manchester Literary and Philosophical Society 7 (1882): 225–48, at 242–3. 13 Letter to Bertrand Russell, 17/05/1905. 14 MacColl, Symbolic Logic and Its Applications (London: Longmans, Green, & Co., 1906) summarizes much of the eight ‘Symbolic Reasoning’ articles. 1

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15 Schröder in particular was critical of the ‘MacColl-Peircean propositional logic’. According to him, logic should not be based upon the calculus of propositions but on the calculi of domains and classes, which was more general. Schröder’s attitude towards MacColl’s work was fluctuating: ‘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.’ S. Rahman and J. Redmond, ‘Hugh MacColl and the Birth of Logical Pluralism’, in Handbook of the History of Logic, vol. 4, British Logic in the Nineteenth Century, ed. D. M. Gabbay and J. Woods (Amsterdam: Elsevier, 2008), 533–604, at 538. 16 MacColl, ‘Review of A Treatise on Universal Algebra with Applications, vol. 1, by Alfred North Whitehead’, Mind 8 (1889): 108–13, at 109. 17 The ‘second’ MacColl, while harshly discarding the attempts of the Booleans to perfect an algebra of logic, extolled 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, ‘Symbolic Reasoning (II)’, Mind 6, n. 24 (1897): 493–510, at 504. Boole’s mistake was to endeavour to make the manifoldness 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’. Ibid., 505. 18 MacColl, ‘Symbolic reasoning (V)’, Mind 12, no. 47 (1903): 355–64, at 364. 19 Similarly, to mean that ‘All men are mortal’, Boole needs to introduce an indefinite variable such that x = vy, i.e. ‘Men are among men those who verify an indeterminate adventitious condition symbolized by v.’ This symbol v creates difficulties because it cannot be detached from its conditions of interpretation. Peirce also criticized it as early as 1870. See Ch. S. Peirce, Chronological Writings of C. S. Peirce, vol. 2 (Indianapolis: Indianapolis University Press, 1984), 387–95. 20 MacColl, ‘The Calculus of Equivalent Statements (II)’, Proceedings of the London Mathematical Society 9, (1878): 177–86, at 180. 21 Ibid., 184. 22 Id., ‘The Calculus of Equivalent Statements (V)’, Proceedings of the London Mathematical Society 28 (1896): 156–83, at 161. 23 Id., ‘Symbolic Reasoning (V)’, 361. 24 Id., ‘Symbolic Reasoning (IV)’, Mind 11, no. 43 (1902): 352–68, at 352. 25 Ibid., 356. 26 Ibid., 354. 27 Id., ‘The Calculus of Equivalent Statements (V)’, 165. 28 Id., ‘Symbolic Reasoning (VI)’, Mind 14, no. 53 (1905): 74–81, at 80. 29 Id., ‘Symbolic Reasoning (IV)’, 355. 30 Ibid., 80. 31 Id., ‘The Existential Import of Propositions’, Mind 14, no. 56 (1905): 295–6, 401–2, 578–80, at, 394. 32 Id., ‘The Existential Import of Propositions’, 396. 33 Jean-Marie C. Chevalier, ‘Some Arguments for Propositional Logic: MacColl as a Philosopher’, Philosophia Scientiae 15, no. 1, Special issue: Hugh MacColl after One Hundred Years, eds. Amirouche Moktefi and Stephen Read, (2011): 127–47, at 137.

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34 MacColl’s strong concept of implication ‘in his early work was defined as relevant and connexive and in his later work as a strict implication’. Shahid Rahman, ‘Ways of Understanding MacColl’s Concept of Symbolic Existence’, Proceedings of the Conference: Hugh MacColl and the Tradition of Logic, eds. Michael Astroh and Stephen Read, Nordic Journal of Philosophical Logic 3, no. 1 (1998), 35–58, on 36. 35 MacColl, ‘Symbolical Reasoning’, 51. 36 ‘Material implication it will appear, applies to any world in which the all-possible is the real, and cannot apply to a world in which there is a difference between real and possible, between false and absurd. Strict implication has a wider range of application. Most importantly it admits of the distinction of true and necessary, of false and meaningless.’ C. I. Lewis, ‘The Calculus of Strict Implication’, Mind 23, no. 90 (1914), 240–7, on 241. As for the conflation of the ‘if-then’ relation on implication, there would be much to be said, but its difficulties do not fall in the scope of the present chapter. It was famously criticized by Quine for involving a use-mention fallacy, especially from the part of Russell in Principia Mathematica. Cf. Matheus Silva, ‘“If-Then” as a Version of “Implies”’, Draft, 2020. 37 Written (a’ + b)η in MacColl’s system. The proof is that the negation of a:b means ‘it is possible that a without necessarily b’, while the negation of (not-a or b) means ‘a and not-b’. 38 MacColl, ‘Symbolical Reasoning’, 51–2. 39 Id., ‘On the Growth and Use of a Symbolical Language’, 243. 40 Id., ‘Symbolical Reasoning’, 51–2. 41 John Venn, Symbolic Logic (London: MacMillan and Co, 1881), 376–8. Cf. Anellis, ‘MacColl’s Influences on Peirce and Schröder’, 103. 42 MacColl, ‘The Existential Import of Propositions’, 390. 43 Id., Symbolic Logic and Its Applications, 47. 44 Id., ‘If ’ and ‘Imply’, Mind 17, no. 65 (1908): 151–2 & 453–5, at 454. 45 Id., ‘Symbolic Reasoning (VI)’, 79. 46 Rahman, ‘Ways of Understanding MacColl’s Concept of Symbolic Existence’, 36. 47 Whereas MacColl adopted a defensive attitude towards any suspicions of plagiarism, insisting that ‘the apparent coincidence of notation, in some few particulars, between him and Prof. Peirce, was entirely accidental’, Peirce avowed that MacColl re-invented everything from scratch: ‘Later in the same year [1877], 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.’ Ch. S. Peirce, Chronological Writings, vol. 3, n. 182. According to every evidence, the two men met at Boulogne-sur-Mer. The newly wed Peirce had indeed planned this meeting, which did take place as is confirmed by a letter from Christine Ladd-Franklin (Letter from Ladd-Franklin to Peirce, 22 November 1902), as the first step of his honeymoon trip to Europe in 1883, showing how much he praised MacColl’s works and expected from their exchange of views. It goes with a conception of science as a living community: ‘the limits of a science are those of a social group’, he would write. Science is not an abstract idea, it is ‘the actual living occupation of an actual group of living men’. Meeting researchers and discussing ideas with flesh and blood people was part of Peirce’s way of thinking. 48 Ibid.

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49 Charles Sanders Peirce, The Chronological Writings of C. S. Peirce, vol. 1 (Bloomington: Indiana University Press, 1982), 337. Unfortunately, Peirce later treated his primary connective, which he called illation, or the so-called claw with which the Barbara syllogism may be expressed as ((S -< M) & (M -< P)) -< (S -2’ and are thus unconditionally valid, that is, invariable. But their immutability stands as long as we keep using the ordinary language, which is not always the case, as the example of non-Euclidean geometries shows. In any case, the meaning of propositions depends on definitions in principle. But it is true that MacColl, who also remarked that logic is ‘independent of the accidental conventions of language’ (‘Symbolic Reasoning (IV)’, 352), left open the possibility of several interpretations. 54 Id., ‘Linguistic Misunderstandings (I)’, Mind 19, no. 74 (1910): 186–99, at 186–7. 55 Id., ‘Symbolic Reasoning (IV)’, 368. 56 Id., ‘Symbolic Reasoning (V)’, 356. 57 Id., ‘Linguistic Misunderstandings (I)’, 190. 58 Id., ‘Review of A Treatise on Universal Algebra with Applications’, 108–9. 59 Id., ‘Symbolic Reasoning (V)’, 355. 60 Id., ‘Symbolical Reasoning’. 61 Id., ‘Symbolic Reasoning (IV)’, 352. 62 Id., ‘The Calculus of Equivalent Statements (V)’, 520. A very similar quote replaces three years later ‘proposition’ with ‘statement’: symbols denote ‘not classes, not properties, not numbers, ratios, regions, or magnitudes, not things of any kind – but complete statements’. MacColl, ‘Review of A Treatise on Universal Algebra with Applications’, 109. A statement is defined as a (combination of) sound(s) or symbol(s) which conveys information. MacColl also explained that since proposition refers more specifically to the subject-predicate form, all propositions are statements but not all statements are propositions. 63 Ibid. 64 Francine Abeles and Amirouche Moktefi, ‘Hugh MacColl and Lewis Carroll: Crosscurrents in Geometry and Logic’, Philosophia Scientiae 15, no. 1, Special issue: Hugh MacColl after One Hundred Years, eds. Amirouche Moktefi and Stephen Read, (2011): 55–76, at 69. 65 MacColl, Symbolic Logic and Its Applications, 2. 66 Id., ‘Review of A Treatise on Universal Algebra with Applications’, 111–12. 67 MacColl, ‘Symbolic Reasoning (II)’, 496. 68 Id., ‘Symbolical Reasoning’, 53.

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69 Russell famously objected that a (genuine) proposition ‘is anything that is true or that is false’, but cannot be sometimes true and sometimes false. ‘An expression such as “x is a man” is therefore not a proposition, for it is neither true nor false’. B. Russell, Principles of Mathematics (London: George Allen and Unwin, 1937), 13. 70 MacColl, ‘Symbolic reasoning (III)’, Mind 9, no. 36 (1900): 75–84, at 80. 71 Id., ‘Symbolic reasoning (VIII)’, Mind 15, no. 60 (1906): 504–18, at 514. 72 Colin Radford, ‘MacColl, Russell, The Existential Import of Propositions, and the Null-Class’, The Philosophical Quarterly 45, no. 180 (1995): 316–31, at 317. 73 Christine Ladd-Franklin, ‘On the Algebra of Logic’, in Studies in Logic, by members of the Johns Hopkins University (Boston: Little, Brown and Company, 1883), 17–71, at 39. 74 MacColl, Id., ‘Symbolic Reasoning (IV)’, 357. 75 Ibid. 76 Id., ‘Symbolic Reasoning (VI)’, 74. 77 Bertrand Russell, ‘On Denoting’, Mind 14, no. 56 (1905): 479–93, at 491–2. 78 Id., ‘Symbolical Reasoning’, 47. 79 Ibid., 58. 80 Anellis, ‘MacColl’s Influences on Peirce and Schröder’, 99.

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Frege’s relation to Aristotle and the emergence of modern logic Erich H. Reck

Gottlob Frege’s works are often taken to mark the beginning of modern logic. More specifically, the year 1879, when Frege’s Begriffsschrift was published, is seen as witnessing its birth. ‘Modern logic’ is here contrasted with traditional Aristotelian logic, especially the theory of syllogism that is central for it. In this chapter, I will reconsider the transition from Aristotelian logic to modern logic in Frege’s and related writings. I will confirm that Frege’s works do, indeed, contain innovations that contributed to a radical transformation of logic. At the same time, I will argue that the story of that transformation is complex, including the fact that Frege’s works contain remnants of Aristotelian ideas that make it not fully modern. The chapter will proceed as follows: In Section 1, I will start with several claims that the publication of Begriffsschrift constituted a ‘revolution’ in logic, while noting some immediate challenges to that claim as well. In Section 2, I will turn to Frege’s central logical innovations and to logicism as the driving force behind them. Next, I will consider Frege’s relationship to Aristotle, both in terms of Aristotelian views Frege rejected explicitly, in Section  3, and features that still tie him to Aristotle’s classical model of science implicitly, in Section 4. In Section 5, finally, I will reconsider the ways in which Frege’s contributions were crucial for a ‘second birth’ of logic, including by being articulated further in the works of later logicians.

1 A Fregean revolution in logic? Since the 1950–1960s, the publication of Frege’s Begriffsschrift has often been seen as the start of modern, post-Aristotelian logic. Crucial roles in assigning a central part to Frege were played by Alonzo Church, W. V. O. Quine and J. v. Heijenoort. As Quine wrote in 1952, ‘[l]ogic is only now emerging from a renaissance such as was undergone by physics centuries ago. […] The logical renaissance might be identified with the publication of Frege’s Begriffsschrift in 1879’.1 Or more pointedly in Quine’s Methods of Logic (1950): ‘Logic is an old subject, but since 1879 it has been a great one.’2 The same perspective shapes Jean van Heijenoort’s influential collection, From Frege to Gödel: A

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Source Book in Mathematical Logic, 1879–1931 (1967). Already its title is indicative, and in the Preface van Heijenoort elaborates: ‘A great epoch in the history of logic opened up in 1879, when Gottlob Frege’s Begriffsschrift was published.’3 While Frege’s works became prominent in the United States in the 1950–1960s, a parallel rediscovery of him took place in Great Britain during the same period, as initiated by J. L. Austin, Peter Geach, Michael Dummett and others.4 In this context, Dummett’s works were particularly influential, especially his book, Frege: Philosophy of Language (published in 1973, but widely circulated already in the 1960s). Like Quine, Dummett presented Frege as providing a radically new beginning in logic, also claiming that it had essentially no precedents. Moreover, Dummett and others suggested that one can find modern logic already fully formed in Begriffsschrift. Or as William and Martha Kneale put it in their well-known book, The Development of Logic (1962): ‘Frege’s work […] contains all the essentials of modern logic, and it is not unfair either to his predecessors or his successors to say that 1879 is the most important date in the history of the subject.’5 Let me highlight two aspects in those remarks: First, Frege’s Begriffsschrift, published in 1879, is flagged as an absolutely crucial work in the history of logic, comparable to Galilei’s, Newton’s and related contributions to physics in the Scientific Revolution. Second, Frege’s book is taken to represent a radical break with the Aristotelian tradition before it. Based on both, it is tempting to think that a revolutionary transformation of logic took place in 1879. This claim is spelled out explicitly by Donald Gillies, a few decades later, in his aptly titled article, ‘The Fregean Revolution in Logic’ (1992). Using the terminology of ‘revolution’ in this context is meant to invoke Thomas Kuhn’s famous book, The Structure of Scientific Revolutions, with its account of similar breaks in the natural sciences. Putting Frege, and Begriffsschrift in particular, in this prominent position is still widespread today. But it has also led to a variety of challenges and ensuing debates. Already in the 1980–1990s, Hans Sluga objected to Dummett’s ahistorical treatment of Frege, especially the claims that there were no important influences on him and that modern logic appeared fully formed in Begriffsschrift (similar to how, in Greek mythology, Athena sprang forth, fully grown and in a complete set of armour, out of Zeus’ skull). More specifically, Sluga argued that Frege’s logicist project had roots in the writings of Hermann Lotze and other German philosophers. This kind of argument was developed further, also in other ways, by Gottfried Gabriel and his co-workers.6 In fact, Frege himself made some connections to Leibniz explicit in his Begriffsschrift, to whom he was led by Friedrich Trendelenburg. This challenge to seeing Frege’s logical contributions as pristine and unprecedented can be extended in two ways. First, his logicism should not only be seen in the context of nineteenth-century German philosophy but also that of nineteenth-century mathematics. As Mark Wilson, Jamie Tappenden and I have noted, Frege’s logicist project was rooted in related developments in geometry and analysis, especially in the works of Gauss, Riemann and their students.7 Second, Susanne Bobzien has emphasized recently that Frege was not unaware of Ancient Stoic logic. In fact, it looks like he borrowed freely from the Stoics, especially with respect to ideas in the philosophy of logic and language, probably via Carl Prantl’s four-volume Geschichte

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der Logik im Abendlande (1855–1870), a widely known history of logic at his time.8 If so, not only Frege’s logicism but also his philosophy of logic has earlier roots. A second challenge to attributing such an exalted place in the history of logic to Frege’s works has come from re-emphasizing the importance of other nineteenthcentury logicians, starting with George Boole and the Boolean school, including Venn, Jevons and Schröder.9 If this suggestion is taken seriously, modern logic started arguably in 1854, with the publication of Boole’s An Investigation of the Laws of Thought. Frege himself responded, of course, that Boolean logic was inferior and less wide-reaching than his own10, even its mature version in Ernst Schröder’s works.11 However, this has been challenged in turn recently, e.g. by pointing out the extent to which another logician in the Boolean school – Charles S. Peirce – came up with aspects seemingly unique to Frege’s logic independently in the early 1880s. The latter encompasses polyadic quantification, including a good notation for it, combined with a treatment of n-place relations, building on work by Augustus De Morgan.12 And as Peirce had a fairly direct influence on the subsequent development of logic, via Schröder, Peano, Russell etc., the question arises how Peircean the ‘Fregean’ revolution in logic was.13 The common story that Aristotelian logic, seen as dominant well into the nineteenth century, was challenged only by Frege, and more generally, the claim that little progress was made in logic before 1879, have been questioned in other respects too. Another recently rediscovered logician is Bernard Bolzano, especially his Wissenschaftslehre (1837). As is well known by now, Bolzano developed a notion of logical consequence that anticipated Tarski’s semantic conception in important respects, a feature not present in Frege’s writings. For this and similar reasons, should the start of modern logic be dated back even further, to 1837?14 Or can one argue that Frege’s went further than both the Boolean logicians and Bolzano in crucial ways? This leads to the question what relevant differences there are in Frege’s logic. As a third kind of challenge to the assumption that modern logic began in 1879 with Frege’s Begriffsschrift, we can return to that book itself. Far from containing modern logic in its full form, it does not even represent Frege’s own mature views about logic. That is to say, there are internal developments to his logic which the common story, with which we began this section, overlooks or downplays misleadingly. In particular, the logical formalism in Begriffsschrift is not yet anchored in a theory of objects and functions as its content, definitely not in its mature form.15 A final point is this: From Begriffsschrift through Basic Laws of Arithmetic (1893/1903) to his later lectures on logic (1910–1914), Frege’s system contains ingredients that are foreign to most of contemporary logic, such as his two-dimensional notation, the judgement stroke and the assumption that quantification in logic is over all objects and all functions, not about restricted and varying domains of them.

2 Frege’s main motives and innovations The arguments summarized in the previous section lead to the conclusion that Frege’s Begriffsschrift was not the completely unprecedented, pristine and full start to modern logic that was assumed previously. Yet it would be wrong to go too far in the other

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direction as well, i.e. to downplay Frege’s contributions too much. So far we have also not considered Frege’s relation to Aristotelian logic in any detail. To prepare the latter, let me now provide some reminders about Frege’s achievements, including ways in which he went beyond Leibniz, Lotze, Trendelenburg, Gauss, Riemann, Boole, Peirce, Bolzano, the Stoics etc. The first thing to observe in this connection is Frege’s main motivation for introducing a new logic: his logicist project. As Frege tells us in the Preface to Begriffsschrift, he was led to introducing his logical innovations, including the novel logical language they are based on, in the service of investigating the epistemological status of arithmetic and in critical dialogue with philosophers such as Leibniz, Kant and Mill. More specifically: My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps.16

Unlike in Boole’s case, say, Frege’s logic was meant to serve a foundational purpose, thereby tying together logic and modern mathematics in a novel way. This is why an approach that allows for ‘gapless’ inferences is so important for him. Frege goes on to remark that, for this specific purpose, he found ordinary language deficient, and that led him to introducing a ‘formula language of pure thought’, one that was ‘modeled upon the formula language of arithmetic’. The primary point of contact between the latter two languages, and a main difference to ordinary language, is ‘the way in which letters are employed’.17 Two aspects are worth highlighting here. First, the main application of Frege’s new logic is mathematical reasoning and mathematical concepts, such as that of following in a sequence. It is with respect to this specific goal that ordinary language is not a sharp enough tool. And traditional logic – with its syllogistic forms of inference and its reliance on the ‘S is P’ form for all statements – is not adequate either. Second, Frege tells us that a crucial feature of his new logic is the use of ‘letters’ in it. I take that to be a reference to what we call ‘variables’, and with them the expression of generality, together with that of relationality. Two further illustrations Frege often mentions in this context, besides following in a sequence, are analysing the concept of the continuity of a function and the concept of limit.18 Again, for capturing such concepts adequately, neither ordinary language nor the tools of traditional Aristotelian logic suffice. What is needed, as Frege worked out clearly, is a relational logic that allows for nested quantification.19 It is that kind of logic Frege starts to introduce in Begriffsschrift. It integrates a treatment of truth-functional connectives, especially of negation and the material conditional (indirectly also conjunction and disjunction), with a treatment of relational and quantificational logic, i.e. the use of variables, constant symbols, nplace function and relation symbols, and nested quantifiers (explicitly the universal quantifier, derivatively also the existential quantifier). Two observations should be added: First, Frege allows for second-order variables and quantification, which play a role in his analysis of mathematical concepts. Second, and as already mentioned,

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the way in which his symbols are interpreted in Begriffsschrift is not yet that of his own mature logic, from after 1890; and even less it is that of standard second-order logic. The reason is that he has not yet introduced several relevant distinctions, such as a sharp separation between objects, first-order functions and concepts, second-order functions and concepts, also between expressions, their senses, and their referents.20 A central aspect of Frege’s logical contributions, also in his own view, is the introduction of his formal language, his ‘conceptual notation’. It is crucial for him — a point Frege emphasizes in distancing his approach from that of the Booleans — that this language allows us to analyse mathematical concepts in a broad and systematic way. As such, it forms a Leibnizian ‘lingua characteristica’. But his logical system also functions as a Leibnizian ‘calculus ratiocinator’, i.e. a means for drawing logical inferences in a formal, mechanizable way. Moreover, it does so in the form of a novel axiomatic approach to inference, very different from Boole’s equational approach, and clearly a model for twentieth-century logic.21 Central here is the explicit list of logical inference rules and logical laws that Frege provides. In Begriffsschrift, he restricts himself to one inference rule, modus ponens, while a number of logical laws provide the rest of the inferential power. In later versions, he varies the rules and laws, since he realizes that several choices are possible. While Frege does not raise the issue in the now standard form, it was shown subsequently (by Bernays, Post, Gödel and others) that the rules and laws for the propositional fragment of his logic form a sound and complete system. Similarly, his rules and laws for the first-order fragment form a sound and complete system, although in the latter case one has to factor in substitution rules that remain implicit in his presentation. In both respects, Frege is again far ahead of his time and deserves special recognition. His own focus is on a kind of ‘relative completeness’, namely the fact that his system allows for the reconstruction of all the usual reasoning in arithmetic and analysis. That is what is supposed to be established conclusively in his later writings, especially in the multiple-volume Basic Laws. Once again, it is his logicist project that is the driving force here. That project is formulated tentatively in Begriffsschrift, and it is then defended in later works, both philosophically and mathematically, starting with The Foundations of Arithmetic (1884). In the Preface to Foundations, Frege highlights three basic principles he intends to follow: (i) to sharply separate the psychological from the logical, (ii) to ask for the meaning of words in context and (iii) to sharply distinguish concepts from objects. Together with his later distinction between sense [Sinn] and reference [Bedeutung], this leads to his mature semantics from the 1890s. In particular, it leads to a sharp break with Aristotelian ‘term logic’ in which several of these distinctions are blurred. To be able to pursue his logicism, Frege also adds another ingredient, somewhat implicitly in Foundations and explicitly in Basic Laws: a theory of classes (or eventually, ‘value ranges’). From today’s point of view, it is questionable whether such a theory can be considered part of ‘logic’, already since Frege’s version falls prey to Russell’s antinomy. Why was it natural for him to do so, at least initially? He does not answer that question explicitly. I would suggest that it is connected with his traditional view of logic as a general theory of concepts and inferences, and for Frege, classes are ‘extensions of

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concepts’. In taking a theory of classes to be part of logic, he was also in line with other logicians at the time, including Boole, Peano and Dedekind.22 Let me highlight a few further details of Frege’s logic that will play a role in discussing the relationship of Fregean to Aristotelian logic more in the next section. Most importantly, there is his replacement of the traditional subject-predicate distinction in logic, and with it the view that all judgements have ‘S is P’ form, by the distinction between function and argument. Well aware that this replacement is an essential departure from traditional logic, Frege justifies it as follows: These deviations from what is traditional find their justification in the fact that logic has hitherto always followed ordinary language and grammar too closely. […] I believe that the replacement of the concepts of subject and predicate by argument and function, respectively, will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts.23

In passages such as these, Frege ties the subject-predicate distinction, and with it traditional logic, to ordinary language and grammar. This connects with his point, already highlighted above, that both traditional logic and ordinary language are deficient for his purposes, especially for the analysis of mathematical concepts. And as he adds in the last sentence in this passage, his alternative function-argument analysis is fruitful in the ‘formation of concepts’, including novel ones. Three other features of Frege’s logic, each of which makes it different from today’s logic, are the following: First, in Begriffsschrift Frege tries to deal with identity and the biconditional together, in a way that does not work and leads to sustained discussions about related topics in his subsequent writings (most famously the sense-reference distinction). Second, Frege’s ‘conceptual notation’ is never meant to be a formal language in the full sense (from a later point of view), since he does not consider its sentences to be uninterpreted. They always express a ‘content’ or ‘thought’ already, even though we can apply derivation rules in a purely formal way. Third, Frege’s introduction of the judgement stroke into his logic, together with his assumption that sentences always express a thought, is an integral part of an understanding of logical inference that is not the one common today. In a Fregean inference, we always move from judgements to another judgement, not just from sentences to another sentence. More particularly, we recognize some thought as true on the basis of recognizing other thoughts as true.24 Let me summarize and round off the current section. Frege’s main motivation for introducing his new logic was to re-analyse mathematical concepts and reasoning in the service of his logicism. This is different from Boole’s case, whose focus was on using mathematics to re-analyse logic; nor can anything similar, or similarly far-reaching, be found in Aristotelian and Stoic logic. Frege realized that reliance on ordinary language and the traditional ‘S is P’ analysis of judgements was insufficient, which led him to his innovative function-argument analysis and to introducing a new logical language, his ‘conceptual notation’. He then developed a unified truth-functional, relational and quantificational logic based on it, including an axiomatic and formally complete deduction system. The latter embodied, quite importantly, the idea of a formal proof in

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the sense of modern logic.25 In its systematic unity, this goes beyond anything Peirce or Bolzano provided as well, although as philosophers of logic they were fellow travellers in some respects. It is such contributions, then, that allow us to see Frege’s works, from 1879 on, as central in the emergence of modern logic.

3 Frege’s criticisms of Aristotelian logic Let me now tie this summary of Frege’s logical innovations more directly to his criticisms of traditional Aristotelian logic. As we will see, his approach to deduction was aimed explicitly at the theory of syllogism in Aristotle’s Prior Analytics, in the sense of pointing out its limitations and replacing it with something essentially stronger. Similarly, his proposals for analysing mathematical concepts was aimed directly at the Aristotelian ‘S is P’ schema. This is the case even though Frege did not discuss either feature of Aristotle’s logic in great detail. There are a number of places in Frege’s writings where he mentions traditional syllogistic reasoning, although he does not give any explicit references to Aristotle’s Prior Analytics. How might he have learned about it? Four answers suggest themselves: (a) As is well known, Frege read several influential logic texts of his time, by Lotze, Sigwart, Mill and others, in which syllogistic was still central. Even in Boole’s innovative mathematical approach, a main goal was to reconstruct syllogistic reasoning, now in an algebraic form. (b) As already noted, apparently Frege studied works on the history of logic too, e.g. Prantl’s four-volume Geschichte der Logik im Abendlande, whose first volume is dedicated largely to Aristotelian logic. Similarly, Frege was familiar with some of Trendelenburg’s works, several of which were dedicated to reviving Aristotelian ideas. (c) Aristotelian logic plays a crucial role in classic philosophical works too, such as Kant’s Critique of Pure Reason, e.g. in its well-known table of judgements. In that connection, we know that, already early in his time as a student, Frege took a class on Kant with Kuno Fischer.26 Finally (d) at the time logic was often simply identified with Aristotelian syllogistic, a view Frege must have grown up with more generally. Frege’s first discussion of relevant themes occurs in his Begriffsschrift, §3, where he rejects the Aristotelian subject-predicate distinction as logically irrelevant, thus as something to be left behind in his logical language. He goes on, in §4, to emphasize that in his new logic, just like in traditional logic, there is an important distinction between ‘universal’ and ‘particular judgments’. But he adds that this should be understood in terms of different ‘contents of judgments’, not different forms of judgement (cf. also §24). Similarly, for Frege there are judgements with a negated content, rather than negative judgements (denials). Along similar lines, he adds that ‘the distinction between categoric, hypothetic, and disjunctive judgements [has] only grammatical significance’. And he distances himself from the traditional distinction, prominent in Kant, between ‘apodictic’ and ‘assertoric’ judgements. He reinterprets the latter as having to do at best with different ‘grounds of judgment’, not with its content. More generally, he puts the alethic modalities aside, including necessity. Overall, in these early sections of Begriffsschrift such traditional distinctions are replaced by his new way of contrasting judgements and their varied ‘contents’.

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In §5, on ‘Conditionality’, Frege then introduces his first truth-functional connective, the material conditional, together with his rule of inference for it: modus ponens. This leads to the following comment about Aristotelian syllogistic: Following Aristotle, we can enumerate quite a few modes of inference in logic; I employ only this one, at least in all cases in which a new judgment is derived from more than a single one. […] An inference in accordance with any mode of inference can be reduced to our case. Since it is therefore possible to manage with a single mode of inference, it is a commandment of perspicuity to do so. Otherwise there would be no reason to stop at the Aristotelian modes of inference; instead one could continue to add new ones indefinitely.27

What Frege hints at in this passage is that, once his new language is in place, we can consider an open-ended list of complex judgements or better, contents of judgements; some of them can serve as, or be derived from, his logical laws; and modus ponens alone allows us then to deal with a plethora of inferences. For illustration, he points ahead: ‘Some of the judgments that take the place of Aristotelian kinds of inference will be listed in §22.’ At that later point in the text, he considers two forms of the inference Barbara (one where the minor premise has a ‘particular content’, the other where the minor premise has a ‘general content’). All syllogistic inferences can be recovered that way in the end, and Frege can go considerably beyond them. The next main reference to Aristotelian logic, closely related to those just mentioned, occurs in its §§11–12 of Begriffsschrift. After introducing his notation for generality (variables and the universal quantifier), Frege shows, in §11, how the four main statement forms from Aristotelian syllogistic can be recovered by him, namely as follows (in updated notation): ∀x(F(x) → G(x)); ∀x(F(x) → ¬G(x)); ¬∀x(F(x) → ¬G(x)); and ¬∀x(F(x) → G(x)) (corresponding to: ‘All F are G’, ‘No F are G’, ‘Some F are G’, and ‘Some F are not G’).28 In §12, he connects this with the ‘square of logical opposition’ from traditional logic, including its representation of ‘contradictory’, ‘contrary’, ‘sub-contrary’ and ‘subalternate’ judgements. These passages, in particular, suggest that Frege is reacting to standard presentations of Aristotelian logic, rather than Prior Analytics itself. And as has been pointed out before, e.g. by Gillies, he does not think through these points very carefully. For example, he overlooks that in his (and current) logic ∀x(F(x) → G(x)) and ∀x(F(x) → ¬G(x)) do not exclude each other, i.e. are not ‘contraries’ (which cannot be true together). The reason is that Fregean logic, unlike traditional logic, allows for vacuously true universal statements. Besides such features of syllogistic inference, another aspect of traditional logic to which Frege pays explicit attention is the view of concepts that underlies them. According to that view, concepts are ‘sums’ of corresponding ‘marks’. In the classic example, the concept ‘human’ is the sum of the marks ‘animal’ and ‘rational’. Concepts can then be compared and systematically ordered by considering their (relatively simple) composition in terms of marks. This is what underlies the Porphyrian Tree, well known from biological taxonomy, in which living things are classified in terms of ‘genus’ and ‘species’ (a topic of special interest to Aristotle). It also underlies the Kantian division of judgements into ‘synthetic’ and ‘analytic’ ones, depending on

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whether the predicate concept in ‘S is P’ is contained in its subject concept or not, i.e. whether all the marks that constitute P are in S or not. Frege emphasizes from early on, in the Preface to Begriffsschrift, that his own view of concepts and concept formation is very different. As he remarks about the latter: ‘Any effort to create an artificial similarity by regarding a concept as the sum of its marks was entirely alien to my thought’ (p. 6). This contrast remains a major theme in Frege’s later writings, and it has wide-ranging consequences. In particular, it is connected with the point that his new logic allows for much more complicated concept formation, as illustrated by the concepts of following in a series, continuity and limit. There is no way of defining such concepts along the traditional ‘marks’ model, since they involve nested quantification essentially, as Frege points out repeatedly. Traditional adherence to the ‘S is P’ model for judgements and the ‘marks’ model for concepts has two other limiting effects, which Frege points out repeatedly in his writings too. First, and as already mentioned, traditional logic involves no clear objectconcept distinction. Second, it conflates several senses of the word ‘is’, namely the ‘is’ of predication (e.g. ‘The Morning Star is a planet.’); the ‘is’ of concept subsumption (‘A whale is a mammals.’); the ‘is’ of identity (‘The Morning Star is the Evening Star.’); the ‘is’ of existence (‘There is an even prime numbers.’); and the ‘is’ of judgement (‘This claim is true.’).29 These are distinguished explicitly in Frege’s logic; he even uses different logical notations for each of them. From a Fregean point of view, such conflations – together with the limits of syllogistic inference – are why Aristotelian ‘term logic’ in its traditional form has remained inadequate and unfruitful in connection with mathematics.30

4 Frege’s debt to the Aristotelian model of science Frege was clearly aware of the basic shape of Aristotelian syllogistic, its four forms of judgement, the underlying theory of concepts and their limits for analysing mathematical reasoning. As the latter was central for his own logicist project, he was critical of Aristotelian ideas in all his explicit references to them, sometimes also more implicitly. However, there is another implicit connection between Frege’s and Aristotle’s approaches that is important too. Namely, Frege remained committed to many parts of the ‘classical model of science’ that goes back to Aristotle.31 Moreover, this commitment connects with aspects in which Frege’s approach to logic remains ‘premodern’, such as his way of thinking about logical inference in terms of judgements, as opposed to sentences. The classical model of science was spelled out first in Aristotle’s Posterior Analytics and influential from then on. It concerns the ideal of a systematic, mature science. Its six main ingredients32 are: (1) All the concepts and judgements in the science concern a specific domain of objects, and only that domain. (2) Among the concepts, some are primitive while the rest are defined in terms of the primitive ones; the primitive ones are specified as such. (3) Similarly, some judgements are primitive and the rest are derived from the primitive ones, and again, the primitive ones are listed explicitly. (4) All judgements in the science are true, necessary and universal. (5) They are

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also known to be true, either directly or through derivations. (6) The concepts are adequately known as well, either directly or through definitions. If a science conforms to these six desiderata, it represents scientific methodology and rationality in a maximal way.33 From Ancient Greece until the nineteenth century, one science was seen as the paradigmatic realization of this model: Euclidean geometry. In The Foundations of Arithmetic (1884), where Frege starts to spell out his logicist approach to arithmetic more fully, he agrees (cf. §§3–4). And in his Basic Laws of Arithmetic, vol. 1 (1893), he comes back to this point, although with one interesting modification: The ideal of a strictly scientific method in mathematics, which I have here attempted to realize, and which might indeed be named after Euclid, I should like to describe as follows: It cannot be demanded that everything is proved, because that is impossible; but we can require that all propositions used without proof be expressly declared as such, so that we can see distinctly what the whole structure rests upon. After that we must try to diminish the number of primitive laws as far as possible, by proving everything that can be proved. Furthermore, I demand – and in this I go beyond Euclid – that all methods of inference employed be specified in advance; otherwise we cannot be certain of satisfying the first requirement. This idea I believe I have now essentially attained.34

Frege’s modification is that, in addition to what was done by Euclid, he is attempting to make all the needed ‘methods of inference’ explicit as well, not just all the basic laws and concepts. Crucially, Euclidean geometry used kinds of inference and proof that went beyond Aristotelian logic, namely its well-known intuitive geometric constructions. Frege’s new logic is meant to replace both Aristotelian logic and these intuitive forms of inference, at least in the case of arithmetic. Frege departs from the classical model of science in another respect as well, but he is less explicit about it. One should remember that in a mature Aristotelian science, not only are that all its judgements true, but they are necessarily true. However, we already saw that Frege rejects necessity, or modal notions more generally, in his logic.35 On the other hand, he retains the demand that all the judgements in a mature science be true. This is intimately connected with his conception of logical inference, where we derive one truth from other truth via rules of inference. It also requires, as Frege is aware, that we rethink what happens in indirect proofs, where we seem to start with what later turns out to be a false assumption. Moreover, for his logicism it requires that we derive the basic laws of arithmetic as truths (rather than treating them as, say, implicit definitions of a relational structure along Hilbertian lines).36 As I interpret Frege, his adherence to the classical model of science — with the two qualifications just mentioned — is connected with two other, closely related, and pre-modern features of his approach. First, he considers logic from a ‘universalist’ perspective.37 This includes that, while every other science restricts itself to a certain domain of objects and concepts, in logic we have no such restriction; we deal with objects and concepts in general. Technically this means that the quantifiers in logical

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laws are not interpreted as restricted in the now standard way. As such, logic is the study of reasoning in general, which explains its universal applicability. Second, the perspective adopted by Frege is a first-person perspective.38 Or put differently, it is the ‘internal’ perspective of a reasoning agent, an agent that uses language to make judgements and to draw inferences. This contrasts with the fully formal and ‘modeltheoretic’ perspective in logic with which we are more familiar today.39 In the literature, it has been argued that Frege adheres to a ‘Euclidean model of rigor’.40 In some respects this is true, I think. But I would suggest that it is more appropriate to see him as holding on to Aristotelian ideals in the end, in ways that go partly against Euclid. This becomes clearer if we do not just consider the aspects of the classical model of science preserved in Frege’s logicism, but also his attempts to lay bare the most basic categories of thinking. The latter include the notions of judgement, thought, object and concept, sense and reference, an ‘objectual’ notion of identity, and a ‘non-model-theoretic’ notion of truth.41 While Frege discards certain Aristotelian assumptions in that context too, such as the approach to concepts in terms of marks, at a deeper level he remains involved in the same project of ‘philosophical logic’.42 At that level, his goal is not so much to replace Aristotle’s approach as to update it, and the update is driven by the same developments in mathematics and logic that underlie Frege’s logicism.43

5 Frege and the emergence of modern logic In which sense, or to what degree, does Frege’s work, and Begriffsschrift in particular, constitute a ‘revolution’ in logic, in something like Kuhn’s sense? As we saw, Donald Gillies made a case for that status explicitly, and it is implicit in claims, by Quine, Dummett and others, that 1879 witnessed the birth of ‘modern logic’. Gillies gives the following two arguments, among others, to support his claim: (a) As typical in a Kuhnian revolution, Frege was an outsider to discussions in logic initially. Until the nineteenth century, these took part mostly in philosophy while Frege (like Boole) was trained as a mathematician. (b) And also typical for a Kuhnian revolutionary, Frege does not spend much time on traditional logic, including on refuting it. After some brief remarks about its limits, he simply replaces it with his own logic and starts to develop the latter. I agree with both of those points. Beyond Gillies, I would add a third feature of Kuhnian scientific revolutions: (c) Such revolutions are typically not punctual affairs, in the sense that one can date them exactly. Instead the crucial changes, while perhaps initiated by one person or work, are drawn out over time; they need to be articulated in substantive ways; and that involves further steps and contributors. In Kuhn’s example, the Copernican Revolution was only fully in place after the works of Kepler, Galileo, Newton etc. Moreover, there were partial anticipations of Copernican ideas before him, e.g. by Oresme. We already encountered one side of this phenomenon in the case of logic, in the form of arguments that the shift to modern logic should be pre-dated to 1854 (Boole) or to 1837 (Bolzano), also that logicians like Peirce, who worked parallel to Frege, should be given their due more. The same holds in a forward-looking direction. Modern logic

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was not present in its final, mature shape in 1879, with Frege’s Begriffsschrift. In fact, it took until the 1940–1950s for it to be fully in place with further steps taken by Russell, Hilbert, Gödel, Tarski etc. Another complication that confirms the drawn-out nature of the ‘Fregean revolution’ in logic concerns the fact that Frege’s work was not well received at first. In fact, initially it was mostly either criticized or ignored. There were crucial exceptions, of course, such as Russell and Wittgenstein, later Carnap and Church. Russell and others also modified Frege’s approach, by working around the impact of Russell’s antinomy on its theory of classes, among others. In the early twentieth century, it was Whitehead & Russell’s Principia Mathematica (1910–1913) that then became central, i.e. the text every logician had to read. And it took until the 1950–1960s for Frege’s works to be rediscovered, through efforts by Church, Carnap, Quine, Dummett etc. It was also during the latter period that Frege’s Begriffsschrift was attributed a central place in the history of logic, but without being read very carefully. As such, it became part of the ‘founding myth’ of modern logic. The 1950–1960s were also the period during which the new logic consolidated itself and took on its classic form. This includes a sharp distinction between object language and metalanguage, the adoption of a meta-logical perspective, the singling out of first-order logic as privileged, and the relegation of a theory of classes, purged of its antinomies, to axiomatic set theory. None of these are present in Frege’s works, not even after Begriffsschrift; in fact, they stand in conflict with some of his core assumptions. In any case, it is along these lines that we get the flowering of proof theory, model theory, recursion theory etc. And in connection with them, Hilbert, Tarski, Gödel, Church, Turing etc., took on the roles that Kepler, Galileo, Newton and others played in the Copernican Revolution. With such clarifications about its drawn-out character in place, it is illuminating to talk about a ‘revolution in logic’, I think, and one in which Frege can be assigned a central role. I do not mean to put too much weight on the terminology of ‘revolution’, however. One can also employ other terminology to highlight the big changes in modern logic. An alternative is to talk about a ‘second birth’ of logic in the late nineteenth century, parallel to a slightly earlier ‘second birth’ of mathematics.44 The latter refers to the radical shift from a more calculational approach to arithmetic and algebra as well as a traditional view of geometry involving intuitive constructions to a novel ‘conceptual’ way of reconstructing and unifying both sides. This new approach to mathematics was pushed forward by figures such as Dirichlet, Riemann, Dedekind, Hilbert and Poincaré. It also involved a ‘new birth of the same subject’, not of something entirely different. Something similar can be said about logic. In fact, Frege’s new approach to logic was in line with this ‘conceptual’ shift in mathematics, even if his views about mathematics lagged behind in some ways.45 One effect of the ‘second birth’ of mathematics that became pronounced over time was a vast extension of the field, not just in terms of new areas, like topology, but also broader approaches to old areas, such as arithmetic and geometry. We find parallel developments in logic in the twentieth century, with various extensions of ‘classical logic’, like modal logic, and with the addition of ‘non-classical’ logics, such as intuitionistic and relevance logic. And of course, logic keeps getting transformed today,

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e.g. with its implementation in computer science and the emergence of approaches like homotopy-type theory. In that context, what became crucial is the kind of ‘tolerance’ to the study of logical systems that Carnap highlighted from the 1930s on, together with the full shift towards a meta-logical perspective. In such respects, Frege lagged again behind. Overall, his various moves beyond traditional Aristotelian logic were milestones in the emergence of modern logic, but he remained pre-modern in part by adhering to an Aristotelian model of science.

Notes 1 2 3 4

5 6 7 8

9 10 11 12 13

Quoted in Terrell Ward Bynum, (ed. and transl.), Frege: Conceptual Notation and Related Articles (Oxford: Oxford University Press, 1972), 236. Willard V. O. Quine, Methods of Logic (New York: Henry Holt and Co., 1950), vii. See the Introduction to this volume for a brief historiographical reflection on Quine’s position on the birth of modern logic. Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, Mass. - London: Harvard University Press, 1967), vi. In Germany, interest in Frege’s works was rekindled in the 1960s too, by Hans Hermes, Günther Patzig, etc., as well as by the publication of scholarly editions of his works, posthumous writings and correspondence. Earlier his contributions to logic had been highlighted already by Heinrich Scholz, in his Geschichte der Logik (Berlin: Junker & Dünnhaupt, 1932; 2nd ed., Abriß der Geschichte der Logik, published in 1967). William Kneale and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), 511. Gabriel and S. Schlotter, Frege und die kontinentalen Ursprünge der analytischen Philosophie (Mentis: Münster, 2017). For references, see Erich H. Reck, ‘Frege, Dedekind, and the Origins of Logicism’, History and Philosophy of Logic 34, (2013): 242–65. Suzanne Bobzien, ‘Frege Plagiarized the Stoics’, in Themes in Plato, Aristotle, and Hellenistic Philosophy. Keeling Lectures 2011–18, ed. F. Leigh (London: Institute of Classical Studies, University of London, 2021), 149–206; earlier also Gottfried Gabriel, Karl-Heinz Hülser and Sven Schlotter, ‘Zur Miete bei Frege – Rudolf Hirzel und die Rezeption der stoischen Logik und Semantic in Jena’, History and Philosophy of Logic 30 (2009): 369–88. Cf. Stanley Burris and Javier Legris, ‘The Algebra of Logic Tradition’, in the Stanford Encyclopedia of Philosophy, ed. E. Zalta (20091; 2021), 1–17. Gottlob Frege, ‘Boole’s Logical Calculus and the Concept-Script [1880–81]’, in Gottlob Frege: Posthumous Writings, ed. H. Hermes et al. (Oxford: Blackwell, 1979), 9–46. Gottlob Frege, ‘A Critical Elucidation of Some Points in E. Schröder, Vorlesungen über die Algebra der Logik [1895]’, in Collected Papers on Mathematics, Logic, and Philosophy, ed. B. McGuinness (London: Basil Blackwell, 1984), 210–28. Cf. Risto Hilpinen, ‘Peirce’s Logic’, in The Rise of Modern Logic: From Leibniz to Frege, eds. D. Gabbay and J. Woods (Amsterdam: Elsevier, 2004), 611–58. Irving Anellis, ‘How Peircean Was the “Fregean” Revolution in Logic?’, Log. Issled 18 (2012): 239–72.

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14 Cf. Göran Sundholm, ‘A Century of Judgment and Inference: 1837–1936. Some Strands in the Development of Logic’, in The Development of Modern Logic, ed. Leila Haaparanta (Oxford: Oxford University Press, 2009), 263–317. 15 Cf. Calixto Badesa and Joan Bertran-San Millán, ‘Begriffsschrift’s Logic’, Notre Dame Journal of Formal Logic 61 (2020): 409–40. 16 Gottlob Frege, Begriffsschrift (Halle: Nebert, 1879); Engl. trans., Begriffsschrift, by S. Bauer-Mengelberg, in Van Heijenoort (ed.), From Frege to Gödel, 1–82, 5. 17 Ibid., 6. 18 See, e.g., Frege’s Lectures on Logic: Carnap’s Student Notes, 1910–1914, eds. and transl. E. Reck and S. Awodey (Chicago: Open Court, 2004), 88–91. 19 I assume Frege saw this more clearly than, independently of, and slightly before Peirce, who extended Boole’s, De Morgan’s, etc., works in that direction too; and I take Frege’s corresponding insights to have grown out of his involvement with the mathematics of the time. See Reck, ‘Frege, Dedekind, and the Origins of Logicism’, 242–65. 20 Cf. Badesa and Bertan-San Milán, ‘Begriffsschrift’s Logic’. 21 Cf. Alonzo Church, Introduction to Mathematical Logic (Princeton, NJ: Princeton University Press, 19441; 199610). 22 Reck, ‘The Logic in Dedekind’s Logicism’, in Logic from Kant to Russell, ed. S. Lapointe (London: Routledge, 2019), 171–88. 23 Frege, Begriffsschrift, 7. 24 Reck, ‘Frege on Truth, Judgment, and Objectivity’, Grazer Philosophische Studien 75 (2007): 149–73. 25 The latter is missing in Richard Dedekind too, who is closest to Frege in several other respects, including in terms of pursuing a logicist project (cf. Reck, ‘Frege, Dedekind, and the Origins of Logicism’ and Id., ‘The Logic in Dedekind’s Logicism’). Of course, the higher-order fragment of Frege’s logic is not complete. 26 Cf. Lothar Kreiser, Gottlob Frege. Leben, Werk, Zeit (Hamburg: Meiner Verlag, 2001), 64–5. Frege was also familiar with Aristotle’s conception of number, which he mentions both in Frege’s ‘Boole’s Logical Calculus and the Concept-Script’ (1880–81) and in his Die Grundlagen der Arithmetik (1884). As some of his references in the latter indicate, he encountered those in Julius J. Baumann, Die Lehren von Raum, Zeit und Mathematik in der neueren Philosophie, vol. 2 (Reimer: Berlin, 1868), which contains a substantive discussion of Kant too. 27 Frege, Begriffsschrift, 17. 28 In Frege’s reconstruction, all of these statements have ‘general content’. Aristotelian, or post-Aristotelian, judgements with ‘particular content’ – e.g. ‘Socrates is mortal’ – can also be captured, of course. Similarly for Aristotle’s Principles of NonContradiction and Excluded Middle. 29 The distinction between the first four sense of ‘is’ plays a role in post-Fregean logic too. For Frege’s separation of the fifth, sometimes called the ‘veridical is’, cf. Irad Kimhi, Thinking and Being (Cambridge, MA: Harvard University Press, 2018). With reference to Aristotle, Kimhi argues against Frege’s separation of the ‘is’ of predication and the ‘veridical is’. 30 The qualification ‘in its traditional form’ points to the fact that one can extend Aristotelian term logic, and ordinary language reasoning, in the needed directions; cf. Fred Sommers, The Logic of Natural Language (Oxford: Oxford University Press, 1892). Having said that, this possibility has had little influence on the philosophy of mathematics, unlike the Fregean approach.

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31 Cf. Willem R. de Jong and Arianna Betti, ‘The Classical Model of Science: A Millennia-Old Model of Scientific Rationality’, Synthese 174 (2010): 185–203. 32 Ibid. 33 Cf. Danielle Macbeth, ‘Frege and the Aristotelian Model of Science’, in Early Analytic Philosophy: New Perspectives on the Tradition, ed. S. Costreie (Cham et al.: Springer, 2016), 31–48. 34 Frege, Die Grundlagen der Arithmetik, 2. 35 In Macbeth (‘Frege and the Aristotelian Model of Science’) this point is connected with interpreting Frege’s overall position as fallibilist. This is compatible with the other desiderata of the classical model, as Macbeth argues. 36 Cf. Patricia Blanchette, Frege’s Conception of Logic (Oxford: Oxford University Press, 2012). 37 Warren Goldfarb, ‘Frege’s Conception of Logic’, in Future Pasts: The Analytic Tradition in Twentieth-Century Philosophy, ed. Juliet Floyd and Sanford Shieh (Oxford: Oxford University Press, 2001), 25–41. 38 Cf. Maria Van der Schaar, ‘Frege on Judgment and the Judging Agent’, Mind 127 (2018): 225–50. 39 The fully formal, model-theoretic perspective started to dominate logic in the 1940– 1950s, while the 1930s were still a transitional period; cf. Georg Schiemer and Erich Reck, ‘Logic in the 1930s: Type Theory and Model Theory’, The Bulletin of Symbolic Logic 19 (2013): 433–74. On the other hand, parts of it were present already in Boole (e.g. domains of interpretation), and Frege fell back behind him in that respect. 40 Tyler Burge, Truth, Thought, Reason: Essays on Frege (Oxford: Oxford University Press, 2005). 41 Cf. Hourya Benis-Sinaceur, ‘Dedekind’s and Frege’s Views of Logic’, in In Memoriam Richard Dedekind (1831–1916), ed. K. Scheel et al., (Münster: WTM Verlag, 2017), 50–62. 42 Cf. Kimhi, Thinking and Being. 43 There are signs that a Fregean ‘internal’, ‘judgment-based’ approach to logic is gaining support again, e.g. in homotopy-type theory. Once more, this is driven by new developments in mathematics (and computer science). Cf. Göran Sundholm, ‘A Century of Judgment and Inference: 1837–1936’, for a corresponding history of logic. 44 Howard Stein, ‘Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth Century Transformations of Mathematics’, in History and Philosophy of Mathematics, ed. W. Aspray and P. Kitcher (Minneapolis: University of Minnesota Press, 1988), 238–59. 45 Reck, ‘Frege, Dedekind, and the Origins of Logicism’.

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Christine Ladd-Franklin’s antilogism Francine F. Abeles

1 Introduction The principal goal of this chapter is to provide the background to Christine LaddFranklin’s logic system – which differed from any previous one – and the remarkable result it contained, her antilogism, a test for the validity of any syllogism. Ladd-Franklin was an exceptional logician, largely unrecognized during her lifetime and not accorded proper recognition even today. To present the entire story, we examine many aspects of her life: in Section 1 her graduate education and early publications; in Section 2 her doctoral dissertation where she presented her antilogism but without proving it, and its reception; in Section 3 an analysis of Susan I. Russinoff ’s proof of Ladd-Franklin’s antilogism, preceded by some additions to points in the history of syllogistic reasoning that Russinoff includes, followed by an analysis of her proof; in Section 4 all of LaddFranklin’s subsequent publications bearing on the syllogism or the antilogism; and in the final Section 5, the professional and personal matters that were important in LaddFranklin’s accomplishments. In March of 1878, Christine Ladd, referring to the positive reception of papers and solutions to problems she published in journals such as the Educational Times and the Analyst, wrote to Professor James J. Sylvester (1814–1897) at Johns Hopkins University, asking if she could study mathematics under his supervision.1 Sylvester, Hopkins’s first professor of mathematics, asked Fabian Franklin (1853–1939), one of his students, to examine Ladd’s work – they later married. Franklin had a good opinion of her work so Sylvester sent Ladd’s request to the university’s president and to its board of trustees. President Gilman wrote to Ladd, telling her that the board of trustees does not support the admission of women as students. However, they would allow her to attend Sylvester’s lectures. Sylvester was able to have the university grant her a $500 a year fellowship for the three years 1879–1882. Although she was not allowed to describe herself as a fellow, she was allowed to attend lectures by all the lecturers. She became a non-matriculated, non-tuition-paying student. After a year of studying with Sylvester, she began attending the classes taught by other instructors, particularly those given by the mathematician William Edward Story (1850–1930), who was the editor in charge of The American Journal of Mathematics from 1878

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to 1882, and Charles Sanders Peirce (1839–1914), a member of the philosophy department who later became her thesis adviser. She enrolled in 1878 and finished in mathematics and logic in 1882. Perhaps the most important of her strengths as a logician derived from her deep knowledge of mathematics. She published mathematical articles as early as in 1877 and 1878: six articles in The Analyst, a journal published by the Mathematics Department of Princeton University which after ten volumes through 1883 became The Annals of Mathematics. Between 1879 and 1881, while she was a graduate student at Hopkins, she published three papers in The American Journal of Mathematics, an important publication that Sylvester became editor of in 1879. The Educational Times and Journal of the College of Preceptors, hereafter (ET), a journal published in London from 1847 to 1918, added a mathematical section in 1848. This section consisted of problems, solutions and, occasionally, brief articles. Space in ET for the mathematics section was so limited that its editor, W. J. C. Miller (1832–1903) established a second publication, Mathematical Questions with Their Solutions from the ‘Educational Times’, hereafter (MQ), where notes, problems and solutions that had appeared in ET were republished semi-annually. Additionally, MQ contained solutions to problems that had appeared in ET without solutions. LaddFranklin published two short notes that appeared in MQ. She was the most prolific female contributor to ET proposing or solving seventy-one problems, the majority of them on three subjects: triangle and circle geometry, loci, and conics. Just three more female mathematicians were active in ET during the same period: Sarah Marks, Belle Easton and Elizabeth Blackwood.2 Founded in 1876, Mind quickly became an important journal in philosophy where many eminent logicians published their work. Logic was not a separate university department. It was included as a subject in philosophy departments. Ladd read the issues of Mind regularly during her years at Hopkins and afterwards to keep abreast of the important issues of her time.

2 Ladd-Franklin’s doctoral dissertation in Studies in Logic and its reception Studies in Logic, edited by Peirce, was a book of articles by Peirce and his Johns Hopkins University students. In the book’s ‘Preface’, referring to Ladd-Franklin’s article, ‘On the Algebra of Logic’, her doctoral dissertation under his supervision, Peirce wrote, Miss Ladd’s article may serve, for those unacquainted with Boole’s ‘Laws of Thought,’ as an introduction to the most wonderful and fecund discovery of modern logic.3

In a note in her article Ladd-Franklin writes that according to Peirce, every algebra of logic has two copulas, one for propositions of non-existence, the other for propositions of existence. Her algebra of logic requires one copula. Is there a disagreement between

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her classification and Peirce’s? No, because as she states, particular propositions denote existence and universal propositions denote non-existence, adding that the quantified copula –< is positive for universal propositions and negative for particular propositions. So, by quantifying the copula, she pairs the type of proposition: particular, universal, with the state of the proposition: existence, non-existence. There is no disagreement between Peirce’s classification and her own classification. The algebra of logic that she created was a variant of the system devised by George Boole (1815–1864), Peirce and Ernst Schröder (1841–1902).4 In her algebra the – fundamental relation is exclusion expressed by the symbol V. The expression a V b – is defined as a is partly b; V and V are symmetrical and intransitive. Inclusion, by contrast, is non-symmetrical and transitive. Changing an inclusion into its equivalent – exclusion requires changing the sign of the predicate. But with V, the quantity of both subject and predicate is universal. She further states that A V B can be read forward or – backward, the same for A V B. From the fact that there is no formal difference between subject and predicate that the advantages of her algebra follow. In his 1918 book, Clarence Irving Lewis (1883–1964) critically examined LaddFranklin’s algebra of logic. In the thirty-five years since the publication of her article, he was the only logician to do so. He remarks that her use of the terms ‘consistent’ and ‘inconsistent’ in the statement, ‘a and b are inconsistent’, i.e. if a is true, b is false; – if b is true, a is false, symbolically: a V b, may be misleading because any two true propositions or any two false ones are consistent, and any two propositions where one is true and the other false are inconsistent in her use of the terms. In his analysis of her calculus, he claims that ‘it is psychologically simpler and more natural to think of – logical relations in terms of V and V ’. But he cautions, ‘[T]hey do not so readily suggest their mathematical analogues in other algebras.’5

3 Russinoff ’s proof of Ladd-Franklin’s antilogism theorem: Background and related developments Providing a general characterization of all the valid syllogisms is the problem Aristotle set out to solve. He focused on arguments that he called categorical statements and described fourteen valid syllogistic forms. Until the nineteenth century little real progress towards achieving Aristotle’s goal was made. In the nineteenth century three figures – Augustus De Morgan, George Boole and Hugh MacColl – were key figures in that progress. And the solution to the problem Aristotle wanted to solve was given by Christine Ladd-Franklin (see ‘Postscript’). De Morgan (1806–1871), a mathematician, saw the syllogism as the central element of logic and focused on the fundamental principles that determine syllogistic inferences. His major contribution was the formulation of a ‘logic of relations’ useful in the theory of the syllogism. George Boole, although he did not particularly like Aristotle’s classification of syllogisms because he found it much too complicated, tried to extend and improve it. But Boole’s algebraic approach was ill suited to Aristotle’s classification.

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Hugh MacColl’s (1837–1909) work on the syllogism was focused on the hypothetical syllogism. Also, he believed that the traditional forms of the syllogism were invalid because the way in which they are stated suggests that the premise and conclusion are asserted. So, if either the premise or the conclusion is false, the syllogism is not valid. Ladd-Franklin’s approach to the solution of Aristotle’s problem was entirely different from any that preceded it. Fundamentally, she gave the algebraic form of Aristotle’s categorical statements in terms of her exclusion relation. Citing the first part of Book I of Aristotle’s Prior Analytics, Russinoff claimed that in addition to providing a theory of categorical syllogistic deduction, Aristotle had also sought a general characterization of the valid syllogisms. The antilogism provides a general characterization of all the valid syllogisms, thereby solving Aristotle’s problem. Comparing syllogistic reasoning with anti-logistic reasoning, in the former the two premises are sufficient for the conclusion; in the latter, the two premises are incompatible with the negative of the conclusion. The antilogism provides a general characterization of all the valid syllogisms, thereby solving Aristotle’s problem. But her algebra of logic and with it her antilogism were never adopted because of the main problem with her exclusion relation – it doesn’t fit with the traditional relations of inclusion and equality. Curiously, Russinoff does not offer any insight into where Ladd-Franklin may have gotten the idea of the antilogism. Perhaps it came from her reading of Aristotle, possibly from De Interpretatione which (freely translated) is: given that p and q together imply r: (1) if p is true and r is false then q is false; (2) if q is true and r is false, p must be false. Preparing the groundwork for her proof of Ladd-Franklin’s antilogism in the third section of her paper, Russinoff discusses the treatment of syllogistic reasoning in classical algebra and suggests, broadly speaking, that syllogistic reasoning consists of ‘eliminating’ the middle term. Citing Lewis as the source for both the postulates of this algebra and for the elimination formula derivable from them, she gives this formula in her treatment of a syllogism with a universal conclusion: ax + b – x = 0 stating that ab = 0 follows. Then she states another formula, described as a theorem of the algebra, in her treatment of a syllogism with a particular conclusion, bc = bc(a + –a) = bca + bc – a. But she provides no explanation for their use. Lewis states that the first of these two formulas is known as the ‘law of elimination resultants’ and provides an explanation that I have simplified below. Every antilogism, an inconsistent triad, corresponds to three valid syllogisms, any two of which will give the contradictory of the conclusion. Writing the two equations of the universals of the antilogism in equation form, we have ax = 0 and b – x = 0. Putting them together, the elimination resultant of ax + b – x = 0 is ab = 0.6 Lewis refers to Boole’s 1854 book, An Investigation of the Laws of Thought, for the claim that formal reasoning is accomplished by the elimination of ‘middle’ terms.7

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Boole published a short book in 1847, The Mathematical Analysis of Logic, where he developed the notion that logical relations could be expressed by algebraic formulas. Boole also developed rules to deal with elimination problems to determine the maximum amount of information obtainable from a given set of propositions. From the mid-nineteenth century on, the central problem of the logic of classes, the elimination problem, was to determine the maximum amount of information obtainable from a given set of propositions. Boole made the solution to this problem considerably more complex in his 1854 book, where he provided the mechanism of a purely symbolic treatment which allowed propositions to have any number of terms, thereby introducing the possibility of an overwhelming number of computations. Boole also developed rules to deal with elimination problems. If the equation f(x)  =  0 denotes the information available about a class x, and we want to find the relations that hold between x and the other classes (y, z, etc.) to which x is related which is symbolized by the expression f(x), Boole, using his laws of calculation, was able to represent algebraically all of the methods of reasoning in traditional classical logic. For example, syllogistic reasoning involves reducing two class equations (premises) to one equation (conclusion), eliminating the middle term, and then solving the equation of the conclusion for the subject term. Ladd-Franklin and Charles Lutwidge Dodgson (Lewis Carroll) (1832–1898) were the first to formulate a mechanical method to implement the elimination of middle terms to solve syllogisms. In 1894, he invented what he called ‘the method of trees’ to demonstrate the validity of the conclusion of complex syllogistic problems (soriteses) involving, for example, twenty-six premises. Its essential feature is that when a conclusion following from a set of premises is assumed to be false, then if reasoning from it together with all its premises results in a contradiction, the original argument is proved to be valid. This is the earliest modern use of the falsifiability tree method to reason efficiently in the logic of classes. An exposition of Lewis Carroll’s, the pen name of Dodgson, tree method, first appeared in Bartley 1977.8 Dodgson must have been aware of Ladd-Franklin’s antilogism because he owned a copy of Peirce’s Studies in Logic and cited it in his unpublished second part of Symbolic Logic. Surely Ladd-Franklin’s ‘inconsistent triad’ appealed to him because like his tree method which uses a ‘reduction ad absurdum’ argument, it involves a similar form of reasoning. Irving Anellis, in a note to my paper ‘Lewis Carroll’s Method of Trees: Its Origins in Studies in Logic’, argued: The trees developed by Carroll in 1894, which anticipate concepts later articulated by Beth in his development of deductive and semantic tableaux, have their roots in the work of Charles Peirce, Peirce’s students and colleagues, and in particular in Peirce’s own existential graphs.9

In 1955, Evert W. Beth (1908–1964) presented a tableau method he had devised consisting of two trees that would enable a systematic search for a refutation of a given (true) sequent, an expression of formulas A1, …, An, → B1, …, Bm such that

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if the An are true, at least one of the Bm is true. Using his method, one can obtain a cut free proof (one that does not use the cut elimination rule, i.e. modus ponens) by systematically building a counterexample for the sequent and finding that at some stage the process becomes blocked; i.e. there is no counterexample, confirming that the sequent is true. A tree is a left-sided Beth tableau in which all the formulas are true. The rules for decomposing this tree, the inference rules, are equivalent to the  rules Gerhard Gentzen (1905–1945) used in the sequences of formulas in his sequent calculus. The modern tree method as a decision procedure for classical propositional logic and for first-order logic originates in Gentzen’s work on natural deduction, particularly his formulation of the sequent calculus known as LK. In his 1934 paper, Gentzen proved that the use of the cut elimination rule can be removed from any logical proof. The cut rule, or modus ponens, is a valid derived rule in natural deduction systems, but it is not obviously a valid derived rule in the tree method, i.e. if there are closed trees for both A and A → B, then it is not obvious that there is one for B as well. Indeed, Gentzen’s LK without the cut rule is the tree method.10 In any natural deduction system, not using the cut rule can increase the length of derivations considerably. The efficiency of the tree method is preserved by adding a version of the cut rule, in the form of the law of the excluded middle, which permits any open branch of a tree to split into two, then allowing any sentence to be appended to the bottom of one of the new branches, and the negation of that sentence to be appended to the bottom of the other. The conclusion of a valid syllogism has no middle term. This ‘elimination’ of the middle term, or ‘cut’, is accomplished by the use of modus ponens. In the fourth section of her paper, Russinoff begins by stating that it is obvious that all triads with the proper form are inconsistent; what must be shown is that every inconsistent triad has that form. She also says that we now define inconsistency and consistency of sets of statements in terms of possible interpretations. A set of statements is inconsistent if and only if there is no possible interpretation under which each member of the set is true. In Ladd-Franklin’s time, the notion of an ‘interpretation’ was just beginning to be considered, so she did not have this tool to work with and therefore could not have proved her antilogism theorem.11 Russinoff chooses predicate logic as the setting for her proof of Ladd-Franklin’s antilogism. A different way to prove the antilogism requires the concept of a propositional function, another notion unavailable in the nineteenth century, as it first appeared in 1902 in Bertrand Russell’s (1872–1970) book.12 Ladd-Franklin was quite familiar with Russell’s work, particularly his and Alfred North Whitehead’s (1861–1947) Principia Mathematica. In fact, in 1918 she published a paper on Russell and symbolic logic and gave a talk on this subject at a meeting of The American Mathematical Society (see Section 4). Given her mathematical background, had she wanted to, she could have proved her antilogism, promoted her discovery and satisfied her critics. But by this time her approach to logic had become more philosophically based and she probably was too much involved with that work to bother. Even so, other logicians did take this opportunity to prove the antilogism. Haskell B. Curry (1900–1982), for one, did just that in 1936 using the notion of an ‘f function’, a category of propositional functions of two variables. His proof (Theorem 6) is also a proof of a valid syllogism.13

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4 Ladd-Franklin’s career 1885–1928 To portray the scope of Ladd-Franklin’s activities, the centrality of the syllogism as the basis of all logical reasoning, and the widening of her interests in logic, we here discuss all of Ladd-Franklin’s papers that deal with the syllogism or antilogism. Papers on other topics are listed in the ‘Further Reading’ section. Ladd-Franklin’s uniqueness as a logician derives from her education both in mathematics and in logic. Her ability to segue from mathematics to mathematical logic to logic in philosophy and psychology is probably the single most important characteristic of her life in logic. In her longest paper, published in 1889, she essentially surveys the current state of the subject in the thirty-five years since Boole published his magnum opus in 1854.14 The problem of logic that Boole solved was that he gave an organized method to handle very many complex premises that in a subject or predicate provide a description of other terms. This process requires the elimination of certain terms, and the ordinary syllogism ‘consists of elimination in the simplest possible case’.15 She discusses symbolism – questions like, what is a symbolic treatment of a subject? Disagreeing with John Venn (1834–1923), her opinion of Boole’s scheme of symbolic reasoning was that it was more complicated than it had to be.16 To substantiate her opinion, she begins by describing what is involved in setting up such a logic system. In her analysis, she mentions De Morgan’s important idea of a limited universe of discourse. In great detail she discusses the eight essential universal and particular propositions involving two terms and the role of the symmetrical and non-symmetrical copulas. The simple symmetrical copulas V and can be inserted anywhere in a logical sum. She worked with the symmetrical affirmative copula; MacColl worked with the non-symmetrical affirmative copula; Boole used the negative symmetrical copula. The symmetrical copulas allow subject and predicate to be freely interchanged; with the non-symmetrical copulas, subject and predicate can change places only when their quality is also changed. She presents the many advantages the symmetrical copula has, emphasizing the advantage gained by not having to deal with the place where terms occur. Casting her net widely she writes, When Mr. Venn said there is only one system of Logic, he seems to have had in mind only Jevons; and Jevon’s work in Symbolic Logic does certainly not amount to a system, but merely to the absence of a system. Since then Mr. Keynes [John Neville Keynes (1852–1949)] has published his treatment of the non-symmetrical copula.17

She describes in detail the sixteen different systems of logic that the type of copula representing the propositions determines. The last part of her paper focuses on four major points. First, the negative symmetrical copula chosen by Boole, ‘no a is b’. She states that Schröder has given this copula its final form in his 1877 paper, ‘Der Operationskreis der Logikkalkuls’ which, oddly, Venn is unaware of, and that his treatment should have superseded Boole’s.18 Schröder has accomplished in a few pages, and with admirable simplicity and closeness to real processes of reasoning, what Boole made a very obscure and tortuous journey of.19

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The fact that it hasn’t, she suggests, is because he didn’t have an English commentator to support him. Then she remarks, without any mention of MacColl, that Keynes has also written the entire logic of the non-symmetrical affirmative copula, all a is b. To this she adds her own development of the logic of the corresponding particular propositions from her dissertation, remarking that Keynes doesn’t see that they have any value, a point of view she vigorously disputes and goes on to argue for. The second point she discusses is the logic of the non-symmetrical affirmative copula, ‘all a is b’ given by MacColl for the first time in a series of three papers from 1877 to 1880.20 Referring to her system and also to MacColl’s she claims they both thought that their systems reflected real, i.e. natural reasoning processes. And she agreed with Schröder’s opinion that MacColl’s calculus was a preliminary stage of Peirce’s algebra of logic. Schröder had accepted MacColl’s priority in the formulation of a propositional logic. [I]t seems incredible that English logicians should not have seen that the entire task accomplished by Boole has been accomplished by Maccoll [sic] 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.21

Citing his important 1880 paper, Ladd-Franklin notes that Peirce has independently worked out the logic of this copula too. She notes that Keynes does not credit MacColl in his reworking of the logic of this copula and that Venn claimed he found nothing important in MacColl’s system.22 After a careful study [of this scheme], aided by a long correspondence with the author, I am unable to find much more in it than the introduction of one more scheme of notation to express certain modifications and simplifications of a part of Boole’s system.23

Her third point is that a system based on the negative non-symmetrical copula would be just the reverse of the logic of the non-symmetrical affirmative copula. Finally, in her fourth point she describes the logic system based on the affirmative copula by Oscar H. Mitchell (1851–1889) who has worked out the algebra of the affirmative symmetrical copula, ‘all but a is b’. In a highly technical Mind article published a year later, she claims that the field of deductive logic has acquired a symmetry and completeness for symbolic logic that should now be reflected in elementary expositions of the subject. She then discusses five areas that require changes.24 Ladd-Franklin’s 1892 article is a refutation of Emily Elizabeth Constance Jones’s (1848–1922) review of Franz Hillebrand’s 1891 long pamphlet of the year before.25 Jones had developed her lengthy review of Hillebrand based on his idea, with which she agrees, that a theory of inference, e.g. the theory of syllogism, depends on a theory of judgement.26 But Ladd-Franklin totally dismisses Hillebrand’s conclusions, stating that his reduction of the syllogism to the two specific forms he gives is both not new

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and not as important as he thinks. Furthermore, she dismisses his idea that either of these syllogistic forms ‘is capable of throwing any light whatever upon either the philosophy or the psychology of a syllogism which is stated in any other of the commonly recognized forms, is nonsense’.27 She is also critical of Jones for stating that Hillebrand has put forth two new inference rules because Ladd-Franklin claims that the first of them originated in MacColl’s 1880 Mind article. She continues with her claim of priority, i.e. her antilogism: The reduction of the syllogism to a particular case of the principles of ‘Understatement’ and of the Laws of Thought, I am obliged to claim for myself.28

In 1892 in a long and detailed review of the first volume of Schröder’s three-volume work on logic, Ladd-Franklin quickly points out the influence of Peirce’s work on him; e.g. Schröder establishes all his formulas using the definitions of sum, product, negative and the axioms of identity and syllogism.29 Schröder uses letters to stand for classes whereas for Peirce they stand for statements. Ladd-Franklin notes that there is no difference between 'statement P implies statement Q' and 'term s implies term t' – as long as universal propositions do not imply the existence of their terms. Schröder had written to her on 17 September 1893 about his plans for further volumes, mentioning that his chapter on relatives was developing into a third volume of the work. Nevertheless, he hoped he would complete volumes 2 and 3 the following year. She also points out two important advantages that the copula, ‘no … is’, has over the copula, ‘all  … is’. Firstly, in solving problems, there is no need to transpose all the terms into the subject because there is no logical difference between subject and predicate. Secondly, the number of theorems in the body of the doctrine is reduced by half because with this copula, one statement represents both itself and its dual opposite in terms of the other copula. Finally, she notes that Schröder’s method to solving problems (first reducing the second member of the statement to zero) is both simpler and better than Peirce’s.30 From 1901 to 1905, Ladd-Franklin was the associate editor for logic in The Dictionary of Philosophy and Psychology.31 In addition to her editorial work, she was the single author or wrote in collaboration with others, of twenty-eight entries on logic or logic-related topics. In the first volume, there are four entries: contraposition, fusion (in logic), hypothetical, knowledge (in logic).32 In the second volume (1902), there are twenty-four entries: Logic, Mark, Mathematics, Middle Term (and Middle), Mood (in logic), Necessary (1) and (2) Sufficient Condition, Negation, Negative, Particular, Petitio Principii, Predication, Proposition, Signification (and Application in logic), Singular or Individual (in logic), Syllogism, the Modern Treatment of the Syllogism, Some (in logic), Symbolic Logic or Algebra of Logic, Sufficient Reason (2) in logic, Term, Theory, Transformation, Uniformity, Universe and Will to Believe.33 In 1908 she gave a paper at an international philosophy conference held in Heidelberg, the only American woman to speak there. This paper is the first one in which she embraces a more nuanced view of the role of symbolic logic, but one

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in  which she maintains the syllogism in a central role.34 Ladd-Franklin’s idea here is that the field of psychology needs a scientific philosophical doctrine that could be provided in a platform from a commission appointed by this congress, so that philosophy can be based on a scientific doctrine. She recommends that members appointed to the commission should believe either that the form of all reasoning is syllogistic or that the laws of thought prove a syllogism. She goes on to list the essential parts of such a doctrine: a theory of reality, a reformed psychology, a theory of truth, a theory of judgement and the ability to determine when propositions are true and when they are not true. To Ladd-Franklin, knowledge is a network of truths that ‘hang together’ and it is the confirmation of their cross-connections, i.e. the conclusions of syllogisms, that provides us with confidence in its validity as an entirety. The two terms of an asserted relation may become part of other relations some of which will be the premises of valid syllogisms. She calls this doctrine, histurgy, and claims the source of the idea came from her article with Fabian Franklin and concludes that it, not the only other known method, induction, is the basis of the validity of our body of knowledge.35 In a paper appearing three years later,36 she is critical of the program and platform for the next meeting of the American Philosophical Association that had appeared in an earlier issue of this journal. She argues that one of the topics, the principle of ‘explicit primitives’, will not be correctly presented by the six ‘realists’ who will be speaking at this meeting. On the other hand, she is pleased that there will be a prepared set of definitions and postulates for these ‘primitives’ and she offers a set of her own consisting of the following: concepts, first principles, axioms, postulates, all of which she refers to as ‘Primitives’. She references both her article, ‘Propositions’ in Baldwin’s DPP, and an article written with Edward V. Huntington.37 The American philosopher at Harvard University, Warner Fite (1867–1955), had published an article where he began by remarking that he knew it might be dangerous to cross words with Mrs Franklin in her supposedly special field of symbolic logic.38 Nevertheless, he proceeded to disagree with her notion that a primitive could be explicit, stating that a primitive is merely an illusion and an explicit primitive is really a contradiction in terms. She responded to him in the first of her two papers from 1912.39 Firstly, she points out that her paper had been written hastily and just for a specific purpose. Then she clarifies what she meant by the term, ‘explicit’. Secondly, she points out that he has mistakenly identified the subject of her paper and that he should instead consult her forthcoming paper, ‘The Implication’, where she is critical of some of his ideas about deductive reasoning and suggests that a deductive system of thought is a type of game in which one tries to see from how small a number of primitive premises all the known-to-be-true propositions can be deduced syllogistically.40 She continues first by referring to her paper of 1908, ‘Epistemology for the Logician’, where she had introduced the notion, ‘histurgy’, and now states, Knowledge starts with inductions, which are based upon facts. After many of these have been accumulated, …, certain pairs of them contain a common term, in such a form that they are capable of constituting the premises of a valid syllogism …

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this conclusion we then submit to the test of fact, simple experiment, or … refined laboratory methods. If, in a given case, the conclusion turns out to be true, the system has received, to this degree, confirmatory evidence. Thus, the closely interwoven tissue of knowledge (hence the name, histurgy).41

Her next paper, from 1912, is her second longest and includes her discussion of the topic, ‘pure reasoning’ and its importance in philosophy, a discipline she characterizes as the most perplexing and difficult of all the sciences. She develops the characteristics that reasoning has, the main one being that a conclusion follows from two or more premises. She discusses mathematical reasoning in terms of ‘necessary’ and ‘sufficient’ conditions that prevent what she calls ‘loose reasoning’. She claims the inevitable form of reasoning is her ‘antilogism’ or inconsistent triad rather than the syllogism and asserts that when a syllogism is expressed as an antilogism, ‘the syllogism, with its numerous modes and figures, becomes one single form, with one simple rule for validity.’ She goes on to say, ‘a good symbolic logic, kept simple, sufficiently elementary, and thoroughly sane, would be really of incalculable value to the philosopher – it has become, in fact, an indispensable tool’.42

Throughout this paper Ladd-Franklin cites many of her own papers as well as those of other philosophers and logicians, both well known and not so well known, including John Neville Keynes, Ernst Schröder, Alfred North Whitehead and Josiah Royce (1855–1916). She is critical of both Giuseppe Peano’s (1858–1932) and Bertrand Russell’s symbolic notation. And, referring to ‘p implies q’ in Russell and Whitehead’s Principia Mathematica, she claims she is unable to attach a consistent meaning to their notation. In a short paper from 1913, she adds the symbol for infinity, ∞, to mean ‘existent things’ or ‘possible states of things’ and gives this new form for her antilogism:43 (a

b) (c

–b) (a V c)

∞.

In her next paper, the first of three papers published in 1918, Ladd-Franklin is highly critical of Russell’s elimination of classes, relations and propositional functions in Principia: his ‘No Classes’ theory by which references to classes (sets) are eliminated, and instead, references to higher order quantification are used.44 But her main concern is with his and Whitehead’s form of logic is that it’s too mathematical for general or philosophical reasoning. She also points out that Russell is inclined to change his views, citing differences between his Principles of Mathematics (1903) and his and Whitehead’s 1910 Principia Mathematica. She is critical of two ideas in the Principia that Russell had learned from Peano. The first is the separation of propositions of the form: s ε M, from x ε G ⊃x, x ε M; the second is that s ε M is different from s ⊂ M; i.e. a member is different from a singleton class. At the end of her paper, she announces that she will be giving a set of ten lectures on symbolic logic for logicians which she distinguishes from symbolic logic for mathematicians.

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In 1918, Franklin Nelson Cole (1861–1926), secretary of the American Mathematical Society published a detailed report of the talk Ladd-Franklin gave at the society’s twenty-fifth summer meeting on Bertrand Russell’s work.45 It contained essentially the same material as in her 1918 paper in The Philosophical Review. In a single page ‘Letter to the Editor’ of Science, dated 8 March 1920, Ladd-Franklin writes that she has come across the best logic-puzzle she has ever seen, an advertisement in a newspaper: We have all known from our youth up that to err is human. If this is so, it must be that all of our competitors are thoroughly human.46

She asks the readers of Science whether or not this is an example of good reasoning and adds that she would like to receive answers to her question. (None are known.) Her final paper, fittingly with the title of the subject she is known best for, appeared in 1928.47 The theme is a familiar one. In his book on logic published in 1921, William Ernest Johnson (1853–1931) on p.  78, (not p.  75 as she claimed), introduced the antilogism as a formal disjunction of two, three or more propositions where each is entertained hypothetically, without mentioning Ladd-Franklin at all.48 She is, of course, annoyed but nevertheless takes this opportunity to describe the advantages of her method of deductive reasoning, referring him to Eugene Shen’s 1927 paper on her antilogism, also to several articles in the DPP, i.e. Proposition, Syllogism, Symbolic Logic, all of which she has authored or co-authored.49 She also includes the fourth (1906) edition of Keynes’ Formal Logic where Keynes has explicitly given her credit for the antilogism term and its use. Ladd-Franklin never abandoned her belief in the advantages of her antilogism as both an elegant way to handle a syllogism and one that reflects a ‘natural’ form of reasoning.

5 Professional and personal matters From her training under Peirce, Ladd-Franklin understood that the methodology of logic could be applied to the sciences, particularly to the competing theories of colour vision which interested her. Furthermore, she believed it was possible to apply the method and logic of scientific reasoning to any field of investigation. Her work in colour vision originated from her interest in the mathematical concept known as the ‘horopter’, the locus of points in external space whose two images formed on the retina of both eyes are seen as one image, i.e. binocular vision. In effect, she decided to pursue two careers: one in colour vision, publishing her first paper on the topic in 1887 in the American Journal of Psychology, and the second in logic, publishing that paper two years later also in The American Journal of Psychology. Nor did she abandon her purely mathematical interest. She published a short article in Messenger of Mathematics on a topic professors Arthur Cayley and James Sylvester were discussing in its pages in 1885–1886. J. W. L. Glaisher edited Messenger of Mathematics, a British journal primarily for young mathematicians that encouraged mathematical research.

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But there were other motives too. Ladd had married Fabian Franklin, a mathematics instructor at Hopkins in 1882, the year she completed the work for her doctoral degree. Two children were born between 1882 and 1884. He took an interest in her work, publishing a paper in 1881, and later two jointly with her, one in Mind in 1888, and the other in the DPP in 1902.50 Disappointed that she could not be registered as a student at Hopkins, which made her ineligible to receive the doctoral degree she had earned in logic and mathematics, Margaret Rossiter suggests that Ladd-Franklin hoped to earn a doctorate in Germany.51 When Fabian Franklin took his sabbatical in 1891–1892, they with their daughter went to Germany. She had to make special applications both to ministries and to professors with whom she wanted to work on her theory of colour vision. She also wrote to Felix Klein, who held the chair in mathematics at the University of Göttingen and had established a research centre there, to apply for admission. German universities at this time did not admit women at all. The German courts ruled that Klein could only admit her to his lectures as an auditor. Perhaps it was this second disappointment in her attempts to pursue a doctoral degree in either mathematics/logic or in colour vision in the German academic world that motivated her decision to work in both fields. In 1901–1902 she was in Europe with her family visiting friends and colleagues. Among others, they included Ernst Schröder and Hugh MacColl. The Franklins returned to Europe in 1908, again meeting many of the leading European philosophers, psychologists and scientists of the time.52 As a married woman, Ladd-Franklin knew that she would never attain a permanent academic position anywhere. In 1886 she had reported to her aunt that she was teaching three times a week at a girls’ school as well as giving private tutorials to three teachers and to three university men. Not until 1904 did she secure a series of two-year lectureships at Hopkins and at Columbia University. She never received a regular appointment. That she would not be discouraged from continuing her work seems  evident from the tenacity she had already exhibited in her quest for higher education studies.

6 Conclusion With her 1889 paper, Ladd-Franklin launched her career as a logician. She had metaphorically thrown down the gauntlet to establish herself as an independent and authoritative voice in the logic community. Four years later she became the first woman inducted into the American Psychological Association (APA). Between 1894 and 1925 she presented ten papers at APA meetings. Even so, she continued to publish papers in her equally important field of colour perception, attaining recognition in that field too by being inducted into the Optical Society of America in 1919. Finally, Ladd-Franklin decided to ask for her degree. She wrote to Frank Goodnow, president of Johns Hopkins University. He responded to her citing ‘the distinguished service which you have rendered to the advancement of knowledge since your residence among us has aided in bringing about this change of policy’.53 In 1926 at the age of seventy-nine, she received her doctoral degree.

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Postscript In a recent article,54 Sara L. Uckelman convincingly argues that the problem LaddFranklin solved was not Aristotle’s more than two-millennia-old problem but William Stanley Jevons’s ‘inverse logical problem’: given certain combinations inconsistent with conditions, to determine those conditions.55 The antilogism, which gives a systematic method for determining conditions inconsistent with certain combinations, is her solution to this problem.

Further Reading De Morgan, Augustus (1847) Formal Logic: Or, the Calculus of Inference, Necessary and Probable, London: Taylor and Walton. Franklin, Fabian (1881) ‘A Point of Logical Notation’, JHU Circular 10, 1879–1882, April 1881: 131. Franklin, F. and Franklin, C. L. (1888) ‘Mill’s Natural Kinds’, Mind, 1349: 83–5. Jevons, William S. (1864) Pure Logic, or the Logic of Quality Apart from Quantity, London: E. Stamford. Jevons, William S. (1880) Studies in Deductive Logic: A Manual for Students, London: Macmillan. Ladd, Christine (1877) ‘Quaternions’, The Analyst, 4: 174–6. Ladd, Christine (1878a) ‘On Some Properties of Four Circles Inscribed in One and Circumscribed about the Other’, The Analyst, 5: 116–17. Ladd, Christine (1878b) ‘The Polynomial Theorem’, The Analyst, 5: 145–7. Ladd, Christine (1878c) ‘Solution to Problem No. 186’, The Analyst, 5: 30–1. Ladd, Christine (1878d) ‘Solution to Problem No. 36’, The Analyst, 5: 191. Ladd, Christine (1878e) ‘Query’, The Analyst, 5: 64. Ladd, Christine (1879a) ‘The Pascal Hexagram’, The American Journal of Mathematics, 2: 1–12. Ladd, Christine (1879b) ‘Note on the Solution of a Congruence of the First degree When the Modulus Is a Composite Number’, Mathematical Questions and Solutions from The Educational Times, 30: 41–2. Ladd, Christine (1879c) ‘Note on Landen’s Theorem’, Mathematical Questions and Solutions from The Educational Times, 31: 39. Ladd, Christine (1880) ‘On De Morgan’s Extension of the Algebraic Processes’, The American Journal of Mathematics, 3: 210–25. Ladd, Christine (1881) ‘Note on Segments Made on Lines by Curves’, The American Journal of Mathematics, 4: 272. Ladd, Christine (1885–6) ‘On the So-Called D’Alembert – Carnot Geometrical Paradox’, Messenger of Mathematics, 15: 36–7. Ladd, Christine (1887) ‘A Method for the Experimental Determination of the Horopter’, The American Journal of Psychology, 1: 99–111. Ladd, Christine (1894) Sophie Germain. An Unknown Mathematician, Century Illustrated Monthly Magazine, 48.6: 946–9. Ladd-Franklin, Christine and Fabian Franklin Papers (1900–19), CLF/FF Papers, Columbia University Rare Books & Manuscript Library. MacColl, Hugh (1880b) ‘Symbolical Reasoning I’, Mind, 5.17: 45–60.

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Moktefi, Amirouche (2019) ‘The Social Shaping of Logic’. In: Dov. Gabbay et al., eds. Natural Arguments: A Tribute to John Woods (London: College Publications): 503–20. O’Connell, Agnes N. and Nancy F. Russo, eds. (1990) Women in Philosophy: A BioBibliographic Sourcebook (Westport, CT: Greenwood Press). Venn, John (1880) ‘On the Forms of Logical Proposition’, Mind, 5.19: 336–49. Venn, John (1881) Symbolic Logic, London: Macmillan and Co.

Notes 1

2 3 4 5 6 7 8 9 10 11

12 13 14 15 16

Hadassah Karp, The Beautiful Scheme: Christine Ladd-Franklin and the Expansion of Logical Methodology, undergraduate thesis (Barnard College, Columbia University, 2019), https://academiccommons.columbia.edu/doi/10.7916/d8–50dk-9m29 (Accessed 29 August 2021). https://educational-times.wcu.edu/ (Accessed 29 August 2021). Charles Sanders Peirce, ‘Preface’, in Studies in Logic (Boston: Little, Brown, and Company, 1883), iii. See the chapters by, respectively, Dunning, Shin, and Peckhaus in the present volume. Clarence Irving Lewis, A Survey of Symbolic Logic (New York: Dover Publications, 1960), 108–10. Ibid., 196. See David Dunnings chapter in this book. William Warren Bartley, III (ed.), Lewis Carroll’s Symbolic Logic, Part II, Advanced (New York: Clarkson N. Potter, 1977). Irving H. Anellis, ‘A History of Logic Trees. Editor’s Note’ [referencing Francine Abeles, ‘Lewis Carroll’s Method of Trees: Its Origins’, Studies in Logic 1, no. 1 (1990): 25–35], Studies in Logic 1, no. 1 (1990): 22–4. Gerhard Gentzen, ‘Untersuchungen über das logische Schliessen, I, II’, Mathematische Zeitschrift 39 (1934/1935): 176–210, 405–431. Dirk Schlimm, who has worked on Beltrami’s and Klein’s models of non-Euclidean geometry, which are interpretations of the axioms that establish their consistency, claims neither would express it this way. He adds that people in the nineteenth century didn’t have the distinction between syntax and semantics which makes it difficult to understand what they meant (private communication). Bertrand Russell, The Principles of Mathematics (Cambridge: Cambridge University Press, 1903). Haskell B. Curry, A Mathematical Treatment of the Rules of the Syllogism (Aberdeen: Aberdeen University, 1936), 213. Christine Ladd-Franklin, ‘On Some Characteristics of Symbolic Logic’, American Journal of Psychology 2, no. 4 (1899): 543–67. Ladd-Franklin, ‘On Some Characteristics of Symbolic Logic’, American Journal of Psychology, 2, no. 4 (1899): 543. For a discussion of Venn’s work on logic see Lukas M. Verburgt, John Venn: A Life in Logic (Chicago and London: The University of Chicago Press, 2022), especially chapters 8 and 9. See also, in this regard, William S. Jevons, Pure Logic, or the Logic of Quality Apart from Quantity (London: E. Stamford, 1864) and John Neville Keynes, Studies and Exercises in Formal Logic, Including a Generalisation of Logical Processes in Their Application to Complex Inferences (London: Macmillan, 1884, first ed.).

204 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32

33

Aristotle’s Syllogism Ibid., 557. Ernst Schröder, Der Operationskreis des Logikkalkuls (Leipzig: B. G. Teubner, 1877). Ibid., 560. Hugh MacColl, ‘The Calculus of Equivalent Statements and Integration Limits’ [first paper] Proceedings of the London Mathematical Society 9 (1877): 9–20; Id., ‘The Calculus of Equivalent Statements’ [second paper], Proceedings of the London Mathematical Society 9 (1878): 177–86; Id., ‘The Calculus of Equivalent Statements’ [3rd paper], Proceedings of the London Mathematical Society 10 (1878): 16–28. On MacColl see Chevalier’s chapter in the present volume. Ladd-Franklin, ‘On Some Characteristics of Symbolic Logic’, American Journal of Psychology 2, no. 4 (1889): 562. Peirce, ‘On the Algebra of Logic’, American Journal of Mathematics 3 (1880): 15–57. John Venn, Symbolic Logic (London: Macmillan, 1894), 372. Ladd-Franklin, ‘Some Proposed Reforms in Common Logic’, Mind 15, no. 57 (1890): 75–88. Ead., ‘Dr. Hillebrand’s Syllogistic Scheme’, Mind 1, no. 4 (1892): 527–30. (Cf. Franz Hillebrand, Die Neuen Theorien der kategorischen Schlüsse (Wien: Hoelder, 1891). On Hillebrand see Cosci’s chapter in this volume). Ead., ‘Review of E. E. Constance Jones’, Elements of Logic as a Science of Propositions’, Mind 15 (1890): 559–63. Ibid., 530. Ibid., 528. Ead., ‘[Review of] Vorlesungen über die Algebra der Logik (Exakte Logik) by Dr. Ernst Schröder’, Mind 1 (1892): 126–32. According to Volker Peckhaus, Schröder’s three volumes of his algebra of logic were published between 1890 and 1905 and its roots, being in the German algebraical ideas of his time, were not derived from Boole’s algebra of logic. For a complete discussion, see Volker Peckhaus, ‘Ernst Schröder und die “pasigraphischen Systeme” von Peano und Peirce’, Modern Logic 1 (1990): 174–205; Id., ‘Hugh MacColl and the German Algebra of Logic’, Nordic Journal of Philosophical Logic 3 (1999): 17–34. James Mark Baldwin (ed.), The Dictionary of Philosophy and Psychology (New York: Macmillan, 1901, 1902). Hereafter DPP. ‘Contraposition’, with Prof. R. Adamson, Glasgow University, p. 227; ‘Fusion (in logic)’, with Prof. E. B. Tichener, Cornell University, and James Mark Baldwin, pp. 398–9; ‘Hypothetical’, pp. 492–3; ‘Knowledge (in logic)’, with Charles Sanders Peirce, p. 603. ‘Logic’, with Peirce, pp. 20–3; ‘Mark’, with Peirce, p. 41; ‘Mathematics’, with Fabian Franklin, p. 47; ‘Middle Term (and Middle)’, with Peirce, p. 77; ‘Mood (in logic)’, with Baldwin, pp. 103–4; ‘Necessary (1) and (2) Sufficient Condition’ p. 143; ‘Negation’, with Peirce and Baldwin, pp. 146–8; ‘Negative’, pp. 148–9; ‘Particular’, with Peirce, pp. 265–6; ‘Petitio Principii’, with Peirce, pp. 287–8; ‘Predication’, with G. F. Stout, University Reader, Oxford, and Baldwin, p. 326; ‘Proposition’, with Baldwin and Stout, pp. 361–70; ‘Signification (and Application in logic)’, with Peirce, p. 528; ‘Singular or Individual (in logic)’, p. 533; ‘Syllogism. The Modern Treatment of the Syllogism’, pp. 632–3; ‘Some (in logic)’, with Peirce, p. 555; ‘Symbolic Logic or Algebra of Logic’, with Prof. Louis Couterat, University of Toulouse, pp. 640–51; ‘Sufficient Reason (2) in logic’, p. 617; ‘Term’, pp. 675–7; ‘Theory’, with Peirce, pp. 693–4; ‘Transformation’, pp. 711–12; ‘Uniformity’, with Baldwin and Stout p. 731; ‘Universe’, with Peirce, p. 742; ‘Will to Believe’, p. 818.

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34 Theodor Elsenhans (ed.), Bericht über den III Internationalen Kongress für Philosophe. ‘Epistemology for the Logician’ (Heidelberg: Carl Winter 1909), 664–70. 35 Fabian Franklin and Ladd-Franklin, ‘Mill’s Natural Kinds’, Mind. 13, no. 49 (1888): 83–5. 36 Ladd-Franklin, ‘Implication and Existence in Logic’, The Philosophical Review 21 (1912): 641–65. 37 Ladd-Franklin, Christine and Edward V. Huntington, ‘Logic, symbolic’, in Encyclopedia Americana, vol. 9 (New York: Grolier, 1905). I have not been able to locate this piece. 38 Warner Fite, ‘Explicit Primitives: A Reply to Mrs. Franklin’, Journal of Philosophy, Psychology, and Scientific Methods 9, no. 6 (1912): 155–8. 39 Ladd-Franklin, ‘Explicit Primitives Again: A Reply to Professor Fite’, Journal of Philosophy, Psychology, and Scientific Methods 9, no. 21 (1912): 580–5. 40 Ead., ‘Implication and Existence in Logic’, The Philosophical Review 21, no. 6 (1912): 641–65. 41 Ead., ‘Explicit Primitives Again’, 584–5. 42 Ead., ‘Implication and Existence in Logic’, 648, 663. 43 Ead., ‘The Antilogism – An Emendation’, Journal of Philosophy, Psychology, and Scientific Methods 10, no. 2 (1913): 49–50. 44 Ead., ‘Symbolic Logic and Bertrand Russell’, The Philosophical Review 29, no. 2 (1918): 177–8. 45 Frank N. Cole, ‘Report by secretary F. N. Cole of Ladd-Franklin’s talk on Bertrand Russell’s ‘Principia Mathematica’ (1910) presented at the Summer Meeting of the American Mathematical Society’, Bulletin of the American Mathematical Society 25, no. 2 (1918): 59–60. 46 Ladd-Franklin, ‘A Logic Test. Letter to the Editor of Science’, Science 51, no. 1321 (1920): 14. 47 Ead., ‘The Antilogism’, Mind 37 (1928): 532–4. 48 William Ernest Johnson, Logic, Part II (Cambridge, Cambridge University Press, 1921). 49 Cf. Eugene Shen, ‘The Ladd-Franklin Formula in Logic: The Antilogism’, Mind 36, no. 141 (1927): 54–60. 50 Fabian Franklin, ‘A Point of Logical Notation’, JHU Circular 1, 1879–1882 (1881): 131. 51 Margaret Rossiter, Women Scientists in America: Struggles and Strategies to 1940 (Baltimore: Johns Hopkins University Press, 1982). 52 Ahti-Veikko Pietarinen, ‘Christine Ladd-Franklin’s and Victoria Welby’s Correspondence with Charles Peirce’, Semiotica 196 (2013): 6. 53 CLF to Frank Goodnow, 16 January 1886, Box 2, CLF/FF Papers; Frank Goodnow to CLF, 6 February 1926, Box 4, CLF/FF Papers. 54 Sara L. Uckelman, ‘What Problem Did Ladd-Franklin (Think She) Solve(d)?’, Notre Dame Journal of Formal Logic 62, no. 3 (2021): 527–52. 55 William Stanley Jevons, Studies in Deductive Logic: A Manual for Students (London: Macmillan, 1880).

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Syllogism and Beyond in the Peano School Paola Cantù

1 Introduction Traditional logic between the end of the nineteenth and the beginning of the twentieth century is often presented as a polemical target, and this for different reasons: from a didactic point of view, it is criticized as an antiquated and useless method of teaching; from a logical point of view as a sterile exercise that cannot account for the developments in mathematics and in other deductive sciences; and from a historical point of view as a wrong development of the true Aristotelian logic. All these ways of presenting logic are obviously aimed at carving out space for a new discipline that struggles to be recognized by philosophers because of its high degree of technical difficulty and its idiosyncratic symbolic language. One would expect the Peano school to excel in similar exercises, given the efforts to popularize the use of symbolic language in the teaching of logic and mathematics. An analysis of the writings of several members of the school proves, however, that its relation to the Aristotelian and syllogistic tradition is much more complex. Firstly, different authors had different levels of access to sources: Peano presumably never read Aristotle, whereas Vailati knew the Aristotelian corpus very well. Secondly, some authors underlined a difference between Aristotelian syllogistic and that of later Scholasticism, whereas others did not. Thirdly, all authors explicitly agreed that mathematical logic was not in opposition to traditional logic, but rather was a generalization or an extension of the latter. So, apart from a different method and an increase in rigour, the objectives and the content of the two approaches to logic were considered partially similar. These results could be taken as supporting the claim that several of Peano’s ‘shortcomings’ with respect to subsequent developments of formal logic can be better understood as symptoms of his adhesion to a traditional view of logic, as claimed by Zaitsev.1 Following Eaton, Zaitsev took traditional logic to be based on (1) the subjectpredicate form of predication so that (2) relations and general propositions are also expressed in the form of singular propositions in subject-predicate form; and (3) the meaning of the terms is prior to that of the propositions in which these terms occur.2 Peano’s shortcomings are then taken to be (1) the use of the same notation for relations

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between classes and between propositions; (2) the lack of a clear distinction between propositional functions and propositions; (3) the systematic use of conditional definitions. Zaitsev argued that Peano’s biggest revolution in logic, i.e. the distinction between membership and inclusion, arose from the problems caused by well-known medieval fallacies. An example is the copula in the subject-predicate form appearing to be sometimes transitive (inclusion) and sometimes intransitive (membership): to explain why seven is a prime number and prime numbers are an infinite set, but seven is not an infinite set, one can say that the term ‘prime number’ has to be considered as an individuum in the first sentence and as a class in the second, and therefore the verb ‘to be’ has to be considered in a composite sense in the first case and in a divided sense in the second case.3 Another example of the influence of traditional logic is, according to Zaitsev, Peano’s treatment of relations and general propositions as singular propositions. This chapter will not discuss in detail all of these claims.4 Instead, it offers further independent arguments to show that the relation of the Peano School to syllogistic logic (either in its Aristotelian form or in successive variations) was more conservative than destructive. Moreover, this chapter extends the inquiry to other members of the school, such as Vailati, Burali-Forti, Pieri and Padoa, allowing for different degrees of conservativeness and destructiveness (Section 2). The relation to syllogistic is examined from the point of view of logical rules of inference: what was the role of syllogism and modus ponens in the school’s logic (Section 3)? Answering this question contributes to a better historical understanding of some key aspects of Peano’s logic, some of which are still the object of debate in recent literature: was there a clear distinction between logical axioms and rules of inference in the Peano School?5 A first methodological problem concerns the ways of distinguishing Peano’s own ideas from those of the other members of the group. For the general purposes of this chapter, it will suffice to refer to published writings, even if further clarifications might come from the investigation of archival sources and correspondence. Notwithstanding the collaborative nature of the Formulaire, it has been claimed that the published editions reflect Peano’s selection of (sometimes contradictory) modification proposals made by members of the school or by readers.6 A second methodological problem concerns how the ideas were often informally circulated between the members of the school prior to their sedimentation in published writings (a good example is offered by the correspondence between Vailati and Giovanni Vacca).7 The present chapter will focus on the sources that are cited in published writings in order to consider not only whether there might be a correlation between the reading of Peirce and the explicit formulation of modus ponens as a rule, but also to distinguish primary from secondary references. Different degrees of knowledge of the sources of Aristotelian logic and the Scholastic tradition might also explain the plurality of views in the school. Peano had no direct knowledge of the sources and often relied on secondary references. Padoa’s sources for syllogistic were mainly indirect: George Boole, Ernst Schröder, Christine FranklinLadd and Burali-Forti.8 Burali-Forti relied (in 1894) mainly on the writings of Peano and Nagy, the first Professor of Logic in Italy. Giovanni Vacca read the Aristotelian corpus in the Greek language and often suggested Aristotelian quotes to Peano, as

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in the case of the identity principle that is quoted in the Formulaire after Aristotle’s Topics (VII,I.15).9 Giovanni Vailati not only quoted most Aristotelian works in Greek, including Prior and Posterior Analytics, Physics, Politics, On Interpretation, Topics and Sophistical Refutations,10 but also devoted several articles to the topic, ranging from mechanics to logic and philosophy (e.g. on the Aristotelian theory of definitions, on the Scholastic precept ‘Distingue frequenter’ and on deductive method).11 Vailati opposed the content of Aristotle’s Organon to successive developments of Scholastic logic: his main sources were Aristotle, Porphyry, Sextus Empiricus and John Stuart Mill. So he claimed (followed by Pieri) that criticism does not apply to Aristotelian syllogistic itself but to inaccurate uses, representations and oversimplifications made by later Scholasticism.

2 Syllogistic versus mathematical logic 2.1 Differences The use of syllogistic as a critical target to show the extension and improvement provided by mathematical logic (or logistics, as it was called at the time) was common in the Peano School. It has been suggested that the disadvantage of this approach was the lack of an adequate symbolism – a consideration that aimed to promote mathematical logic rather than attacking syllogistic logic. In this chapter, I will mention several differences between syllogistic and mathematical logic that have been highlighted by the members of the Peano School. They insist on (a) an enlargement and at the same time a simplification of the rules of reasoning; (b) a greater variety of relations, calculi and definitions that mathematical logic allows us to express; (c) the substitution of a symbolic language for natural language; and (d) foundational virtues such as simplicity, rigor and generalization, which are enhanced by mathematical logic. (a) Peano claims that many reasoning rules used in mathematics do not have the form of a syllogism. In the Dizionario di Matematica written in collaboration with Vacca, Vailati and Padoa, he writes: ‘[Mathematical Logic] has in common with the Aristotelian Logic only the syllogism. The classifications of the various modes of syllogisms, when they are exact, have in mathematics little importance. In the mathematical sciences one encounters numerous forms of reasoning irredeemable to syllogisms’,12 for example, substitution, modus ponens and other properties of deduction (see below Section 3). Besides, many modes of syllogism are not really useful: ‘Scholastic logicians consider other modes of syllogisms which can be transformed one into the other by conversion. They are of little importance.’13 Finally, some syllogistic forms are considered fallacious unless the existential import is explicitly stated.14 (b) Mathematical logic extends the investigation of relations because not all mathematical relations can be expressed as a logical product of classes: membership, but also other mathematical relations that are classified according to their formal properties. For example, the relations ‘father of ’ and ‘different from’ are neither reflexive nor symmetric nor transitive; ‘being perpendicular to’ is a symmetric

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relation between straight lines that is neither reflexive nor transitive; ‘greater than’ is a transitive relation that lacks reflexivity and symmetry.15 Mathematical logic includes a calculus of propositions which ‘is more general and goes further’, because it investigates not only inclusion and predication between concepts, but all sorts of relations.16 Mathematical logic also extends Aristotle’s theory of definitions. The definitions by genera and species are expressed as a logical product of classes (intersection), but other kinds of possible operations between classes are also considered, for example the addition between classes (union). Moreover, the theory of definition is extended from the definition of a single term to the definition of a sentence including the term (implicit definition), and definability becomes a relative rather than an absolute property: no term is definable in isolation, but only definable with respect to a given set of terms taken as primitive. In mathematical logic, several different possible definitions of the same term can be given, and ‘essential’ properties are not privileged.17 (c) Syllogistic and mathematical logic both study forms of reasoning, but Aristotelian logic studies those that are proper to natural language, whereas mathematical logic studies the forms that are proper to the deductive sciences, especially mathematics. Syllogistic logic is expressed in natural language terms, whereas mathematical logic is expressed in mathematical symbols. So they differ in notation.18 There are several advantages in the use of symbolic rather than natural language. Natural language formulations are ambiguous, whereas symbolical formulations are not. The sentences composed by linguistic terms belong to a given language, whereas the mathematical symbols are a universal, language-independent ideography.19 The new method allows to observe, as if under a microscope, very small differences of natural language, so that a microscopy of thought can be achieved.20 (d) Mathematical logic is more rigorous, both in the analysis of concepts and in deductions.21 It makes accessible to people of average intelligence what would otherwise be accessible only to geniuses. Symbolic language and mathematical logic are a tool and a science that do not teach us how to reason but rather allow us to represent concepts that would hardly be detectable using ordinary language, just as the microscope and bacteriology do not teach us how to see but allow us to see bacteria that would not be visible otherwise. Unlike Scholastic logic, which is ‘a sumptuous and cumbersome dress which the professor discards as soon as the lesson is over’’, ideographic logic ‘is the secret and faithful companion of his work, to which he has recourse unceasingly when he wants to probe the rigor of his reasonings or those of others’.22 Pieri claimed that mathematical logic eliminates the opposition between logic of quantity and logic of quality, the former being identified with the mathematical method and the second with the syllogistic method, an opposition that had already been recognized by Pascal and questioned by Leibniz.23 Mathematical logic includes metamathematical results, such as the investigation into the equivalence of two systems: can the primitives of a system be explicitly defined by means of the primitives of the other system? Can the primitive propositions of the former be deduced from the primitive propositions of the latter? Finally, the analysis of propositions is simplified because the complex distinctions between affirmative and negative, general and particular, categorical and

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hypothetical, can all be reduced to a fundamental distinction between conditional and existential propositions.24 Mathematical logic is a simplification of syllogistic logic, since it reduces all different forms to just three. Using the commutativity of conjunction and other properties of the negation as well as the refusal of existential import in particular propositions, Padoa claims that all valid syllogisms might be reduced to the collective syllogism (Barbara), to the singular syllogism (Socrates is a man) and to the sorites or polysyllogism (chains of inferences consisting in repeated applications of Barbara).25

2.2 Similarities Notwithstanding the many differences, some similarities were also highlighted. If mathematical logic is an extension of traditional logic, the similarities with mathematical logic relate first and foremost to their common part: (i) a formal approach to the calculus of classes and the investigation of syllogism as a rule of reasoning. Additional similarities concern (ii) the common grounding principle and (iii) the distinction between singular and universal propositions. (i) Mathematical logic ‘assimilates what was good in Scholastic logic’; it contains the traditional logic as a part.26 But it is also an evolution and an improvement of syllogistic logic: ‘Logistics, in short, is not another logic, different from the traditional one, but rather an evolution and improvement of the formal logic of Aristotle and the Scholastic logic: I would almost say that it is the modern form of deductive logic, which contains all of the Aristotelian and Scholastic logic, although far exceeds them.’27 Mathematical and syllogistic logic are similar because they both contain a calculus of classes. In mathematical logic the latter ‘is more elementary and deviates less from classical logic’.28 Both are formal: the difference between arithmetic and logic concerns the matter, but not the form, which amounts to a given order of logical relations.29 They are both sciences in themselves.30 (ii) They both ground syllogism in the principle ‘dictum de omni’ rather than in the logical principles of identity, contradiction and excluded middle. Pieri remarks that for this reason deductive method has heuristic power, even if it asserts in the conclusion what is implicitly contained in the premises. Recalling Vailati’s analogy with Michelangelo Buonarroti’s image of a statue that is already implicit in the marble stone from which the carver extracts it, Pieri attacks the simplistic conception of logic which considers only identity, contradiction and excluded middle as basic principles, ignoring the fact that Aristotle grounded syllogism in the principle ‘dictum de omni’ (see also Section 3).31 (iii) Peano’s mathematical logic re-established the Scholastic distinction between singular and universal propositions as he introduced two distinct symbols for belonging (of an individuum to a class) and inclusion (of a class into another class).32 What we would now consider as the Peano school’s main innovation is explicitly mentioned by Padoa among the similarities with Scholastic logic. In this sense, Zaitsev’s claim concerning the influence of the Scholastic distinction between the divided and the composite sense of the copula had already been made by the members of the Peano school themselves (see Section 1).

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2.3 Dismissals and strong appraisals It is possible to identify a number of negative characterizations of Scholastic logic put forward by members of the Peano School. They can be found mainly in Padoa’s writings, who uses the language of syllogistic to compare it with mathematical logic in an effort to promote the latter. Padoa’s key paper, in this regard, appeared in the Revue de Mètaphysique et de Morale in French and was part of Couturat’s project to renew the French teaching of logic, which was mainly based on traditional syllogistic logic.33 Padoa’s comparison aims to show I. the higher generality of mathematical  logic and II. the uselessness of syllogistic logic in teaching and in mathematical proofs. I. Challenging giants and demolishing false beliefs. Given that so many people still believed in the power and generality of syllogistic, Padoa introduced syllogistic logic and its complex distinctions. Syllogistic treatises are like windmills to be faced and knocked down by adopting their own language: The analysis I am going to make of syllogistic logic (and which is not in the Formulaire) will be tiresome and not very pleasant; […] I must therefore face and tilt at windmills; indeed, as most men still believe that they are giants, if I avoided them, someone might think that I am afraid of them! So, I will adopt the language of the Scholastic philosophers. 34

As mentioned above in Section  2.2, mathematical logic simplifies the number of syllogisms: a greater number of modes do not imply a greater generality. On the contrary, the huge number of syllogisms in Scholastic logic ‘is only an illusion due to the imperfect means of expression used by Scholasticism’.35 II. Stop teaching syllogistic logic! Courses on syllogistic logic are useless: ‘the aim I have in mind is precisely to persuade future teachers of Logic not to tire their students with useless lessons on this subject’.36 Padoa recalls here an argument often made by Peano, namely that Scholastic logic is of no use in mathematical proofs: ‘It is well known that Scholastic logic is of no appreciable use in mathematical demonstrations, since the latter never mention the classifications and rules of the syllogism, and on the other hand make use of reasoning, which is quite convincing, but not reducible to the forms considered in Logic.’37 If Padoa is the most critical, Vailati is the most appreciative: he praises the deductive method, the theory of definitions, syllogism and the sensus compositi vs sensus divisi distinction. I will say something about the first two issues here. The last point has been mentioned already in Section 1 and will be reconsidered in Section 3. A. The deductive method. Vailati discussed Aristotle’s views on deduction and induction with the aim of proving that Aristotle considered not only deduction but also induction as a warranty of truthfulness. Vailati remarked that the necessity involved in deduction concerns the relation between premises and conclusions and not the certainty of the conclusions which should not be trusted any more than those reached by induction.38 The main difference between modern science and Aristotle does not

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concern what deduction is, but how it is used: whereas Aristotle considered deduction from false premises in rare cases, such as the reductio ad absurdum in mathematics and the ad hominem arguments in rhetoric, modern science is different because it considers the fruitfulness of applying deduction to premises that are less certain and evident than the conclusions to be drawn from them. It is exactly because deduction is ‘a good conductor of evidence and certainty, and of passing on untouched, to the conclusions achieved by means of it, all the credibility and authority possessed by the corresponding premises’39 that it took such a long time before scientists applied it to anticipate experience. Even the criticism of deduction as a passage from the general to the particular is not directly attributed to Aristotle, but to Porphyry, who, in choosing the tree as an instrument of representation, favoured the asymmetrical metaphor of going up from the conclusions to the premises or down from the premises to the conclusions.40 B. The theory of definitions. Vailati praises the subtleties of the Aristotelian theory of definitions and defends them against its reception and transformation by Porphyry and later Scholasticism, who reduced it to the genus-differentia specifica form, ignoring Aristotle’s Topics.41 Peano remarks that even if some mathematical definitions have this form (square = quadrilateral ∩ equilateral ∩ equiangular), most mathematical definitions do not (e.g. definitions based on operations such as 2 = 1 + 1, or ‘composite number = number bigger than 1 multiplied by a number bigger than 1’, or ‘e = limit of …’). Operations cannot be conceived of as genera because they are not classes but functions and thus do not belong to the Aristotelian category of ousia, but to that of pros ti.42 According to Vailati, the Aristotelian theory of definitions is more complex and based on four possible meanings of the expression ‘the As are Bs’: definitions, when B occurs among the properties that define B (definitions and genus), and general propositions asserting that B is a property shared by all objects that are A, even if B is not part of the definition of A (proprium if no object that is not A has B, accident otherwise). Vailati’s insistence on the importance of Aristotle’s theory of definitions is connected to the analysis of paralogisms that constituted an obstacle to the development of geometry. Vailati analysed Saccheri’s claim that Borelli’s definition of parallels is a fallacy of complex definition. The latter occurs when a thing is defined by the simultaneous possession of several properties whose compatibility is not previously verified.43 The Aristotelian theory of definitions can prevent us from making mistakes induced by an erroneous misunderstanding of the role of definitions and axioms. Peano follows Aristotle (Post. An. I.10,9) when it comes to the idea that definitions do not have existential import because in mathematics one often introduces definitions of things that might not exist (the biggest prime number, the limit, derivative or integral of a function) and that definitions should proceed from the known to the unknown (Topics VI,4.2) and thereby introduce new symbols by means of the symbols already introduced.44 Aristotle is also quoted with reference to the eliminability rule: in a definition the definiens should always be replaceable by the definiendum. Eliminability is a criterion of adequacy of a definition: whenever the replacement is problematic, the definition is inadequate: ‘In order for the inexactness of the definition to become manifest, the concept must be placed in place of the name’ (Top.6.4).45

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3 Syllogism as a logical rule In Section  2 I have pointed out how the members of the school generally felt that mathematical logic was an extension of traditional logic, and more specifically a way to simplify and generalize Aristotle’s syllogistic theory. Yet the overview presented in Section 2 faces methodological objections, as it might well have been the result of a rhetorical strategy on the part of the Peano School: an appeal to tradition. A detailed analysis of the school’s relation to syllogistic must deal with these and other historical and methodological difficulties: (a) Peano’s actual practices might differ from his explicit assertions, (b) Peano’s views might differ from those of other members of the school, (c) different views might occur at different stages of development of his logic and (d) distinct views might be related to sources discussed in different writings or by different members of the Peano School. So, in the following section, I will investigate the actual mathematical practice of the school to ascertain whether the conservative approach described in Section 2 corresponds to an effective use of syllogism in logic and mathematics and to determine whether the appeal to syllogistic reasoning was propulsive or regressive with respect to the transformation of logic. I will thus present in chronological order several writings by Peano and by his collaborators, investigating whether and how the syllogism in Barbara and the modus ponens (which is never called by this name) are introduced and used in proofs and what sources are cited in each case. Concentrating on syllogism and modus ponens will allow us to discuss a question that is controversial in the literature and which might benefit from a detailed analysis of the actual logical and mathematical practice. From Frege onwards, the Peano School has often been accused of not having understood the crucial role of rules in logical calculi: e.g. syllogism was considered a logical axiom rather than as an inference rule.46 The main objective of this section is on the contrary to show that syllogism was explicitly introduced both as a logical identity and as an elimination rule already in 1898 by Peano, and in 1891 by Vailati, who considered the dictum de omni as a transitive property of the relation of deduction. Besides, it will be argued that the interest in Aristotle’s original theory of syllogism is accompanied by a focus on the search for other rules of inferences, progressively distancing the group from the influence of the logical tradition which presented the principles of identity and non-contradiction as fundamental logical axioms. The analysis of syllogism as a transitive property of deduction suggested the investigation of other properties of deduction. If the modus ponens is associated with Peirce in the historical notes added to the Formulaire in 1895, it emerged independently in the logical and mathematical practice of the Peano School. It was definitely added to the list of logical inferences, only once its formulation acquired a meaning not only in the logic of propositions, but also in the logic of classes; i.e. only once the distinction between membership and inclusion was introduced. Like many other properties of the deductive relation, syllogism and modus ponens are listed among the logical axioms or propositions but are also considered among the most frequent rules used in mathematical proofs.

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3.1 Peano’s early writings and Burali-Forti’s Logica matematica In the Calcolo geometrico (1898) Peano does not quote Aristotle but rather Leibniz, Cayley, Schröder, Peirce, Clifford, Jevons and Liard.47 Peano was heavily influenced by Schröder, but here he quotes only the Operationskreis, which does not include a section on syllogistic reasoning, even if Schröder mentions a general principle corresponding to universal syllogism (Barbara) in the calculus of classes.48 Leibniz’s influence concerned the theory of permutations and especially the calculus of the number of sentences that can be produced on n classes by means of a certain number of primitive symbols and the related problem of the number of the modes of categorical syllogism. Peano first introduces a logic of classes, defining identity for classes (when they have the same elements), logical conjunction (the intersection of the extensions of the classes), logical disjunction (the union of the extensions of the classes), the negation (the complement of a class), the symbols for ‘nothing’ (a class with no elements) and for ‘all’ (the universe), the symbol ‘smaller than’ used to express the inclusion of a class in another.49 Peano then relates the calculus of classes to a calculus of propositions by the addition of an abstraction operator symbol ‘x’: so that given a proposition α and an indeterminate element x, x: α is the class formed by all entities for which the proposition α is true. Next, he simplifies the notation for the case in which the condition is satisfied for whatever x. Instead of writing [x: a] < [x: b], which reads ‘the class of the x satisfying condition α is part of the class satisfying condition β’, he simply writes α and < respectively.’ Propositions 5’’, 6’’, 5’’’ and 6’’’ say that ‘from a system of logical equations, all true, and containing the same sign =, < or >, new equations can be deduced that are also true, if one multiplies both members by the same factor or adds the same term, or

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sums them member by member, or multiplies them member by member’.53 These are syntactic transformations that are independent from the meaning of the symbols (A,B indicate classes or the conditional propositions defining classes) and can be used to derive different formulations of the same logical equation. As an example, Peano gives the general proposition ‘Every A is B’, which is indicated by A