Truth, Meaning, Justification, and Reality: Themes from Dummett 9783110459135, 9783110458312

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Michael Frauchiger (Ed.) Truth, Meaning, Justification, and Reality

Lauener Library of Analytical Philosophy

Edited by Wilhelm K. Essler, Dagfinn Føllesdal, and Michael Frauchiger

Volume 4

Truth, Meaning, Justification, and Reality Themes from Dummett Edited by Michael Frauchiger

ISBN 978-3-11-045831-2 e-ISBN (PDF) 978-3-11-045913-5 e-ISBN (EPUB) 978-3-11-045839-8 ISSN 2198-2155 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Truth, Meaning, Justification, and Reality Themes from Dummett The foundations for this collection of original essays, which has taken several years to complete, were laid at the 4th International Lauener Symposium in honour of Sir Michael Dummett, one of the most influential, creative, judicious and committed analytic philosophers in the second half of the 20th and the early 21st centuries. The contributors to this volume, including some of Dummett’s most vivacious, stand-alone former students, attempt to get a broad variety of vital Dummettian themes in perspective. The revised contributions critically and perspicuously reflect on various concerns of Dummett’s groundbreaking work in philosophy of language, philosophy of mathematics and logic, metaphysics and epistemology. The essays sometimes direct towards aspects of Dummett’s pioneering work in the history of analytical philosophy, particularly his interpretations of the works of Frege and of Wittgenstein, which in conjunction with Dummett’s own highly original and clear ideas on truth and meaning have shaped decisive contemporary debates concerning notably the distinction between realism and anti-realism. Further, the volume includes a cheerfully serious excursion into popular, non-academic philosophy by Michael Dummett himself, a speech he presented upon accepting the 2010 Lauener Prize for an Outstanding Oeuvre in Analytical Philosophy. To boot, the collection features a biographical sketch, a laudatio and an epilogic piece which reveal less known facets of Dummett’s many-sided work and activities such as his innovative contributions to philosophical theology and political philosophy of immigration and asylum, and beyond that, his extraordinary ability to bring philosophical aspects to bear on cultural, social and political issues by putting his ideas into effect in major fields of public life. What appears to link the contributions to this collection is the respect they demonstrate for the intellectually and morally serious, empathic personality of this powerful philosopher, who additionally was an untiring, vigorous and warm-hearted philosophico-religious campaigner for racial justice and humanity. The book brings together contributions by Michael Dummett, Eva Picardi, Crispin Wright, Timothy Williamson, Ian Rumfitt, Daniel Isaacson, Dag Prawitz, Dale Jacquette, Alex Burri, Michael Frauchiger.

Acknowledgements I am very grateful to those who have contributed to the collection on hand, which has been in preparation for some time. They all have chipped in generously elaborated papers and a number of them have shown persevering commitment to the project of this book on Dummettian themes by thoroughly thinking the germ ideas through they had originally presented during the intense and rich discussions at the 2010 Lauener Symposium in honour of Michael Dummett’s philosophical oeuvre. I owe special gratitude to Dan Isaacson for travelling together with Michael and Ann Dummett from Oxford for the Lauener Symposium in Bern, thereby assisting me in making arrangements that minimized the amount of walking they were required to do and consequently enabling Dummett’s active participation in the Lauener Symposium in his honour. Michael Dummett was visibly aged and physically debilitated at the time—one and a half years, as we now know, before his demise—but he said he was pleased to be receiving the Lauener prize, and he told me that his health was somewhat improved and that he therefore expected that he should be able to take part in the award ceremony and the symposium. He made clear that he did not feel up to writing a paper for the symposium and had abandoned all attempt to do any more original work in philosophy, but that he could make a closing speech. We owe a great debt of gratitude to Michael Dummett for the optimistic spirit of collaboration that he showed regarding his participation in the Lauener Symposium on themes from him under the exceptionally difficult physical circumstances of his old age. Further, sincere thanks are due to Gertrud Grünkorn, Florian Ruppenstein, Johanna Wange and Maik Bierwirth at De Gruyter, for their patient support and cooperation in guiding the book to final publication. I am also grateful to Wilhelm K. Essler and Dagfinn Føllesdal, my fellow coeditors of the book series “Lauener Library of Analytical Philosophy” (edited on behalf of the Lauener-Stiftung) as well as to my fellow members of the Foundation Council of the Lauener-Stiftung (Dagfinn Føllesdal, Stephan Hottinger, Alex Burri, Dieter Jordi as well as the late Dale Jacquette) for their patient encouragement.

Contents Daniel Isaacson  Michael Anthony Eardley Dummett: A Biographical Sketch | 1 Ian Rumfitt Dummett Laudatio | 13 Michael Dummett  Closing Speech to Lauener Symposium | 25 Eva Picardi Michael Dummett’s Interpretation of Frege’s Context Principle: Some Reflections | 29 Crispin Wright What Was Frege’s Mistake? | 63 Dale Jacquette Dummett on Truth-Conditions, Frege’s Analysis of Sentence Meaning, and the Slingshot Argument | 81 Dag Prawitz To Explain Deduction | 103 Ian Rumfitt Prospects for Justificationism | 123 Timothy Williamson Dummett on the Relation between Logics and Metalogics | 153 Alex Burri Residues of Realism | 177 Michael Frauchiger  Putting Reality into Perspective by Understanding Theoretic Propositions, Theistic Thoughts and Political Dealings | 189 Contributors | 247 Index | 249

Daniel Isaacson

Michael Anthony Eardley Dummett: A Biographical Sketch* Michael Dummett was born at 56 York Terrace, London, his parents’ home, on 27 June 1925, and died on 27 December 2011 at 54 Park Town, Oxford, the home where he and his wife Ann had lived since 1957 and brought up their children. He was the only child of George Herbert Dummett (1880–1970), a silk merchant, who also later dealt in rayon, and his wife, Mabel Iris née Eardley-Wilmot (1893–1980), whose father, Sir Sainthill Eardley-Wilmot, had been InspectorGeneral of Indian Forests, and after whom Michael was given Eardley as his middle name. Michael’s father had two sons and a daughter by a previous marriage.

Early life and education At the age of ten Dummett was sent as a boarder to a preparatory school, Sandroyd, in Cobham, Surrey. In September 1939, at the onset of the Second World War, he began his secondary education at Winchester College, having come top of the election roll for Scholars. After a compulsory year on the ‘classics ladder’, he opted for science, but was ‘deeply disappointed’ by it (Dummett

|| University of Oxford || This biographical sketch is a revised and expanded version of the author’s entry on Michael Dummett in the Oxford Dictionary of National Biography, Oxford University Press, January 2015; online edition January 2016 http://www.oxforddnb.com/view/article/104464, by permission of Oxford University Press. I am grateful to a number of people who have helped me in writing about Michael Dummett’s life, especially Michael’s daughter Suzie, who has answered endlessly many questions, always quickly and cheerfully, with special insight and great encouragement, and to Michael Screech for his memories of Michael at the Bedford Japanese course and the Wireless Centre in Delhi. I am grateful also to Suzanne Foster, the Archivist at Winchester College, for her always immediate answers to questions about Dummett’s time there, and I am grateful to Alex May, Research Editor at The Oxford Dictionary of National Biography, who provided key documents, excellent advice, and cheerful encouragement while I was writing the ODNB entry for Michael Dummett.

DOI 10/1515-9783110459135-001

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(2007), p. 4) and switched to history. In 1943 he obtained a history scholarship to Christ Church, Oxford, but—now eighteen and with the war still raging—went instead into the Royal Artillery, under which auspices he was sent on a sixmonth ‘short course’ at Edinburgh University. There he contacted the Catholic Chaplaincy and underwent instruction by the Chaplain, Father Ivo Thomas, and was received into the Catholic Church in February 1944. He took the confirmation name Anthony, after St Anthony of Padua, and used it as a middle name thereafter, in addition to the middle name from his birth certificate. A child of irreligiously Anglican parents, and himself a declared atheist at fourteen, the deep religious faith of his conversion remained central throughout the rest of his life, though not always without struggle (Dummett (2007), pp. 5–6). Dummett was then sent for six weeks of army basic training, after which he went on a six-month course in Bedford to be taught to read and translate Japanese, and then to the Wireless Experimental Centre outside Delhi to translate intercepted Japanese messages. When the war with Japan ended, he was sent to Malaya as part of Field Security. He wrote that ‘it must have been in Malaya that a passionate hatred of racism was first born in me. I learned of the means by which the British masters of pre-war colonial Malaya had maintained and acted out the myth of white racial superiority’ (Dummett (2007), p. 8), though Michael Screech, who was on the Bedford course and at the Wireless Centre with Dummett, remembered him expressing anger about racism already at that time. Dummett was by then a heavy smoker, as he remained throughout his life, and Michael Screech recalled that tapping the end of a cigarette many times before lighting it came to be called ‘dummetting’ by those around him. Dummett was demobilized in 1947, and went up to Christ Church. He felt that after four years in the army he had forgotten much of the history he had learned, and decided instead to read for the Honour School of Philosophy, Politics and Economics. He was ‘soon captivated by philosophy’ (Dummett (2007), p. 9). He chose to do a paper established by John Austin called ‘The origins of Modern Epistemology’, available for the first time in the year he took it. Candidates were expected to study four texts from a list of seven, one of which was Frege’s Foundations of Arithmetic, newly translated from the German by Austin for this purpose. Dummett later wrote of Frege’s Grundlagen der Arithmetik, ‘I thought, and still think, that it was the most brilliant piece of philosophical writing of its length ever penned’ (Dummett (2007), p. 9). Dummett’s ensuing work on Frege transformed understanding of Frege’s logic and philosophy.

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Early career After taking first class honours in PPE finals in the summer of 1950, Dummett was appointed to a one-year Assistant Lectureship in Philosophy at the University of Birmingham. That October he sat the fellowship examination at All Souls College, Oxford, and was elected, with immediate effect, but nonetheless fulfilled his commitment to Birmingham, rushing back to Oxford during term to pernoctate as required by All Souls. The first project Dummett set himself as a Prize Fellow at All Souls was to read all the published work of Frege, most of which at that time had been neither translated nor republished. He also visited the Frege archive in Münster to study what survived of Frege’s unpublished work. Despite his passion for Frege, Dummett began his philosophical career thinking of himself as a follower of Wittgenstein, arising from the impact of the arrival in Oxford during his last year as an undergraduate of typescripts of The Blue and Brown Books and of notes of Wittgenstein’s classes on philosophy of mathematics, and also from his philosophical contact and ensuing friendship with Elizabeth Anscombe, then a Fellow of Somerville College, to whom he had been sent for tutorials in that year. By 1960 he no longer considered himself a Wittgensteinian (Preface to Dummett (1978), p. xii). Late in life he wrote, ‘I should like to come to terms with Wittgenstein: I am sure I have not yet’ (Dummett (2007a), p, 54). On 31 December 1951, in his second year as a Prize Fellow, Dummett married Ann Chesney (1930–2012), who had taken finals in History from Somerville College that year. She was the daughter of the actor Arthur William Chesney. Fifty years later Dummett wrote of Ann, ‘she has been my constant support and delight throughout my life’ (Dummett (2007), p. 10). They had seven children, four sons and three daughters, of whom two, a son and daughter, died in infancy. Early in his All Souls Fellowship, Dummett had the idea of doing a second B.A. in Mathematics, but the Warden of All Souls refused permission, on the grounds that it would disgrace the College if he failed to obtain a First, and he settled for some tutorials with John Hammersley, an applied mathematician in Oxford, later a fellow of Trinity College, Oxford. Dummett was awarded a Harkness Fellowship to spend the academic year 1955–56 at the University of California, Berkeley, studying logic and mathematics. Ann and their two young children managed to join him there for seven months, on his very limited stipend. He learned a great deal from Leon Henkin, Raphael Robinson, John Myhill, Paul Halmos, and others (but not Alfred Tarski, who was away that year). He also at that time came to know Donald Davidson at Stanford, and they

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remained firm friends and philosophical interlocutors to the end of Davidson’s life.

Mid-career and anti-racism While in Berkeley Dummett became closely involved with the American civil rights movement. He noted later that ‘at that time the United States was the most racist country in the world after South Africa’ (Dummett (2007), p. 11). He and Ann joined the National Association for the Advancement of Colored People and attended a rally in San Francisco addressed by Dr Martin Luther King. Part of the duty of a Harkness Fellow was to travel in the United States, and after Ann and the children returned to England, Dummett devoted himself to visiting black Americans during that summer of 1956. He travelled to Montgomery, Alabama, where the boycott organised by Martin Luther King to force, by peaceful means, repeal of the law segregating blacks on the city’s bus system was in progress. He there met Dr King, whom he admired greatly. In 1957 Dummett was elected to a further seven years as a Fellow of All Souls. In the next year he spent one term on his own at the University of Ghana, lecturing on the philosophy of time, one of his developing philosophical interests. While in Ghana he was offered a position in the Philosophy Department at Berkeley, which he accepted after consulting Ann by post, but when reunited they decided they did not want their children to grow up ‘in an environment alien to us which we did not truly understand’ (Dummett (2007), p, 16), and Dummett withdrew his acceptance. In 1959 Dummett published his paper ‘Truth’ in the Proceedings of the Aristotelian Society, a seminal work and his single most important paper. It contained within it the seeds of a great deal of his later philosophy. It adumbrated the opposition between realism and anti-realism, as Dummett characterised these positions, in terms of bivalence and the law of excluded middle, and surveyed a variety of contexts, both mathematical and non-mathematical, in which this opposition arises. A connection between these considerations and Wittgenstein’s dictum that meaning is use was sketched. This heady mixture of ideas took decades to explore, and led to new understanding of the nature of logic and meaning. Dummett’s notion of anti-realism in ‘Truth’ was inspired by the mathematical Intuitionism of L.E.J. Brouwer (as Dummett explained, ‘What I have done here is to transfer to ordinary statements what the intuitionists say about mathematical statements’ (Dummett (1959), p. 160)). Students of Dummett’s philosophy have sometimes been puzzled by his espousal of both

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Frege, a fervent realist, and Brouwer, a passionate anti-realist. Dummett explains this seeming cognitive dissonance as follows: Remarking that he felt ‘strongly drawn’ to intuitionist philosophy of mathematics (Dummett (2007), p. 15), he notes that ‘This may seem inconsistent with my devotion to Frege, who was a determined realist’. Dummett resolves this apparent inconsistency by noting that what attracted him to Frege was not his realism ‘for which I thought, he never really argued, but which he simply took for granted, but the clarity of his thought: much of his thinking was perfectly compatible with a constructive view of mathematics.’ (loc. cit.) Further, Reflecting on my rationale for intuitionistic mathematics as an exemplification of Wittgenstein’s dictum about meaning as use, it struck me that the metaphysical conceptions accompanying both Platonist and constructive conceptions of mathematics were not the foundations of those conceptions: they were merely pictures illustrating them. One could not argue from the metaphysical pictures, because there was no independent ground for accepting one or the other. The core of the different conceptions lay in the divergent views of what the meanings of mathematical statements must consist in: to adopt one or the other view was to make one or the other picture natural. (op. cit., p. 17)

Another clash between Frege and Brouwer is Frege’s fierce opposition to psychologism, which Brouwer’s Intutionism exemplifies. Dummett resolved this tension by recasting intuitionist mathematics and logic on a completely antipsychologistic basis, in Dummett (1977). The guiding principle of Dummett’s anti-realist programme was that ‘the concept of truth-values determined by reality independently of us should be abandoned. The notion of a statement’s being true should be replaced by that of its being shown to be true: the content of a statement consists in what is needed to show it to be true.’ (Dummett (2007), p. 18) Dummett’s subsequent interest ‘concentrated ever more on the question whether global anti-realism, modelled after intuitionist mathematics, was sustainable.’ (loc. cit.) In his 1969 paper, ‘The reality of the past’ he accepted that ‘There are a number of reasons for doubting whether global anti-realism is coherent’ (p. 250). We naturally think that what is true at a certain time remains true, regardless of whether or not the evidence that showed it to be true at that time is later irretrievably lost. Dummett early and late considered that statements about the past were for him the most serious impediment to anti-realism as a comprehensive philosophy; ‘I think that without doubt the thorniest problem for one who wishes to transfer something resembling the intuitionist account of the meanings of mathematical statements to the whole of discourse is what account he can give of the meanings of tensed statements’ (op. cit. pp. 250–251). He returned to this problem again and again.

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In early pursuit of his anti-realism versus realism project, Dummett submitted to Oxford University Press, around 1958, the manuscript of a book on deviations from the law of excluded middle, which was accepted by John Austin, ‘but he told me that the style needed improvement. Though this annoyed me at the time, I am very glad he did: while trying to revise it stylistically, I became dissatisfied with it and did not resubmit it. I should be ashamed of it now if it had been published.’ (Dummett (2007), pp. 14–15) In 1962 Dummett applied for and was appointed to the Oxford University Readership in Philosophy of Mathematics, in succession to Hao Wang (who had succeeded Friedrich Waismann, the first incumbent), which he held in conjunction with a Fellowship of All Souls. Between 1960 and 1966 he was regularly a visiting professor in the Philosophy Department at Stanford for the summer quarter (in part to earn money so that he could take his family on holiday). During one of those visiting appointments, in 1964, he gave a course of lectures as a preliminary version of a book he hoped to write surveying every variety of realism or denial of realism, but when he returned to Oxford that summer he and Ann decided ‘that the time had come for organised resistance to the swelling racism in England’ (Dummett (2007), p. 18), and he put this project on hold, along with a book on Frege he had been planning. For the next four years, Dummett devoted every moment he could spare to the fight against racism, while keeping up with his heavy teaching commitments. He and Ann were deeply engaged both in organizational activity to combat racism as a trend in British government and society, and in work on behalf of individuals threatened by racist policies and attitudes. This latter included intervening to stop persons of colour, attempting to enter the country, from being deported straight back to the country from which they were fleeing. A telephone call at any hour of the day or night would alert Dummett to such a case, and transform him from philosopher to activist, telephoning the Chief Immigration Officer to obtain a stay of immediate deportation, then dashing to the airport to argue the case, often successfully. Dummett’s organizational work against racism included a role in founding the Oxford Committee for Racial Integration, participation in the turbulent and ultimately self-destructive Campaign Against Racial Discrimination, and playing a key role in founding the Joint Council for the Welfare of Immigrants. He chaired its inaugural meeting at the Dominion Theatre in Southall in September 1967. The JCWI continues to do important work to the present day, and Dummett maintained his association with it to end of his life. Its website contains an obituary of Dummett by Habib Rahman, its then Chief Executive, who remembered Dummett as ‘an extremely compassionate person and a fierce opponent of racism in our society’, and de-

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clared, ‘He will be sorely missed and fondly remembered by us for his uncompromising struggle for equal rights for migrants and refugees in Britain.’ https://www.jcwi.org.uk/2012/01/06/sir-michael-dummett. Ann is also memorialized by an obituary on the JCWI website. Dummett later described this time as ‘the most exhausting period of my life’. (Dummett (2007), p. 21) One could say that he had anti-racism in his genes, as seen by the fact (discovered, to his great delight, by his daughter Suzie) that Sir John Eardley Eardley-Wilmot, grandfather of his mother’s father, campaigned for the abolition of slavery, and is among those depicted in the painting by Benjamin Robert Haydon, hanging in the National Portrait Gallery, of the Anti-Slavery Society convention of 1840. During this period of maximum commitment to anti-racism, Dummett not only continued to be a generous and inspiring teacher but also played a key role in establishing mathematical logic within Oxford University. This resulted in the creation in 1965 of a University Lectureship in Mathematical Logic associated with a Fellowship at All Souls, to which John Crossley was appointed, and, together with Crossley, Dummett then established a new Oxford undergraduate degree in Mathematics and Philosophy, with philosophy of mathematics, and mathematical logic as bridge subjects. A significant number of the best graduates in philosophy from Oxford since that time have come from this course. Dummett also later was instrumental in the establishment, in 1971, of a Professorship in Mathematical Logic, to which Dana Scott was appointed. These two posts were within the Philosophy Sub-Faculty, but came to be based in the Mathematical Institute, where they now constitute the core of a world renowned group in mathematical logic. In 1968 Dummett was elected a Fellow of the British Academy. He resigned his Fellowship in 1984, in protest at the British Academy’s failure, as he saw it, to stand up to the Thatcher government’s attack on British universities by its cuts to spending on higher education and research, as he explained in a letter published in The Guardian on 19 June 1984, p. 14 under the headline ‘When an academy leaves academics in the lurch’. He gave as an additional reason for his resignation that the British Academy ‘is run in a thoroughly undemocratic fashion’. In 1995 he accepted re-election to the British Academy as a Senior Fellow.

Later career The period in which Dummett gave the fight against racism highest priority among all his commitments came to an end in 1968. As he explained, ‘by 1968 Britain had become irretrievably identified by the black people living here as a

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racist society …. The alienation of racial minorities is now so great that a white ally in the struggle can, except in special circumstances, play only the most minor ancillary part. It was only at the stage at which … I felt that I no longer had any very significant contribution to make that I thought myself justified in returning to writing about more abstract matters of much less importance to anyone’s happiness or future’. Dummett offered this account of his return to writing philosophy in the Preface to his first book, Frege: Philosophy of Language, published in 1973, to great critical acclaim. Dummett went on to publish eight further books in philosophy and three volumes of essays. In the same year as Dummett published his first book, his wife Ann published A Portrait of English Racism, about which Dummett later said, ‘I would rather have written that book than any of the many I have written’ (Dummett (2007), 24). In 1974 Dummett applied for and was elected to a Senior Research Fellowship at All Souls, and resigned as Reader in the Philosophy of Mathematics, in order to have more time for research, but also to be free to work more broadly than specifically in the philosophy of mathematics. In this period he had embarked on a series of major philosophical papers pursuing lines of research adumbrated in ‘Truth’, beginning in 1973 with his British Academy lecture ‘The justification of deduction’, and continuing with ‘The philosophical basis of intuitionistic logic’ (given as a lecture in 1973, published in 1975), ‘What is a theory of meaning?’ (lecture in 1974, published in 1975), ‘What is a theory of meaning? (II)’ (1976), and his William James Lectures at Harvard in 1976, ‘The logical basis of metaphysics’ (published in an expanded and revised form as a book in 1991). In 1977 Dummett published Elements of Intuitionism, a remarkable accomplishment mathematically, philosophically, and pedagogically. He there established that intuitionist mathematics and logic can indeed be cast in the form of Dummettian anti-realism, as foreshadowed in ‘Truth’—a completely different basis from the psychologism by which Brouwer had argued for intuitionist mathematics. In addition to the originality and clarity of his exposition of intuitionist mathematics and logic in that book, Dummett also established intuitionistically the completeness of negation-free intuitionist logic, a best possible result in light of the result by Gödel and Kreisel that the completeness of Heyting’s predicate calculus intuitionistically implies Markov’s Principle, which is not intuitionistically valid. In 1979 Dummett was elected to the Wykeham Professorship of Logic, and moved from All Souls, which had been his academic home for twenty-nine years, to New College, to which the Wykeham chair is attached. The question in that election was not whether Dummett would be elected but whether he would

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accept, which entailed giving up his Research Fellowship at All Souls, with its very limited formal demands, which he had held for five years and which could have continued for thirteen years more, until retirement. His doing so was a selfless act of loyalty to Oxford Philosophy. Almost immediately he was called upon to supervise substantially more than fifteen graduate students at a time. This was in part because professors had a statutory obligation to do a lot of graduate supervision, but mostly because his publications were now setting the agenda for important philosophical developments, and graduates flocked to Oxford to study with him. In 1982 Dummett was awarded a Humboldt-Stiftung Research Prize which he used for four months at the University of Münster, working on Frege. In 1985 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. He spent the academic year 1988–89 in Stanford as a Fellow of the Center for Advanced Study in the Behavioral Sciences, during which he finished two major books begun earlier, Frege: Philosophy of Mathematics, and The Logical Basis of Metaphysics (both published in 1991).

Final years Michael Dummett retired from Oxford in 1992, at the compulsory age of sixtyseven. He gave many lectures in retirement, including the Gifford Lectures at St Andrews University in 1997, which he published as Thought and Reality in 2006. His aim in those lectures and the ensuing book was ‘to describe the conception of the world—of reality—that would be proper to one who accepted the version of anti-realism that has been associated with me, namely a generalisation to all language of the intuitionist understanding of mathematical language, which I have never for long more than provisionally accepted. It turned out very Berkeleyan, with a strong asymmetry between past and future, something to which I am temperamentally averse.’ (Dummett (2007), p. 31) In 2003 he gave the John Dewey Lectures at Columbia University, published in 2004 as Truth and the Past, in which he continued the struggle with which he had been engaged ever since his paper ‘Truth’ between the pull toward a global anti-realism and a countervailing pull toward a realist view on statements about the past, as he had explored in ‘The reality of the past’, in 1969, where his final sentence had been, ‘Of course, like everyone else, I feel a strong undertow towards the realist view: but then, there are certain errors of thought to which the human mind seems naturally prone.’ He now attempted again to find a tenable antirealism for statements about the past. His assessment of his earlier attempt in (1969)

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was that ‘the conclusion that I reached was the most disappointing possible. Antirealism about the past was not incoherent; but it was not believable, either. I have been perplexed by this matter ever since.’ (p. 45). He characterizes the position he attains in Truth and the Past as follows: An account along purely justificationist lines would embody an untenable antirealist concept of the past. The account that has here been offered of our understanding of the past tense is still justificationist in character, but the theory has been revised in a realist direction. Adopting it does not demand of the justificationist that he repudiate his general principles: he will still hold that a statement about the past can be true only in virtue of an actual or possible direct verification of it. But he will take a more realist attitude to whether such a direct verification was or could have been carried out. In doing so, his theory of meaning has been modified so as to approach realism more closely.

As Dummett notes, ‘The position I have adopted in this book is greatly at variance with those I express in my [at that time] not yet published Gifford Lectures of a few years ago. In those, I did not embrace antirealism about the past.’ (Dummett (2004), p. x) Dummett received many honours, including five honorary degrees, the Lakatos Prize, for his book Frege: Philosophy of Mathematics, in 1994, and the Rolf Schock Prize for Logic and Philosophy in 1995. In 1999 he was knighted ‘for services to Philosophy and to Racial Justice’ (London Gazette, 30 Dec 1998). In 2010 he was awarded the Lauener Prize for Analytical Philosophy. Dummett continued to attend the philosophy of mathematics seminar in Oxford until the spring of 2010. He died on 27 December 2011, at the age of 86, and was buried in Wolvercote Cemetery on 17 January 2012 after a Requiem Mass at St Aloysius Church. Ann died on 7 February 2012, and they were commemorated together in a memorial service in New College Chapel on 2 June 2012.

Dummett’s other interests, and his place in philosophy Dummett played an important role, as had been described, with his wife Ann, in combatting racism in Britain, for which he wrote or co-wrote five pamphlets and an article. He published two articles on nuclear warfare and deterrence. He had a strong interest in voting systems, and published significant work on this topic, both theoretical and practical, in the form of two books and three articles, and had occasion to make use of his own expertise when he presided, as Sub-

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Warden, over the election of a new Warden of All Souls. He published twentyone articles or pamphlets arising from his deep commitment to Catholicism. He pursued a passionate side interest in the games played with tarot cards, and in the cards themselves, about which he published six books and nearly forty articles. He wrote a practical book on English grammar and style, motivated by his experience as an examiner shortly before he retired. He loved the blues, and was proud to have heard Billie Holiday sing, in a small bar on the South Side of Chicago in 1956 (Dummett (2007), p. 13). Michael Dummett was one of the most important analytic philosophers during the second half of the twentieth century. He published his philosophical ideas in nine books and nearly a hundred articles, including one book and three articles in mathematical logic. Dummett’s philosophical publications have given rise to a vast literature of responses, including the ultimate accolade, a volume devoted to The Philosophy of Michael Dummett in the Library of Living Philosophers.

Publications by Michael Dummett cited in this biographical sketch Auxier, Randall E. and Lewis Edwin Hahn (eds) (2007), The Philosophy of Michael Dummett, The Library of Living Philosophers Volume 31, Open Court Publishing Company, Chicago and La Salle, Illinois. Dummett, Michael (1959), ‘Truth’, Proceedings of the Aristotelian Society 59, pp. 141–162. Dummett, Michael (1969) ‘The reality of the past’, Proceedings of the Aristotelian Society 69, pp. 239–258. Dummett, Michael (1973), Frege: Philosophy of Language, Duckworth, London. Dummett, Michael (1973a), ‘The justification of deduction’, British Academy, 1973; republished in Proceedings of the British Academy 59 (1975), pp. 201–232. Dummett, Michael (1975), ‘The philosophical basis of intuitionistic logic’, H.E. Rose and J.C. Shepherdson (eds), Logic Colloquium 73, North Holland, Amsterdam, pp. 5–40. Dummett, Michael (1975a), ‘What is a theory of meaning?’ in Samuel Guttenplan (ed,), Mind and Language, Oxford University Press, pp. 97–138. Dummett, Michael (1976) ‘What is a theory of meaning? (II)’ in Gareth Evans and John McDowell (eds,) Truth and Meaning, Oxford University Press, pp. 67–127. Dummett, Michael (1977), Elements of Intuitionism, Oxford University Press; second edition 2000. Dummett, Michael (1978), Truth and Other Enigmas, Duckworth, London. Dummett, Michael (1991), Frege: Philosophy of Mathematics, Duckworth, London. Dummett, Michael (1991a), The Logical Basis of Metaphysics, Gerald Duckworth & Co, London. Dummett, Michael (2004), Truth and the Past, Columbia University Press. Dummett, Michael (2006), Thought and Reality, Oxford University Press.

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Dummett, Michael (2007), ‘Intellectual autobiography’, Randall E. Auxier and Lewis Edwin Hahn (eds), op cit., pp. 3–32. Dummett, Michael (2007a), ‘Reply to Brian McGuinness’, in Randall E. Auxier and Lewis Hahn (eds), op. cit., pp. 51–54.

Ian Rumfitt

Dummett Laudatio On at least four occasions in his life, Gottlob Frege began to compose what his contemporaries would have called a Logik – not a textbook of formal logic, but a systematic overview of his central contentions in the philosophy of logic and related areas of metaphysics and epistemology. The first sentence of the earliest surviving Logik is: ‘Das Ziel des wissenschaftlichen Strebens ist Wahrheit’. Our honorand this evening, Professor Sir Michael Dummett, is famous throughout the world for his work on Frege and on truth. But it is the other part of Frege’s sentence that is the key to the man. For the mainspring of Michael’s life is, exactly, das wissenschaftliche Streben; he is, par excellence, one who strives after knowledge, one who sets himself to discover the truth. Perhaps I can best convey what I mean by this by recollecting my first impressions of him. These date from 1983 or 1984 – a time of austerity in British universities in general (plus ça change), and in Oxford in particular, that had left the philosophical professoriate there severely depleted. Charles Taylor had returned to Montreal to become, if not a philosopher king, then at least a philosopher kingmaker; Dana Scott had retreated to Pittsburgh to take refuge among the computer scientists; and Richard Hare had perplexed everyone by opting for early retirement in the swamps of Northern Florida. That left Michael and his much-missed colleague, Sir Peter Strawson, to do what they could as the only professors of contemporary philosophy remaining in Oxford. What each of them did was superb, but it was superb in very different ways. Peter Strawson’s lectures were – to borrow an epithet from Plutarch – Apollonian. Formally gowned in the splendour of the Examination Schools, Peter would take a topic from pretty well any area of philosophy: truth, meaning, conditionals, scepticism about the external world, Kant’s paralogisms. He would address the topic by reading out a text in which perfectly balanced sentences perfectly expressed judicious assessments of the key theses. He would deftly steer the argument, with seeming inevitability, to some invariably sensible conclusion. One would leave the Schools at the end of the academic hour with the euphoric illusion that, whilst there were no doubt certain further points

|| All Souls College, Oxford University Text of a laudatio delivered in honour of Sir Michael Dummett on Wednesday 26 May 2010 at the Haus der Universität in Berne. DOI 10/1515-9783110459135-002

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of detail to be filled in, one had heard all that really needed to be said on the topic that had been addressed. Michael’s classes were entirely different. They took place in the comparative informality of the Philosophy Lecture Room in 10 Merton Street, a scuffed, scruffy space which then contained, in addition to a battered old table and some backbreakingly uncomfortable chairs, a large number of metal ashtrays on three-foot stilts. The lecture would be preceded by Michael’s placing one of these ashtrays on each side of the double blackboard. During the lecture proper, he would pace between these ashtrays, occasionally writing something on the blackboard, but always ensuring that the column of ash protruding from his cigarette holder made it into the tray rather than onto the floor. The effect was mesmerizing, but what of the content? Unlike Peter Strawson’s lectures, Michael’s would resist summary. There would, naturally, be a topic, or at least a point of departure – often a question left over from the previous class. But what we heard and saw was a philosopher at the height of his powers wrestling publicly with a philosophical problem that he precisely did not regard as settled. Of course, the lectures were prepared: indeed, only a thinker who has pondered a philosophical question intently would risk wrestling with it in public. But the thinking was being done afresh; the wrestling was for real, not for show, for the conclusion was not predetermined. What we were being shown, then, was exactly das wissenschaftliche Streben, the quest for philosophical understanding. Pursuing Plutarch’s dichotomy and calling these classes Dionysian risks giving quite the wrong impression: it would be hard to think of events that were less bacchanalian. But at a deeper level the parallel is apt. Whatever else they may have been, the Dionysian Mysteries were rites of fertility, and what we were being allowed to glimpse was precisely a process through which vital new responses to philosophical questions might break out of the soil of perplexity and grow. How, though, had Michael reached the point of being able to give philosophical classes at this level? In the words of one of his favourite poets, ‘a shilling life will give you all the facts’. Equally, though, in the words of one of his favourite philosophers, all the facts constitute the entire world. So let us renounce any attempt at completeness and settle for a thumbnail sketch. Michael Dummett was born in London on 27 June 1925; his father was a merchant in silk and rayon; his mother bred Alsatian dogs. His intellectual gifts became evident early on, and at the age of thirteen he won a scholarship to Winchester College, the most cerebral of the great schools of England. There, after a brief flirtation with the natural sciences, he focused on history, and it

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was in that subject that he won a scholarship to Christ Church, Oxford, in 1943. He was, however, unable to take up his place immediately, for, having turned eighteen, he was called up for military service and joined the Royal Artillery. Michael’s four years in the army were formative in a number of important respects. The Artillery sent him first to Edinburgh to take courses in (inter alia) basic mechanics, but whilst the course failed to teach him how to repair a jeep, he took advantage of his proximity to the Catholic Chaplaincy at Edinburgh University to receive instruction in the faith, and he was received into the Church in February 1944. He remains a Catholic to this day. Following his basic training, Michael was transferred to the Intelligence Corps and, having been taught Japanese, he was sent out to India to translate intercepted wireless messages, emanating chiefly from the fighting in Burma. After nine months of this exacting work, Japan surrendered, whereupon Michael was sent further east, to Malaya, to join a Field Security section in the Military Administration that had assumed responsibility for the government of that country following the Japanese retreat. Michael much enjoyed this part of his service. He loved the way that the different peoples of the peninsula – Malays, Chinese, and Indian Tamils – cooperated and intermingled. Even fifty years later, one could meet fellow members of the British Military Administration who took pleasure in recalling Michael’s sartorial contribution to the more general synthesis, with his army jacket worn over a local sarong. In April 1946, though, the BMA gave way to the Malayan Union, a temporary restoration of the pre-war colonial government. Whilst sympathetic to the traumas of the colonists – many of whom had had suffered appallingly in Japanese jails – Michael was nevertheless repelled by their assumption of racial superiority over the peoples whom they were governing. The hatred of racism which was born then has never left him, and was to have important consequences. A further, oblique, consequence of his war service was Michael’s turn to philosophy. On being demobilized in September 1947, he was keen to take up his Christ Church scholarship at the start of the new academic year the following month. But he felt he had forgotten so much of his history that he could not excel in that subject and so, despite having read little in any of its three elements, he switched to Philosophy, Politics, and Economics as soon as he arrived at Oxford. He was quickly captivated by philosophy, particularly as a result of the teaching of Elizabeth Anscombe, who was then a young research fellow at Somerville College and who became a lifelong friend. It was through her that his engagement with Wittgenstein’s ideas began. His First Class in PPE Finals in the summer of 1950 brought him the offer of an assistant lectureship at the University of Birmingham; but his election to a Prize Fellowship of All Souls’ that

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autumn began a twenty-nine year association with that college and, after discharging his duties in Birmingham for a year, he settled in Oxford. He married Ann, who is with us this evening, and he began to develop the philosophical ideas for which he is known. What is remarkable about the stream of papers that emerged from his pen in the 1950s is how fully the mature philosopher is present in them. To be sure, they are, stylistically, of their time, and hence rather different from Michael’s later papers. His essays of the fifties are short and highly compressed: in those days, readers were trusted to pick up hints about how theses might further be elaborated. They lack footnotes: readers were expected to know the writings to which an author alluded. But they are, quite unmistakeably, essays by Dummett. Each starts with an interesting philosophical question – for example, whether an effect can precede its cause (Dummett 1954); whether it is possible to construct the physical world from qualia (Dummett 1957) – on which Michael brings to bear a powerful and imaginative philosophical and logical intelligence. And yet, just as in his classes thirty years later, there is no sense of the analysis being forced to support a predetermined conclusion. Rather, Michael follows the argument wherever it leads. He reaches conclusions, but never pretends that those conclusions are the last word on the topic. To the contrary, he is aware that his conclusions raise as many questions as they answer and must for that reason be provisional. Although Michael’s writings became longer and more ambitious, he has always kept this exploratory spirit alive in his philosophizing. It is hard to keep it alive, perhaps especially when one’s work receives as much attention as Michael’s has. There is a perennial temptation to build a system around which the wagons can circle, at the cost of freshness and vitality. But Michael has resisted the temptation: even the books and papers he has published in the past decade give his readers the rare pleasure of starting out upon an intellectual journey without knowing where they are going to end up. Everything that Michael published in the 1950s repays study. In 1959 alone, there appeared two important formal papers on modal logic (one co-authored with John Lemmon) and the essay ‘Wittgenstein’s Philosophy of Mathematics’ which for the first time laid out clearly the powerful and disturbing argument that underlies Wittgenstein’s Remarks on the Foundations of Mathematics (see Dummett 1959a, Dummett and Lemmon 1959, and Dummett 1959b). But if one had to choose a single paper from that decade, it would be another essay published in that exceptionally fertile year, the one entitled ‘Truth’ (Dummett 1959c). This stands alongside Quine’s ‘Truth by Convention’ as a signal example of a short essay (as originally published, it is only twenty pages long) in which a whole philosophy is latent. Indeed, there are perhaps several subtly different

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philosophies struggling to emerge from it, so that the reader is left with the demanding task of trying to decide which of the possible developments is best. The central argument of ‘Truth’ – as completed in his later writings – is so important for the future development of Michael’s thinking that it may be helpful to sketch it here. The notion of truth, Michael claims, is tied to that of understanding, for to understand a statement is to know under what conditions it would be true. In general, however, this knowledge must be implicit, and Michael holds that an ascription of implicit knowledge to a speaker is vacuous unless it amounts to attributing to him a disposition, the possession of which may be fully manifest in his behaviour. Now knowledge of the conditions under which a statement is true cannot always be cashed out as a fully manifest disposition. At least, this is so if our conception of truth is the usual realist one, whereby a statement may be true in circumstances where no one can recognize that it is true. For the only plausible candidate to be a disposition, possession of which amounts to knowing a statement’s truth conditions, is the disposition to assent to it in circumstances where one recognizes that it is true. And if a statement may be true in circumstances where no one can recognize that it is true, then a speaker’s implicit knowledge that a statement is true in, and only in, certain conditions will not be fully manifest in his disposition to assent to it in the circumstances in which it may be recognized as true. On the basis of this argument, Michael concludes that we must reject the usual realist notion of truth. A statement cannot be true except in circumstances in which it is in principle possible for someone who understands it to recognize that it is true. Similarly, a statement can be false only if it is in principle possible for someone to know that it is false. We thus reach a substantial ‘epistemic constraint’ on the notions of truth and falsity, a constraint which constitutes a form of antirealism. The constraint has implications for logic itself. Of an arbitrary statement, it cannot be assumed that it is in principle possible for someone to know that it is true, or to know that it is false. Given the epistemic constraint, then, it cannot be assumed that an arbitrary statement is either true or false. The standard semantics for classical logic, though, relies precisely upon this assumption, the Principle of Bivalence. So, insofar as classical logic needs classical semantics for its foundation, or otherwise entails the correctness of Bivalence, classical logic must itself be revised. Like all the best philosophical arguments, this one is clear in outline, but bewilderingly complex in its details and its ramifications. As Michael always acknowledged, its roots lie in Wittgenstein’s idea that meaning is use, but it replaces that somewhat nebulous formula with a far more precise philosophical thesis. And the idea that these considerations might in the end force one to

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revise classical logic is radically unWittgensteinian; it is a striking example of Michael’s willingness to follow the argument wherever it leads. Fifty years on, it remains a hotly debated question whether the argument of ‘Truth’ is cogent. In any case, Michael next set himself to investigate whether the stark anti-realism of ‘Truth’ can be defended across the board. As the argument of that paper shows, a touchstone for realism – with respect to a certain field of inquiry – will be whether statements in the field conform to the Principle of Bivalence. In important papers that appeared in the 1960s, Michael began to explore whether this condition is met by mathematical statements (Dummett 1963) and by statements about the past (Dummett 1969). One result of this path-breaking work – which promised a fresh approach to hoary debates about realism in various areas of inquiry – was Michael’s appointment (in 1962) as Reader in the Philosophy of Mathematics at Oxford, in succession to Friedrich Waismann and Hao Wang. The line of inquiry that Michael had indicated captured the imagination of some of the brightest young philosophers of the 1960s, a couple of whom are with us this evening. But, having set the ball rolling, Michael himself rather disappeared from philosophical view. With the exception of a short encyclopaedia article that he had completed earlier, and a paper read to the Aristotelian Society in 1969, he published nothing in the years from 1965 to 1972. The reason was that he was devoting all the time he had spare from teaching to combating the racism that was then disfiguring British life. This work began locally, with the foundation of the Oxford Committee for Racial Integration, but this body soon affiliated with the Campaign Against Racial Discrimination, and Michael came to play a national role as a member of the latter organization’s council. His work against racism was more direct, though, and far more taxing, than this description would imply. Immigration into Britain in the 1960s was not regulated by entry clearance certificates. Rather, would-be immigrants simply arrived at an airport and were put back on the next plane if the duty immigration officer refused them entry. It was, however, possible to intervene on a refused immigrant’s behalf if one could do so before he was put on the plane. Knowing this, Michael set up a network of informants at Heathrow who were encouraged to telephone him at any hour of the day or night to tell him of someone refused entry who had no one to speak for him. Michael would then drive to the airport and make representations on the refused person’s behalf. These representations were often availing, but it is little wonder that Michael has described these years as ‘the most exhausting period of my life’. The pressure on him was eventually alleviated by the Joint Council for the Welfare of Immigrants, which Michael helped to found in 1967. He chaired its

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inaugural meeting and remained an active trustee until very recently. The Council is still going strong, and Michael regards it as the most significant legacy of his anti-racist work. An important Nebenarbeit started in this period. During a holiday in France, one of Michael and Ann’s sons acquired a pack of cards which included the rules of a form of tarot. Michael quickly became fascinated by the game, but soon discovered that (in England at least) little was known about its history. With his characteristic desire to get to the bottom of things, he set about remedying the deficiency, spending long hours in the Bodleian and other libraries, and corresponding with scholars abroad. He has written that his interest in the history of tarot provided a vital diversion when he was primarily occupied with the fight against racism. But his work in the field has gone from strength to strength and has led to several path-breaking publications (see especially Dummett 1980, Dummett and Mann 1980, Dummett and McLeod 2004). By the early 1970s, the Joint Council for the Welfare of Immigrants was firmly enough established that Michael felt able to return to writing philosophy, and in 1973 his first book appeared, Frege: Philosophy of Language (Dummett 1973). The book not only set forth a coherent and systematic interpretation of Frege’s ideas in the philosophy of language and philosophy of logic; it also brought out the continuing relevance of those ideas by relating them to key theses of the leading contemporary American philosophers of languages – Quine, Davidson, and Kripke. The book was immediately recognized as a masterpiece and it is a significant literary, as well as philosophical, achievement: Michael’s rolling periods, reminiscent of Milton’s prose, carry the reader through its seven hundred pages of intricate philosophical analysis. I shall not try to summarize its argument tonight, but I would urge those who have not read it for a while to return to it, for it contains riches which many have forgotten but which could help advance contemporary discussions. Those engaged in the current debate about relativism, for example, might benefit from studying pages 392–400. In 1974, Michael was elected to a Senior Research Fellowship of his college, and the extra time for thinking and writing that this post afforded allowed him to return to a question that had been raised by the argument of ‘Truth’ – namely, how someone might justify reasoning in accordance with the logical rules that he follows. He lectured on ‘The Justification of Deduction’ (Dummett 1975) to the British Academy, a body of which he had been a Fellow since 1968, and the topic was the focus of the William James Lectures he delivered at Harvard in 1976. A book of these lectures promised to the Harvard University Press, however, was again delayed. In 1979, Michael vacated his Research Fellowship

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at All Souls’ in order to take up the Wykeham Chair of Logic at Oxford. His accession coincided with the election of a government which was set on retrenching university funding, so Michael found himself not only shouldering an increasingly heavy burden of graduate supervision, but also called upon to do what he could to protect individuals and departments left vulnerable in the new climate. There were, however, real compensations. Michael had been instrumental in founding, in 1969, the Joint Honour School of Mathematics and Philosophy at Oxford, and much of his teaching in the 1970s had been directed to ensuring that the fledgling degree survived its infancy. As professor, he had the pleasure of overseeing the Joint School’s growth into adolescence and young adulthood, and last year he was guest of honour at the celebrations to mark its fortieth anniversary. Nothing has done more than the creation of this Joint School to raise the level of logical expertise in British philosophy; many of those at this week’s symposium are either its graduates or current or former teachers in it. Despite the burdens of his chair, Michael continued to publish. In the early 1980s he defended his interpretation of Frege against the criticism it had inevitably attracted (Dummett 1981). We need not dwell on this: his interpretation has endured, whilst the alternatives propounded by those critics have not. But none of them could accuse Michael of failing to respond to their points. I well remember a series of seminars in Oxford in the 1980s where he responded at length – his forbearance only occasionally tinged with exasperation – to his two main local critics, whose chief complaint (it is now amusing to recall) was that Michael had made Frege appear to be far more interesting philosophically than he really was. The 1980s also saw another extra-curricular project come to fruition. In tandem with a former student, Robin Farquharson, Michael had around 1960 made pioneering advances in the theory of voting (Dummett and Farquharson 1961). A full-length treatise on Voting Procedures appeared in 1984, to be followed thirteen years by a simplified exposition which, he hoped, politicians might be able to understand (Dummett 1984, 1997). With electoral reform now a real possibility in Britain, it is greatly to be hoped that politicians and journalists seriously interested in the subject will revisit these writings. Aggregating individual preferences so as to yield collective decisions presents genuine problems; Michael’s work in this area shows how these problems may be addressed in a fair-minded way that transcends the calculation of short-term party advantage. A sabbatical year at Stanford in 1988–89 finally gave Michael the time he needed to work up his Harvard lectures for publication, and the resulting book, The Logical Basis of Metaphysics (Dummett 1991a), is the most complete state-

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ment of his case against classical logic and in favour of the rival intuitionistic logic that he prefers. Initially, it seemed that the philosophical community was determined to greet this book with awestruck silence: when I gave classes on it at Oxford in the late 1990s, very few of the graduate students there had studied it. However, more recent researchers – notably, in continental Europe and Scandinavia – have acquired the mastery of proof theory that is needed to engage with it productively, and in the past ten years The Logical Basis of Metaphysics has established itself as the essential point of reference for modern discussions of the conflict of logical laws and of the philosophy of logic more generally. The year of 1991 also saw the appearance of the second part of Michael’s study of Frege, Frege: Philosophy of Mathematics (Dummett 1991b). Michael had long intended to imitate the plan of Frege: Philosophy of Language and relate Frege’s foundational contributions to the philosophy of mathematics to a whole range of modern debates in the field. He came to see, though, that because Frege had no inkling of the sort of non-subjective anti-realism that Michael espouses, he could not be portrayed as contributing to the debate that Michael regards as central in the philosophy of mathematics. Accordingly, he abandoned his original plan and composed instead a lucid and perceptive commentary on the whole of Frege’s masterpiece, Die Grundlagen der Arithmetik, and on some related passages from Frege’s later writings. The result is the most accessible of Michael’s philosophical books, although accessibility is not bought at the cost of philosophical excitement. The central chapters powerfully challenge the ‘neo-Fregean’ programme, initiated by Crispin Wright, which itself did so much to reinvigorate the philosophy of mathematics in the 1980s. Michael retired from his chair in 1992 but he has continued to think and to write. Indeed, a book that he says will be his last philosophical work, The Nature and Future of Philosophy, was published only a few weeks ago (Dummett 2010). In 1997, he gave the Gifford Lectures at St Andrews and, in 2003, the Dewey Lectures at Columbia. Both sets of lectures address a fundamental problem for the antirealist theory of meaning that Michael favours – namely, how to treat statements about the past – but the solutions each proposes are radically different. It is characteristic of Michael that he should, all the same, have published both sets of lectures within a couple of years (Dummett 2004, 2006). As always, the aim is not to create a system, to be defended come hell or high water. Rather, it is to explore interesting philosophical theses; to see where they lead; to seek – in the true spirit of serious inquiry – the optimal elaboration of diverse promising ideas. In retirement, much deserved honours came thick and fast. In 1995, the Royal Swedish Academy awarded Michael the Rolf Schock Prize for Logic and Philosophy; he was only the second recipient, Quine having been the first. Also in 1995,

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he was elected to a Senior Fellowship of the British Academy: he had resigned from that body eleven years earlier in protest at what he saw as its failure to defend academic values against Mrs Thatcher’s assault on the universities. He had since 1985 been a Foreign Honorary Member of the American Academy of Arts and Sciences; in 1990 he accepted election to the Academia Europaea. In 1999, he was knighted; it greatly pleased him that the citation praised his contributions to social (and particularly, racial) justice, as well as the philosophy. In 2007, he had the rare honour of seeing a volume devoted to his work appear in The Library of Living Philosophers (Auxier and Hahn, eds, 2007). That last distinction brought with it a great deal of work, for he had to write replies to an exceptionally large number of critical and interpretative essays. But the result was a volume that reads as though it were the record of an extraordinarily interesting seminar on his philosophy. In a few moments, Michael will receive another signal honour, the 2010 Lauener Prize for an Outstanding Philosophical Oeuvre. As I hope my description will have shown, there is no one living who deserves the honour more. Much of his work has, perforce, gone unmentioned, and there has been too little time this evening to assess or even describe properly what has been mentioned. But his achievement is, quite patently, colossal, his oeuvre a monumental contribution to the philosophy of the past century. I am certain that it will last – that is, will continue to be discussed. Indeed, there is more and better discussion of it today than there was twenty years ago: it has taken a while for people to come to terms with Michael’s ideas and to see how they can best be taken forward. That is as it should be, for thinkers who can find something genuinely new and insightful to say about the central problems of philosophy are rare in any age. A little over two thousand years ago, and about four hundred and fifty miles southeast of here, a great poet found himself contemplating another body of work that he was sure would last. He turned out to be right about that and, although Michael himself would be far too modest to say them, or even to think them, five of the magnificent lines that have resounded down the centuries come irresistibly to mind as we salute Michael’s achievement. Let me conclude, then, by saying them: Exegi monumentum aere perennius regalique situ pyramidum altius, quod non imber edax, non Aquilo impotens possit diruere aut innumerabilis annorum series et fuga temporum. (Horace, Odes, Book III, xxx)

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References Auxier, R.E. and L.E. Hahn, eds. 2007. The Library of Living Philosophers: The Philosophy of Michael Dummett. Chicago and La Salle: Open Court. Dummett, M.A.E. 1954. ‘Can an effect precede its cause?’ Proceedings of the Aristotelian Society, supplementary volume 28: 27–44. Dummett, M.A.E. 1957. ‘Constructionalism’. The Philosophical Review, 66: 47–65. Dummett, M.A.E. 1959a. ‘A propositional calculus with denumerable matrix’. The Journal of Symbolic Logic, 24: 97–106. Dummett, M.A.E. 1959b. ‘Wittgenstein’s philosophy of mathematics’. The Philosophical Review, 68: 324–348. Dummett, M.A.E. 1959c. ‘Truth’. Proceedings of the Aristotelian Society, 59: 141–162. Dummett, M.A.E. 1963. ‘The philosophical significance of Gödel’s Theorem’. Ratio, 5: 140–155. Dummett, M.A.E. 1969. ‘The reality of the past’. Proceedings of the Aristotelian Society, 69: 239–258. Dummett, M.A.E. 1973. Frege: Philosophy of Language. London: Duckworth. Dummett, M.A.E. 1975. ‘The justification of deduction’. Proceedings of the British Academy, 59: 201–232. Dummett, M.A.E. 1980. Twelve Tarot Games. London: Duckworth. Dummett, M.A.E. 1981. The Interpretation of Frege’s Philosophy. London: Duckworth. Dummett, M.A.E. 1984. Voting Procedures. Oxford: Oxford University Press. Dummett, M.A.E. 1991a. The Logical Basis of Metaphysics. London: Duckworth. Dummett, M.A.E. 1991b. Frege: Philosophy of Mathematics. London: Duckworth. Dummett, M.A.E. 1997. Principles of Electoral Reform. Oxford: Oxford University Press. Dummett, M.A.E. 2004. Truth and the Past. New York: Columbia University Press. Dummett, M.A.E. 2006. Thought and Reality. Oxford: Oxford University Press. Dummett, M.A.E. 2010. The Nature and Future of Philosophy. New York: Columbia University Press. Dummett, M.A.E. and R. Farquharson. 1961. ‘Stability in voting’. Econometrica, 29: 33–43. Dummett, M.A.E. and E.J. Lemmon. 1959. ‘Modal logics between S4 and S5’. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 5: 250–264. Dummett, M.A.E. and S. Mann. 1980. The Game of Tarot: From Ferrara to Salt Lake City. London: Duckworth. Dummett, M.A.E. and J. McLeod. 2004. A History of Games Played with the Tarot Pack: The Game of Triumphs. Lampeter: Mellen Press.

Michael Dummett

Closing Speech to Lauener Symposium Prologue to Speech Leave well alone. After the brilliant sequence of papers to which we have listened, would it not be better to leave it at that – to leave well alone? I feel some impulse to do so, but I have been asked to round off our symposium, and so I SHALL DO SO: BUT I shall yield to the impulse to the extent of speaking for far less time than the 50 minutes allotted to me. After leaving school in 1943, I spent 4 years in the Army. Some of this was spent at a place north of Delhi, where I translated from Japanese, which I had been taught in Bedford. When the war ended, I was sent to Malaya, which I loved. Malaya was the south-east Asian Switzerland. It was inhabited by Malays, by Chinese and by Indian Tamils, by Muslims, Buddhists and Christians. At least when I was there, these different peoples lived together in great harmony, which was why I loved the country. I returned home and was demobilised just in time to take up the History scholarship to Christ Church, Oxford, which I had won while at school. After 4 years in the Army, I had forgotten most of my history, and so decided to read P.P.E. (Philosophy, Politics and Economics) instead. While at Oxford, I met Elizabeth Anscombe, and persuaded her to act as my tutor in philosophy for a little time. She was a marvellous teacher: she would not let you get away with anything. If you wrote what you thought she would agree with, she was still as fierce as ever. I encountered Frege by taking a paper devised by J. L. Austin, which included his Grundlagen der Arithmetik among twelve works you had to study. I was overwhelmed by that book: it was

|| Editor’s note: This is the closing speech Professor Sir Michael Dummett gave at the 4th International Lauener Symposium on Analytical Philosophy, which took place in his honour in Bern on 27 May 2010. Straight after his speech, Prof. Dummett entrusted the then president of the Lauener Foundation, Prof. Wilhelm K. Essler, and myself with the original manuscript of his speech for inclusion in the present volume. This text is a true copy of Dummett’s original typescript with handwritten corrections and additions. We deem this very late piece of philosophical writing by Dummett to be of considerable historical value. Dummett acknowledged the popular sense of “philosophy” and, though he was above all a keen, accurate analytic philosopher, he very occasionally engaged himself in popular philosophy as well: a graceful, inspiring and slightly puzzling sample of this is the present speech, which he has contributed to this volume dedicated to his philosophical oeuvre and influence.

DOI 10.1515/9783110459135-003

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the clearest and most forceful piece of philosophical writing on which I had ever set eyes. And so I was captured by philosophy. I sat for the All Souls’ College exam, and was lucky enough to be elected. I spent my first few years there reading through all of Frege’s works – in German, because little had then been translated. I spent 29 years as a Fellow of All Souls, before I became Professor of Logic and transferred to New College. But in all this time my professional concern was with philosophy.

Speech The word “philosophy” has two senses, an academic and a popular one; at least it does for practitioners of analytic philosophy. In my professional career I taught and wrote analytic philosophy. I also of course did some nonprofessional things. I married my dear wife Ann, and together we brought up five offspring, of whom we are both very fond; I cannot call them ‘children’ any longer – the youngest has just celebrated his 50th birthday. (We had two other children, who both died in infancy.) I threw myself into the battle against racism, as also has my wife; J.C.W.I. – the Joint Council for the Welfare of Immigrants, which I took part in founding – is now over 40 years old. I was an active member of C.A.R.D. – the Campaign Against Racial Discrimination – which was troubled by internal dissension due to ideological disagreements, and I thought that these would be avoided in a body concentrating on some one sector of the large field; this was one of the motives for founding J.C.W.I.; and so it has proved. In addition to these concerns, I studied and wrote books about the history of the game of Tarot (Tarock). I first encountered the game while on holiday in France with my family, and it has fascinated me ever since. Those of you who enjoy playing card games are strongly recommended to try one of the many forms of Tarock. It will disabuse you of the superstition that Contract Bridge is the only worthwhile form of card play. So now you expect me to talk philosophy. I will do so, but in the popular, not the academic, sense. My subject will be that of prizes. Some of you will be familiar with the writings of the author of Alice’s Adventures in Wonderland and of Through the Looking-Glass, the great Oxford writer whose pen-name was Lewis Carroll. His true name was C. L. Dodgson. Members of the governing bodies of other Oxford colleges are called Fellows; those of Christ Church are confusingly called students. Dodgson was a student of Christ Church. When the college acquired a water-well somewhere, there was discussion what inscription they should place on it. When it came to Dodgson’s turn to make a suggestion,

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he said, “Leave well alone”. Those two great books of his are read by every child born and brought up in England, at least if their parents were born and brought up in England. Those who know those books will recall that one of the characters, the Dodo, declared that everyone should have a prize. Was he ridiculing the idea of a prize when he said this? – for surely it is integral to the idea of a prize that not everyone gets a prize. No, he was not, for he gave as his reason for saying that everyone should have a prize that everyone had won. “Everyone has won”, he said, “and all must have prizes”. I am very much in sympathy with the Dodo’s declaration; but we cannot endorse it literally. We could all name individuals who do not deserve prizes; and we can all think of whole categories of people membership in which should debar anyone from receiving a prize. But suppose we restrict the range of our generalisation to academics – philosophers, scientists, mathematicians and scholars. Well, then it begins to look more plausible. I am thinking of their activity as researchers rather than as teachers; please forgive the oddity of speaking of research in philosophy. The great majority of academics do the best of which they are capable: do they not, then, all deserve prizes? It does not follow, you may say. Abbott’s best may not be as good as Bishop’s best. But how do we judge whether one piece of work is better or less good than another piece of work? We ought to judge in accordance with how difficult each was to do. But we don’t: we judge by the effect each has on us. To judge by how difficult it was to make the discovery or to devise the theory would be an extremely hard task, almost an impossible one. We should have to put ourselves in a position to conceive how things appeared before the discovery was made or the theory propounded, and then estimate the difficulty of imagining and executing the experiment that led to the discovery, or of constructing the theory that explained our observations or predicted new observations. We do not do this. Instead, in salient cases we evaluate the discovery or the theory by how it affects our conception of the world in which we live. This is how we judge of the really great advances, such as those associated with Copernicus, Newton, Darwin and Einstein: they altered the way in which we think of ourselves or of the entire universe. The extent to which a new discovery or a new theory has the capacity to do this depends on how heavily it impinges on our existing conception of ourselves or of the universe. Only the most far-reaching theory or discovery has the power to affect us so fundamentally. But our estimate of the value of one less encompassing is still not a perception of a feature intrinsic to it alone. In all cases it depends upon its interaction with our own thought. It may prompt us to frame new ideas which we can apply elsewhere. I do not mean only that it may be

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good to borrow from. A new theory may incite us to try to refute it; an alleged new discovery may arouse our scepticism. The attempt to disprove it may lead us to devise new arguments or to introduce new concepts. Whether we accept it or reject it, if it stimulates new thoughts in us, by touching on matters that concern us or questions we have asked, we shall prize the proposal relatively highly; we might even award the proponent a prize.

Eva Picardi

Michael Dummett’s Interpretation of Frege’s Context Principle: Some Reflections Abstract: In discussing Frege’s Context Principle Michael Dummett often employed the contrast between a thin and a robust notion of reference, on the one hand, and between an idle and an operative conception of that notion, on the other. Is the contrast between idle and operative only a different formulation of the contrast between thin and robust? I seem to detect a difference. The thin vs. robust contrast covers a family of different cases, and does not provide a clearcut criterion of application. My positive suggestion is that only a notion of reference governed by the Principle of Compositionality counts as operative, and might have satisfied GGA’s logical demands. However, in GGA Frege also appeals to a sophisticated version of the Context Principle in order to prove that the new names generated by the abstraction operator are referential. The coexistence of these two principles creates a tension in the structure of Frege’s argument. On the other hand, in GLA, where for the first and only time the Context Principle is formulated, no semantic theory comparable to the one present in GGA is in the offing, and this is why GLA lends itself to a variety of interpretations, ranging from moderate platonism to tolerant reductionism.

Setting the Scene “The Context Principle: Centre of Frege’s Philosophy” is the title of the address delivered by Michael Dummett at the Frege Colloquium which took place in Jena in 1993. The title leaves little room for doubt as to the conclusion he had reached in 1993: the Grundgedanke encapsulated in the Context Principle (= CP) informs the whole of Frege’s philosophy, from Die Grundlagen der Arithmetik (= GLA) of 1884 – where the principle is stated for the first and only time – to the Grundgesetze der Arithmetik (= GGA). The first volume of GGA was published in 1893, shortly after the publication of the seminal essays “Function und Begriff” (1891 = FuB), “Über Sinn und Bedeutung” (1892 = SuB) and “Über Begriff und Gegenstand” (1892 = BuG). In these essays Frege offers a philosophical account

|| Università di Bologna

DOI 10/1515-9783110459135-004

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of the new technical notions by means of which he had revised and extended his Begriffsschrift (= BS) of 1879. The second volume of GGA was published in 1903, accompanied by an Afterword, where Frege tries, unsuccessfully, to circumscribe the disruptive effects brought about on his foundational program by Russell’s discovery that a contradiction is derivable in the formal language of GGA. The blame was put on Basic Law V, which connects concepts to their value-ranges by providing a criterion of identity for value-ranges in terms of the equivalence of the concepts of which they are the extension (and vice versa). Already in 1885, in his review of GLA, Cantor had warned Frege against an unqualified use of the notion of Umfang, possibly a residue of old-fashioned logic-books. In his reply, Frege (1885) rectifies Cantor’s misunderstanding of the explicit definition of number presented in GLA, which involves the Umfang of a concept of second level, i.e. “equinumerous to the concept F”, and not the Umfang of a concept of first level F. He also reminds his unsympathetic reviewer that little hinges on that notion, and backs his statement with a reference to the notorious footnote appended to § 68 of GLA, were we are told that a concept is logically prior to its Umfang, and that everybody knows what the Umfang of a concept is. And, at any rate, instead of “extension (Umfang) of a concept” one can simply say “concept”. In retrospect, Frege’s notorious footnote, if read right, can be seen as vindicated. In section 40 of Der logische Aufbau der Welt Rudolf Carnap, who attended, and took notes of, Frege’s lectures in Jena between 1910 and 1914, claims that GLA is not affected by Russell’s contradiction, provided that we construe classes (i.e. Umfänge) as quasi-objects (Quasigegenstände) within a broadly constructivist framework. We know that Frege was aware that his Begriffsschrift of 1879 is not hostage to Russell’s contradiction (cf. Űber Schoenflies: Die logischen Paradoxien der Mengenlehre (1906), in Frege (1969), Nachgelassene Schriften (= NS), p. 191; Posthumous Writings (= PW), p. 176); whether he was also aware that the bulk of the derivation of (an equivalent of) Peano Axioms sketched in GLA §§ 72–83 was consistent, as Carnap claims, is a different question. On the other hand, Wright (1983) and Boolos (1998) have shown that Hume’s Principle (Frege’s criterion of identity of numbers put forward in GLA §§ 62–63) construed in a robustly impredicative fashion, adjoined to secondorder logic, suffices for the derivation of (a suitable translation of) Peano Axioms, as Frege himself had done both in GLA and in GGA. The key to understand what goes on in GLA is to view Frege Arithmetic as a theory whose underlying logic is the second-order logic put forward by Frege in his Begriffsschrift of 1879, extended by the addition of an Axiom expressing Hume’s Principle. The notion of extension of a concept does not play a significant role in GLA, and as Boolos

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(1987) points out, the only extensions of which Frege claims the existence in GLA are those of the concepts of second level mentioned in the Reply to Cantor, and these extensions of higher level concepts can be construed in such a way as to be immune from Russell’s contradiction. Thus Frege’s footnote receives further vindication, this time, however, from the platonist side. Both Carnap’s and Wright’s interpretations of Frege’s GLA rely heavily on the CP and on definitions by abstraction. One may well wonder how come that the CP lends itself to such variety of interpretations. My surmise, in broad agreement with Dummett, is that in GLA there is no vestige of a semantic theory comparable to the one outlined in GGA, and this is why there is room for different interpretations. The notion of reference plays a crucial role in Dummett’s interpretation of Frege’s CP. In his writings he often employed the contrast between a thin and a robust notion of reference, on the one hand, and between an idle and an operative conception of that notion, on the other. Is the contrast between idle and operative only a different formulation of the contrast between thin and robust? I seem to detect a difference. My positive suggestion is that only a notion of reference governed by the Principle of Compositionality counts as operative, and might have satisfied GGA’s logical demands. On the other hand, in GLA, where for the first and only time the Context Principle is formulated, no semantic theory comparable to the one at work in GGA is in the offing, and this is why the notion of reference implicit in GLA lends itself to a variety of interpretations, ranging from moderate platonism to tolerant reductionism.

A Generalized Context Principle? With GGA we enter a completely new world, and Frege’s reservations with regard to the dispensability of Umfänge no longer hold. Value-ranges (the heirs of Umfänge) play a crucial role in GGA, numbers being a variety of value-ranges. In his letter to Russell of 28 July 1902 Frege asks: “How do we apprehend logical objects? I have no other answer but this: we apprehend them as extensions (Umfänge) of concepts, and more generally as value-ranges of functions” (Frege, Wissenschaftlicher Briefwechsel (= WB), p. 223). And then he repeats almost literally the content of the footnote appended to GLA § 68, mentioned above: the bearer of number is a concept, and instead of the concept one may just as well take its Umfang. This is side A of Basic Law V, and with a judicious choice of concepts does not lead to contradiction. The problematic side of Basic Law V is side B, and probably in 1902 Frege had not yet hit on the way out offered in the Afterword to GGA. Frege then mentions (tacitly: Peano’s) définitions par abstrac-

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tion, with which Russell was familiar, and points out that he too in GLA had made use of that style of definition. He then rashly jumps to the conclusion that definitions by abstraction present the same difficulties as the conversion (Umsetzung) of the generality of an equality into an identity of value-ranges, which forms the content of Basic Law V (WB, p. 223–24). But not all definitions by conceptual abstraction share the fate of Basic Law V, Hume’s Principle being a case in point. Since the publication in 1983 of Wright’s book Frege’s Conception of Numbers as Objects the specialized literature on the significance of Hume’s Principle has reached a considerable level of complexity and intricacy. Hume’s Principle is stated in § 62 of GLA, and is interpreted by Wright as a conceptual truth. What is usually meant by “Frege’s Theorem” is the result obtained by adding to full impredicative second-order logic what Dummett in Frege: Philosophy of Mathematics (= FPM) calls the Original Equivalence and Wright (1983) calls Hume’s Principle (cf. also Parsons 1965). In Dummett’s opinion the Original Equivalence is not an implicit definition of the numerical operator (“the number of” operator), but a test that any correct definition of the cardinality operator should pass (Dummett, FPM, p.201). According to Dummett the most important reason why numbers had to be logical objects is that in reckoning numbers among the objects, i.e. as belonging to the domain of the individual variables “he [Frege] was enabled to spin the infinite sequence of natural numbers out of nothing, as it were” (FPM, pp. 132–133). Hume’s Principle is satisfiable only in an infinite domain: according to Hale and Wright, the infinity of the number series can be seen to follow from a constitutive principle for the identity of cardinal numbers, and this is an important contribution to the epistemology of arithmetic. Dummett evinces no inclination to see matters thus, for reasons connected to his mistrust of impredicative principles and definitions. Or so it seems to me. Does the Context Principle survive in GGA in content if not in formulation? In Dummett’s opinion it does: The Context Principle, as one relating to sense, amounts to the conceptual priority of thoughts over their constituents: the constituents can be grasped only as potential constituents of complete thoughts. The principle governed Frege’s philosophy from start to finish (Dummett, FPM , p. 184)

This much is agreed on all hands. What is controversial is (a) whether in GLA the CP also applies at the level of reference, and (b) whether the CP is still operative in GGA. Can we construe §§ 29–33 of GGA I as informed by a Generalized CP (= GCP)? In GLA Frege focuses on statements of identity issuing from judgments of recognition (Wiedererkennungsurteile), whereas in GGA the CP applies to all

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sentences that can be built in the formal language, and this is why it should be qualified as “generalized”. Interpreters of Frege’s philosophy divide over this issue: for example, Resnik (1976) denies that this is the case, while at the same time providing an insightful account of the many good ideas displayed by Frege in the proof of referentiality offered in those sections of GGA. Heck (2012) considers Frege’s inductive argument in good standing, but, alas, inconsistent. In 1995 Dummett’s considered answer is in the affirmative. However, the use of the Generalized Context Principle, as it stands, is viciously circular, and Dummett’s positive suggestion is that a reformed CP should find the means for circumscribing in a non-circular way the domain of (first and second order) quantification: The stipulations governing the primitive functors […] could be determinate only if the domain, consisting wholly or largely of value-ranges, was determinate; but the domain was in the process of being determined by fixing the Bedeutungen of the value-range terms, and so the procedure went round in a circle (Dummett (1995), p. 18).

Unsurprisingly perhaps, Dummett’s paper ends on a note of caution, indeed agnosticism: The realist interpretation could be jettisoned without abandoning the context principle itself, but only if that principle, as here understood, can be shown to be coherent; and this remains in grave doubt. And yet, it is hard to see how it can be abandoned, so strong is the motivation for it. The alternative is an apprehension of objects, including abstract objects, underlying, but anterior to, an understanding of reference to them, or, indeed, a grasp of thought about them: and this is a form of external realism too coarse to be entertained. I am therefore forced to conclude without either endorsing the central feature of Frege’s philosophy or rejecting it; I can do no more than to say lamely that the issue is one of whose resolution is of prime importance to philosophy. (Dummett (1995), p. 19)

One is left with the impression that even if a non-circular way of circumscribing the domain of the quantifiers and the Principle of Comprehension by means of predicativist restrictions were available, the employment of the GCP, as a means of laying down the semantic interpretation of a fundamental mathematical theory, would still be open to grave doubts. Possibly, too small a fragment of mathematics can be reconstructed within a Fregean predicativist framework (cf. Heck (1996), and Burgess (2005)). As I said, the issue of logical abstractionism, with or without reference to the CP, has been the object of a lively debate during the last thirty years or so, thanks to the revival of Neo-Fregeanism initiated by Crispin Wright (1983), and owing also to the technical results of George Boolos (1996), Wright himself and many others. Unlike Dummett, I am not skeptical about the philosophical significance of principles of abstraction, though I am well aware that they can be

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very tricky, as the cases of Basic Law V and of Boolos’s (1998, p. 181) anti-zero demonstrate. This is the “bad company objection” leveled by Dummett (1998) against abstraction principles. Fine (2002) has added an impressive list of logical and philosophical perplexities in store for abstractionism and for the CP as a principle governing definitions. What stands in the way of Wright’s platonist interpretation of GLA? In Dummett’s opinion only a robust notion of the reference can do justice to Frege’s realism: To justify a realist interpretation of a given theory, even if it be an internal version of realism, it is not enough that we should be entitled to ascribe a reference to the terms of that theory; the notion of reference, as so ascribed, must be sufficiently robust to bear the weight of a realist interpretation. This surely requires that the notion of reference should play a genuine role within the semantic theory; that is, that determining the reference of a term should be a step in determining the truth-value of a sentence containing it. But, if the referentiality of the term was justified by appeal to the context principle, its reference is semantically idle, since it was secured by specifying the truth-value of sentences of which the term is part, or the reference of more complex terms of which the term is part, by some means not involving the identification of the referent of that term. (Dummett (1995), p. 18, my emphasis)

But how, exactly, Frege’s conception of sense and reference connects with the issue of realism?

Reference and Realism As I said, in many of his writings Dummett employs a distinction between a thin and a robust conception of reference, on the one hand, and an idle vs. an operative conception, on the other. These distinctions feature also in Dummett (1995), where, in addition, a parallel is drawn between a metaphysical conception of reference and a conception of reference internal to language, in the spirit of the Internal Realism put forward by Hilary Putnam in Reason, Truth and History. Dummett now argues that Frege’s conception of reference is internal to language both in GLA and in GGA, but for different reasons, having to do with the lack of a distinction between object language and meta-language in GLA, which makes it impossible to ask questions of reference in the formal mode, and with its presence in GGA, according to a very subtle construal of that distinction offered by Dummett (1995, p. 16). I do not find the parallel with Putnam’s internal realism particularly helpful, and, as it stands, the contrast between a thin and a robust conception of

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reference in the case of number words also appears elusive. Indeed, if, following the suggestion put forward by Dummett in The Interpretation of Frege’s Philosophy, we construe Frege’s notion of Bedeutung as semantic value, and ascribe a semantic value to incomplete expressions, there seems to be little point in denying a semantic value also to names of abstract objects. But appearances are deceptive: lest we give the impression of stipulating abstract objects into existence, or of treating ascription of reference to number words as a mere façon de parler, more has to be said about the way in which the referents in question play an operative role in determining the truth-conditions of the relevant sentences. The ascription of semantic value to incomplete expressions is one of the main changes Dummett’s interpretation of Frege underwent since the publication of Frege: Philosophy of Language in 1973. And it is, as I argued elsewhere (Picardi 2008), a change for the better. Nor does Dummett wish to deny a semantic value to number words, for he is out of sympathy with nominalistic strictures. But he is convinced that by appeal to the CP only a thin notion of reference is available, a notion too weak to satisfy the requirements laid down in GGA. As I said, the discussion of this issue in Dummett (1995) is made more elusive by his appeal to the notion of reference featuring in Putnam’s Internal Realism, as opposed to the notion of reference implicit in Metaphysical Realism. The contrast between the Internal vs. Metaphysical, as applied to reference, must be kept apart from a similar distinction which Dummett draws as regards the role of the meta-language in GGA with respect to the object language. He suggests that in GGA Frege was trying to explain in the meta-language what the formulae of the object language were to mean, not from without, but from within the object language itself (Dummett (1995), p. 16). But Frege’s notion of Bedeutung does not play just the role of a theoretical tool useful for laying down in the meta-language the semantics of a formalised language such as GGA. The notion of Bedeutung is supposed to account for our actual grasp of the thoughts conveyed by the sentences of our language. It has a dual role. Our understanding of the thought expressed by a sentence requires that we have a conception of what it is for a given name or singular term to designate a certain object, what it is for a certain predicate expression to pick out a property of the objects that fall under a given concept: for the objects falling under a concept have as properties those indicated by the characteristic marks of the concept in question. By introducing the notion of functional application Frege thought that he could avoid getting involved in the issue of whether and how, exactly, universals relate to particulars, how properties attach (inherence, instantiation, being part of) to the objects that fall under a given concept: what matters is that we, in judgement, can map objects onto truth-values. And this is

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why then notion of truth and assertion play a paramount role in Frege’s epistemology. Anyone who performs an act of judgement must implicitly recognize the existence of two logical objects, the True and the False (SuB, p. 34). Frege certainly had no qualms in pleading the cause of what is objektiv unwirklich. Concepts are reason’s proper study. Hopefully, many of our concepts capture objective features of the world. But I think that Frege would have emphatically resisted the suggestion that our grasp of concepts is causally dependent on the world’s being a certain way, and if metaphysical realism entails that, he most certainly would have been out of sympathy with this brand of metaphysical realism. However, being out of sympathy with metaphysical realism does not turn Frege into an internal realist à la Putman: the opposition between metaphysical and internal conceptions of reference simply does not apply to Frege’s notion of Bedeutung. According to Dummett, the acceptance of truth-conditional semantics together with the endorsement of the principle of bivalence, are necessary conditions of semantic realism. Frege’s conception of semantics is committed moreover both to a realistic conception of reference, characterized by the application of the name-bearer prototype to expressions belonging to all the significant syntactic categories out of which sentences are built, including sentences themselves, and to a notion of truth conceived as an objective feature that sentences possess or fail to possess, independently of our ability to come to know, even in principle, that their truth-conditions obtain or fail to obtain. But there are varieties of truth-conditional semantics that while using syntax as a guide to semantic interpretation, reject the building-block conception of sentences and thoughts underlying Frege’s account of compositionality as put forward after 1890. One may reject Frege’s contention that the sense of a linguistic expression is the way in which its reference is given to those that grasp the thought expressed by the sentence belonging to a language of which one has full mastery, either because one is committed to Quine’s thesis of the indeterminacy of reference, or because one subscribes to the meta-semantic claim according to which the notion of reference plays only a subsidiary role in the theory of meaning (Donald Davidson is a case in point). At the opposite end, are the upholders of causal theories of reference: they reject Frege’s account of the way sense determines reference as marred by descriptivism. Would Quine’s or Davidson’s conception of reference count a thin, in Dummett’s sense? Dummett (FPM, p. 211, footnote) draws a parallel between Putnam’s internal realism and Davidson’s claim “that the notion of reference – unlike that of truth – is internal to the theory of meaning”, but I don’t find this parallel convincing. Davidson always declined to enter the realism/antirealism dispute, whereas in 1981

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Putnam’s endorsement of internal realism takes place against the background of a dispute between externalist and internalist conceptions of rationality. The question “What objects the world consists of?” makes sense only within a theory or description (Putnam (1981), p. 49). Dummett’s mention of Putnam’s internal realism suggests that the conception of reference implicit in metaphysical realism counts as robust, but “too coarse to be entertained”. The thin vs. robust distinction invites such comparisons, but in my opinion does not address Dummett’s real concern, which centres round the question: Is the CP, as put forward in GLA, compatible with Frege’s later characterization of the sense of an expression, and of a singular term in particular, as the canonical means of determining its referent? Frege’s realist conception of truth-conditions is, according to Dummett, inseparable from his conception of the way in which the truth-conditions of a sentence are determined by the semantic values of its constituents and components. In his article “Realism” Dummett (1982) examines the issue of realism with reference to reductionism, a philosophical programme often associated with anti-realism. As an example of reductionism Dummett mentions the suggestion put forward by Frege in GLA § 64, that the concept of direction is obtained by re-conceptualizing the content of the sentence “Line a is parallel to line b” in such a way as to obtain “The direction of line a is the same as the direction of line b”. The equivalence relation is parallelism and the objects obtained by abstraction are directions, be they conceived as a conservative extension of the pre-existing ontology, or, as Neo-Fregeans suggest, as becoming discernable in the ontology thanks to the services of logical abstraction. Dummett describes this as a situation in which we have a method for translating statements containing terms for and quantification over directions into ones containing only terms for and quantification over straight lines. The grasp of this method of translation is integral to our understanding of statements about directions. Dummett does not dispute that, provided that the word “direction” has a pattern of inference typical of singular terms, and that we have laid down determinate truth-conditions, and shown, moreover, that they are satisfied, we are justified in ascribing reference and ontological significance to it. He has no sympathy for the reductionist who claims there are really no such things as directions. And yet we are presented here with a model of explanation for statements about directions which is an abandonment of a realistic view concerning such statements: Even if, relying on Frege’s principle that a term has a reference only in the context of a sentence, we continue to ascribe reference to terms for directions, we do not need to invoke the notion of reference, as applied to such terms, in order to explain how a sentence containing such a term is determined as true or false: the determination of the truth-value

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of the sentence does not proceed via the identification of an object as the referent of the term. […] Suppose that we have the simplest kind of statement about a direction, a sentence formed by inserting in the argument-place of a suitable one-place predicate a term for a direction. Then, under the given account of the meanings of statements about directions, the canonical means by which we establish the atomic sentence as true or as false is not by identifying some direction as being that to which the term refers, and then determining that the predicate is true of it; it is by first translating the sentence into a statement about lines, and then determining, by whatever are the appropriate means, the truthvalue of the resulting statement. […] And, because this is so for atomic sentences about directions, it is so also for sentences involving quantification over directions: we shall determine such sentences as true or as false not by considering their truth-values as the values of infinitary truth-functions whose arguments are the truth-values of their instances, but by first applying our translation scheme to obtain some statement about lines, and then determining the resulting statement as true or as false. (Dummett (1982) 1993, pp. 240–241)

Suppose that, for argument’s sake, we grant Dummett’s contention that under the model of explanation governed by the CP, the reference of the word “direction” is not semantically operative. The crucial question is: why Frege’s conception of the sense of a singular term as the canonical or most direct way in which its referent is given to a thinker or speaker should bear the unmistakable mark of realism? Constructivist and procedural interpretations of this notion are perfectly legitimate. Dummett would counter this objection by pointing out that after 1890 Frege conceived of the Bedeutung of a singular term as being given in advance and independently of our present means of recognition and calculation. After 1890 contextual definitions are banned, and only explicit definitions count as legitimate means of fixing the reference of a newly introduced expression. A different objection that could be levelled against Dummett’s conception of the way in which the word “direction” acquires a reference, is that he has overlooked Quine’s distinction between the legislative and the discursive. In “Carnap and Logical Truth”, Quine raises many objections against Carnap’s conception of truth by convention. Conventionality is a trait of events and not of sentences: Carnap’s distinction between the empirical and the conventional “refers to the act, and not to its enduring consequences, in the case of postulation as in the case of definition” (Quine (1966), p. 112). Applied to our present case, Quine’s distinction between the legislative and the discursive intimates that once the word “direction” becomes entrenched in the language, we lose track of the way in which it was originally introduced: the contrast between an idle vs. an operative conception of reference disappears. Quine’s objection is a powerful one, and here I will content myself with pointing out that Dummett is interested in what understanding the proposition in which the concept of direction is deployed consists in, and not in what the word “direction” refers to in the

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context of a sentence. I will briefly return to this cluster of issues in the last section of my paper. Unsurprisingly perhaps Dummett’s characterization of reductionism fits very well the philosophical programme of Rudolf Carnap, the champion of reductionisms of all hues. Carnap’s masterpiece, Der logische Aufbau der Welt, draws generously on Frege’s and Russell’s conceptions of definitions in use and by logical abstraction. Carnap construes Frege’s contextual dictum as applied to class names as a means of introducing in the language names of quasi-objects. Such names do not stand for classes; rather, in the context of a sentence, indicate the quasi objects made available at a given stage by the constructional method (Konstitutionssystem). A quasi object is an autonomous complex (ein selbständiger Komplex), not an aggregate or a collection, and Frege was perhaps the first to emphasize that a class should not be conceived as an aggregate or a plurality of objects. Carnap claims that the explicit definition of the cardinality operator offered by Frege in § 68 of GLA is not prey to the contradiction, provided that we construe class-names in the right way. The function of class names is to encode, as it were, the salient properties of the concept of cardinality: The cardinal number 5 is a quasi object, just as the cl5 is; the symbol “5” does not designate a proper object, but serves to make statements about those properties which all possible classes of five objects have in common. Thus, we see that the indicated definition of cardinal number does not replace the cardinal numbers by other schematically construed entities, which have a certain formal analogy with cardinal numbers, but that this definition meets precisely the arithmetical concept itself. It is the only rarely articulated, but frequently tacit, conception of classes as wholes or collections which has obscured this fact. (Carnap (1928) 1967, p. 69; Carnap’s emphasis)

In the bibliographical references attached to section 40 Carnap points out that his conception agrees with that of Hermann Weyl. I cannot here enter a closer examination of Carnap’s interpretation of GLA. What I would like to suggest is that Carnap’s construal of definitions in use (contextual definitions) of number words and class names is a legitimate interpretation of what Frege is up to in GLA, and can be taken as an example of the stance of the tolerant reductionist described by Dummett in chapter 15 of FPM, and contrasted with the austere reductionist and the robust interpretation of the notion of reference provided by Wright (1983). The tolerant reductionist recognises that ‘“31” refers to an object’ can be construed as equivalent, in the formal mode, to “There is such a number as 31”, and hence as uncontroversially true. What he denies, however, is that the notion of reference, as so used, is to be understood realistically (FPM, p. 191), where a “realistic construal” is one in harmony with the conception of

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sense outlined above. As far as contextual definitions are concerned, Dummett endorses the intermediate position of the tolerant reductionist. In Wright’s robust interpretation of GLA contextual definitions “succeed in conferring upon sentences containing the terms contextually defined senses which warrant our viewing them as having just that semantic structure which their surface forms suggest” (FPM, p. 191). As a matter of fact, in later writings Wright and Hale have qualified the claim that Hume’s Principle and definitions by abstractions in general suffice to secure reference to abstract objects. The referential status of a singular term does not entail the existential claim that the object referred to exists. Free logic is the appropriate background of Neo-Fregeanism. What NeoFregeans are committed to is the conditional statement that if numbers exist, they are what satisfy Hume’s Principle (cf. Wright and Hale (2000)). In his writings after 1982 Dummett has modified and refined his conception of realism and antirealism, but I believe that in the essay of 1982 we have the clearest statement of what he considers the essential features of the conception of reference characteristic of Frege’s semantics. It is worth stressing that Dummett’s argument has nothing to do with the worry that pure abstract objects cannot be pointed at on a par with concrete objects – as some interpreters have suggested. A different explanation of what is implicit in the contrast between thin and robust conception of reference can be gleaned from Dummett’s “Reply to Peter Sullivan”. Here, Dummett points out that in the case of empirical statements of identity such as “Hesperus is Phosphorus” it may well require elaborate astronomical investigations to find out whether the identity holds, but that there is no need to specify in advance the domain over which the statement in question is defined; we are confident that a determinate answer to the identity statement will be forthcoming. In the case of fundamental mathematical theories, on the other hand, a circumscription of the domain is of the essence, for we do not have “a sufficiently definite conception of what objects belong to the domain of such theories” (Dummett 2007, p. 791). And this is why, among other things, Dummett believes that an intuitionistic understanding of the quantifiers, and of logic in general, is preferable to a classical one. To sum up: Dummett’s perplexities concerning the robustness of a notion of reference licensed by the CP stem from two sources: the first has to do with the notion of sense, conceived as the canonical means of determining the referent of a singular term; the second concerns the failure of the CP to provide a definite conception of the abstract objects belonging to a given domain, unless we are given a previous specification of the domain. And even if these two failures could be remedied, Dummett’s agnosticism with regard to the CP as a means of

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laying down the semantic interpretation of a fundamental mathematical theory remains unabated (Dummett 1995).

Interlude: Some Tenets of Frege’s Conception of Logic and Language Before examining what it is for a notion of Bedeutung to play an operative role in the semantic interpretation of a formal language such as GGA, I would like to recall three tenets of Frege’s conception of logic and language that Dummett (1981) has helped us to disentangle: (a) the CP, formulated with great emphasis in GLA, but, in Dummett’s opinion, at work in a subtle way also in GGA; (b) the principle of compositionality which, after 1890, Frege applied at the level of both sense and reference, and which informs his conception of thought and meaning from 1890 onwards; and (c) the principle of extraction of a (complex) predicate from a sentence under a certain conception [Auffassung] of its content. We encounter principle (c) for the first time in § 9 of BS, it puts in an appearance in § 70 of GLA; and, with considerable shifts of emphasis, persists throughout Frege’s philosophical development. Indeed, in retrospect, Frege came to consider it as THE distinctive feature of his conception of logic. It may be a good idea to spell out these principles which in the recent literature on the CP often get confused or assimilated. So, let us briefly recall Frege’s formulations of the extraction principle (c), the CP (a), and the principle of compositionality (b), beginning with the last one. Frege’s “Third Logical Investigation” opens with the following observation: It is astonishing what language can do. With a few syllables it can express a incalculable multitude of thoughts […]. This would be impossible if we could not distinguish parts in the thought corresponding to the parts of a sentence, so that the structure [Aufbau] of the sentence can serve as a picture of the structure of the thought (Frege (1923), p. 36, 1984 p. 390).

To be sure, the application of the part-whole picture to thoughts is metaphorical. Strictly speaking, it is only sentences that have parts. A key-word in the quoted passage is “structure”. For, as Frege goes on to remark, “the question now arises how a thought comes to be constructed, and how the parts are so combined together that the whole amounts to more than the parts taken separately”. To answer this question Frege appeals to the contrast between saturated and unsaturated parts of a thought, where saturation and combination are not

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meant as temporal processes. “It is natural – he goes on to say – to suppose for logic in general that combination into a whole always comes about by saturation of something else.” (Frege (1923), p. 136, (1984), p. 390) This is in a nutshell, Frege’s answer to the ancient problem (or an aspect of the problem) of the unity of a proposition (or thought). It rests squarely on his insight into the functional character of concepts. And concepts are (among other things) the predicates of possible judgments, possibly rules for discerning and grouping individuals. And judgment, as Frege clearly saw, relates to truth, and truth relates to assertion and justification. Possibly, every assertion contains a predication (i.e. its content is amenable to the function-argument analysis), but predication and assertion must be kept apart: when a sentence occurs as the antecedent of a conditional or in the formulation of questions or in the laying down of definitions, we are dealing with instances of predication, but not of assertion. At the level of reference principle (b) asserts that the semantic value of a sentence depends solely on the semantic values of its components words and the way they are put together. The principle of compositionality constitutes the basis of (almost) all varieties of truth-conditional semantics. Also the principle of extraction of a complex predicate from a sentence (what Ricketts (2012) calls the principle of “logical segmentation”) is involved in the issues of predication and of the unity of the proposition, which I for my part consider two separate issues. In one of its latest formulations this principle reads: What is distinctive about my conception of logic is that I begin by giving pride of place to the content of the words ‘true’, and then immediately go on to introduce a thought as that to which the question ‘Is it true?’ is in principle applicable. So I do not begin with concepts and put them together to form a thought or judgement. I come by the parts of a thought by analysing the thought. This marks off my concept-script from a similar invention of Leibniz and his successors, despite what the name suggests; perhaps it was not a happy choice on my part. (Frege (1919) 1979, p. 253)

This is by no means a “rational reconstruction” ex post of the spirit that had informed his logical work. In an unpublished essay, written around 1882 and known as the “Anti-Boole”, young Frege, after having given an example of what he means by the several “decompositions” to which the same equation, e.g. 24 = 16, is amenable, goes on to say: “Instead of putting a judgement together out of an individual as a subject and an already previously formed concept as predicate, we do the opposite and arrive at a concept by splitting up the content of possible judgement.” (Frege (1880/81), in NS, p. 18; PW, p. 17) This line of reasoning is present also in “Über Begriff und Gegenstand”, marred, however, by the paradox of the concept horse (cf. Picardi (2008)).

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Frege’s logic is from the very beginning conceived as a symbolic language suited to represent the structure of sentences which express judgeable contents. It carries with it from the very beginning an essential appeal to the notion of truth. In § 9 of BS Frege’s goal is to show that the argument-function distinction is to be preferred to the traditional analysis in terms of subject and predicate. To the isolation of the characteristic pattern discernible in an array of sentences Frege attributed the utmost importance. The extraction principle helps to highlight the different decompositions into function and argument to which a sentence gives rise, and is an essential ingredient of the compositional account of quantified sentences. After 1890 Frege simply took it for granted that concepts, relations, functions are the Bedeutung of different complex predicates resulting from different decompositions. Although the principle (a) is stated by Frege as a methodological maxim which, together with the distinction between concept and object, between the logical (and objective) and the psychological (and subjective) should guide the inquiry into the concept of number, its significance is by no means confined to the philosophy of arithmetic. In the Preface of GLA the CP is formulated as the injunction “never to ask for the meaning of a word in isolation, but only in the context of a sentence”. In GLA §§ 60–62 the principle is applied to arithmetical statements, and in the closing sections of GLA §§ 106–107 the significance of the principle is underlined, in spite of its failure to secure a definition of the numerical operator. The CP is appealed to also in § 46, perhaps the most important section of the book and, at any rate, the one he refers back to in his preface to GGA as encapsulating the gist of GLA. It is to concepts, and not material aggregates that number ascriptions apply. A second-level predicate containing a numeral as a component part is ascribed to a concept of first level. Sentences in the context of which number-words occur in adjectival position are of the utmost importance. GLA § 46 opens with an application of the context principle which is rarely commented on: It should throw some light on the matter to consider number in the context of a judgment [im Zusammenhange eines Urtheils] which brings out its basic use [Anwendungsweise]. While looking at one and the same external phenomenon, I can say with equal truth both ‘It is a copse’ and ‘It is five trees’, or both ‘Here are four companies’ and ‘Here are 500 men’. Now what changes from one judgment to the other is neither any individual object, nor the whole, the agglomeration of them, but rather my terminology [Benennung]. But that is itself only a sign that one concept has been substituted for another. This suggests as the answer to the first of the questions left open in our last paragraph, that the content of a statement of number is an assertion about a concept. (GLA, § 46, emphasis added)

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This fundamental insight is not endangered by Russell’s paradox, and Frege will often refer to it in his post-1906 writings. But numerically definite quantifiers alone do not suffice for establishing the claim that numbers are (logical) objects. According to the CP, as stated in GLA, a content (Inhalt) can be secured to number words by securing a sense to a special class of sentences containing such words. By paying attention to the logical form of equalities we should be led to the realization that number words play, in the context of a sentence, the syntactic role of singular terms, and not the role of predicates or attributes (GLA § 60). The major changes with respect of the BS of 1879, apart from the sensereference distinction and the new interpretation of identity, is that the notion of Inhalt bifurcates into the thought expressed and its truth-value. A sentence is now conceived as a complex proper name of one of the two truth values. The rules of substitution salva veritate licensed by Leibniz’s Law of identity of indiscernibles are applicable to sentences as well. To sum up: what (a), (b) and (c) have in common is the priority allotted to a sentence over its component words in the order of explanation. The assimilation of sentences to names of truth-values seems to make the CP redundant. But it still lingers in GGA, for what matters is the close connection between principles (a) and (b) and the concepts of truth and reference. For also sentences construed as complex names of truth-values express thoughts, and in GGA I § 32 we are told that the sense of a sentence is the thought that its truth-conditions are realized; each component part contributes via its sense to the thought that is being expressed. To be sure there are interpreters of Frege’s philosophy, such as Joan Weiner, Thomas Ricketts, Warren Goldfarb, and many others, who deny that Frege’s work contains the guidelines of a semantics for a formal (as well as) natural language of some kind. Ricketts (2012) in particular interprets the CP as a version of the segmentation principle put forward in BS, and, as such, belonging to syntax (for an extensive discussion of this issue cf. section 4.5 of Heck (2012)) . But a discussion of the tenets of what, after the model of “the New Wittgenstein”, might be called “the New Frege” must be postponed to a different occasion.

Compositionality and the Context Principle At first sight, the principle of compositionality suggests an order of explanation, leading from the parts to the whole, different from the CP, leading from the whole to the parts. Or, to put it very roughly, the principle of compositionality

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takes us from reference to truth, while the CP, if all goes well, takes us from truth to reference (cf. Fine (2002)) However, I believe that the Principle of Compositionality can be reconciled with (a) and (c), provided that by “parts” we understand the logically significant units discerned in a sentence as a consequence of having applied either principles (a) or (c), or both. The parts thus discerned can, indeed must, occur as independent components of new sentences and contribute to the truth-value of new sentences via their semantic value. More to the point: the principle of compositionality presupposes that the semantic values of expressions belonging to the different logical categories out of which a sentence is composed is already settled, and this is why we can think of the truth-value of a sentence as determined by the semantic values of its subsentential components and the way they are put together. The Bedeutung of a sub-sentential component is operative if it plays a role in determining the truthconditions of the sentences in which it occurs. However, also at the time of GLA and, with much greater emphasis, at the time of GGA Frege strove for a construal of number words as semantic units that could be detached from the sentential context in which they occur and enter as components into new sentences. In GGA the referentiality of names of value-ranges is not settled by the informal stipulation offered in GGA I § 9. And it is at this stage that a version of the GCP makes its appearance. In § 29 of GGA I Frege stipulates under what conditions an expression of GGA counts as referential, and in § 31 attempts to show that all the expressions that can be constructed in GGA by means of the eight primitive functions are, in fact, referential. The case of names of value-ranges has to be handled with special care, since the abstraction operator introduces new objects in the domain, in addition to the two truth-values, whose significance (both philosophical and logical) is taken by Frege as already understood by anyone who is familiar with the practice of making judgments (SuB, p. 34, KS, p.149). Had this proof been successful, names of value-ranges would have been admissible as featuring in the formulation of the logical axioms and definitions of GGA. And this is what Frege took himself as having established, in spite of the indeterminacy or under-determinacy that he detects in value-ranges. For as soon as the formal language of GGA is extended by means of a function not reducible to the ones already known, new stipulations are needed in order to establish which values the new function is to have for value-ranges as arguments. And, this Frege surmises, can be regarded “as a determination of the value-ranges as well as that of the function” (GGA I, § 10). It looks as if logical objects acquire more definite features as soon as the language is extended by means of new functions in which they can occur as suitable arguments. And the function itself becomes

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more determinate in the process. Whether the procedure described by Frege can be made coherent, I am not in a position to judge. On the face of it, it looks viciously circular. However, the procedure is in accordance with the GCP. The GCP is appealed to in order to make the interpretation of functional expressions, as applied to a given domain of objects, more determinate. Is a function capable of different interpretations (determinations) in different domains? Have we not always been told that Frege does not envisage different domains, but only a universal one? Possibly Frege is here thinking of the additional stipulations needed to fix the Bedeutung of e.g. the signs for addition and multiplication when the language is extended to fractions, irrational numbers, complex numbers (cf. GLA, §§ 96–104; GGA II, § 67), an issue which will become of paramount importance in his controversy with Giuseppe Peano over the requirements that definitions should satisfy if they are to count as logically admissible. After 1890 only explicit definitions, which satisfy the requirement of completeness (GGA I, § 33; GGA II, §§ 66–67), are considered logically admissible. Had the stipulations concerning value-ranges been successful in securing a Bedeutung to names of value-ranges, we would have earned an entitlement to use names of value-ranges as standing for independent logical objects. Above all, we would have shown how number words contribute via their reference to the truth-conditions of the sentences which can be constructed in the formal language. It seems to me that Dummett’s interpretation of Frege’s semantic realism does justice to Frege’s diagnosis of what amending his conception of logical objects along Russellian lines would have entailed. And it also accounts for the tension between the CP, as stated in GLA, and Frege’s mature conception of semantics, which is governed by the Principle of Compositionality. This line of argument is fully explicit in the Afterword to GGA II. To appreciate the connection between the CP and the principle of compositionality let us pause to look at the following statement. In his Afterword Frege writes (p. 255): […] presumably what is left is to consider class names as pseudo-proper names which would have in truth no reference. They would have to be viewed only as parts of a sign which would have a reference [Bedeutung] only as a whole [footnote 1: Cf. § 29, vol. I]. It may of course be considered advantageous for certain purposes to form distinct signs that agree in some part without thereby rendering them composite. The simplicity of a sign demands no more than that distinguishable parts in it do not autonomously have a reference. In this case, even what we are accustomed to regard as a number-sign would not in fact be a sign, but the non-autonomous part of a sign. An explanation of the sign “2” would be impossible; instead, one would have to explain many signs which contain “2” as a non-autonomous [unselbständig] component, but are not to be thought as composed logically out of “2” and another part. It would be inadmissible to allow a letter to stand in for such a non-autonomous part; for as far as the content is concerned there would be no composition at all. The generality of arithmetical propositions would thus be lost. It would

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also not be comprehensible how one could talk of the cardinal number of a class, of the cardinal number of cardinal numbers. (GGA II, p. 255, trans. 2013, p. 255, emphasis added)

I interpret the above-quoted passage as follows. At the beginning of the Afterword we are reminded that an object is what can be the argument of a function of first level. Value-ranges too are suited for that role, on a par with individual objects (under a generous construal of this notion). The allusion to GGA I § 29 indirectly confirms the conjecture that the CP is still operative in GGA. The ontological independence of value-ranges is attested by the circumstance that number-words are suited to occupy a position where individual variables (“letters” in Frege’s terminology) can occur and be bound by a quantifier. As Dummett (1981, p.381) points out “To have reference and to be capable of replacement by a bound variable is one and the same thing”. “To be is to be the value of a (bound) variable” is one of the two famous dicta formulated by Quine in “On What There Is”. The other dictum is “No entity without identity”: as regards the latter, it is Quine himself who, in “Identity, Ostension and Hypostasis”, recognises his indebtedness to Frege. It is on individual variables bound by quantifiers that the ontological status of numberwords rests. However, unlike Quine, Frege did not confine quantification to first-order quantification, and unlike Quine gave pride of place to the category of singular terms as linguistic devices apt to fulfil a referential role. Frege was a realist concerning logical objects all right, and he was a realist in the sense of semantic realism. Semantic realism, as construed by Dummett, amounts to the assumption that a sentence which is neither ambiguous nor vague as well as devoid of indexical features is determinately either true or false. Tertium non datur. Would it not have been possible for Frege to continue to endorse semantic realism as far as statements belonging to number theory are concerned without at the same time requiring a realist conception of the referents of names of value-ranges? Unfortunately, Russell’s contradiction shattered his conviction that the stipulations of § 31 of GGA I provide a reference (a truth-value) for all the sentences which could be constructed in the language. Semantic realism was crushed (cf. letter to Russell of 22 June 1902, WB, p. 213), and logical objects, conceived as extensions of concepts, could not survive the catastrophe of the contradiction. Logic turns out to be barren, for names of value-ranges turn out to be empty names. An indirect bonus of the demise of extensions after 1906 is that the paradox of the concept horse evaporates. Possibly, those who are enthralled by the paradox of the concept horse may not rest content with this observation. However, the text of the set of lectures delivered at Jena in the spring of 1914, which Carnap attended, survives in the Nachlass under the name of Logik in der

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Mathematik. In BuG Frege argued that the predicate of first level “being realized” (erfűllt sein) can be applied to those objects of a very special kind (Gegenstände ganz besonderer Art) which are the referents of expressions such as “the concept horse” (BuG, p.201, KS, p.174). Frege’s interpreters have put forward various conjectures as to what these objects of a very special kind could possibly be. The obvious conjecture is that in the sentence “The concept horse is realized” the object referred to is none other than the Umfang of the concept horse. This conjecture receives indirect confirmation in Frege’s 1914 Lectures. To the question how sentences such as “There is a positive number” and “There is a cube root of 1” should be construed, Frege now answers: In each case it is asserted of the concept that is not empty, but satisfied (erfullt). It is indeed strictly a mistake to say ‘the concept positive number is satisfied’, for by saying this I seem to make the concept into an object, as the definite article ‘the concept’ shows. It now looks as if ‘the concept positive number’ were a proper name designating an object and as if the intention were to assert of this object that it is satisfied. But the truth is that we have no object at all here. (NS, p. 269; PW, pp. 249-250, my emphasis)

And even if these objects of a very special kind differed from extensions of concepts, they would share the same fate of their more popular relatives. In forming these pseudo-names we have slipped inadvertently into fiction: what in BuG counted as a true sentences, on closer inspection, turns out to be neither true nor false. Possibly utter nonsense: we are under the impression of thinking a singular thought, but there is no object for our thought to be about. It is unclear whether in 1914 Frege’s conception of sentences with empty names in referential position is the same as the one put forward in SuB. The only comment we find in Frege’s unpublished papers concerning the origin of Russell’s paradox is that grammar is to be blamed for seducing us into thinking that the definite article in front of a concept-word signals the existence of a logical object, a class, for which it stands, even though nothing whatsoever corresponds to it. As diagnosis of the origin of the contradiction it must strike everybody as very poor. As far as I know, Frege never put forward a better diagnosis in the articles he saw to the press after 1906.

The Context Principle in GLA In GLA Frege does not appeal to the role of variables bound by quantifiers, but to the grammatical distinction between singular and plural and to the function of the definite article, which, when placed in front of a number word, allows

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one to speak of “the 1”, “the 2”, etc., thus paving the way for an understanding of identity as a relation between objects, and not between conceptual contents or conventional signs. A complaint similar to the one voiced in the Afterword to GGA II, as regards the failure to secure to number-words an independent meaning (Bedeutung), was expressed at the close of § 56 of GLA, where, after having explained how to render number ascriptions by means of (what we today call) numerically definite quantifiers, he points out that it is an illusion to have defined the number 0 or the number 1 once we have bestowed a sense to phrases such as “the number 0 belongs to”, “the number 2 belongs to”, etc., for here the number words occur only as components of larger sentences: “We have no authority to pick out the 0 and the 1 here as self-subsistent objects that can be recognized as the same again.” (GLA, p. 68). Frege also points out that the recursive definition does not provide a way to define the “number of” operator for variable n. Moreover this kind of definition makes it impossible to decide whether Julius Caesar is a number, an objection which will be rehearsed in § 66, this time with reference to directions. The definition by abstraction of the concept of direction, that is, “The direction of line a is identical with the direction of line b iff lines a and b are parallel”, fails to determine the concept of direction, for it gives us no clue how to decide whether England is the same as the direction of the Earth’s axis. However, the decisive objection as regards numerically definite quantifiers is that from the statements that to the concept F belongs the number 2 and that to the concept G belongs the number 2, we are not in a position to say that the number of Fs equals the number of Gs, that the same number belongs to both concepts. The notion of “just as many Fs as Gs” is not available at this stage of inquiry, nor have we been told what “recognisable as the same again” means. §§ 62–63 of GLA provide a brilliant answer to both questions, which, however, Frege considers unsatisfactory. He then opts for the explicit definition of the cardinality operator. Frege is fully aware that in his explicit definition of the cardinality operator the sense of “extension (Umfang) of a concept” is taken as already known, and that this assumption may be considered objectionable. However, he does not deem it necessary to counter such objection, for he attaches “no decisive importance even to bringing in the extensions of concepts at all” (GLA, § 107). In GLA, § 68 he explicitly instructs the reader that in the definition of the cardinality operator the word “gleichzahlig” does not receive its meaning from its linguistic composition (sprachliche Zusammensetzung), but must be regarded as an arbitrarily selected expression which receives its Bedeutung only from the definition. In his letter to Husserl of 24 May 1891 Frege remarks in passing that in GLA, § 97 he would now use “bedeutungsvoll” instead

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of “sinnvoll” (WB, p. 96). Also in §§ 100–102 he would now use “Bedeutung” instead of “Sinn”. Unsurprisingly, perhaps, the sections in question deal with definitions which fix the Bedeutung (and not the sense) of complex and imaginary numbers by means of arbitrary stipulations – a practice to which Frege richly helped himself (cf. Dummett (1991b)). In GLA, § 55 Frege casts doubts – which according to Dummett (FPM, pp. 99–108) are entirely spurious – on the recursive definition by means of which numerically definite quantifiers can be introduced in the language. Dummett points out that the adjectival strategy which construes number words as quantifiers can be carried out to a considerable extent, and that Frege is prevaricating on his readers by rendering the logical imports of sentences like “The number of Jupiter Moons is four” as identity statements. Already Quine (1967) in Methods of Logic, Section 44, offered a reconstruction of Frege’s numerically definite quantifiers, which has many features in common with Carnap’s (1928) reconstruction of class names in Frege’s GLA. Quine shows how Frege’s adjectival strategy can be implemented by drawing on resources made available by Russell’s theory of types. The suggestion offered by Dummett differs from Quine’s, but both have the merit of showing in detail that Frege’s dismissal of the adjectival strategy at that stage of inquiry is premature. However, in GLA, § 60 Frege seems to have forgotten the complaint voiced in § 56 according to which the adjectival strategy fails to secure an independent Bedeutung to number words, for they would have to be regarded “als Teile von Zeichen, die nur als Ganze eine Bedeutung hätten”. In § 60 we are confronted with a fresh start, and the CP is restated with great emphasis. Here we are told that “Es genügt, wenn der Satz als Ganzes einen Sinn hat; dadurch erhalten auch die Theile ihren Inhalt”: But we ought always to keep before our eyes a complete proposition. Only in a proposition have the words really a meaning (Bedeutung) […] It is enough if the proposition taken as a whole has a sense; it is this that confers on its parts also their content (Inhalt). (GLA § 60, trans. Austin, emphasis added)

This section suggests that if a sentence, considered as a whole, is endowed with sense, this should be sufficient for its component parts to receive a content as well. Should the word “content” be interpreted as alluding to what was later to become “Bedeutung” or as synonymous with “sense”? The second interpretation is certainly on the safe side. And yet, if we survey the structure of the argument, it becomes apparent that what Frege is aiming at is to secure a referent (of whatever kind) for number words. Frege’s complaint in the Afterword of GGA II seems directed against the specific formulation that the CP receives in § 60 of

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GLA. The reference in footnote 1 of the Afterword to GGA I § 29 confirms the conjecture that a version of the CP is operative also in GGA. Frege’s use of the word “Inhalt” is ambiguous both before and after 1890, and generally, when applied to singular terms, indicates the thing signified by the sign, and not the sense of the sign. In § 65 of GLA Frege says that Leibniz’s Law offers a definition of identity, and that the rules of substitution salva veritate are constitutive of its very meaning – a bold statement indeed, and an easily falsifiable one in non-extensional contexts. In my opinion the appeal to Leibniz’s law shows that, as far as singular terms are concerned, the distinction between Sinn and Bedeutung is already operative in GLA, possibly in an inchoate form, together with the new interpretation of identity, very similar to the one which he was to employ after 1890. I believe that in 1884 Frege had already moved to a conception of identity as governed by Leibniz’s Law (cf. GLA § 65), and very different from the one to be found in § 8 of BS. In GLA Frege seems to take it for granted from the very beginning that proper names and number words flanking an identity sign stand for objects. As far as sentences and predicate expressions are concerned, on the other hand, I fully concur with Dummett (1995) when he says that in GLA Frege deploys only a generic conception of the significance of an expression. What speaks in favor of my interpretation is that Frege himself, both in FuB and in the Preface to GGA, stresses the continuity between GLA and his later work. As far as the notion of Inhalt is concerned, also in Frege’s post-1890 writings there is a lingering ambiguity. For example, in FuB after stigmatizing the unsatisfactory way in which mathematicians tend to characterize the essence of a function, Frege introduces the contrast between form and content, and reminds the reader of the contrast between sign and thing signified which he had emphasized in GLA. He then goes on to criticize formal theories of arithmetic: We there have talk about signs that neither have nor are meant to have any content, but nevertheless properties are ascribed to them which are unintelligible except as belonging to the content of the sign. So also here, a mere expression, the form for a content, cannot be the heart of the matter; only the content itself can be that. Now what is the content (Inhalt), the Bedeutung of ‘2.23 + 2 = 18’? The same as ‘18’ or ‘3.6’. What is expressed in the equation ‘2.23 + 2 = 18’ is that the right-hand has the same Bedeutung as the left-hand one. I must here combat the view that e.g. 2 + 5 and 3 + 4 are equal (gleich) but not the same (dasselbe). This view is grounded in the same confusion of form and content, of sign and thing signified. (FuB, p. 2, KS, p. 126, CP, p. 138, English translation modified)

And in the footnote appended at the close of this section Frege refers to GLA and to his article, published in 1885, on formal theories of arithmetic. Here “Inhalt” is just another word for “Bedeutung”, and my surmise is that also in GLA, as far

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as singular terms are concerned, the word “Inhalt” is used to indicate “the thing signified”, the referent of a sign. A few pages later we come across Frege’s notorious statement that in converting the generality of an equality into an equality of value-ranges, a principle that “is indemonstrable and must be taken as a fundamental law of logic”, we are confronted with the same sense expressed in a different way (FuB, p. 11, KS, p. 130). In FuB Frege seems torn in opposite directions: the use of the word “Umsetzung”, which we may translate as “conversion”, suggests a mere syntactic device; on the other hand, what the resulting sentence expresses is a fundamental law of logic, which in GGA will become Basic Law V. In GGA Frege is more cautious, and the two sides of Basic Law V no longer count as synonymous. However, the informal explanation of the smooth breathing (the abstraction operator) offered in GGA I § 9 closely follows the pattern of FuB, including the use of the word “Umsetzung”. The suggestion that we are confronted with the same sense expressed in different ways is reminiscent of the metaphor of re-carving a given content (Inhalt) in different ways, thereby apprehending a new concept, employed in GLA § 64. Here “Inhalt” refers to the content of the judgment “Line a is parallel to line b”, which in the post-1890 terminology will be called a “thought”. However, what the argument is meant to show is that the content of the identity statement “The direction of line a is the same as the direction of line b”, is not, or not only, that of an analytically equivalent reformulation of the content of the first judgment (unless the judgment about lines already contains an implicit reference to directions, as Neo-Fregeans surmise). In the identity statement the semantic role of the newly introduced expressions is that of singular terms purporting to stand for the thing signified. Perhaps, as Dummett (1982) suggests, the word “direction” is here capable only of a thin construal. However, a tolerant reductionist acknowledges that a new ontological commitment is undertaken which was absent from the original judgment. Dummett (1995) stresses that in GLA Frege is working with the generic notion of the significance of an expression, and that the use of notion of “Inhalt”, as Frege himself pointed out, fails to draw a sharp line between what after 1891 was allotted either to Sinn or to Bedeutung. In 1903, on the other hand, there is no ambiguity left, and Frege’s complaint is that under Russell’s suggested reading number words would not receive a reference (Bedeutung) as detachable parts of sentences, and for Frege this amounts to no reference at all. Should we come to the conclusion that in GLA Frege was operating with a thin notion of reference? To be sure, at the close of § 60 of GLA we read:

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The self-subsistence which I am claiming for number is not to be taken to mean that a number word signifies something when removed from the Context of a proposition, but only to preclude the use of such words as predicates or attributes, which appreciably alters their meaning.

It would be a mistake, I believe, to take this observation as suggesting that in GLA Frege was aiming at a holophrastic reading of sentences containing number words, say, in a Quinean spirit. Christian Thiel reads this section of GLA as suggesting that number-words belong to a third category of expressions, different from proper names and predicates. These names of a third kind stand for objects of a third kind, which can occur only in certain sentential contexts (Thiel (1976): pp. 250–53). What Frege is alluding to is the contrast between an adjectival and a substantival use of number words in mixed statements of number. As a matter of fact, however, also in GLA Frege is eager to show that number words should be construed as detachable and recognizable parts of the sentences in whose context they occur. Failing that, they would not be available for flanking the identity sign and as being subject to Leibniz’s law. What makes a decisive difference between GLA and GGA – as Dummett never tires of pointing out – is that by 1893 Frege had outlined a semantics for his formal language, whereas not a vestige of a semantic theory can be obtained from GLA (cf. also Heck 2007). I have alluded to Quine’s holophrastic construal of the utterances of the natives we come across in chapter 2 of Word and Object, because Dummett’s contrast between a thin and a robust notion of reference may be interpreted in a way reminiscent of Quine’s and Davidson’s conception of reference. One can be a (scientific) realist in the sense of Quine while at the same time allotting to the notion of reference only a subsidiary role – as the theses of the inscrutability of reference and of ontological relativity make clear. Semantic realism is compatible with a thin notion of reference, whereas metaphysical realism is not. However, according to Quine, an ontological commitment to abstract entities is unavoidable when it comes to number theory, and is justified by its entrenchment in the fabric of science. In “Reality without Reference” Davidson argues that the notion of reference helps only to provide a smoother application of Tarski’s Convention T in constructing an empirical theory of (radical) interpretation for sentences belonging to a natural language (Davidson (1984)). He, too, however has no qualms about accepting an ontology of events, should they prove useful in uncovering the logical form of action sentences. Why should a Neo-Fregean be reluctant to treat number words as referring to abstract objects if the latter can be purchased at so cheap a price? Because the ambition of Neo-Fregeans is to offer an a priori justification of the logical laws and definitions, which, together with the conceptual truth delivered by Hume’s Principle, may warrant

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the ascription of reference to number words. The justification in question is not only a priori, but it should also qualify as analytic (cf. Wright and Hale (2000)), in a wider sense of analyticity encompassing also conceptual truths. As I said, in chapter 15 of FPM Dummett argues, both against radical nominalists, such as Hartry Field, and moderate platonists, such as Crispin Wright and Bob Hale, that a tolerant reductionist should try to occupy a middle ground between these two extremes. Dummett suggests that we abandon nominalistic scruples, and take singular terms at face value, as purporting to refer to abstract objects, provided that they occur in an articulated corpus of true statements. However in his opinion we should refrain from endowing such terms with robust reference. To a tolerant reductionist only a thin notion of reference is available, nor would he ask for more. Possibly a weaker (and in this sense “thin”) notion of reference is one that does not permit the elimination of the contextually defined term – the numerical operator – by means of the definiens in all sentential contexts. From this point of view also Wright’s notion of reference would count as thin. This weaker notion of reference might have saved Frege’s foundational programme at the time of GLA, along the lines suggested by Wright (1983). Bearing in mind Frege’s insistence that the concept is logically prior to its extension, and considering also his claim in GLA that the notion of extension (Umfang) is possibly dispensable, would he not have rejoiced, in retrospect, to be presented with such a brilliant result? Unlike Ruffino (2003), I believe that the author of GLA would have been pleased with such result. The author of GGA, on the other hand, saw things differently, and my conjecture is that only a semantics governed by the principle of compositionality can account for the realist notion of reference apt to satisfy Frege’s demands. From the point of view of a molecularist semantics, the advantage of Hale’s and Wright’s proposal is that it draws attention to a single a priori principle, namely Hume’s Principle, without incurring the arrogance of an outright stipulation of Peano Axioms as providing an implicit definition of the primitive notions of the theory (cf. Wright and Hale (2007)). Besides, one may embed Hume’s Principle in a predicativist framework and investigate how strong the resulting arithmetic turns out to be (cf. Heck (1996) and Linnebo (2004)). Also Carnap’s (1928) interpretation of Frege’s conception of classes captures important features of GLA, and indirectly supports Dummett’s diagnosis that only a thin conception of reference is appropriate to GLA. I find it difficult to decide which way we should go: GLA seems hospitable both to Carnap’s and to Wright’s interpretations, for it is semantically underdetermined. To sum up: what I have tried to show in this section of my paper is that in GLA the CP was meant to apply not only at the level of sense, but also at the

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level of the thing signified. In the absence of a full-fledged semantic theory, it is difficult to assess how weak or strong the notion of reference, as applied to singular terms, is required to be. In GLA there is no distinction between objectlanguage and meta-language, and this contributes to make matters more difficult to assess. According to Dummett, it is in GLA that for the first time in the history of philosophy an epistemological question, with an ontological question standing behind it, is transposed into a linguistic key. The linguistic turn was taken in GLA (Dummett 1993, chapter 2). In his account, Dummett draws attention to the Kantian flavor of the question Frege asks at the beginning of § 62, “How, then, are numbers given to us, if we cannot have any ideas or intuitions of them?” Frege’s answer is: Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number word occurs. That, obviously, leaves us still a very wide choice. But we have already settled that number words are to be understood as standing for self-subsistent objects. And that is enough to give us a class of propositions that must have a sense, namely those which express our recognition of a number as the same again. If we are to use the symbol a to signify an object, we must have a criterion for deciding in all cases whether b is the same as a, even if it is not always in our power to apply this criterion. (GLA, § 62, Austin transl. p. 73)

The linguistic turn is an attempt at bypassing traditional metaphysical disputes, such as the dispute between realism vs. idealism, idealism vs. nominalism, transcendental idealism vs. empiricism, etc. This is, at any rate, how Frege’s work struck Carnap and Wittgenstein. Perhaps in 1884 the exact status of the notion of reference (“the thing signified by a sign”) as applied to number words was not Frege’s most pressing concern.

A different interpretation of the Principle of Compositionality Are there candidates other than reference that can play the role of semantic value in the case of names of logical objects? A possible candidate could be the inferentialist account put forward by Brandom (2002, 2008). As is well known, Brandom’s hero is the author of the Begriffsschrift, for it is in this book that he detects the seeds of an expressivist conception of logic in harmony with his inferentialist conception of semantic value. Perhaps it is not by chance that Frege’s masterpiece of 1884 sits uncomfortably in Brandom’s characterization. For, on the one hand, GLA is arguably the work in which the notion of content is

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given the richest inferential treatment, but, on the other hand, it is undeniable that making explicit the inferential potential enshrined in the concepts of arithmetic as a plant in a seed is conceived by Frege as a step in the process of establishing his claim that numbers are objects – a distinctly objectivist claim. Since in his paper “Frege’s Technical Concepts” Brandom (1986) disagrees with Hans Sluga concerning a purely non-relational reading of the referential idiom employed by Frege, something needs to be said concerning the way in which GLA fits in with the inferentialist framework. However, Brandom (1996) believes that Frege’s attempt to establish that numbers are (logical) objects is doomed to failure because of the high standards of individuation he set on abstract objects, and perhaps this is why he pays so little attention to GLA. But there are also proposals for reforming Neo-Logicism coming from a constructivist standpoint. For instance, there is Tennant’s suggestion that we construe the crucial equivalences discussed by Frege in GLA as introduction and elimination rules à la Gentzen (cf. Tennant (2013)) – a suggestion which I find very promising. The semantic value of a term would be given by its introduction and elimination rules, possibly in a framework of free logic. Also Dummett’s own “compositional” interpretation of the CP suggests a departure from truth-conditional semantics. In chapter XVI of FPM (pp. 202– 203) he interprets the principle of compositionality thus: It is meaningless to speak of grasping the sense of an expression conceived as standing on its own, independently of any sentence in which it occurs. Its sense just is its contribution to thoughts expressed by sentences of which it is part; to regard the expression as standing on its own, independently of any sentence, is to destroy the whole conception of its possessing a sense. The escape from this dilemma requires us to regard sentences, and the thoughts they express, as ordered by a relation of dependence: to grasp the thoughts expressed by certain sentences, it is necessary first to be able to grasp those expressed by other, simpler, ones. To grasp the sense of a given expression requires us to be able to grasp the thoughts expressed by certain sentences containing it; if it did not, we should be able to grasp the sense in isolation, contrary to the context principle. Not, however, of all sentences containing it, but only of certain ones: those of a particular simple form, characteristic for the expression in question.

In the case of definitions this amounts to the requirement that they should be linearly ordered, and different orderings are surely conceivable. Dummett’s compositional interpretation of the CP requires that we discern a pattern of relative dependence among sentences, and that this pattern be faithful to relations of conceptual priority among them. In Dummett’s reformulation, the Principle of Compositionality closely connects meaning and understanding (cf. also Dummett’s The Logical Basis of Metaphysics (= LBM), Chapter 10). Following in Wright’s (1983) footsteps, I suggest that look at Frege’s argumentative strategy

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from a Wittgensteinian perspective. We survey the collection of sentences which may lay claim to counting as basic and on which a molecularist semantics governed by the compositionality principle alluded to above can be built. If we are satisfied that in the use of these basic sentences the reference of number words plays an operative role, we do not need to repeat the test of referentiality when it comes to more complex sentences. The meaning of number-words was not fixed by means of an explicit definition, but by surveying its role in those sentences that can be constructed on the ground floor, as it were. Unfortunately this thought experiment is short-lived. Wittgenstein’s philosophy of mathematics is not hospitable to the notion of reference, be it thin or robust, idle or operative. Elsewhere (Picardi 2010) I have drawn a parallel between Frege’s and Wittgenstein’s different ways of understanding the CP at the time of the Tractatus (TLP) and of the Philosophical Investigations, respectively. The Tractatus, while endorsing the CP, emphatically denies that in arithmetical equations the symbol of identity should be construed along the lines suggested by Frege (1893). In fact, propositions of mathematics do not express thoughts (TLP 5.21), and in this respect differ dramatically from propositions in which names occur as representatives of objects in states of affairs. In the Philosophical Investigations the notion of reference hardly plays a role when it comes to assess the significance of mathematical statements: objective standards of correctness should replace truth-conditions. Proofs, rules, algorithms and conventions are all we have, and all we need. And yet, the collection of basic sentences (e.g., “There are 8 dogs in the yard”, “8 is the number of dogs in the yard”, “There are just as many dogs in the yard as planets of the solar system”, “The number of dogs in the yard equals the number of planets of the solar system”), which Frege selects in order to find out what Number is, wouldn’t by and large coincide with the collection of sentences that someone interested in a descriptive elucidation of the grammar of statements of number would opt for? Once we gain a bird’s-eye view of the body of sentences that fall within its scope, light dawns from above. To use Wittgenstein’s terminology, this survey may be part of a grammatical enquiry into the interlocking language games in which number words are used. Consider a parallel. Suppose that one were to offer an account of what words for sensations mean. The natural way to proceed is to single out two classes of basic sentences: the ones in which we self-ascribe sensations and the ones in which we ascribe them to other people. There may be disagreement over the conceptual priority of sentences in the first person over sentences in the third person, much as there may be disagreement over the conceptual priority of the adjectival use of number words over their substantival use. But words for

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sensations, like number words, occur also in contexts which belong to neither category, and the question arises whether a uniform semantic account is possible or whether we should simply rest content with listing various ways in which words for sensations are used in our language. Sometimes we are confronted with a new application of a word or concept, and we wonder whether the meaning of a given word has been extended to cover a new case or whether it has changed altogether. But since often no sharp line exists between a new concept and a new application of an old concept, it is up to us how to describe things. Officially, Wittgenstein refused to acknowledge that there are sentential contexts which are more basic than others and which should be singled out as such in a grammatical survey of the way we talk about sensations. But in practice this is how he proceeded. If what I have said is on the right track, does not Frege’s survey of the basic sentences in whose context number words occur provide an example of a grammatical investigation which should have attracted Wittgenstein’s approval? Well, as a matter of fact, I am convinced that it did, although he always rejected Frege’s conception of numbers as objects.1

References Boolos, G. 1990: “The Standard of Equality of Numbers”. In: Boolos (ed.), Meaning and Method: Essays in Honor of Hilary Putnam, pp. 26–78. Cambridge, MA: Harvard University Press, repr. in: Boolos 1998. Boolos, G. 1996: “On the Proof of Frege’s Theorem”. In: A. Morton, D. Stich (eds.), Benacerraf and his Critics, Cambridge, MA: Harvard University Press, pp. 143–159, repr. in: Boolos 1998. Boolos, G. 1998: Logic, Logic, and Logic. With Introductions and Afterword by John P. Burgess, ed. by Richard Jeffrey, Cambridge, MA: Harvard University Press. Brandom, R. 1986: “Frege’s Technical Concept”. In: L. Haaparanta and J. Hintikka (eds.), Frege Synthesized, Dordrecht: Reidel. Brandom, R. 1996: “The Significance of Complex Numbers for Frege’s Philosophy of Mathematics”. In: Proceedings of the Aristotelian Society, vol. 96, pp. 293–315. Burgess, J. 2005: Fixing Frege. Princeton and Oxford: Princeton University Press. Cantor, G. 1885: “Die Grundlagen der Arithmetik”. In: Deutsche Literaturzeitung 6, pp. 728– 729, repr. in: Abhandlungen mathematischen und philosophischen Inhalts, pp. 440–441, ed. by E. Zermelo, 1932, repr. Hildesheim: Olms, 1966, pp. 440–441.

|| 1 I would like to thank Joachim Schulte and Andrea Sereni for helpful suggestions and advice on an earlier draft of this paper.

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Carnap, R. 1998 (1st edition 1928): Der logische Aufbau der Welt. Hamburg: Felix Meiner Verlag. Engl. Transl. by Rolf George, The Logical Structure of the World, Open Court Classics, 2003. Davidson, D. 1977: “Reality without Reference”, Dialectica 31. Repr in: Davidson, Inquiries into Truth and Interpretation, Oxford: OUP, 1980, pp.215–225. Demopoulos, W. (ed.) 1995: Frege’s Philosophy of Mathematics. Cambridge, MA: Harvard University Press. Dummett, M. 1973: Frege: Philosophy of Language. London: Duckworth, second ed. 1981. Dummett, M. 1981: The Interpretation of Frege’s Philosophy. Cambridge, MA: Harvard University Press. Dummett, M. 1982: “Realism”, Synthese, 52, pp. 55–112, reprinted in Dummett 1993: The Seas of Language, Oxford: Clarendon Press, pp. 230–276. Dummett, M. 1991a: Frege: Philosophy of Mathematics. London: Duckworth. Dummett, M. 1991b: “Frege and the Paradox of Analysis”. In: Dummett, Frege and Other Philosophers, Oxford, Clarendon Press, pp. 17–96. Dummett, M. 1991c: The Logical Basis of Metaphysics. London: Duckworth. Dummett, M. 1993: Origins of Analytical Philosophy. London: Duckworth. Dummett, M. 1995: “The Context Principle: Centre of Frege’s Philosophy”. In: I. Max, W. Stelzner (eds.), Logik und Mathematik: Frege-Kolloquium Jena 1993, Berlin: de Gruyter, pp. 3–19. Dummett, M. 1998: Neo-Fregeans in Bad Company? In: M. Schirn (ed.), The Philosophy of Mathematics Today, Oxford: Clarendon Press. Dummett, M. 2007: “Reply to Sullivan”: In: R. Auxier, L. Hahn (eds.), The Philosophy of Michael Dummett, The Library of Living Philosophers, Chicago and LaSalle, IL: Open Court, pp. 786–799. Field, H. 1980: Science without Numbers. Oxford: Blackwell. Fine, K. 2002: The Limits of Abstraction, Oxford: Clarendon Press. Frege, G. 1879: Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: Nebert. Engl. Transl. by T. W. Bynum: Conceptual Notation and Related Articles, Oxford: Clarendon Press, 1972. Frege, G. 1884: Die Grundlagen der Arithmetik. Breslau: Koeber. Centenarausgabe edited by Christian Thiel: 1986. Hamburg: Meiner. Eng. trans. by J. L. Austin: The Foundations of Arithmetic, Oxford: Blackwell, 1974. Frege, G. 1885: “Erwiderung auf Cantors Rezension der ‘Grundlagen der Arithmetik’”. In: Deutsche Literaturzeitung 6, p. 1030, repr. in: Frege, Kleine Schriften, p. 112. Frege, G. 1891: Function und Begriff, Jena, Pohle, repr. in: Frege, Kleine Schriften, pp. 125–142. Frege, G. 1892: “Über Sinn und Bedeutung”. In: Zeitschrift für Philosophie und philosophische Kritik, vol. 100, pp. 25–50, repr. in: Frege, Kleine Schriften, pp. 144–162. Frege, G. 1893, 1903: Grundgesetze der Arithmetik, vol. I 1893, vol. II 1903. Jena: Pohle. Repr. Darmstadt: Wissenschaftliche Buchgesellschaft, 1962. Eng. trans. Basic Laws of Arithmetic, trans. and ed. by P. A. Ebert and M. Rossberg with C. Wright, Oxford: OUP, 2013. Frege, G. 1906: “Antwort auf die Ferienplauderei des Herrn Thomae”, repr. in: Frege, Kleine Schriften, pp. 323–328. Eng. ed., pp. 341–350. Frege, G. 1919: “Aufzeichnungen für Ludwig Darmstaedter”. In: Frege, Nachgelassene Schriften, pp. 253–257.

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Frege, G. 1923: “Logische Untersuchungen. Dritter Teil: Gedankengefüge”. In: Beiträge zur Philosophie des deutschen Idealismus 3 [1923–1926], in Kleine Schriften, pp. 378–394. Eng. ed., pp. 330–406. Frege, G. 1967: Kleine Schriften. Ed. by I. Angelelli, Hildesheim: Olms, second ed. 1990. Eng. ed. by B. McGuinness: Collected Papers on Mathematics, Logic, and Philosophy, Oxford: Blackwell, 1984. Frege, G. 1969: Nachgelassene Schriften. Ed. by H. Hermes et al., Hamburg: Meiner. Eng. trans. by P. Long and R. White: Frege, Posthumous Writings, Oxford: Blackwell, 1979. Frege, G. 1976: Wissenschaftlicher Briefwechsel. Ed. by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel, A. Veraart, Hamburg: Meiner. Eng. ed. abridged by B. McGuinness, trans. by H. Kaal: Philosophical and Mathematical Correspondence, Oxford: Blackwell, 1980. Haaparanta, L., Hintikka, J. (eds.) 1986: Frege Synthesized. Dordrecht: Reidel. Hale, B. 1987: Abstract Objects. Oxford, Blackwell. Hale, B., Wright, C. 2000: “Implicit Definition and the A Priori”. In: Boghossian, C. Peacocke (eds.), New Essays on the A Priori, Oxford: Clarendon Press, pp. 286–319; repr. In B. Hale, C. Wright (2001). Hale, B., Wright, C. 2001: The Reason’s Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics. Oxford: Clarendon Press. Hale, B., Wright, C. 2004: “Logicism in the Twenty-first Century”. In S. Shapiro (ed.), The Oxford Handbook of Mathematics and Logic, Oxford: OUP, pp. 166–202. Heck, R. 1996: “The Consistency of the Predicative Fragments of Frege’s Grundgesetze der Arithmetik”, History and Philosophy of Logic, 17, pp. 209–220. Heck, R. 2007: “Frege and Semantics”. In: D. Greimann (ed.), Frege’s Conception of Truth, Amsterdam: Rodopi, pp. 27–64 (Grazer Philosophische Studien, vol. 75). Heck, R. 2012: Reading Frege’s Grundgesetze, Oxford: OUP. Linnebo, Ø. 2004: “Predicative Fragments of Frege Arithmetic”. In Bulletin of Symbolic Logic 10, pp. 153–174 Linnebo, Ø. 2009: “Frege’s Context Principle and Reference to Natural Numbers”, in S. Lindstrom et al. (eds.), Logicism, Intuitionism, and Formalism, Synthese Library 341, pp. 47–69. MacBride, F. 2003: “Speaking to Shadows: A Study of Neo-Logicism”. In: British Journal for the Philosophy of Science 54, pp. 103–163. Parsons, C. 1965: “Frege’s Theory of Number”. In: M. Black (ed.), Philosophy in America, Ithaca, NY: Cornell University Press, pp.180–203, reprinted with a Postscript in: C. Parsons, Mathematics in Philosophy: Selected Essays, Ithaca, NY: Cornell University Press, 1983, pp. 150–175. Picardi, E. 2008: “Frege and Davidson on Predication”. In: M.C. Amoretti, N. Vassallo (eds.), Knowledge, Language and Interpretation: On the Philosophy of Donald Davidson, Frankfurt am Main: Ontos, pp. 49–79. Picardi, E. 2010: “Wittgenstein and Frege on Proper Names and the CP”. In D. Marconi, P. Frascolla, A. Voltolini (eds.), Wittgenstein: Mind, Meaning and Metaphilosophy, London: Palgrave, pp. 166–187. Picardi, E. 2012: “Construction and Abstraction”. In: G. Löffladt (ed.), Mathematik – Logik – Philosophie: Ideen und ihre historischen Wechselwirkungen, Frankfurt am Main, Wissenschatlicher Verlag Harri Deutsch, pp. 305–320. Potter, M., Ricketts, T. (eds.) 2012: The Cambridge Companion to Frege, Cambridge: Cambridge University Press.

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Putnam, H. 1981: Truth, Reason, and History. Cambridge: Cambridge University Press. Quine, W. V. O. 1953: From a Logical Point of View. Cambridge, MA: Harvard University Press. Quine, W. V. O. 1967: Methods of Logic, Third Edition (First Edition 1952). London: Routledge and Kegan Paul. Quine, W. V. O. 1960: Word and Object. Cambridge, MA: MIT Press. Quine, W. V. O. 1966: “Carnap on Logical Truth” (1954). Reprinted in Quine, The Ways of Paradox and Other Essays, New York: Random House, pp. 100–125. Resnik, M. 1967: “The Context Principle in Frege's Philosophy”. In: Philosophy and Phenomenological Research, vol. 27, pp. 356–365. Resnik, M. 1976: “Frege’s Context Principle Revisited”. In: M. Schirn (ed.), Studien zu Frege Studies on Frege III, Stuttgart: Frommann-Holzboog, pp. 35–49. Ricketts, T. 2012: “Concepts, Objects, and the Context Principle”. In: M. Potter, T. Ricketts (eds.), The Cambridge Companion to Frege, pp. 149–219. Ruffino, M. 2003: “Why Frege would not have been a neo-Fregean”. In: Mind, vol. 112, pp. 51– 78. Schirn, M. (ed.) 1996: Frege: Importance and Legacy. Berlin: de Gruyter. Schirn, M. (ed.) 1998: The Philosophy of Mathematics Today. Oxford: OUP. Sullivan, P. 2007: “Dummett’s Case for a Constructive Logicism”. In: R. E. Auxier, L. E. Hahn (eds.), The Philosophy of Michael Dummett, La Salle, IL: Open Court, 2007, pp. 155–167. Tennant, N. 2013: “Logicism and Neologicism”. In: Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. Thiel, C. 1976: “Gottlob Frege: Die Abstraktion”. In M. Schirn (ed.) Studien zu Frege Essays on Frege, vol. I, Stuttgart: Frommann-Holzboog, pp. 243–264. Wittgenstein, L. 1921: Tractatus Logico-Philosophicus. Repr. in: Werkausgabe, Frankfurt am Main: Suhrkamp, 1984. Eng. trans. by David Pears and B. McGuinness, London: Routledge and Kegan Paul, 1961. Wittgenstein, L. 1953: Philosophische Untersuchungen, Philosophical Investigations, edited by G. E. M. Anscombe and R. Rhees, Oxford: Blackwell, fourth ed. by P. M. S. Hacker and Joachim Schulte, Malden, MA: Wiley-Blackwell, 2009. Wright, C. 1983: Frege’s Conception of Numbers as Objects. Aberdeen: Aberdeen University Press.

Crispin Wright

What Was Frege’s Mistake? “How did the serpent of inconsistency enter Frege’s paradise?” asks Michael Dummett at the start of chapter 17 of Frege: Philosophy of Mathematics. And in the final chapter he suggests an answer: that Frege’s major mistake – the key to the collapse of the project of Grundgesetze – consisted in … his supposing there to be a totality containing the extension of every concept defined over it; more generally [the mistake] lay in his not having the glimmering of a suspicion of the existence of indefinitely extensible concepts.1

The diagnosis is repeated in the essay, “What is Mathematics About?”, where Dummett writes that Frege’s mistake … lay in failing to perceive the notion [of a value-range] to be an indefinitely extensible one, or, more generally, in failing to allow for indefinitely extensible concepts at all.2

Now, claims of the form, Frege fell into paradox because…….

are notoriously difficult to assess even when what replaces the dots is relatively straightforward. Paradoxes of any depth are usually complex and seldom involve moves that, once exposed, allow of straightforward identification as clearcut “mistakes”. The paradox attending Basic Law V is no exception. Diagnostic offerings have included –

|| New York University and University of Stirling This article overlaps with my article, “How did the Serpent of Inconsistency enter Frege’s Paradise?” forthcoming in Philip Ebert and Marcus Rossberg (eds.): Essays on Frege’s Basic Laws of Arithmetic. Oxford: Oxford University Press. I retain the copyright for this article and authorise De Gruyter to utilise it as part of the present volume on themes from Dummett. || 1 Dummett 1991, p. 317. 2 Dummett 1993, p. 441.

DOI 10/1515-9783110459135-005

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(A) Unrestricted quantification: Frege fell into paradox because he allowed himself to quantify over a single, all-inclusive domain of objects (Russell, Dummett). (B) Impredicative objectual quantification: Frege fell into paradox because he allowed himself to define courses-of-values using (first-order) quantifiers ranging over those very courses-of-values (Russell, Dummett). (C) Impredicative higher-order quantification: Frege fell into paradox because he allowed himself to formulate conditions on courses-of-values using (higherorder) quantifiers ranging over those very conditions (Russell, Dummett). (D) Inflation: Frege fell into paradox because he adopted an axiom – Basic Law V – which is inflationary, i.e. defines its proper objects by reference to an equivalence on concepts that partitions the higher-order domain into too many cells (Boolos, Fine). And while it is indeed clear that Frege did do all these things – and prior to that, clear, or clear enough, what it is to do them – the diagnoses presented are nevertheless problematic. Contra (A), for example, there are multiple instances where unrestricted (objectual) quantification seems both intelligible and essential to the expression of the full range of our thoughts. Contra (B) and (C), while impredicative quantification of both first and higher orders is indeed essential to the generation of the paradox, it is also essential to a range of foundational moves in classical mathematics and, in so far as it may seem objectionable, the objections seem more properly epistemological than logical. Contra (D), there is no straightforward connection, in a higher-order setting, between unsatisfiability and inconsistency; and it is salient in any case that the actual derivation of the contradiction from Frege’s axiom nowhere implicitly depends upon an assumption of the classical range of the second-order variables but would go through on, for example, a substitutional interpretation of second-order quantification. However with Dummett’s quoted proposal: (E) Frege fell into paradox because he didn’t have even a glimmering of a suspicion of the existence of indefinitely extensible concepts,

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matters may seem yet worse. This diagnosis may seem not to get so far as proposing any definite account of Frege’s “colossal blunder” (as Dummett elsewhere characterises it3) at all, even a controversial one. What exactly did Frege do, or fail to do, because he failed to reckon with the indefinite extensibility of extension or course of values? What indeed exactly is indefinite extensibility? The notion continues to be met with the kind of scepticism which George Boolos espoused when he roundly rejected Dummett’s diagnosis, opining that it was “To his credit, [that] Frege did not have the glimmering of a suspicion of the existence of indefinitely extensible concepts” [my emphasis].4 Indefinite extensibility has been connected in recent philosophy of mathematics with many large issues, including not just the proper diagnosis of the paradoxes, but the legitimacy of unrestricted quantification, the content of quantification (if legitimate at all) over certain kinds of populations, the legitimacy of classical logic for such quantifiers, the proper conception of the infinite, and the possibilities for (neo-)logicist foundations for set-theory. But my project here must be limited: I shall first address a problem that obscures the usual intuitive characterisations of the notion of indefinite extensibility, and offer thereby what I believe to be the correct characterisation of the notion.5 En passant, we shall review some issues about the “size” of indefinitely extensible concepts. Finally we will scrutinise the connections of the notion as characterised with paradox. A full enough plate.

1 The intuitive characterisations and the problem of circularity Dummett’s conception of indefinite extensibility, and the suggestion that it is playing some kind of devil’s part in the paradoxes, is of course anticipated in Russell. Following an examination of the standard paradoxes, the latter’s [1906] concludes: … the contradictions result from the fact that … there are what we may call selfreproductive processes and classes. That is, there are some properties such that, given any

|| 3 Dummett 1994, p. 243. 4 Boolos 1998, at p. 224. I should observe, though, that, in context, Boolos is assuming that an indefinitely extensible concept comes with a prohibition on unrestricted quantification over its instances – something that Dummett repudiates in his response. 5 Here I draw on Shapiro and Wright 2006.

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class of terms all having such a property, we can always define a new term also having the property in question. Hence we can never collect all of the terms having the said property into a whole; because, whenever we hope we have them all, the collection which we have immediately proceeds to generate a new term also having the said property.

For comparison, Dummett [1993, p. 441] writes that an indefinitely extensible concept is one such that, if we can form a definite conception of a totality all of whose members fall under the concept, we can, by reference to that totality, characterize a larger totality all of whose members fall under it.

According to Dummett, an indefinitely extensible concept P has a “principle of extension” that takes any definite totality t of objects each of which has P, and produces an object that also has P, but is not in t (see also Dummett [1991, pp. 316–319] where he cites the above passage from Russell). But what does “definite” mean in that? Presumably a concept P is Definite for Dummett’s purpose in those passages just if it is not indefinitely extensible! If so, then Dummett’s remarks won’t do as a definition, even a loose one, since they appeal to its complementary “definite” to characterise what it is for a concept to be indefinitely extensible. And Russell, of course, does no better by speaking unqualifiedly of “any class of terms all having such a property”, since he is taking it for granted that classes, properly so regarded, are “wholes” or “have a total” – that is, presumably, are Definite. Notice that it would not do just to drop any reference to definiteness, or an equivalent, in the intuitive characterisation. If the suggestion had been, for example, that an indefinitely extensible concept is one such that, for any given totality all of whose members fall under the concept, we can, by reference to that totality, characterise a larger totality all of whose members fall under it,

then the usual suspects would fail the test – if we took set, for instance, as the target concept and then picked as the first mentioned “totality” simply the sets themselves, there would be no “larger” totality of sets to extend into. And if we then stipulated that attention should be restricted to proper sub-totalities, then all concepts would pass the test. This problem of implicit circularity in the intuitive characterisation of indefinite extensibility is a serious one. Indeed, it is the major difficulty in forming a clear idea of the notion, and one I propose to solve here. But it would be premature to lose confidence in the notion of indefinite extensibility because of it. The three concepts targeted by the classic set-theoretic paradoxes – Burali, Cantor, and Russell – surely present a salient common pattern:

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(1) Ordinal. Think of the ordinals in an intuitive way, simply as order-types of well-orderings. Let O be any Definite collection of ordinals. Let O′ be the collection of all ordinals smaller than or equal to some member of O. O′ is wellordered under the natural ordering of ordinals, so has an order-type – γ. So γ is itself an ordinal. Let γ′ be the order-type of the well-ordering obtained from O′ by tacking an element on at the end. Then γ′ is an ordinal number, and γ′ is not a member of O. So ordinal number is indefinitely extensible.6 (2) Cardinal. Let C be any Definite collection of cardinal numbers. Assign to each of its members a set of that exact cardinality, and form the union of these sets, C′. By Cantor’s theorem, the collection of subsets of C′ is larger than C′, so larger than any cardinal in C. So cardinal number is indefinitely extensible. (3) Set/class. Dummett writes Russell’s concept class not a member of itself provides a beautiful example of an indefinitely extensible concept. Suppose that we have conceived of a class C all of whose members fall under the concept. Then it would certainly involve a contradiction to suppose C to be a member of itself. Hence, by considering the totality of the members of C together with C itself, we have specified a more inclusive totality than C all of whose members fall under the concept class not a member of itself.7

Observe that it follows that set itself is indefinitely extensible, since any Definite collection – set – of sets must omit the set of all of its members that do not contain themselves. To be sure, the argumentation involved in these cases is not completely incontestable. Someone could challenge the various set-theoretic principles (Union, Replacement, Power-set, etc.) that are implicitly invoked in the constructions, for instance. But I think it reasonable to agree with Russell and Dummett that the examples do exhibit some kind of “self-reproductive” feature which the notion of indefinitely extensibility gestures at. The question is whether we can give a more exact, philosophically robust characterisation of it.

|| 6 As Dummett [1991, p. 316] puts it, “if we have a clear grasp of any totality of ordinals, we thereby have a conception of what is intuitively an ordinal number greater than any member of that totality. Any [D]efinite totality of ordinals must therefore be so circumscribed as to forswear comprehensiveness, renouncing any claim to cover all that we might intuitively recognise as being an ordinal.” 7 Dummett 1993, at p. 441.

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2 Indefinite extensibility and the ordinals: Russell’s Conjecture and ‘small’ cases We can make a start by following up on a suggestion of Russell himself. Russell [1906, p. 144] wrote that it “is probable” that if P is any concept which demonstrably “does not have an extension”, then “we can actually construct a series, ordinally similar to the series of all ordinals, composed entirely of terms having the concept P”. The conjecture is in effect that if P is indefinitely extensible, then there is a one-to-one function from the ordinals into P. If Russell is right, then any indefinitely extensible concept determines a collection at least as populous as the ordinals – so, one might think, surpassing populous! And in that case one might worry whether the connection made by Russell’s Conjecture is acceptable. For Dummett at least has characteristically taken it that both the natural numbers and real numbers are indefinitely extensible totalities in just the same sense that the ordinals and cardinals are, with similar consequences, in his opinion, for the understanding of quantification over them and the standing of classical logic in the investigation of these domains. Moreover in the article [Dummett 1963] which contains his earliest published discussion of the notion, Dummett argues that the proper interpretation of Gödel’s incompleteness theorems for arithmetic is precisely to teach that arithmetical truth and arithmetical proof are also both indefinitely extensible concepts – yet neither presumably has an even more than countably infinite extension, still less an ordinals-sized one. (The ordinary, finitely based language of second-order arithmetic presumably suffices for the expression of any arithmetical truth.) It would be disconcerting to lose contact with perhaps the leading modern proponent of the importance of the notion of indefinite extensibility so early in the discussion. But then who is mistaken, Russell or Dummett?8 The issue will turn out to be important for the proper understanding of indefinite extensibility. To fix ideas, consider the so-called Berry paradox, the

|| 8 It is relevant to recall that Russell [1908] himself, in motivating a uniform diagnosis of the paradoxes, included in his list of chosen examples some at least where the “self-reproductive” process seems bounded by a relatively small cardinal. For instance the Richard paradox concerning the class of decimals that can be defined by means of a finite number of words makes play with a totality which, if indeed indefinitely extensible, is at least no greater than the class of decimals itself, i.e. than 2‫א‬0. Was Russell simply unaware of this type of example in 1906, when he proposed the conjecture discussed above? Or did he not in 1906 regard the Richard paradox and others involving “small” totalities as genuine examples of the same genre, then revising that opinion two years later?

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paradox of “the smallest natural number not denoted by any expression of English of fewer than 17 words”. Here is a statement of it. Define an expression t to be numerically determinate if t denotes a natural number and let C be the set – assuming there is one – of all numerically determinate expressions of English. Consider the expression: “The smallest natural number not denoted by any expression in C of fewer than 17 words”. Assume that this is a numerically determinate expression of English. Then contradiction follows from that assumption, the assumption that the set C exists, and the empirical datum that b has 16 words (counting the contained occurrence of ‘C’ as one word). The analogy with the classic paradoxes may look good, a principle of extension seemingly inbuilt into a concept leading to aporia when applied to a totality supposedly embracing all instances of the concept. But, as emerges if we think the process of “indefinite extension” through, there are complications. To see why, let an initial collection, D, consist of just the ten English numerals, “zero” to “nine”. Count ‘D’, so defined, as part of English, and consider “the smallest natural number not denoted by any member of D of fewer than 17 words”. Call this 16-worded expression “W1”. Its denotation, clearly, is 10. W1 is a numerically determinate expression of English, but not in D. Let D1 be D∪{W1}. Count ‘D1’ as an English one-word name. Now repeat the construction on D1, producing W2. Let D2 be D1∪{W2}. Count ‘D2’ as an English one-word name. Do the construction again. Keep going … How far can you keep going? Well, not into the transfinite. For reflect that 0 to 9 are all denoted by single-word members of D; 10 is denoted by the 16-worded “the smallest natural number not denoted by any member of D of fewer than 17 words”; 11 is denoted by the 16-worded “the smallest natural number not denoted by any member of D1 of fewer than 17 words”; 12 is denoted by the 16-worded “the smallest natural number not denoted by any member of D2 of fewer than 17 words”; and so on. So every natural number is denoted by some expression of English of fewer than 17 words. So the “the smallest natural number not denoted by any expression in C of fewer than 17 words” has no reference – and hence is not a numerically determinate expression after all, contrary to the assumptions of the paradox. This result does not dissolve the Berry paradox, since it depends on assumptions about English – specifically, that it may be reckoned to contain all the series of names, D, D1, D2, etc., and that these can be reckoned to be oneword names – which may be rejected. The point I am making, rather, is that, when the relevant assumptions about what counts as English are allowed, the construction shows that while there is indeed a kind of indefinite extensibility about the concept, numerically determinate expression of English, it is a bounded

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indefinite extensibility, as it were: indefinite extensibility up to a limit – in this case the first transfinite ordinal, ω. When the limit is reached, the result of the construction is a (presumably) definite collection of entities that does not in turn admit of extension by the original operation. So the targeted concept will not be indefinitely extensible, at least not in the spirit of Dummett’s and Russell’s intuitive characterisations. Consider another example. As noted above, Dummett [1963] contends that Gödel’s incompleteness theorem shows that arithmetical truth is indefinitely extensible. That should mean that given any Definite collection C of arithmetical truths, one can construct a truth – the Gödel sentence for C – that is not a member of C. This is apt to impress as a problematic claim, at least if sethood suffices for Definiteness, and if the former concept is understood as governed by its classical mathematics. For following Tarski, we may give a straightforward explicit definition of arithmetical truth. Assuming a well-defined set of arithmetical sentences, it should then follow by the Aussonderungsaxiom that there is a set – a Definite collection – of all arithmetical truths. But there is no “Gödel sentence” for this set. Still, it is clear enough what Dummett has in mind. It is straightforward to initiate something that looks like a process of “indefinite extension”. Just let A0 be the theorems of some standard axiomatisation of arithmetic. For each natural number n, let An+1 be the collection An together with a Gödel sentence for An. Presumably, if An is Definite, then so is An+1, and, of course, An and An+1 are distinct. Unlike the case of the Berry paradox, this construction can indeed be continued into the transfinite. Let Aω be the union of A0, A1, … Arguably, Aω is Definite. Indeed, if A0 is recursively enumerable, then so is Aω. Thus, we can obtain Aω+1, Aω+2, … and so on. Then we take the union of those to get A2ω, and onward, “Gödelising” all the way. On the usual, classical construal of the extent of the ordinals, however, this process too cannot continue without limit, but must “run out” well before the first uncountable ordinal. Let λ be an ordinal and let us assume that we have obtained Aλ. The foregoing construction will take us on to the next set Aλ+1 only if the collection Aλ has a Gödel sentence. And that will be so only if Aλ is recursively axiomatisable. But clearly it cannot be the case that for every (countable) ordinal λ, Aλ is recursively axiomatisable. For there are uncountably many countable ordinals but only countably many recursive functions.

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3 Indefinite extensibility explicated Let’s take stock. Russell’s Conjecture, that indefinitely extensible concepts are marked by the possession of extensions into which the classical ordinals are injectible, still stands. At any rate some apparent exceptions to it, like numerically determinate expression of English (when “English” is understood to have the expressive resources deployed above) and arithmetical truth, are not really exceptions. For the principles of extension they involve are not truly indefinitely extensible but stabilise after some series of iterations isomorphic to a proper initial segment of the ordinals – at least if the ordinals are allowed their full classical extent. That said, though, the point remains that Russell’s Conjecture, even should it be extensionally correct, is certainly not the kind of characterisation of indefinite extensibility we should like to have. If Russell’s Conjecture were the best we could do, it would be a triviality that the ordinals themselves are indefinitely extensible. What is wanted is a perspective from which we can explain why Russell’s Conjecture is good, if indeed it is – equivalently, a perspective from which we can characterise exactly what it is about ordinal that makes it the paradigm of an indefinitely extensible concept. So let’s step back. An indefinitely extensible totality P is intuitively unstable, “restless”, or “in growth”. Whenever you think you have it safely corralled in some well-fenced enclosure, suddenly – hey presto! – another fully P-qualified instance pops up outside the fence. The primary problem in clarifying this figure is to dispense with the metaphors of “well-fenced enclosure” and “growth”. Obviously a claim is intended about sub-totalities of P and functions on them to (new) members of P. But, as we observed, the intended claim does not concern all sub-totalities of P: we need to say for which kind of sub-totalities of P the claim of extensibility within P is being made. If we could take it for granted that the notion of indefinite extensibility is independently clear and in good standing and picks out a distinctive type of totality, then we could characterise the relevant kind of sub-totality exactly as Dummett did – they are the sub-totalities that are, by contrast, Definite. For the indefinite extensibility of a totality, if it consists in anything, precisely consists in the fact that any Definite sub-totality of it is merely a proper sub-totality. But at this point the clarity and good standing of the notion of infinite extensibility may not be taken for granted. Here is a way forward. Let us, at least temporarily, finesse the “which subtotalities?” issue by starting with an explicitly relativised notion. Let P be a concept of items of a certain type τ. Typically, τ will be the (or a) type of individual

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objects. Let Π be a higher-order concept – a concept of concepts of type τ items. Let us say that P is indefinitely extensible with respect to Π if and only if there is a function F from items of the same type as P to items of type τ such that if Q is any sub-concept of P such that ΠQ then (1) (2) (3)

FQ falls under the concept P, it is not that case that FQ falls under the concept Q, and ΠQ′, where Q′ is the concept instantiated just by FQ and by every item which instantiates Q (i.e., ∀x[Q′x ≡ (Qx ∨ x = FQ)]; in settheoretic terms, Q′ is (Q∪{FQ}) ).

Intuitively, the idea is that the sub-concepts of P of which Π holds have no maximal member. For any sub-concept Q of P such that ΠQ, there is a proper extension Q′ of Q such that ΠQ′. This relativised notion of indefinite extensibility is quite robust, covering a lot of different examples. Here are three: (Natural number) Px iff x is a natural number; ΠQ iff the Qs (i.e the instances of Q) are finite in number; FQ is the successor of the largest instance of Q. So natural number is indefinitely extensible with respect to finite. (Real number) Px iff x is a real number; ΠQ iff the Qs are countably infinite. Define FQ using a Cantorian diagonal construction. So real number is indefinitely extensible with respect to countable. (Arithmetical truth) Px iff x is a truth of arithmetic; ΠQ iff the Qs are recursively enumerable. FQ is a Gödel sentence generated by the Qs. FQ is a truth of arithmetic and is not one of the Qs. So arithmetical truth is indefinitely extensible with respect to recursively enumerable. And naturally the three principal suspects are covered as well: (Ordinal number) Px iff x is an ordinal; ΠQ iff the Qs exemplify a wellordering type, γ (which since Q is a sub-concept of ordinal, they will) FQ is the successor of γ. So ordinal number is indefinitely extensible with respect to the property of exemplifying a well-ordering type. (Cardinal number) Px iff x is a cardinal number; ΠQ iff the Qs compose a set. FQ is the power set of the union of a totality containing exactly one exemplar set of each Q cardinal. So cardinal number is indefinitely extensible with respect to the property of composing a set.

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(Set) Px iff x is a set; ΠQ iff the Qs compose a set. FQ is the set of Qs that are not self-members. So set is indefinitely extensible with respect to the property of composing a set. The relativised notion of indefinite extensibility should impress as clear enough, but it does not, of course, shed any immediate philosophical light on the paradoxes. Our goal remains to define an unrelativised notion of indefinite extensibility that still covers ordinal number, cardinal number, and set but somehow illuminates why they are associated with paradox while natural number, real number and arithmetical truth are not. So what next? Three further steps are needed. Notice to begin with that the listed examples sub-divide into two kinds. There are those where – helping ourselves to the classical ordinals – we can say that some ordinal λ places a lowest limit on the length of the series of Π-preserving applications of F to any Q such that ΠQ. Intuitively, while each series of extensions whose length is less than λ results in a collection of Ps which is still Π, once the series of iterations extends as far as λ the resulting collection of Ps is no longer Π, and so the “process” stabilises. This was the situation noted with numerically determinate expression of English in our discussion of the Berry paradox, and is also the situation of the first three examples above. But it is not the situation with the principal suspects: in those cases there is no ordinal limit to the Π-preserving iterations. With ordinal number, this is obvious, since the higher-order property Π in that case just is the property of having a well-ordering type. Indeed, let λ be any ordinal. Then the first λ ordinals have the order-type λ and so they have the property. The “process” thus does not terminate or stabilise at λ. With set and cardinal number, we get the same result if we assume that for each ordinal λ, any totality that has order-type λ is a set and has a cardinality. Let’s accordingly refine the relativised notion to mark this distinction. So first, for any ordinal λ say that P is up-to-λ-extensible with respect to Π just in case P and Π meet the conditions for the relativised notion as originally defined but λ places a limit on the length of the series of Π-preserving applications of F to any sub-concept Q of P such that ΠQ. Otherwise put, λ iterations of the extension process on any ΠQ “generates” a collection of Ps which form the extension of a non-Π sub-concept of P. Next, say that P is properly indefinitely extensible with respect to Π just if P meets the conditions for the relativised notion as originally defined and there is no λ such that P is up-to-λ-extensible with respect to Π. Finally, say that P is indefinitely extensible (simpliciter) just in case there is a Π such that P is properly indefinitely extensible with respect to Π. My suggestion, then, is that the circularity involved in the apparent need to characterise indefinite extensibility by reference to Definite

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sub-concepts/collections of a target concept P can be finessed by appealing instead at the same point to the existence of some species – Π – of sub-concepts of P/collections of Ps for which Π-hood is limitlessly preserved under iteration of the relevant operation. This notion is, to be sure, relative to one’s conception of what constitutes a limitless series of iterations of a given operation. No doubt we start out innocent of any conception of serial limitlessness save the one implicit in one’s first idea of the infinite, whereby any countable potential infinity is limitless. Under the aegis of this conception, natural number is properly indefinitely extensible with respect to finite and so, just as Dummett suggests, indefinitely extensible simpliciter. The crucial conceptual innovation which transcends this initial conception of limitlessness and takes us to the ordinals as classically conceived is to add to the idea that every ordinal has a successor the principle that every infinite series of ordinals has a limit‚ a first ordinal lying beyond all its elements – the resource encapsulated in Cantor’s Second Number Principle. If it is granted that this idea is at least partially – as it were, initial-segmentally – acceptable, the indefinite extensibility of natural number will be an immediate casualty of it. (Critics of Dummett who have not been able to see what he is driving at are presumably merely taking for granted the orthodoxy that the Second Number Principle is at least partially acceptable.)

4 Indefinite extensibility: Burali-Forti and Cantor Very well. Roughly summarised, then, the proposal is that P is indefinitely extensible just in case, for some Π, any Π sub-concept of P allows of a limitless series of Π-preserving enlargements. Since the series of Π-preserving enlargements is limitless, any such concept P must indeed allow of an injection of the ordinals into its instances, so Russell’s conjecture is confirmed by this account. It is immediately striking, though, that there seems to be nothing immediately paradoxical about indefinite extensibility, so characterised. Why should a concept in good standing not be sufficiently “expansive” to contain a limitlessly expanding series of Π sub-concepts without ever puncturing, as it were? I’ll return to this below. Still, there is a connection with paradox nearby. For example, in case P is ordinal, and ΠQ holds just if the Qs exemplify a well-order-type, it seems irresistible to say that ordinal is itself Π. After all, the ordinals are well-ordered. But then the relevant principle of extension, F, kicks in and dumps a new object on us that both must and cannot be an ordinal – must because it corresponds, it

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seems, to a determinate order-type; but cannot because the principle of extension always generates a non-instance of the concept to which it is applied. Thus runs the Burali-Forti paradox. The question, therefore, is why we have allowed our intuitive concept of ordinal to fall, fatally, within the compass of the relevant Π/F pair? For that, it may seem, is the key faux pas. Well, but what option do we have? There is no room for question whether the ordinals are well-ordered. But to be well-ordered is to have an order-type, and we have identified the ordinal numbers with ordertypes. The only move open, it seems, is to deny that every well-ordered series is of a determinate order-type, has an ordinal number. Specifically, it seems we have to deny that ordinal itself determines a well-ordered series of a determinate order-type and so has an ordinal number. But the price of that denial is that before we can assure ourselves of the existence of any particular limit ordinal, we need first to know that its putative predecessors are not ‘all the ordinals there are’. And this price will be exacted right back at first base, when the issue is that of justifying the existence of ω, the limit of the finite ordinals. In short, the pressure that induces the faux pas is just the pressure to allow the ordinals to run into the Cantorian transfinite in a principled fashion in the first place. The Burali-Forti paradox, and the more general predicament of ordinal number that it brings out, thus seems aptly described as indeed exactly a paradox of indefinite extensibility. How close is the comparison provided by cardinal number and Cantor’s paradox? These remarks of Dummett [1991, pp. 315– 316] suggest that he regards the situation as a tight parallel: … to someone who has long been used to finite cardinals, and only to [finite cardinals], it seems obvious that there can only be finite cardinals. A cardinal number, for him, is arrived at by counting; and the very definition of an infinite totality is that it is impossible to count it. … [But this] prejudice is one that can be overcome: the beginner can be persuaded that it makes sense, after all, to speak of the number of natural numbers. Once his initial prejudice has been overcome, the next stage is to convince the beginner that there are distinct [infinite] cardinal numbers: not all infinite totalities have as many members as each other. When he has become accustomed to this idea, he is extremely likely to ask, ‘How many transfinite cardinals are there?’. How should he be answered? He is very likely to be answered by being told, ‘You must not ask that question’. But why should he not? If it was, after all, all right to ask, ‘How many numbers are there?’, in the sense in which ‘number’ meant ‘finite cardinal’, how can it be wrong to ask the same question when ‘number’ means ‘finite or transfinite cardinal’? A mere prohibition leaves the matter a mystery. It gives no help to say that there are some totalities so large that no number can be assigned to them. We can gain some grasp on the idea of a totality too big to be counted … but once we have accepted that totalities too big to be counted may yet have numbers, the idea of one too big even to have a number conveys nothing at all. And merely to say,

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‘If you persist in talking about the number of all cardinal numbers, you will run into contradiction’, is to wield the big stick, but not to offer an explanation.

However, I think the parallel is questionable. It is true that we only get the indefinitely extensible series of transfinite cardinals up and running in the first place by first insisting on one-one correspondence between concepts as necessary and sufficient for sameness, and hence existence, of cardinal numbers in general – not just in the finite case – and that the conception of cardinal number as embracing both the finite and the spectacular array of transfinite cases thus only arises in the first place when it is taken without question that concepts in general – or at least sortal concepts in general: concepts that sustain determinate relations of one-one correspondence – have cardinal numbers, identified and distinguished in the light of those relations. That is how the intuitive barrier to the question, how many natural numbers are there, is overcome. And it is also true that that at least loosens the lid on Pandora’s box: for the intuitive barrier to the question, how many cardinal numbers are there, is thereby overcome too. But loosening the lid isn’t enough to trigger paradox. Hume’s principle, identifying the cardinal numbers associated with sortal concepts in general just when those concepts are bijectable, encapsulates exactly the “resistanceovercoming” move that Dummett is talking about. And it generates, indeed, not merely a cardinal number of cardinal numbers but the universal number “Antizero”, the number of absolutely everything that there is. But it does not spawn any paradox, as far as it goes. It is a consistent principle; at least, it is consistent in classical second-order logic. To get the paradox – Cantor’s paradox – out of the notion of cardinal number that Hume’s principle characterises, we need to embed it in a set-theory containing the associated principles sufficient to generate Cantor’s theorem itself: unrestricted Union, an exemplar set for any given set of cardinals, and a set of all cardinals. None of that baggage is entailed just by the assumption that every sortal concept has a cardinal number, identified and distinguished from others by relations of one-one correspondence. Moreover, the notion of cardinal number is not needed at all to spring that paradox. Given only a universal set, and unrestricted power set, standard moves in naive set-theory will allow us to prove both that its power set is injectable into the universal set (via a unit set mapping, e.g.) and that there can be no such injection (via the diagonalisation in Cantor’s theorem). This is already a paradox. But it is a paradox for the (naïve) notion of set. Cardinal number, as extended into the transfinite via a criterion of one-one correspondence, is not in play. Someone could reject that extension and still have to confront the antinomy. The core of Cantor’s paradox can indeed be assumed under our template for a paradox of indefinite extensibility: simply take P as object (or self-

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identical), ΠQ as the Qs compose a set and F as the power-set operation. Consider any such Π concept, Q. The reasoning of Cantor’s theorem shows that some of the members of FQ cannot be instances of Q. This immediately gives a contradiction when P itself is taken to be Π, i.e. when we assume a universal set. But no assumptions about cardinal number are involved. It is true that, as illustrated earlier, cardinal number is indefinitely extensible with respect to set when the appropriate assumptions about sets – Union, Power and Replacement – are made, and that this is enough for a paradox of indefinite extensibility if cardinal number is itself assumed to determine a set. But this should impress as a frameup, rather than an insight. The real problem is with the set-theoretic assumptions involved. Notice, incidentally, that if we deny that set, and cardinal number themselves determine sets, then we obtain – or at least I know of no reason to doubt that we obtain – examples of the possibility shortly canvassed earlier: concepts that are indefinitely extensible but with whose indefinite extensibility no paradox is associated. The philosophical justifiability of that denial is, naturally, entirely another matter.

Basic Law V If the foregoing is correct, the cases of two of the ‘principal suspects’, ordinal number and cardinal number, are different. The former is unquestionably guilty; the jury should take more time on the latter. When comprehension principles are accepted for the ordinals that both ensure that every well-ordered collection has an ordinal and provides for unlimited applicability of successor and limit, ordinal number is essentially both indefinitely extensible and susceptible to a paradox of indefinite extensibility qua satisfying the relevant trigger concept, Π. When comprehension for the cardinals is determined by Hume’s principle, it takes set-theoretic assumptions to make a case that cardinal number is indefinitely extensible, and further set-theoretic assumptions to make a paradox out of that. These assumptions have no evident intrinsic connection with cardinal number. So what, finally, about course-of-values as fixed by Basic Law V? Is it fair, in the light of the account of indefinite extensibility now on the table, and its connection with paradox, to attribute the antinomy that Russell discovered to the indefinite extensibility of the notion that Law V characterises? I don’t think so. There is certainly a paradox of indefinite extensibility in the offing. Here is how it goes. Restrict attention to the case of courses-of-values

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whose ranges are concepts and whose values truth-values – to the case of extensions of concepts – so the axiom becomes, in effect: (∀P)(∀Q)({x:Px}={x:Qx} ↔ (∀x)(Px ↔ Qx)) Extensionality and Naïve Comprehension can be read off straightaway: extensions are identical just when their associated concepts are co-extensive; and every concept has one. (Proof: take ‘P’ for ‘Q’, detach the left-hand-side of the biconditional, and existentially generalise on one occurrence of ‘{x:Px}’.) So absolutely any concept of extensions is associated with its own extension. Take P then as extension itself, and Π as has an extension. Let Q be any subconcept of P. By Law V, Q has an extension. Define membership in one of the natural ways.9 Consider the concept R: Qx and not: xεx. Form the extension of this concept, r. Choose this for FQ. Suppose Qr. Do we have rεr? If so then, r falls under R and is hence a Q that is not a member of itself. But Qr and not: rεr is in turn is the condition for being a member of r. Contradiction. So not Qr. Take Q′ as the concept: Qx ∨ x=r. Referring back to the three conditions listed earlier for our initial, relativised notion of indefinite extensibility, the foregoing completes a case for saying that extension is indefinitely extensible with respect to has an extension. Paradox is then immediate when we reflect that by Basic Law V, ΠP, i.e. that the Law requires that extension itself has an extension (compare: that there is a set of all sets). But although it exploits a similar trick, that is not quite the paradox that Russell discovered. Paradoxes of indefinite extensibility, as now understood, turn essentially on the application of the principle of extension, F, to the concept P itself – an application made possible by P’s satisfaction of the higherorder trigger concept, Π. That doesn’t happen in the reasoning from Basic Law V that Russell found – or a least, that Frege took him to have found. The key resource for that reasoning is simply the license, granted by Basic Law V, to take it that every monadic open sentence with an objectual argument place has an extension, and hence in particular that x is not self-membered has an extension. The assumption that extension has an extension is not at work in the brewing. It is true that, as with ordinal number, the paradox flows very directly from comprehension principles that go right to the heart of the intended notion. But in the case of Law V, and in contrast to ordinal number, the comprehension princi-

|| 9 For instance stipulate that x is a member of y just if x satisfies every P of which y is the extension.

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ple concerned has to do not with an intuitive vision of the desired extent and structure of the relevant population of objects, but simply with the most straightforward view – absolutely integral to Frege’s philosophy of mathematics and his treatment of mathematical existence – of the relation between concepts and their associated logical objects. Dummett’s writings on this topic are shot through with the idea that the contradictions are the symptom of a deeper philosophical mistake, that Russell’s paradox is, as it were, a carbuncle on the face of an edifice that betrays a deeper underlying malaise. For Dummett, the indefinite extensibility of fundamental mathematical domains is a philosophically vital fact about them, and one gets the impression almost that he regarded the paradox as a fitting nemesis for Frege’s failure to understand and acknowledge this fact. But he nowhere says what Frege should have done differently if he had recognised the fact, nor how it would have helped. I am sceptical about the diagnosis, for the reasons I have given. But I do not wish to reject the idea that the contradiction marks a fundamental problem in the way that Frege is thinking about mathematical ontology. There is nothing wrong with the deflationary answer to our leading question that says: Frege fell into paradox because he failed to think through the implications of the full repertoire of open sentences that fall within the range of higher-order quantifiers – failed, if you like, to reckon with the expressive resources, and especially those of diagonalisation, that come with classical, impredicative higher-order logic. He simply “didn’t think of that kind of case”. But the real question is: why did he adopt an axiom that put his project at risk from that very understandable oversight? And the answer is because the simple correlation postulated by Basic Law V encapsulated a vision of mathematical ontology that was absolutely integral to his logicism: that the mathematical objects of arithmetic and analysis are simply the logical objects that are the Fregean surrogates of functions. That is why the paradox went right to the heart of his philosophy of mathematics, and why his reaction to it was eventually one of despair.

References Boolos, G. 1993: “Whence the Contradiction?”, Proceedings of the Aristotelian Society Supplementary Volume 67, 211–233. Reprinted in his [1998] at pp. 220–236. Boolos, G. 1998: Logic, Logic and Logic, Cambridge, Mass.: Harvard University Press. Dummett, M. 1963: “The Philosophical Significance of Gödel’s Theorem”, Ratio 5, 140–155. Dummett, M. 1991: Frege: Philosophy of Mathematics, Cambridge, Massachusetts: Harvard University Press.

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Dummett, M. 1993: The Seas of Language, Oxford: Oxford University Press. Dummett, M. 1994: “Chairman’s Address: Basic Law V”, Proceedings of the Aristotelian Society 94, 243–251. Russell, B. 1906: “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society 4, 29–53. Russell, B. 1908: “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics 30, 222–262. Shapiro, S. and Wright, C. 2006: “All Things Indefinitely Extensible”, in A. Rayo and G. Uzquiano (eds.), Absolute Generality, Oxford, New York: Oxford University Press, 255– 304.

Dale Jacquette

Dummett on Truth-Conditions, Frege’s Analysis of Sentence Meaning, and the Slingshot Argument Abstract: Michael Dummett’s interpretation of Frege’s theory of sentence meaning is supposed to open the door to slingshot argument objections, reducing truth-conditional semantics to the absurdity of implying that every true sentence has the same truth-maker. Dummett’s account has further been alleged to conflict with Frege’s criticism of synonymous meaning, according to which any two sentences with the same truth value express the same thought in the sense of designating the same objectified truth value, the True or the False. I defend Dummett’s interpretation of Frege, emphasizing the unique semantic analysis of sentence meaning it provides by interposing the fixing of truth-conditions as mediating between a sentence’s sense and its reference to respective reified truth values. The distinction blunts the force of slingshot arguments in a characterization of truth conditions generally, and of truth-making states of affairs in particular for the Fregean truth value reference of true sentences. The solution thereby vindicates Dummett’s clarification of Frege’s theory of sentence meaning and the individuation of propositions or Fregean Gedanken. My conclusion is that Dummett interprets Frege correctly, and that Frege’s analysis of sentence meaning avoids the counterintuitive consequences of slingshot argument reasoning for any kindred correspondence theory of truth.

1 Dummett’s Interpretation of Fregean Sentence Meaning Frege famously distinguishes between three ‘levels’ of meaning, sense (Sinn), reference (Bedeutung), and associated idea, mental image, or subjective connotation (Bild, Vorstellung). Of these, only Sinn and Bedeutung are supposed to be objective features of meaning to be included in a scientific semantics, while Bild and Vorstellung are meant to be excluded as subjective, variable, accidental, and in many instances even capricious, and hence extraneous to an effort at

|| 1953–2016

DOI 10/1515-9783110459135-006

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articulating the exact requirements for a scientific theory of meaning.1 Michael Dummett in his exposition of Frege’s philosophy of language interpolates a third objective requirement for sentential meaning, which he identifies as a truth-condition or truth-maker in the broad sense of whatever fact or state of affairs must exist in order for a sentence to be true. If Dummett is right that Frege is also committed to the category of truth-conditions for sentential meaning, then a sentence’s characterization of its truth-conditions is an essential part of what the sentence means. The semantics for sentential meaning must then accordingly include the states of affairs by which a sentence is made true in case it is true, and otherwise made false. For convenience, we shall flag the category of sentential Sinn with the subscript, Sinns, to distinguish it from the Sinn of names, nominal Sinn, to be represented as Sinnn, and similarly for Bedeutungs and Bedeutungn. Dummett connects Frege’s concept of sentential Sinns indirectly to sentential Bedeutungs, mediated by the semantic category of truthconditions, typified in the case of true sentences by truth-making facts or states of affairs.

2 Sense, Truth-condition, Reference Dummett strongly hints at the mediation of truth-conditions between Sinns and Bedeutungs in two important passages that I discuss at length and ultimately defend against several recent critics of Dummett’s exposition of Frege’s philosophical semantics. Dummett first writes: For [Frege], our grasp of the sense of a sentence consists in our understanding of what has to be the case for it to be true, and, if the sentence is not a [logically] decidable one, this merely means that that in virtue of which it is true, if it is true, is not something we can effectively recognize as obtaining whenever it obtains. Thus we can grasp what it is for a universally quantified statement to be true, even when the domain of quantification is an unsurveyable or infinite totality, so that we are able, if at all, to determine its truth only indirectly or inconclusively.2

We can speak alternatively of truth-makers as the states of affairs that would make a corresponding sentence true were they to obtain, allowing for truthbreakers in the contrary case, as the nonexistence of whatever facts would need

|| 1 Frege (1970), p. 56. 2 Dummett (1981), pp. 514–515.

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to exist in order for a sentence to be true.3 We officially combine the two concepts as Dummett appears to do, in company with many other semantic theorists as truth-conditions. Dummett in the second key proposition for his interpretation of Fregean sentential meaning now explains: And, since our grasp of the senses of such sentences is related to an understanding of truth-conditions which we do not pretend to be, in general, effectively recognizable by us, we are fully entitled to maintain that, whenever definite truth-conditions have been laid down, the sentence must be determinately either true or false, quite independently of whether we know the truth-value or have any means of ever discovering it.4

Dummett, accordingly, interprets Frege’s later (post-1890s) semantics of sentential meaning as establishing an objective human-knowledge-independent connection between three indispensable components: (1) the sentence’s sense (Sinns); (2) its truth-conditions; and (3) its truth-value (Bedeutungs). As this schema suggests, Dummett understands Frege to connect a sentence’s sentential Sinns and sentential Bedeutungs indirectly by means of the sentence’s sensedetermined sense-reference-mediating truth-conditions. The idea is that a sentence’s Sinns describes its truth-conditions, the fact or state of affairs that would need to exist in order for the sentence to be true, which in turn determine the sentence’s sentential Bedeutungs. The sentence’s sentential Bedeutungs is always one of Frege’s two reified, objectified truth values, the True or the False, depending, respectively, on whether or not the sentence’s truth-conditions exist.5 The sentence, ‘Bertrand Russell smoked a pipe’, for example, has as its Sinns a description of the state of affairs in which Russell smoked a pipe, which describes the sentence’s truth-condition. If the truthcondition obtains as the sentence’s truth-maker, if it is an existent fact that Russell smoked a pipe, as we have reason to believe, then the sentence is true and its sentential Bedeutungs is the True, while otherwise the sentence is false and its sentential Bedeutungs is the False. Such, in any case, is my understanding of how Dummett interprets Frege, which agrees also with how I understand the basic structure of Frege’s philosophical semantics.

|| 3 See Jacquette (2010a), pp. 165–177. Also Jacquette (2010b), pp. 153–163. 4 Dummett (1981), p. 515. 5 Frege (1970), p. 63: ‘Every declarative sentence concerned with the reference of its words is therefore to be regarded as a proper name, and its reference, if it has one, is either the True or the False’.

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3 Criticisms of Frege and of Dummett on Frege Dummett in the two passages cited seems to recognize something that is present and vitally important, but somewhat inconspicuous in Frege’s original texts. It is the idea that truth-conditions mediate between sentential Sinns, as a compositional function of the meaningful words a sentence contains, and the sentence’s reference or Bedeutungs to the True or the False. I now want to suggest that several recent criticisms of other aspects of Dummett’s interpretation of Frege’s philosophy of language do not take sufficient notice of the three-part semantic theory that Dummett ascribes to Frege. Such critics of Dummett are consequently led into supposing that Dummett has either misinterpreted Frege, or that Frege is embroiled in previously unappreciated logical and semantic difficulties. The major objection to Dummett’s account of Frege’s theory of sentence meaning prominently involves an application of the so-called slingshot argument to establish the homogeneous sentential referential meaning of all like-truth-valued sentences, and the exact identity requirements for sentences, sentence synonymy, and the proper analysis of sentence meaning. Dummett’s interpretation in this light has been alleged by Dalia Drai to leave Frege open to a trivializing objection based on the slingshot argument, by which any version of the correspondence theory of truth is supposed to be rendered vacuous in virtue of having the same single fact or state of affairs serve as truth-maker for any true sentence whatsoever.6 Similarly, James Levine, responding in part to Drai, argues that Dummett’s interpretation saddles Frege with the supposedly questionable implication that every Gedanke has a unique ultimate semantic analysis.7 It should be historically and theoretically interesting as a result to concentrate on the question of whether critics like Drai and Levine have correctly understood Dummett on Frege, and whether Frege, so interpreted, has correctly understood the relation between sentences and their meaning within his later sentential sense-reference theory. I defend Dummett’s account of Frege against both Drai and Levine, and, emphasizing the ultimacy of unique semantic analyses of sentence meaning, I argue that, properly understood, Dummett’s interpretation of Frege’s three-part theory of sentence meaning entirely avoids the detrimental consequences of slingshot arguments. Dummett correctly interprets Frege’s theory of sentence meaning, if I am right,

|| 6 Drai (2002). 7 Levine (2006).

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and Frege’s theory of sentence meaning, in elaborating an explanation of truthconditions and of truth-making states of affairs for the truth-values of sentences, entirely avoids the slingshot objections of such critics as Drai and Levine. The defense of Dummett’s interpretation of Frege at the same stroke also vindicates Frege’s theory of sentence meaning and the individuation of Gedanken in what I understand to be the larger context of Dummett’s interpretation of Frege.

4 Sentential Bedeutungs versus Truth-Maker Semantics My primary intuition, the starting-place for this inquiry, is that even if all true sentences have the same Fregean sentential Bedeutungs (the True), it does not follow, as in Donald Davidson’s application of the slingshot, that they do not have distinct truth-conditions, and, in particular, that they do not have distinct truth-makers, or that therefore their Fregean sentential meaning is ultimately indistinguishable in the case of any two identically truth-valued sentences. The relationship between the Fregean Sinns of sentences, their Bedeutungs, and their truth-conditions, appears intuitively to be more complex than the slingshot argument recognizes. We begin with two true sentences: P = Snow is white. Q = Grass is green. THEN: Bedeutungs(P) = the True Bedeutungs(Q) = the True It nevertheless does not follow logically from these assumptions alone that therefore: Truth-maker(P) = Truth-maker(Q).

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There is no Fregean reason why two Gedanken with the same sentential Bedeutungs cannot have distinct truth-makers; nor, equivalently, is there any internal Fregean reason why the truth-makers of P and Q should be identical. To see why, we can first reflect that even Fregean nominal Sinnn needs external assistance, in the form of ontic factors beyond itself, in order to determine Fregean nominal Bedeutungn We take to heart the fact that for Frege, although introductory prospecta often say so unqualifiedly, it is not strictly the case that Fregean intension or Sinn all by itself determines Fregean extension or Bedeutung, even where the nominal Sinnn and nominal Bedeutungn of Fregean Eigennamen or proper names (including definite descriptions), to say nothing yet of Sätze, sentences, propositions or Gedanken, are concerned. Let us consider in this vein the case of existent states of affairs in the presumably more fundamental Fregean nominal Sinnn → Bedeutungn relation. A proper name like ‘Pegasus’ for Frege, purporting to name something that does not actually exist, has a Sinnn, but lacks any (or has null) nominal Bedeutungn.8 This fact about the Bedeutungn of the proper name ‘Pegasus’ is not exclusively a result of the proper name’s sense, Sinnn(‘Pegasus’), but also, and equally importantly, because of the extra-linguistic fact that no entity in the extant domain of existent objects has the particular properties spelled out by the application, Sinnn(‘Pegasus’) = {winged, horse, etc.}. Thus, even the nominal Bedeutungenn of non-referring Fregean proper names are determined, not by their nominal Sinnen alone, but are partly determined in association also with and against a background, in some cases, of the principles of logic, mathematics, or conceptual analysis, and, often, even typically, with the logically contingent facts of the world also entering into the Fregean establishment of nominal Bedeutungenn, as partially determining factors.

|| 8 Frege, (1970), p. 58: ‘It may perhaps be granted that every grammatically well-formed expression representing a proper name always has a sense. But this is not to say that to the sense there also corresponds a reference. The words ‘the celestial body most distant from the Earth’ have a sense, but it is very doubtful if they also have a reference. The expression ‘the least rapidly convergent series’ has a sense but demonstrably has no reference, since for every convergent series, another convergent, but less rapidly convergent, series can be found. In grasping a sense, one is not certainly assured of a reference.’

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5 Parallelism of Nominal and Sentential Sinn and Bedeutung Significantly, for the present argument, Frege’s tight parallelism between the semantics of proper names and sentences is documented by his 1891 letter to Edmund Husserl, where he graphically presents the relationship between nominal Sinnn and Bedeutungn and sentential Sinns and Bedeutungs.9 Frege depicts the analogy in this way: Sentence

Proper name

Concept word

↓

↓

↓

Sense of sentence

↓

Sense of proper name

Sense of word

↓

↓

Reference of

Reference of

Reference of

sentence

proper name

concept word → falls under

(Truth-value)

(Object)

(Concept)

Object which

concept

What is important for Frege’s theory of nominal meaning is that the logically contingent world facts by which the question of whether a proper name has nominal Bedeutungn in addition to nominal Sinnn, though determined by, are altogether ontically distinct from and logically independent of, both the name’s nominal Sinnn and nominal Bedeutungn. The point is that if the logically contingent states of affairs in the world enter into the determination of nominal Bedeutungn, so that there is no unmediated determination of nominal Bedeutungn by nominal Sinnn, then we might reasonably and naturally expect as a result that there should also be no direct fact- or state-of-affairs- unmediated determination of sentential Bedeutungs by sententional Sinns. If this expectation is fulfilled, as in Dummett’s interpretation of Frege’s theory of sentential meaning, then slingshot arguments offered as objections against Frege’s semantics on Dummett’s interpretation, properly explicated, cannot possibly succeed. An appeal to the facts of logic, mathematics, conceptual analysis, and the actual state of the world constitute a third indispensable element of the objective meaning of Fregean proper names. This third objective requirement is not to be confused with what Frege describes as the third level of meaning in the

|| 9 Frege (1979), p. 96.

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subjective poetic coloring or connotation of meaning, which he maintains is a psychologistic factor that is generally present but philosophically irrelevant to the requirements of an objective ‘scientific’ theory of meaning.10 If there is a third element involving world facts or states of affairs in determining the Fregean nominal Bedeutungn of a Fregean proper name in conjunction with its Fregean nominal Sinnn, mediating between Sinnn and Bedeutungn, then why should we not consider the likely possibility that the same might also be true for the relation between truth-making facts mediating between Fregean sentential Sinnes and Fregean sentential Bedeutungens? That is the strategy to be pursued in what follows in defense of Dummett’s interpretation of Frege’s theory of sentential meaning, and hence ultimately of Frege, against slingshot objections. The logical possibility that the Fregean Sinns of a sentence or Fregean Gedanke, its sentential Bedeutungs, and its truth-conditions, are so related that sentential Sinns alone might not simply determine a sentence’s sentential Bedeutungs (as the True or the False), just as the Fregean nominal Sinnn of a Fregean proper name alone does not generally determine the proper name’s nominal Bedeutungn, makes it possible to enrich Frege’s parallelism between Fregean nominal Sinnn, Bedeutungn, and Fregean sentential Sinns and Bedeutungs, in both cases mediated by the existence-determining or truth-making facts among the actual world’s existent states of affairs. We shall first maintain that the Fregean sentential Bedeutungs of a Satz or Gedanke is identical with its truth-value, but not with its truth-condition; that is, with whatever, if anything, it is that makes the Satz or Gedanke to possess a certain truth-value or sentential Bedeutungs, as the True or the False. What sort of a picture of the relationship between Fregean sentential Sinns, its truth-maker

|| 10 Frege (1970), pp. 60–61: ‘We can now recognize three levels of difference between words, expressions, or whole sentences. The difference may concern at most the ideas, or the sense but not the reference, or, finally, the reference as well. With respect to the first level, it is to be noted that, on account of the uncertain connexion of ideas with words, a difference may hold for one person, which another does not find. The difference between a translation and the original text should properly not overstep the first level. To the possible differences here belong also the colouring and shading which poetic eloquence seeks to give to the sense. Such colouring and shading are not objective, and must be evoked by each hearer or reader according to the hints of the poet or the speaker. Without some affinity in human ideas art would certainly be impossible; but it can never be exactly determined how far the intentions of the poet are realized. In what follows there will be no further discussion of ideas and experiences; they have been mentioned here only to ensure that the idea aroused in the hearer by a word shall not be confused with its sense or its reference.’

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or truth-conditions, and sentential Bedeutungs, might we plausibly consider? We investigate the following possibility in a simplified illustration: Schematic Example: Satz S = Fregean Semantic Compositionality: Sinns(S) = f(Sinnn(T1), Sinnn(T2), Sinnn(T3)) REMARK 1: Either Fregean Semantic Compositionality as formulated above is manifestly false, or it must be instead understood as involving ordered inputs to the compositional function f, as in: Sinns(S) = f The reason is that if we do not deliver ordered inputs to the compositional function f, then, as in the problem of Bradley’s Regress, rediscovered by Bertrand Russell and Ludwig Wittgenstein, and given that love, unfortunately, is not a symmetric relation, there can be no satisfactory way of recovering the intuitively distinct senses of the evidently logically nonequivalent sentences, ‘Romeo loves Juliet’ and ‘Juliet loves Romeo’.11 Such a reformulation of the Fregean Semantic Compositionality principle in turn is possible only if functions with ordered inputs are intelligible. It might appear that functions with ordered inputs are often encountered, necessitated by the requirements of mathematics and mathematical logic. Perhaps it is so, but the intelligibility of functions taking ordered inputs is less than obvious, besides being both logically and ontically problematic.

|| 11 The difficulty arises in connection with Russell’s posthumous book, Theory of Knowledge, which develops the so-called multiple relation theory of judgment that Wittgenstein rejected due to the problem of (F.H.) Bradley’s Regress. The problem is that an infinite regress is launched if the terms ‘Romeo’, ‘Juliet’, ‘loves’ are said to occur in a certain order by another sentence, which equally stands in need of analysis as the original ‘Romeo loves Juliet’, whereas without the necessary ordering there is no way adequately to distinguish the meaning of ‘Romeo loves Juliet’ from ‘Juliet loves Romeo’. For a discussion of the problem and extensive bibliography see Jacquette (1992–1993), pp. 193–220.

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The ontic difficulty is that functions are supposed to be abstract, and hence outside of space and time. If a function takes ordered input, however, then it appears that the function must operate within time, working sequentially with each input in turn, taking it as an argument and producing a particular output, and then taking the next input in the ordered sequence, making it the argument for the function and producing a particular output, before proceeding in each case to the next item in the ordered input set. That certainly seems like something that would not be possible in the abstract. An abstract function accepting ordered input, it appears, must first choose the first-ordered item in the ordered set to do something with or to, and then choose the second-ordered item, etc. We are left to ask what such an ordering is supposed to mean, and what it is supposed to amount to, what difference it could make, if the abstract correlations established by such a function do not boil down to the putative formalization of a procedure that gets undertaken one step at a time. A sausage mill, for example, is not a function, because it operates in real time, nor is a computer chip a function, although both of these things among many others can model or instantiate functions to which they are analogous, which presumably is not the same thing, by receiving one or more inputs and producing a single output. I am thus far not impressed with the objection that functions with ordered input are indispensable to arithmetic. The difficulty is that there do not seem to be any mathematical functions that require ordered input rather than being reducible to a function of non-ordered functions of non-ordered input, all of which, so to speak, occur in a single stroke and in the abstract, rather than sequentially over any expanse of time. Take minus, as in 3–2, for example. There is obviously a difference between 3–2 and 2–3, but the function need not be interpreted as first taking the first number in –(3,2) and then taking a second number and subtracting the second number from the first, but rather as the function minus-2 applied to 3. A function, after all, is nothing more than the correlation of input with output, and we can apply minus-2 to any number whatsoever with distinct output in all but special cases, and, in the abstract, correlate minus-2(0), minus-2(1), minus-2(minus-2(0)), etc. Meanwhile, minus0, minus-1, minus-2, minus-3, etc. can all be defined as the more generalized minus function recursively applied sequentially to the individual whole numbers. Thus, we need not say in theory that there are so many minus functions, rather than a single minus function, for we can obviously define minus-n or –n for any number n, in which a generalized minus function is applied only to a single non-ordered argument from the function's domain, such as the whole numbers. The reduction nevertheless seems to avoid the need to admit that even the minus function logically or mathematically requires ordered input in order

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to preserve the foundations of arithmetic. We could try to justify the reduction, on the grounds that the question itself is purely theoretical, purely philosophical, and that we do achieve a reduction of two-place ordered functionality for 1place non-ordered (and non-orderable) functionality. Similarly for truth functions, which need not be construed as having ordered input. It is not as if we have →(p,q) for an ordered input of p and q in →; rather, we have as input the unordered truth value of p and the truth value of q correlated with the truth value of p → q. The order of that input to the truth function so construed is irrelevant, it does not need to be ordered, reflected in the fact that if we are setting up the truth table for p → q, then it does not matter whether we begin with a leftmost column for the truth values of p, or a leftmost column for the truth values of q. That is to say that there is no specific order of input to a truth function. I set these scruples aside provisionally and against my own better judgment in considering Frege’s Semantic Compositionality Thesis. As far as I can see, Semantic Compositionality depends essentially on the possibility of defining a meaning function with ordered input so as to recover specific sentential sense from the nominal senses of a sentence’s component meaningful words in a specifiable grammatical order, ‘Romeo loves Juliet’ as opposed to ‘Juliet loves Romeo’, for a semantic function applied to the otherwise unordered set, f(Romeo,Juliet,loves). If Semantic Compositionality presupposes ordered input to a semantic Sinns function applied to the nominal Sinnen of meaningful terms contained in a given sentence, and if no functions unlike sausage mills can intelligibly be applied to ordered input, then Frege’s Semantic Compositionality Thesis is false. That, however, is an independent issue distinct from the questions raised by critics of Dummett’s interpretation of Frege’s account of sentential meaning on which the present study is focused. REMARK 2: Fregean Semantic Compositionality is also arguably at odds with Frege’s lifelong methodological principle articulated in his 1884 Grundlagen der Arithmetik, that: ‘The meaning of a word must be inquired after in propositional context, not in isolation’ (nach der Bedeutung der Wörter muss im Satzzusammenhange, nicht in ihrer Vereinzelung gefragt werden).12 REMARK 3: Frege insists that an individual word, a name for an object or a predicate

|| 12 Frege (2007), p. 17 (1884 edition, p. xxii).

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or concept term representing a concept, has meaning only in the context of a sentence. But… Frege also insists in the Semantic Compositionality thesis that the meaning of a sentence is composed functionally from and is equivalently analyzable as a function of the meanings of the individual meaningful words by which the sentence is constructed. These, as we have independently observed above, would need to be taken in a specific order and according to whatever grammatical principles enable the language to express at least the most elementary sentences describing the state of the world, attributing some kind of property to some kind of object. We would then apparently need to understand the meaning of such a sentence before we could understand the meanings of the meaningful words it contains, and at the same time we would also need to understand the meanings of the meaningful words a sentence contains before we could understand the meaning of the entire sentence. Such an explanation would be bootstrapping of a particularly audacious sort; although, perhaps that is exactly what is wanted somewhere or other in a largely formal theory of meaning. We can nevertheless ask: Is there a less dramatic more down-to-earth way of understanding Frege’s concept of the functional composition and functional analysis of sentential meaning from and into the most elementary meaningful terms that avoids some of these potential embarrassments?

6 Solution to Frege’s Circularity Problem It may be enough at this point to say with sufficient emphasis that for Frege the meaning of the meaningful words in a sentence is contextually determined by their contribution to the meaning of the sentence as a whole. With the meanings of individual words contextually fixed within the sentence, we can reconstruct the meaning of the entire sentence as determined by and analyzable, if permissible, as an ordered-input function of the contextually meaningful words of which the sentence is constructed or composed, in a specific order and under an applicable grammatical rule to propose the existence of a corresponding truthmaking state of affairs by virtue of its sentential Fregean Sinns. Sentential Sinns, so defined, identifies the sentence’s truth-condition, which, in light of the prevailing state of the world, makes the sentence true or false, and thereby determines the sentence’s sentential Bedeutungs, as the True

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or the False. Let us therefore consider from the ground up a Frege-friendly semantic theory that relates all three elements of an objective semantics that Dummett seems to attribute to Frege: Fregean sentential Sinns and Bedeutungs, mediated by the determination of truth-conditions as fixed by a sentence’s Fregean Sinns and the relevant facts or existent states of affairs by which the world is constituted. The task is to relate the Fregean Sinns of a sentence analyzable as a function of the ordered nominal Sinnen of the sentence’s meaningful object and concept words (proper names and predicates, the latter eventually construed by Frege as unsaturated functors).13 We propose to do so in the following way, proceeding on the basis of Dummett’s three-part interpretation of Frege’s theory of sentence meaning: Truth-condition Determination by Fregean sentential Sinns: Determines(Sinns(S),Truth-condition(S)) — ∀x[x = Sinns(S) ↔ ∃y[y = Truth-condition(S) ∧ Determines(x,y)]] Typically and informally, Sinns(S) determines the truth-maker for S by expressing the senses of sentence S (analyzed compositionally as a function of the senses of the component meaningful terms contained in the sentence) by describing or otherwise identifying the fact, state of affairs, or circumstances that would need to exist in order for S to be true. Then we have: Truth-Conditions (Truth-maker) — E!(Truth-maker(S)) ↔ Bedeutungs(S) = the True (Truth-breaker) — ⇁E!(Truth-maker(S)) ↔ Bedeutungs(S) = the False This is the final step in the interpolation of truth-making facts between a sentence’s compositionally analyzable Fregean Sinns and its Fregean Bedeutungs as the True or the False. Despite important differences with Frege, it is worth observing that Wittgenstein in Tractatus Logico-Philosophicus 4.023, offers an

|| 13 Frege (1891).

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account of truth-conditions, without the same terminology, effectively, truthmakers and truth-breakers, that is similar to the proposed three-part interpretation of Frege’s sentential semantics, when he maintains: The proposition determines reality to this extent, that one only needs to say “Yes” or “No” to it to make it agree with reality. Reality [Die Wirklichkeit] must therefore be completely described by the proposition. A proposition is the description of a fact. As the description of an object describes it by its external properties so propositions describe reality by its internal properties.14

7 Slingshot Objections to Frege’s Semantics Consider, then, the notorious slingshot argument. If two sentences have the same truth-value, then they have the same Fregean Bedeutungs (the True or the False). This, we already know, is supposed to be true in Frege’s theory of sentential meaning, for it is a matter about which he is perfectly explicit.15 What is now further argued is that the slingshot argument also shows that the correspondence theory of truth is undermined and intuitively falsified, by virtue of there being just one distinctly corresponding fact or existent state of affairs in the world corresponding to every true sentence. Suppose, as Dummett’s critics do, that Fregean sentential Sinns immediately determines Fregean sentential Bedeutungs, without mediation by Sinnsdetermined truth-makers or truth-conditions more generally, together with the facts or states of affairs of the world, by which the relevant truth-makers exist or fail to exist, or, equivalently, the relevant truth-conditions are satisfied or fail to be satisfied. Then slingshot arguments, in addition to showing that all true sentences have the same Fregean sentential Bedeutungs, also show that all true sentences have identically the same truth-maker. Thus, Drai, in her essay, ‘The Slingshot Argument: An Improved Version’, argues:

|| 14 Wittgenstein (1922). 15 See the essays on Frege’s concept of truth in Burge (2005), esp. ‘The Concept of Truth in Frege’s Program’, pp. 77–82 and ‘Frege on Truth’, pp. 83–152 (including the ‘Postscript’).

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The slingshot argument aims to establish Frege’s thesis that the reference of sentences is their truth-value. The thesis and the argument have far reaching implications in philosophy of language and metaphysics. First, the thesis can be used against the correspondence theory of truth. According to this theory a sentence is true iff the fact it states obtains. If the relation of a sentence to the fact it states respects some minimal theoretical constraints, then the slingshot argument shows that all true sentences state the same fact. Given a broadly intuitive notion of fact, the slingshot argument shows that the correspondence theory is wrong…16

Slingshot arguments or prototypes thereof are generally attributed to Frege himself, to Kurt Gödel, and others.17 Donald Davidson, in his 1967 essay, ‘Truth and Meaning’, introduces the concept in explicit application to Frege’s theory of sentential meaning: The device proposed by Frege to this end has a brilliant simplicity: count predicates as a special case of complex singular terms. Now, however, a difficulty looms if we want to continue in our present (implicit) course of identifying the meaning of a singular term with its reference. The difficulty follows upon making two reasonable assumptions: that logically equivalent singular terms have the same reference, and that a singular term does not change its reference if a contained singular term is replaced by another with the same reference.18

I do not think I understand, or, not to put too fine a point on things, I am reasonably sure that I do not understand, what Davidson means by the phrase ‘logically equivalent singular terms’, since to my knowledge logical equivalence is a matter of shared truth-values, and singular terms have no truth-values. Perhaps Davidson means that such terms must be co-referential. If this is Davidson’s objection, however, then what is the interest in showing that coreferential terms have the same Fregean nominal reference or Bedeutungn? After all, that is simply what it means for the terms in question to be ‘co-referential’ in Frege’s sense, and we could hardly expect Frege to dispute or disown that basic fact. Davidson nevertheless continues:

|| 16 Drai (2002), p. 194. 17 Church attributes to Frege’s ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik, 100, 1892, pp. 32–33, the honor of having originated the slingshot argument, and it has become customary to mention him as the argument’s originator. See Church (1943), pp. 298–304. Føllesdal (1983), pp. 91–98. Gödel (1944), pp. 125–153. Barwise and Perry (1981), pp. 387–404. See also Perry’s more recent (1996), pp. 95–114. Neale (1995), pp. 761–825. Also, Neale (1990); and Neale (2001). Young (2002), pp. 121–132. 18 Davidson (1967), p. 19.

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But now suppose that ‘R’ and ‘S’ abbreviate any two sentences alike in truth-value. Then the following four sentences have the same reference: (1) R (2) ιx(x = x . R) = ιx(x = x) (3) ιx(x = x . S) = ιx(x = x) (4) S19

Davidson seems to misdescribe slingshot reasoning in several ways. It is not that ιx(x = x . R) = ιx(x = x) and ιx(x = x . S) = ιx(x = x) as Davidson’s (2) and (3) would have it, but rather only that: Bedeutungs(ιx(x = x . R)) = Bedeutungs(ιx(x = x)) = Bedeutungs(ιx(x = x . S)) = Bedeutungs(ιx(x = x)). The conclusion is then supposed to be that therefore there is only one truthmaker for every true sentence, the Great Fact, as further explained in Davidson’s later essay, ‘True to the Facts’: Since aside from matters of correspondence no way of distinguishing facts has been proposed, and this test fails to uncover a single difference, we may read the result of our argument as showing that there is exactly one fact. Descriptions like ‘the fact that there are stupas in Nepal’, if they describe at all, describe the same thing: The Great Fact. No point remains in distinguishing among various names of The Great Fact when written after ‘corresponds to The Great Fact’. This unalterable predicate carries with it a redundant whiff of ontology, but beyond this there is apparently no telling it apart from ‘is true’.20

The important question to ask is whether Davidson is right about any of this. Is it true to say that there is only one Great Fact that serves as a conglomerated truth-maker for all true sentences, just because all true sentences are supposed by Frege to have the same sentential reference or Bedeutungs? Does Fregean sentential Bedeutungs = Fregean sentential truth-condition or truth-maker? I understand Frege as saying or implying no such thing. I also understand Dummett as interpreting Frege rather differently, interposing truth-conditions between Fregean sentential Sinns and Fregean sentential Bedeutungs. I read Frege as maintaining that sentential references ≠ sentential truth-condition, but that sentential sense fixes, determines or identifies a corresponding sentential truth-condition, and that a sentential truth-condition, together with the facts or existent states of affairs in the world, in turn fixes, determines, or identifies a

|| 19 Ibid. 20 Davidson (1969), p. 42.

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sentence’s Fregean sentential references as the True or the False. On such a (hopefully) Frege-friendly analysis of sentence meaning, there is no single truth-maker or Great Fact as Davidson proclaims by virtue of which all true sentences are made true. We can only foist such a mistaken interpretation onto Frege if we fail to observe the more complicated internal structure of Frege’s own three-part distinction in objective sentential meaning, as I believe Dummett also recognizes, between Fregean Sinns, truth-conditions, and truth-makers in particular, and Fregean Bedeutungs, in the proper semantic analysis of each meaningful sentence type or Fregean Gedanken. We cannot be so badly led astray unless we willfully conflate Fregean sentential Bedeutungens with the Fregean sentential Sinns-determined truth-conditions by which particular sentences in conjunction with the prevailing facts of the world have the True or the False as their Fregean sentential Bedeutungens.

8 Resilience of Dummett’s Fregean Semantics This is certainly a mistake that Frege does not make, if Dummett has correctly interpreted him, and if I in turn have correctly interpreted Dummett. Nor is it a mistake that contemporary logicians and semantic theorists or philosophers of language need to make in anything resembling a neo-Fregean theory of meaning, or in adopting some version of a truth-maker correspondence theory of truth. We next compare Levine’s criticism of Dummett’s interpretation of Frege as being committed to the proposition that each distinct Gedanke admits of a ‘unique ultimate analysis’ in Levine’s essay, ‘Analysis, Abstraction Principles, and Slingshot Arguments’. Levine argues that: Frege’s views regarding analysis and synonymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege’s criterion of synonymy, according to which two sentences express the same thought if one cannot understand them without regarding them as having the same truth-value.21

|| 21 Levine (2006), p. 43.

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Levine raises several provocative questions about Frege on Dummett’s interpretation. What would a ‘unique ultimate analysis’ of a sentence’s meaning be like in a Fregean framework as Dummett understands it? The analysis of the objective component of a sentence’s meaning for Frege would naturally include the sentence’s sentential Sinns and its sentential Bedeutungs. The interesting further question is how we get from one to the other, from Fregean sentential Sinns to Fregean sentential Bedeutungs, immediately or mediately? Certainly Sinns cannot do the job alone, else sentential truth and falsehood would always be a consequence of internal sentential sense-meaning alone. What, then, if Levine is right about Dummett’s interpretation of Frege, does a unique ultimate Fregean analysis of a sentence’s meaning look like? We can present it as the sentence’s sentential Sinns, a (charitably interpreted) ordered function of the nominal Sinnen of all sinnvolle Wörter in the same grammatical order as that prescribed by the meaning-building ordered function by and from which the sentence’s Fregean sentential meaning is supposed to be composed and into which it is supposed to be fully analyzable. If we follow this model, then a unique ultimate analysis of a sentence’s meaning (if, indeed, Dummett’s Frege is so committed, if Dummett’s critics are correct at least in this), would be simply: 1. 2. 3. 4. 5. 6.

Satz S = Sinnn(T1) = {P1} ∧ Sinnn(T2) = {P2} ∧ Sinnn(T3) = {P3} Sinns(S) = f ∀x[x = Sinns(S) ↔ ∃y[y = Truth-condition(S) ∧ Determines(x,y)]] E!(Truth-condition(S)) ↔ Bedeutungs(S) = the True ⇁E!(Truth-condition(S)) ↔ Bedeutungs(S) = the False

The relation posited by proposition (4) mediates between a sentence S’s semantically compositional Fregean sentential Sinns and its Fregean sentential Bedeutungs as the True or the False, not directly, but only in conjunction with the prevailing facts of the world by which either the truth-condition of S exists or fails to exist. If proposition (4) is true, and if we assume that different true Fregean Sätze and Gedanken can have different Fregean sentential Sinnes, then it does not follow from the fact that The Davidsonian Great Fact exists and makes all Fregean Sätze and Gedanken true that The Davidsonian Great Fact is therefore the massive unified truth-maker of every true sentence, a factive Parmenidean One. If the open philosophical question that the slingshot argument is supposed to help answer is whether there are distinct truth-makers for sentences distinct in meaning, then we cannot use proposition (4) in any slingshot argument con-

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text to conclude as Davidson does that The Great Fact is the truth-maker of every true sentence, unless we are prepared implausibly to deny that some syntactically distinct true sentences have distinct sentential Sinnes. Moreover, we are reminded that on the proposed account, assuming again that functions with ordered input are permitted, propositions (1)-(3) above would hold: 1. 2. 3.

Satz S = Sinnn(T1) = {P1} ∧ Sinnn(T2) = {P2} ∧ Sinnn(T3) = {P3} Sinns(S) = f

From this it follows that any sentence, regardless of its syntax, vocabulary, grammar, and lexicon, has a unique ultimate Fregean analysis of its sentential meaning or Sinns, as Dummett is credited with or accused of maintaining, provided that the terms of which the sentence is composed have distinct nominal Sinnen. We thereby account for the possibility of expressing the same sentential meaning or Sinns by means of different synonymous words within the same language and in different languages with different grammatical conventions, for sentences S = and S* = , provided only that, as proposition (2) explicitly requires, the nominal Sinnn of these terms are such that either Sinnn(T1) = Sinnn(T4) = {P1} ∧ Sinnn(T2) = Sinnn(T5) = {P2} ∧ Sinnn(T3) = Sinnn (T6) = {P3} (or occurring in another one-one, one-many or many-one correlation to accommodate grammatical differences of expression within a language and between languages). We are guaranteed a unique ultimate Fregean analysis of the meaning of distinct sentences when and only when, as often happens, both within and outside a Begriffsschrift formal language, the sentences compositionally have distinct sentential Sinnes. For such situations, we can say, more formally if somewhat less elegantly: 7.

∀S,S*[Fregean-Sentential-Analysis(S) = Fregean-SententialAnalysis(S*) ↔ Sinns(S) = Sinns(S*)]

We can then suppose from Frege’s discussion, in turn, continuing with the previous numbering of propositions to pursue the same line of thought: 8. ∃S,S*[Sinns(S) ≠ Sinns(S*)] 9. ∀S,S*[[S = ∧ S* = ∧ Tj ∈ ∧ Sinnn(Tj) = Sinnn(Ti) = {P1} ∧ … ∧ Tk ∈ ∧ Sinnn(Tm) = Sinnn(Tk) = {Pm}] ↔ Sinns(S) = Sinns(S*)]

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The model provides a unique ultimate (‘privileged’) Fregean analysis of the sentential meaning of every distinctly meaningful Satz, or alternatively of every Gedanken, of every one with a distinct Fregean sentential Sinn. Where it follows from proposition (7) again: 10. ∀S,S*[Fregean-Sentential-Analysis(S) ≠ Fregean-SententialAnalysis(S*) ↔ Sinns(S) ≠ Sinns(S*)] This equivalence, I propose, is precisely what we should want from an adequate semantic analysis of sentential meaning, both in Frege and in Dummett’s interpretation of Frege.22

9 References Barwise, Jon and Perry, John. 1981. ‘Semantic Innocence and Uncompromising Situations’. Midwest Studies in the Philosophy of Language, 6, 387–404. Burge, Tyler. 2005. Truth, Thought, Reason. Oxford: The Clarendon Press, 2005. Church, Alonzo. 1943. ‘Carnap’s Introduction to Semantics’. The Philosophical Review, 52, 298–304. Davidson, Donald. 1967. ‘Truth and Meaning’, Synthese 17, 1967, 304–323. Davidson, Donald. 1969. ‘True to the Facts’, The Journal of Philosophy, 66, 748–764. Drai, Dalia. 2002. ‘The Slingshot Argument: An Improved Version’. Ratio, 15, 194–204. Dummett, Michael. 1981. Frege: Philosophy of Language, 2nd ed. Cambridge: Harvard University Press. Føllesdal, Dagfinn. 1983. ‘Situation Semantics and the “Slingshot” Argument’. Erkenntnis, 19, 91–98. Frege, Gottlob. ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und philosophische Kritik, 100, 1892; trans., ‘On Sense and Reference’, in P.T. Geach and Max Black, Translations from the Philosophical Writings of Gottlob Frege (Oxford: Basil Blackwell, 1970), (all references to ‘On Sense and Reference’ to this edition and translation). Frege, Gottlob. 1891. Funktion und Begriff (Vortrag gehalten in der Sitzung vom 9. Januar 1891 der Jenaischen Gesellschaft für Medizin und Naturwissenschaft). Jena: Hermann Pohle. Frege, Gottlob. 1979. Letter to Edmund Husserl, 1891, in Frege, Philosophical and Mathematical Correspondence, edited by Brian McGuinness, trans. Hans Kaal Oxford: Basil Blackwell. Frege, Gottlob. 2007. The Foundations of Arithmetic: A Logical-Mathematical Investigation into the Concept of Number, trans. and introduction with critical commentary by Dale Jacquette

|| 22 I am grateful to Michael Frauchiger and the Lauener Foundation for inviting me to participate in this historic meeting to honor Michael Dummett, and to members of the audience on the occasion for their valuable comments and criticisms, many of which I have incorporated into this improved version of my original presentation.

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(of Die Grundlagen der Arithmetik: Eine logisch-mathematische Untersuchung über den Begriff der Zahl, 1884). New York: Longman Library of Primary Sources in Philosophy. Gödel, Kurt. 1944. ‘Russell’s Mathematical Logic’, in Paul Arthur Schilpp, editor, The Philosophy of Bertrand Russell (Evanston and Chicago: Northwestern University Press, 125–153. Jacquette, Dale. 1992–1993. ‘Wittgenstein’s Critique of Propositional Attitude and Russell’s Theory of Judgment’. Brentano Studien, 4, 193–220. Jacquette, Dale. 2010a. Logic and How it Gets That Way. Durham: Acumen Books. Jacquette, Dale. 2010b. ‘Truth Breakers’, Topoi: An International Review of Philosophy, 29, special issue on Logic, Meaning, and Truth-Making States of Affairs in Philosophical Semantics, guest edited by Dale Jacquette, 153–163. Levine, James. 2006. ‘Analysis, Abstraction Principles, and Slingshot Arguments’. Ratio, 19, 43–63. Neale, Stephen. 1990. Descriptions. Cambridge: The MIT Press. Neale, Stephen. 1995. ‘The Philosophical Significance of Gödel's Slingshot’. Mind, 104, 761– 825. Neale, Stephen. 2001. Facing Facts. Oxford: Oxford University Press. Perry, John. 1996. ‘Evading the Slingshot’, in Philosophy and Cognitive Science: Categories, Consciousness, and Reasoning (Dordrecht, London, Boston: Kluwer Academic Publishing, 1996), Philosophical Studies Series 69, 95–114. Wittgenstein, Ludwig. 1922. Tractatus Logico-Philosophicus, ed. C.K. Ogden. London: Routledge & Kegan Paul. Young, James O. 2002. ‘The Slingshot Argument and the Correspondence Theory of Truth’. Acta Analytica, 17, 121–132.

Dag Prawitz

To Explain Deduction Abstract: The Justification of Deduction is the title of one of Michael Dummett’s essays. It names also an important theme in his writings to which he returned in the book The Logical Basis of Metaphysics. In the essay he distinguishes different levels of justification of increasing philosophical depth. At the third and deepest level, the focus is on explaining deduction rather than on justifying it. The task is to explain how deduction can be both legitimate and useful in giving us knowledge. I suggest that it can be described as essentially being the task to say what it is that gives a deduction its epistemic force. It is a fact that deduction has such force, consisting in its capacity to provide grounds for assertions and thereby extend our knowledge, but it is a fact that has to be explained. What is it that gives a deduction this capacity? This task is more challenging than is usually assumed. Obviously, it is not the validity of its inferences, as this is usually understood, which gives a deduction its epistemic force. Truth conditional theory of meaning does not seem to have any satisfactory solution to offer, and I argue that nor have inferential theories of meaning, which take the meaning of sentences to be determined by inference rules accepted in a language. In the last part of the paper, I sketch a different approach to the problem. The main idea is here to give the concept of inference a richer content, so that to perform an inference is not only to make a speech act in which a conclusion is claimed to be supported by a number of premisses, but is in addition to operate on the grounds for the premisses with the aim of getting a ground for the conclusion. I suggest that it is thanks to such operations that deductions provide grounds for their final conclusions.

|| Stockholms Universitet This paper was written more than five years ago, shortly after Michael Dummett received the Lauener prize. In the years that followed I have continued to work on the ideas presented here and have developed them further, particularly in my essay Prawitz (2015). The paper is nevertheless published now in this volume as it was originally written, only somewhat updated, since it contains more references to Michael Dummett’s work than my later essay and could perhaps be read as an easy introduction to that essay. I am grateful to professor Cesare Cozzo for reading the manuscript and making helpful remarks.

DOI 10/1515-9783110459135-007

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1 Dummett on the justification of deduction To explain deduction is an important theme in Michael Dummett’s writings. His first publication on this theme is the essay The Justification of Deduction1, read at the British Academy, and constituting its annual philosophical lecture 1973. Here he considers three levels at which the question of a justification of deduction arises. At the first level we justify an inference by showing how its conclusion can be obtained from its premisses by a chain of simpler inferences that we accept. It is a form of justification that is philosophically unproblematic, but it leaves us with some set of unjustified basic inferences. At the second level we try to justify semantically such a set of basic forms of inferences by showing that they are valid. In such a soundness proof, as it is called, we must of course make use of some inferences, and typically we use all the inferences of the set that are to be justified and some additional ones. One may question the point of such a procedure, and part of Dummett’s essay is devoted to a discussion and defence of the philosophical significance of such a justification. But there is a third and deeper level of justification where we are not concerned with particular forms of inference but want an explanation of how deductive arguments are possible at all. It is the third level that I am interested in here. As Dummett remarks the task is not to persuade a sceptic who doubts the possibility of deduction and demands a justification before he uses it; that would be an impossible task, since a justification certainly requires deduction. The task is instead to reach an understanding of “what it is that we are doing when we reason”. We may thus speak of an explanation rather than a justification of deduction. What we have to explain according to Dummett is, in particular, how the practice of deduction can be at the same time both legitimate and useful. Deduction is useful in being capable of extending our knowledge, but as Dummett says, philosophers have often had difficulties in explaining this because of their tendency to account for the legitimacy of deduction by stressing the narrowness of the gap between premisses and conclusion, saying for instance that in a logically valid inference the premisses already contain the conclusion.

|| 1 Dummett (1974).

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In the preface to the volume Truth and other enigmas from 1978, in which the essay is reprinted, Dummett remarks that he has included the essay “not as being, as it stands, a satisfactory treatment of the topic, but as containing some ideas which may be found to suggest fruitful lines of thought”. He says that the essay concerns a subject “that had long been on my mind, and about which I have subsequently thought and written, though not yet published, a good deal”.2 He returns to the subject in the book “The Logical Basis of Metaphysics” from 1991, devoting six of the books fifteen chapters to it. Three of them stay more or less within the realm of ideas of the essay The Justification of Deduction – one having the same title as that essay, one being titled “Circularity, Consistency, and Harmony”, and one “Holism”. The other three chapters enter into what Dummett calls “Proof-Theoretic Justification of Logical Laws” and give essentially a presentation, development, and discussion of the notion of valid argument that I introduced as an attempt to make precise Gentzen’s ideas about what justifies his rules of natural deduction.

2 Relation to theory of meaning The justification of deduction has to provide for deductive inference, Dummett (1974, pp. 25–26) writes, “what a theory of meaning must provide for every component of our practice in the use of our language, an understanding of the way it works: we seek not merely a description of our practice, but a grasp of how it functions”. The justification of deduction should thus be seen as constituting a part of a theory of meaning, and it is an important part, because, as he emphasizes: “Deductive inference is an integral component of a linguistic practice … It is not an isolable subpractice, like fictional narrative, … it is connected at start and finish with the ordinary assertoric use of language. It is required to start from statements whose assertion is warranted, and it serves as a warrant for asserting the conclusion.”3 The ability to harbour an explanation of how deduction works becomes in this way a requirement on a theory of meaning. If the meaning of an assertion is given without reference to our inferential practice, we must nevertheless ac-

|| 2 Dummett (1978, p. l). 3 Dummett (1991, p. 193).

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count for how that practice is related to other criteria for when the assertion is warranted. For instance, the meaning of an observational sentence may be explained in terms of the kind of observations that have to be made in order for it to be right to use the sentence assertorically, but we must then explain why such an assertion can also be warranted when arrived at by deductive inference. Similarly, the meaning of an arithmetical identity may be explained in terms of the computation that warrants its assertion, but it must then also be explained how it comes that the truth of the identity, for instance one of the form a + b = b + a, can also be established by deductive reasoning instead of computation. But even when the meaning of an assertion is explained by how it can be proved deductively similar problems arise. For instance, an intuitionist may explain the meaning of a disjunction A ∨ B by saying that it is proved by proving either A or B. But this cannot be a complete explanation of the meaning of disjunction, because even a constructivist wants to allow the inference of A(n) ∨ B(n) for a natural number n after having established the universal generalization ∀x(A(x) ∨ B(x)) by an inductive proof, which may contain neither a proof of A(n) nor a proof of B(n). To allow for that he must therefore somehow modify or supplement the meaning explanation. He may preferably distinguish between direct or canonical and indirect or non-canonical ways of establishing the truth of a proposition and say that the truth of a disjunction is established directly by proving one of the disjuncts. To establish the truth of the disjunction via a proof of the universal generalization is to count as an indirect proof. But as Dummett remarks, one must then ask why one should allow also such indirect proofs – does not that distort the meaning of the assertion as first explained?

3 The epistemic force of inference One of Michael Dummett’s great and well-known contributions to philosophy is to have raised questions about the validity of specific logical laws, in particular, the law of the excluded third, and to have shown in what terms a discussion of the validity of logical laws can be carried on. His insistence on the need of a general justification of deduction in the sense described above, explaining within a theory of meaning the function of deduction, has received less attention. I strongly agree with him on the need of such an explanation, but I am not convinced by the proposals he makes for how to arrive at it. I think that Dummett’s main point about what should be explained about deductions can be summarized simply in saying: philosophy should explain why and how some inferences have conclusive epistemic force. An inference

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has such a force when it provides a (binding) ground for its conclusion given (binding) grounds for its premisses. I have called such an inference legitimate4 – it is the kind of inferences that can be legitimately used in a deductive proof – but it would be natural to refer to them as valid, if it was not that there is a traditional notion of valid inference that is defined very differently. In this context where we are concerned with deduction, we are only interested in binding grounds or conclusive evidence, and I shall therefore usually drop the attributes “binding” and “conclusive”. For something to deserve to be called a deduction, it should proceed by making legitimate inferences and one should have grounds for any initial premisses. Hence, a deduction provides us with a ground for its final conclusion, and in cases where we did not already have such a ground, it thereby extends our knowledge; I take knowledge to require grounds for holding propositions to be true. The main problem is therefore to explain the fact that some inferences have the epistemic power to supply the conclusion with a ground when applied to premisses for which grounds are already available. Once this is explained, which is in short to explain why certain inferences are legitimate in the sense I have given to this term, one has also explained how deduction can be both legitimate and useful, which for Dummett is the main task of a justification of deduction at the third level. Dummett proposes a quite different explanation of why deduction can extend our knowledge. The suggestion is in terms of the discernment of patterns and is made already in his first publication on the subject, The Justification of Deduction. It is developed a little further in the chapter with the same title in the book The Logical Basis of Metaphysics. Referring to Frege’s idea about the extraction of concepts from thoughts, Dummett’s proposal is that after recognizing the truth of what is asserted in the premisses of an inference, a pattern may be discerned which one did not need to have discerned in order to recognize the truths in question, but which was there to be discerned. To see the pattern will in general require a creative act, and by performing it one is able to arrive at a conclusion containing a thought that was not present in the premisses. Dummett acknowledges that Frege’s idea is not quite sufficient for explaining the fruitfulness of deduction, but after having outlined the idea he leaves the proposal, saying that Frege’s “explanation is surely along the right general lines”. I do not want to deny the interest of this idea of Frege, but I think it does not really address the problem raised by Dummett. It seems that what Dummett || 4 Prawitz (2011).

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wants to explain with this reference to Frege is not only how deduction can extend our knowledge but also how it can give us knowledge that was not even implicitly present when recognizing the truths asserted in the premisses. He would probably say that, for instance, an inference by conjunction introduction does not genuinely extend our knowledge because if we had grounds for the premisses, we would already know implicitly that the proposition asserted in the conclusion was true. I think on the contrary that an inference by conjunction introduction may constitute a genuine extension of our knowledge. We are seldom able to put together all that we know: for instance, we may have learned that the proposition p is true and that the proposition q is true, but we may never have put together these two pieces of knowledge; that could be evident from our behaviour – if we had been aware of the fact that both p and q were true and therefore also the conjunction p & q, we would not have acted as we did. To make a conjunction introduction inference is exactly to put together our knowledge of the truth of two propositions and then draw the conclusion that their conjunction is true, which may constitute an important advance of our knowledge, from which further inferences could be made. Of course, I do not want to deny that the extension of our knowledge is of quite a different character when obtained by inferences representing indirect means of establishing the truth of a proposition, and I agree completely with Dummett that the difficult task is to explain how there can be indirect means of arriving at truth. To explain the latter is however the same as explaining how inferences that do not represent the direct way of arriving at a truth can nevertheless have epistemic force. My main dissatisfaction with Dummett’s proposals for how to justify or explain deduction is however that I find a proof of soundness, what he calls a justification on the second level, to be quite unsatisfactory or almost irrelevant when it comes to explaining how an inference can have epistemic force. A soundness proof establishes the validity of certain inferences, and to begin with Dummett takes the notion of validity that is in question here in its traditional sense of preserving truth. However, the fact that an inference preserves truth does not imply that the inference has any epistemic force. A proof of the soundness of the inference together with grounds for its premisses would give a ground for its conclusion, but the epistemic force then depends on the soundness proof, while the inference in question may lack any such force. To call an inference valid when it preserves truth under all interpretations of its non-logical terms is a misnomer: it requires not only too little but also too much – for instance, it makes inference by mathematical induction invalid since

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such an inference does not preserve truth under any reinterpretation of the term natural number. Dummett is certainly open to different notions of truth, and, of course, if we vary the notion of truth, we also vary what it is for an inference to preserve truth and to be valid in that sense. But it seems likely that the preservation of truth, when that notion is taken in a reasonable way, will never guarantee preservation of grounds, which is what is required if the inference is to have epistemic force. As is well known, Dummett has been especially interested in notions of meaning and truth explained in terms of how assertions are justified, for instance by proving them, and, as already mentioned, in the book The Logical Basis of Metaphysics he devotes much attention to what he calls a prooftheoretic justification of logical laws. I shall return to these ideas in section 5.3 and shall state there why I think that even the soundness proof that use the notion of valid inference defined in that context does not give the explanation that we want of why inferences can have epistemic force.

4 The project How are we then to carry on the project that Dummett embarked on, to explain deduction, that is, in particular to explain how deduction is able to extend our knowledge? As argued above, I find this to consist essentially in the task to explain how some inferences have the epistemic power to justify their conclusions by providing grounds for them, given that grounds for their premisses are available. In short, this is the task to explain why some inferences are what I call legitimate. I take it to be a fact that some inferences are legitimate, and I expect that an explanation of this fact should proceed as explanations do generally in deducing the explanandum from definitions or properties of the concepts involved. Inference and ground are of course the two main concepts that occur in the definition of legitimate inference, but I agree with Dummett that the explanation must be a part of a theory of meaning, and the notion of the meaning of a sentence is then to be expected to play a major role in the project. One should not expect the deduction of the fact that some inferences are legitimate to be very deep or complicated. The reason why the problem of explaining how inferences can have epistemic force is often overlooked is certainly that their epistemic force seems to be so obvious and is expected to be easily deduced from the concepts involved, if one cares to do so. That the problem has in

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fact no immediate solution generally agreed upon depends in my view upon insufficient analysis of the concepts involved. If they were understood rightly, the answer to why some inferences are legitimate may be easy. The demonstration that a certain inference is legitimate is the philosopher’s concern, so to say. The person who makes a legitimate inference from premisses for which she has grounds is thereby warranted when she asserts the conclusion without doing anything more. She does not need to demonstrate that the inference is legitimate. To demand that would be to start on an infinite regress. But what we expect of a philosophical explanation of the fact that certain inferences are legitimate is a characterization of legitimate inferences from which these facts can be deduced. Before continuing this project some preliminaries about inferences may be useful. Deductions as well as (deductive) arguments, as I understand the terms, consist of a number of inferences chained to each other in one of the usual ways, that is, linearly or in form of a tree. They start from some assertions made either categorically or as assumptions. When we make an inference we draw a conclusion from some premisses. The conclusion is again an assertion, made either categorically or under some given assumptions. An inference may discharge an assumption, and the scopes of assumptions are supposed to be indicated somehow; similarly it may bind a free variable occurring in the sentences above its conclusion. In what I call a closed deduction or argument there is no undischarged assumption or unbounded variable. It is a closed deduction that is supposed to deliver a ground for its final conclusion. This is guaranteed if a deduction is defined as a deductive argument such that its initial premisses have grounds and its inferences are legitimate. It is sometimes objected to this description of inferences. One objection is that an inference may not involve the assertion of a conclusion but may consist instead in the rejection of a previous belief. It is of course true that the ultimate result of an argument may be such a rejection, but it makes little harm in this context to take the belief in question as an assumption. What occurs can then conveniently be seen as obtained in two steps, namely a first step in which a contradiction is asserted under certain assumptions, and a second step in which the negation of one of these assumption is asserted and the assumption is discharged. There is thus no serious limitation in describing a deductive argument as a chain of inferences starting from and resulting in a number of assertions, provided that we do not restrict ourselves to categorical assertions but allow that assertions are made under assumptions. When we make an inference we normally do not claim that the inference is valid. Yet there is at least an implicit claim to the effect that the premisses

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somehow epistemically support the conclusion typically indicated by inserting expressions such as “hence” or “therefore”. I shall come back to the question whether an inference is something more than a complicated speech act in which a number of assertions are made and it is claimed that one is supported by the others; I will suggest that it is.

5 Meaning and recognition of truth Dummett has argued for the close connection between, on the one hand, classical logic and truth-conditional theories of meaning which explain the meanings of sentences with reference to conditions that the world has to satisfy for the sentences to be true and, on the other hand, intuitionistic logic and theories of meaning which instead explain meanings with reference to how sentences are established as true. There is no general agreement about this. For instance, Per Martin-Löf takes linguistic meaning to be explained from an intuitionistic point of view without any reference to epistemic matters.5 I shall not enter into that discussion here, but I am inclined to agree with Dummett’s dictum: “Once we have allowed the two notions, that of truth and that of the means by which truth is recognized, to be sundered at the outset, we shall never find a means to connect them up again”. What is to count as a ground for an assertion must of course depend on the meaning of the assertion, and I find it difficult to see how this connection is to be made if there is no connection at the outset between meaning and the means by which truth is recognized or what is counted as conclusive justification of an assertion, which at the end comes to the same. I think therefore that a discussion of the concept of epistemic ground better starts in a theory of meaning. I shall consider here two kinds of theories that connect meanings with epistemic grounds, so-called inferential theories of meaning and the BHK-explanation of the logical constants.

5.1 Radical inferentialism The traditional picture that the acceptance of an inference depends on the meanings of the sentences involved is turned upside down, at least to some extent, by what has been called inferentialism or inferential theories of mean-

|| 5 See e.g. Martin-Löf (1998) and Prawitz (2012).

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ing, which take the meaning we attach to a sentence to be determined by the inferences involving the sentence that we accept. A most radical theory of that kind, first suggested by Carnap for formalized languages, takes linguistic meaning to be determined by all the inferences that the speech community accepts. As Dummett points out, this is to adopt a holistic picture of language and to trivialize the attempt to explain deduction, saying simply “we speak as we choose to speak”; all inferences that we accept are said to be legitimate in virtue of the meanings of the involved sentences, which are determined simply by our using them in deductive inferences treated as legitimate, and we are free to let our sentences mean what we like. According to such a picture, if we enrich our language by adding new concepts with their forms of inferences and are then able to deduce sentences that occur already in the previous language but could not be deduced there, we are changing the meaning of these sentences. This is to deny that we may get new knowledge by these deductions. For instance, going up to a higher order language in mathematics, we may be able to prove sentences that occurred already in the first order language but could not be proved there. This is ordinarily understood as gaining knowledge concerning questions that were possible to express already in the poorer language. In contrast, on the holistic picture, the meanings of these questions have now been changed; it is not the old questions that are settled but new ones. This is not a credible description of the actual situation, and seems to me to be a decisive argument for rejecting this picture of language.

5.2 Moderate inferentialism An inferential theory of meaning that is more restrictive in what kind of inference rules it takes to be meaning constitutive has been worked out by Cesare Cozzo (1994). He considers what he calls argumental rules obtained by generalizing the notion of inference rule, but lets only immediate rules be meaning constitutive, that is, rules not derivable from other rules. A main idea is that a rule contributes to determining the meaning of an expression only when the expression does not belong to a language fragment that must already be mastered if one is to understand the rule. It requires that one can establish a transitive and non-symmetric presupposition relation between the words of a language: the understanding of a word may require the understanding of some other words, and this presupposition is sometimes reciprocal but not always. There is thus a strict, partial order between words of a language: a word w1 comes before (precedes) a word w2 in this order, if w1 is presupposed by w2 but

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not vice versa. The meaning of a word is now taken to be determined only by the immediate argumental rules of the language that concern the word, a notion defined in two steps: a rule immediately concerns the words that are mentioned in the formulation of the rule and presuppose all the (other) words mentioned by the rule, and a rule concerns a word w if it immediately concerns a word presupposed by w. For instance, a rule by which one can infer that x is a parent of y from the premiss “x is a father of y or x is a mother of y” does not immediately concern “or”, given that the understanding of “or” does not presuppose the understanding of “parent”, but may naturally be taken to immediately concern all the other words mentioned by the rule. The meaning of a sentence is taken by Cozzo to be determined by its syntactic structure and the meanings of the words occurring in the sentence. The theory of meaning is thus compositional, and the meaning of a sentence is independent of rules concerning words that are not presupposed by the words of the sentence. Hence, unlike the situation in radical inferentialism, the meaning does not change when new words are added, even if thereby new possibilities arise for deducing the sentence. However, all immediate inference rules are taken, as in radical inferentialism, to be meaning constitutive. This is in my view to see too many rules as meaning constitutive. Consider for instance inference by mathematical induction, where we infer that all natural numbers have a certain property after having deduced that the property holds for 0 and for n+1 if it holds for the natural number n. This is in first order arithmetic an instance of an immediate inference rule; it cannot be derived from other rules. According to the present view, the inference rule is simply postulated in that context and contributes to the meaning of the conclusion. A rival view is that the meaning of the conclusion, among other things the meaning of the term natural number, is already fixed independently of the inference rule of mathematical induction, and that an inference according to that rule is legitimate in virtue of that meaning of the term natural number; in other words, the ground for the conclusion is seen as being obtained by the inference, being constructed from the grounds for the premisses, not as being postulated to exist as a part of what gives meaning to the term natural number. If the theory of meaning and the concept of ground can be developed so as to bolster this rival view, this seems preferable. We should not take an inference rule as determining the meanings of involved words, I think, if its acceptance can be explained by showing that its instances indeed transfer grounds from the premisses to the conclusion because of how the meanings of the involved words are already determined.

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5.3 Gentzen’s explanation of meaning and the validity of arguments Gentzen was the first to suggest an inferential theory of meaning where only some of the basic inference rules were taken as meaning constitutive. More precisely, as is now well known, he suggested that the introduction rules of his natural deduction should be considered as determining the meanings of the logical constants concerned. Furthermore, he observed that the elimination rules could be justified by noting that they were in accord with the meanings determined by the introduction rules. This can be made precise by saying that if the major premiss of an elimination inference is inferred by introduction, then by applying certain reduction operations a deduction of the conclusion can be obtained already from the deductions of the premisses without using the elimination inference. These ideas are the ones that I tried to generalize by defining a notion of validity of (deductive) arguments in general6, proceeding by arbitrary inferences, which was taken up by Dummett (1991), as mentioned above (section 3). In this generalization some inferences are to be picked out as determining the meanings of the sentences occurring in their conclusions, and the other not meaning constitutive inferences are to be assigned reduction operations in order to justify them in a way analogously to how Gentzen’s elimination rules were justified. The way in which the meaning constitutive inferences are thought of as determining the meanings of different sentences is by the stipulation that they constitute the direct or canonical ways in which the sentences are established as true. An argument that ends with such an inference is said to be canonical or in canonical form and should thus be legitimate, transfer grounds, since, according to the meaning of the sentence occurring in the conclusion, this is how the sentence is directly established as true. Formally, this is reflected in defining a closed canonical argument as valid, if its immediate subarguments are valid. If we want we may identify the truth of a sentence with it having a closed valid canonical argument. As to inferences that are not meaning constitutive, the idea is that they are justified depending on whether the reduction operations assigned to them really transform or, in other words, reduce them to canonical form. Formally, this is reflected in defining a closed non-canonical argument as valid, if it

|| 6 Prawitz (1973) or (1974).

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reduces (relative to the assigned reduction operations) to a valid closed canonical argument ending with the same sentence. Having defined the notion of valid argument, an inference may be defined as valid when the result of attaching it at the end of valid arguments for the premisses yields a valid argument for the conclusion. Identifying the truth of a sentence with it having a closed canonical argument, a valid argument as now defined still has the property to preserve truth. It is now easy to show that all derivable inferences in intuitionistic predicate logic are valid in this sense. The question is whether this soundness proof gives us what we want. Being in possession of a valid closed canonical argument for an assertion, one is in possession of what makes the asserted sentence true, and one may think that one then has a ground for the assertion. But note that the validity depends on the validity of the arguments for the premisses of the last inference, and they may be non-canonical or open. When in possession of a closed noncanonical argument, one has in fact the means to transform it to a canonical argument and thus to come in possession of what makes the asserted sentence true. But regardless of whether this can be considered to be in the possession of a ground for the assertion, it is not the inference itself that provides one with this possible ground. By a new deductive argument we may be able to prove that the assigned operations really bring the argument into canonical form, and hence that, in fact, the argument is valid and the asserted sentence is true, but we are then back in the situation that it is not the inference in itself that has epistemic force – the ground or the knowledge of the truth of the asserted sentence comes primarily from another deduction. We must conclude that to be in possession of a valid closed argument cannot in general be taken to have a ground for the assertion that occurs as its conclusion, and hence that the soundness proof does not show the inferences to have epistemic force. Dummett defines the notion of valid argument somewhat differently from how it is described above, dropping the justifying operations from the argument and requiring from a valid closed non-canonical argument only that there exists an effective operation that takes it to canonical form. A valid argument then contains no information of how to transform it to canonical form and is even farther away from constituting a ground for asserting its conclusion.7

|| 7 For a further discussion of how Dummett’s and my notions are related, see Prawitz (2006). Another development of the notion of valid argument is presented by Schroeder-Heister (2006). Concerning the relation between valid arguments and BHK-proofs, which are taken up in section 5.4, see Prawitz (2016).

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Although a valid argument (in its different versions) contains anyway more information than an argument proceeding by inferences that are valid in the sense this is traditionally defined, it does not give us the account we are now asking for, namely an explanation of what it is that gives a deduction its epistemic force.

5.4 Heyting and the BHK-explanation The situation is much the same if we build a meaning theory on the well-known BHK-explanation of the logical constants, which was presented as spelling out ideas of Heyting in particular. He saw the meaning of a proposition as determined by an intended construction and an assertion of a proposition as stating that the intended construction had been found. The BHK-explanation supplements Heyting’s explanations by defining inductively what counts as a construction of a first order compound statement in terms of what counts as a construction of its immediate parts. As originally presented, it does not pay attention to how the construction is denoted, but if that is done a distinction has to be made, as in the case of arguments, between presenting a construction in canonical and non-canonical form, where a non-canonical presentation of a construction should contain information of how the canonical presentation is found. Here we have again a theory of meaning that explains the meaning of a sentence in terms of what has to be accomplished in order to use it warrantedly in making an assertion. If we make use of the notion of truth, although foreign to Heyting, and count a proposition as true when there is a construction of it, we can say that the meaning of a proposition is explained in terms of how its truth is recognized. But the explanation is not be counted as an inferential theory of meaning, because, in spite of the fact that what Heyting calls constructions are called “proofs” in the BHK-explanation, they are not built up by inferences but consist rather of mathematical objects; for instance, in the case of an implication A → B, the required construction is an operation or function that applied to a construction of A gives as value a construction of B. Although this theory of meaning is not concerned with inferences and proofs as usually understood, it can define an inference to be valid when a construction of its conclusion can be found given constructions of its premisses; again, it preserves truth as that notion is now understood. It is easily shown that all inferences derivable in intuitionistic first order logic are valid in this sense. But again such a soundness proof does not show that the inference is legitimate. It is true that if the possession of the construction intended by a proposition p is

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counted as a ground for asserting p (which I think is reasonable, as I shall argue in the next section), the validity of an inference together with grounds for the premisses (in the form of constructions of them) will guarantee the existence of a ground for the conclusion. However, it is not the inference itself that provides this ground for the conclusion. How a construction of the conclusion is to be found from constructions of the premisses has in general to be shown by a proof. There is no limit on how difficult that task can be; a valid inference is surely not in general legitimate. Although the last two meaning theories that we have considered do not clarify what it is that gives a deduction epistemic force, they have two features that seem valuable for this project: the meaning of a sentence is given in terms of a certain direct or canonical way in which its truth is established, and to assert the sentence warrantedly one must either have established its truth in that way or have accomplished something that can be transformed to the canonical way of establishing the truth of the sentence. These features will come back in some form in what follows.

6 Grounds for assertions To get any further we must make precise what is to be understood as a ground for an assertion. For an everyday example of the use of this notion, we may consider so-called observation sentences: to have made the relevant observation would commonly be taken to have a ground for uttering the sentence with assertive force. For a mathematical example we may consider arithmetical identities: to have made the relevant calculations may be counted as grounds for asserting that two terms are identical. For the assertions of compound propositions in mathematics we are used to take deductions as grounds. However, it would of course be circular to define a deduction as a chain of legitimate inferences, a legitimate inference as one that transmits grounds for the premisses to a ground for the conclusion, and then a ground for an assertion as a deduction of the assertion. For the present project we must seek a notion of ground for an assertion that is independent of the concept of deduction. Heyting’s ideas about meaning supplemented with the BHK-explanation of the logical constants seem to offer a theory of meaning where this is possible. If we, following Heyting, take assertions as saying that the construction intended by the asserted proposition has been found, we must regard the possession of

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the construction as justifying the assertion and thus as constituting a ground for the assertion. One may note that this could be characterized as a truth-conditional theory of meaning if we identify the truth of a sentence with the existence of the intended construction. Its crucial feature for the present project is that what makes a sentence true, its truth-maker to use an expression that Martin-Löf and Sundholm have favoured in this context, is not a certain ontological state that may be inaccessible to us but something that we can be in possession of. This is what makes it possible to identify the possession of the truth-maker of a sentence with having a ground for the corresponding assertion, which seems to be an attractive outcome. I therefore suggest that we take as grounds for the assertion of compound sentences their constructions, that is, what is referred to as “proofs” of the sentences in the BHK-explanation. The constructions, which I shall henceforth refer to as grounds, may be seen as abstract objects built up by various operations. To speak of them we shall of course use certain notations or terms. They will have to be distinguished with respect to whether they are in canonical or noncanonical forms, but this distinction is not to be applied to the grounds themselves. For instance, a ground for asserting a conjunction A & B is defined to be obtained by bringing together two grounds g and h for asserting A and for asserting B, respectively, by applying an operation which we may call conjunction grounding or, if we want to carry over Gentzen’s terminology for deductions to grounds, conjunction introduction. The result of applying this operation may be written &I(g,h). Similarly, a ground for asserting an implication A → B is defined to be obtained by applying an operation called implication introduction to an open ground g(x) for asserting that B is true under the assumption that A is true; g(x) is then an unsaturated ground that becomes a ground g(h) for (categorically) asserting B when saturated by a ground h for asserting A. This operation binds the open occurrences of x in g(x), and the result of applying it may be written →Ix(g(x)), where the variable x placed after the operator is to indicate that the operator binds it. The variable has to be typed to indicate what grounds it ranges over, and the compound terms also receive types thereby. This explanation agrees of course with the meaning attached to compound sentences from an intuitionistic point of view, but there is not necessarily something particularly intuitionistic about defining meaning in terms of grounds for assertions. What is taken as ground may of course differ depending on whether we intend a classical or an intuitionistic reading. For instance, in the case of

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disjunction and existential quantification what are counted as grounds is much weaker at a classical reading compared to an intuitionistic reading. To think of meaning as determined by grounds for assertion may be alien to a realistic conception of meaning and may not result in a bivalent notion of truth, but these questions may be left open at this stage.

7 What is an inference? One source of difficulties that is met when trying to account for the epistemic force of deductions is, I think, that too little content is given to what it is to make an inference. If to make an inference is only to make assertions organized as premisses and conclusion, indicated for instance by inserting “because” before the premisses or “hence” before the conclusion, then it is no wonder that it is difficult to explain why the mere making of an inference makes one justified in asserting the conclusion. Phenomenologically we experience the making of an inference as involving something more than a speech act of that kind. To draw a conclusion could be described as a thought process in which, starting from some truths that have already been recognized or are only assumed, we come to see that something else is also true or is true under the assumptions made. In logic we do not want to fall back on inner, mental processes of that kind when describing inferences, but should try to represent them as acts of some kind that go beyond mere speech acts. Having defined a ground for an assertion along the lines sketched above, we get the possibility to regard an inference as an act in which one transforms given grounds for the premisses with the aim of getting a ground for the conclusion. As exemplified above, a ground for the assertion of a compound sentence is built up by applying an operation to other grounds. This operation, which has to be taken as primitive, is such that when applied to the grounds for the premisses of an instance of the corresponding introduction rule, the result is precisely the ground for the conclusion that the explanation of its meaning stipulated to be required in order for the conclusion to be asserted warrantedly. An inference by introduction may consequently be seen as the act of applying the operation in question to grounds for the premisses of the inference. For other inferences we can define operations that applied to grounds for the premisses yield a ground for the conclusion. For instance, we can define two operations, which we may call conjunction elimination, &E1 and &E2, on grounds for the assertion of a conjunction by the two equations

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&Ei(&I(g1,g2)) = gi (i = 1 or 2). They will transform a ground for the assertion of a conjunction to a ground for the assertion of one of the conjuncts. Thus, they will transform a ground for a premiss of a conjunction elimination inference to a ground for the conclusion of the inference. Similarly, we define an operation implication elimination, →E, on two arguments, one of which is to be a ground for asserting an implication A → B, while the other is to be a ground for asserting A, by the following equation →E(→Ix(g(x)),h) = g(h). It will transform grounds for the premisses of an inference by modus ponens to a ground for the conclusion of the inference. I suggest that we see an inference as an act that is individuated, not only by assertions occurring as premisses and conclusion and possibly the discharge of assumptions and the binding of variables, but also by an operation defined on possible grounds for the premisses. To carry out the inference act or to make the inference is then not only to make a speech act but is in addition to apply the operation in question to the given grounds for the premisses. To apply the operation is to transform the grounds for the premisses according to how the operation is defined. For instance, if the operation is →E, then to apply it to given grounds →Ix(g(x)) and h for the premisses is to transform them to g(h), which is to saturate the open ground g(x) by the ground h. It is represented at a syntactical level by substituting the term h for the variable x in the term g(x). If the ground for the major premiss is given by a term t that is not in canonical form, we have to write the result of applying the operation as →E(t,h) and cannot immediately compute its value. But if t is a ground for the assertion of an implication, we know that it can be written in canonical form and that the value can then be computed in the described way. When an inference is understood in this way, I define it as deductively valid when a ground for the conclusion is actually obtained by applying the operation.

8 Conclusions If one accepts the proposed meaning theory with its concepts of ground and justification and the view of inferences that have been sketched above, it follows at once from how the concepts have been defined that a deductively valid

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inference is legitimate. Consequently, on the assumption that to be justified in making an assertion is to have ground for it, if one performs a deductively valid inference and is justified in making the assertions occurring as premisses, one becomes justified in making the assertion occurring as conclusion. In the manner suggested in section 3, this explains the epistemic force of a valid inference as now defined. We may illustrate by an example. If we are justified in asserting A and A → B categorically, we are in possession of grounds for the assertions. By definition, a ground for the assertion of an implication A → B is defined to have been obtained by implication introduction from a ground g(x) for asserting that B is true under the assumption that A is true. It can thus be written →Ix(g(x)). If we then make an inference by modus ponens, which now means that we do not only assert B but also apply the operation →E to the ground →Ix(g(x)) and a ground h for the assertion of A, we obtain thereby the construction g(h), which is a ground for asserting B. To be justified in asserting B, it has not been required that we have proved that g(h) is a ground for B. It is certainly easy to prove that g(h) is a ground for asserting B by using modus ponens. However, the epistemic force of modus ponens does not consist in this possibility but in the fact that when making an inference by modus ponens, having grounds for the premisses, one carries out an operation by which one actually gets a ground for the conclusion. Defining a deduction as a chain of valid inferences in the new sense, we get that a deduction is justifying its conclusion, not in the sense of constituting a ground for the conclusion, but in the sense of providing such a ground, since the operation associated to the last inference of the deduction has as output a ground for the conclusion.

References Cozzo, Cesare 1994, Meaning and Argument: A Theory of Meaning Centred on Immediate Argumental Role, Almqvist & Wiksell International, Stockholm. Dummett, Michael 1974, The Justification of Deduction (from The Proceedings of the British Academy, LIX (1973)), British Academy, London. Dummett, Michael 1978, Truth and other enigmas, Duckworth, London. Dummett, Michael 1991, The Logical Basis of Metaphysics, Duckworth, London. Martin-Löf, Per 1998, “Truth and knowability: On the principle C and K of Michael Dummett”, in Truth in mathematics, (eds) H. G. Dales and G. Olivieri, Clarendon Press, Oxford, pp. 105– 114.

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Prawitz, Dag 1973, “Towards a foundation of a general proof theory”, in Logic, Methodology, and Philosophy of Science IV, (eds) Suppes P. et al, North-Holland Publishing Company, Amsterdam, pp. 225–250. Prawitz, Dag 1974, “On the idea of a general proof theory”, Synthese 27, pp. 63–77. Prawitz, Dag 2006, “Meaning Approached via Proofs”, Synthese 148, pp. 507–524. Prawitz, Dag 2011, “Proofs and Perfect Syllogisms”, in Logic and Knowledge, (eds) C. Cellucci et al., New Castle upon Tyne, Cambridge Scholars Publishing, pp. 385–402. Prawitz, Dag 2012, “Truth and Proof in Intuitionism”, in Epistemology versus Ontology, (eds) P. Dybjer et al, Springer, Cham, pp. 45–67. Prawitz, Dag 2015, “Explaining deductive inference”, in Dag Prawitz on Proofs and Meaning (Outstanding Contributions to Logic vol 7), (ed) Heinrich Wansing, Springer, Cham, pp. 65–100. Prawitz, Dag 2016, “On the relation between Heyting’s and Gentzen’s approaches to meaning”, in Advances in Proof-Theoretic Semantics (Trends in Logic vol 43), (eds) T. Piecha and P. Schroeder-Heister, Springer, Cham, pp. 5–25. Schroeder-Heister, Peter 2006, “Validity concepts in proof-theoretic semantics”, Synthese 148, pp. 525–571.

Ian Rumfitt

Prospects for Justificationism Justificationism in the theory of meaning is the thesis that a statement’s content is given by the grounds for asserting it.1 It was Michael Dummett who coined the term ‘justificationism’, and for half a century he espoused the thesis it designates (although not always under that name). The main conclusion of his early paper, ‘Truth’, is that we should ‘no longer explain the sense of a statement by stipulating its truth-value in terms of the truth-values of its constituents, but by stipulating when it may be asserted in terms of the conditions under which its constituents may be asserted’ (Dummett 1959, 17–18). The context makes it clear that this is a formulation of justificationism: a speaker may assert a statement when, and only when, he has sufficient grounds for his assertion; complaints that the assertion would be impolite, or would break a confidence, are not to the point. Forty-three years later, in his Dewey Lectures of 2002, we find Dummett still propounding ‘a justificationist theory of meaning, the meaning of a form of statement is constituted by what is needed to establish it as true’

|| All Souls College, Oxford University For comments on a draft of this paper, I am very much indebted to Christopher Peacocke. Note added, June 2016. I wrote this paper in 2010, and returned to the prospects for justificationism in chapter 5 of my book, The Boundary Stones of Thought: An Essay in the Philosophy of Logic (Oxford: Clarendon Press, 2015). My conclusions there are more pessimistic than those reached in the present essay. I remain convinced, however, that the evidentialist theory of meaning adumbrated here – more particularly, the bilateral evidentialism of § 4 below – is the justificationist’s best chance of providing a satisfactory account of the determinants of content. How optimistic or pessimistic he should be, I leave readers of the present essay and my book to judge. || 1 By a ‘statement’, I understand an ordered pair whose first element is a declarative type sentence (identified as belonging to a language), and whose second element comprises all the contextual features that may be relevant to assessing the truth or falsity of an utterance of that type sentence. (Thus if a statement’s first element is the English type sentence ‘You are ill’, its second element will comprise the addressee and the time of utterance.) Because they belong to languages, statements in this sense are not Fregean thoughts: a statement has, or expresses, a propositional content; it is not itself such a content. But it makes sense to predicate truth or falsity of statements simpliciter. It also makes sense to speak of a statement’s being asserted and denied. A statement will be asserted (denied) when the relevant type sentence is asserted in a context where the speaker commits himself to its truth (falsity).

DOI 10/1515-9783110459135-008

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(Dummett 2003, 53). In his Gifford Lectures, published in 2006, Dummett also defends the doctrine. There he remarks that, for a justificationist, grasping a statement’s content ‘will consist…of an ability to recognize evidence for [the statement] when presented with it, and to judge correctly whether or not it is outweighed by any given piece of counter-evidence’ (Dummett 2006, 59). In those two sets of lectures, Dummett explores very different ways in which a justificationist might deal with statements about the past.2 Some contemporary philosophers, however, regard justificationism as a non-starter among theories of meaning – or, at best, as a runner that falls at far lower hurdles than those presented by discourse about the past. Thus, in discussing ‘assertabilityconditional semantics’ (his preferred term for justificationism), Timothy Williamson writes that this tradition in the theory of meaning began with one more or less working paradigm: Heyting’s intuitionistic account of the compositional semantics of mathematical language in terms of the condition for something to be a proof of a given sentence. The obvious and crucial challenge was to generalize that account to empirical language: as a first step, to develop a working assertabilityconditional semantics for a toy model of some small fragment of empirical language. But that challenge was shirked. Anti-realists preferred to polish their formulations of the grand programme rather than getting down to the hard and perhaps disappointing task of trying to carry it out in practice. The suggestion that the programme’s almost total lack of empirical success in the semantics of natural languages might constitute some evidence that it is mistaken in principle would be dismissed as crass (Williamson 2007, 282).

In this paper, I take up the challenge of developing a workable justificationist semantics for empirical discourse, although I will not take it up in quite the way that Williamson suggests. He expects the justificationist to respond to his challenge by generalizing the Heyting semantics for the language of intuitionistic mathematics, whereas I shall argue (in § 2) that Heyting’s theory limps even as an account of the intuitionists’ use of mathematical statements, and is certainly no model for a theory of meaning for empirical discourse. Consequently, the rest of the paper will be devoted to elaborating an alternative form of justificationist semantics – one I call bilateral evidentialism. In a single paper, I only have room to sketch how the bilateral evidentialist can deal with a few of the problematic constructions that natural languages present. But I hope to say enough to cast doubt on Williamson’s conclusion that justificationists ‘proceed as if Imre Lakatos had never promulgated the concept of a degenerating research programme’ (op. cit., 284). Whilst a great deal of further work is needed to discover

|| 2 For further discussion of this topic, see Peacocke 2005 and Dummett 2005.

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whether justificationism can yield a satisfactory semantics for natural language, it is premature to deem it to be a non-starter. In arguing that it is premature to set justificationism aside, I am of course siding with Dummett, but there is an important respect in which my position would be uncongenial to him. As the passage quoted from Williamson hints, Dummett supposes that a justificationist semantics will be modelled on the Heyting semantics for the language of intuitionistic mathematics. For this reason, Dummett’s arguments for justificationism extend to give him arguments against the unrestricted validity of classical logic. Although, as we shall see, bilateral evidentialism does not compel its adherents to accept full classical logic, it differs from Dummett’s semantics in accommodating that logic without strain. Since many people regard the forced deviation from classical logic as a demerit of Dummett’s version of justificationism, they may see this as an advantage of bilateral evidentialism.

1 The obligations of minimalism about truth First, though, I need to say something about why justificationism might be thought to be a plausible position in the theory of meaning, or at least, why it might be interesting to see how far it can be developed; for the merits, or interest, that a philosophical theory is perceived to have perforce constrain its elaboration. Dummett’s main argument for justificationism is that it is the only theory of meaning which makes possible a non-circular account of what it is to understand a statement. The salient contrast is with the more familiar truthconditional theory, whereby a statement’s content is constituted by the conditions under which it is true. According to Dummett, a theory of meaning is of interest only if it is a theory of our knowledge of meaning: as he puts it, a theory of meaning is a theory of understanding. On this conception, the key thesis of the truth-conditional theory is the claim that understanding a statement is a matter of knowing under what conditions it is true. In general, however, this knowledge must be implicit, and Dummett holds that an ascription of implicit knowledge to a speaker is vacuous unless it amounts to attributing to him a disposition, the possession of which may be fully manifest in his behaviour. Now knowledge of the conditions under which a statement is true cannot always be cashed out as a fully manifest disposition. At least, this is so if our conception of truth is the usual realist one, whereby a statement may be true in circumstances where no one can recognize that it is true. For, Dummett sup-

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poses, the only plausible candidate to be a disposition, possession of which amounts to knowing a statement’s truth conditions, is the disposition to assent to it in circumstances where one recognizes that it is true. And if a statement may be true in circumstances where no one can recognize its truth, then a speaker’s implicit knowledge that a statement is true in, and only in, certain conditions will not be fully manifest in his disposition to assent to the statement in the circumstances in which it may be recognized as true. By contrast, Dummett claims, knowledge of the conditions in which we have grounds for asserting the statement may be fully manifest in a speaker’s behaviour: such knowledge will be manifest in the speaker’s disposition to assert the statement when he has such grounds, and to refrain from asserting it when he does not. Critics have challenged this argument at a number of points. In the chapter from which I have already quoted, Williamson complains that the demand for a non-circular account of understanding ‘is motivated more by preconceived philosophical reductionism than by the actual needs of empirical linguistics’ (2007, 282). In other writings, he has argued that Dummett’s manifestability requirement is so strong that even a justificationist cannot meet it: if knowledge of a statement’s grounds is to be fully manifest in behaviour, he claims, then whenever we have such grounds, we must be in a position to know that we have them, and Williamson has an argument to show that this condition is almost never met.3 For present purposes, however, we need not evaluate these objections. For the argument for justificationism that I wish to consider differs fundamentally from the main Dummettian argument just outlined: the argument I have in mind does not even assume that a theory of meaning must be a theory of understanding. In approaching this second argument for justificationism, we may begin by observing that ‘By virtue of what does a statement have the content that it has?’ is a good philosophical question. In Pierre’s mouth, the statement ‘La neige est blanche’ has, or expresses, a propositional content. We can specify that content by saying that Pierre’s statement expresses the thought that – or, more simply, says that – snow is white. However, it is surely not a brute fact that his statement has this content: the statement says that snow is white by virtue of other facts about it. Accordingly, it must make sense to ask what these other facts are. That is: it must make sense to ask, ‘By virtue of what does this statement say that snow is white?’ It falls to the philosopher of language – and, more particularly, to the theorist of meaning – to answer this question and similar ones. No

|| 3 For the argument in question, see Williamson 2000, chap. 4, esp. 110–113. For criticism, see Berker 2008.

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doubt any plausible answer will advert to facts about speaker-hearers who understand the statement under consideration. All the same, the question is not directly about understanding. To ask ‘By virtue of what does a statement say what it says?’ is not to ask ‘What does someone who understands the statement know?’ The latter question belongs to what one might call the theory of understanding, or the epistemology of language. The former question, by contrast, is not epistemological but constitutive. One might say that it belongs to the metaphysics of content, or the metaphysics of meaning. In this paper, I shall understand justificationism to be an answer to this constitutive question about content. I shall understand it, in other words, to be a thesis in the metaphysics of meaning. So understood, the thesis comes to this: a statement has the content that it has by virtue of having the grounds that it has. More precisely: a statement has the content that it has by virtue of the fact that any speaker who understands it, and who respects the general norms of assertion, will produce assertive instances of it only when he is in possession of certain grounds. As we have seen, this is not how Dummett understands justificationism; for him, justificationism is a contribution to a theory of understanding. However, even when justificationism is taken to answer the metaphysical question about content, we may still discern an analogue of the rivalry that preoccupies him. Our question is: ‘By virtue of what does a statement say what it says?’ And many philosophers hold that the best answer to this question is framed in terms of truth conditions. That is, they hold that Pierre’s statement says that snow is white by virtue of the fact that it is true if and only if snow is white. Or, more precisely, they hold that Pierre’s statement says that snow is white by virtue of the fact that anyone who understands it knows that it is true if and only if snow is white. There are, then, truth-conditional accounts of the constitution of linguistic content to which justificationism (as I understand it) is an alternative and a potential rival. Moreover, although Dummett does not directly engage with our metaphysical question, one of his arguments shows that a whole class of philosophers must reject the truth-conditional answer to it, so that they, at least, have reason to investigate seriously the justificationist alternative to that answer. To see which class of philosophers I have in mind, let us first remark that a similar constitutive or metaphysical question may be asked about truth itself. Given any candidate for assessment as true or as false – given any truth-bearer – we can ask: ‘What is it for that truth-bearer to be true?’ Now a perennially popular answer to this question about truth is some form of deflationism or minimalism. Any truth-bearer – whether it is a statement, an individual assertive speech action, a belief, or whatever – must have or express a propositional content: a

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statement (or assertion, or belief, or…) is always a statement (or…) that things are thus-and-so. The kernel of minimalism about truth is the claim that what it is for a statement (or…) that things are thus-and-so to be true is, simply, that things should be thus-and-so. For example, what it is for a statement that snow is white to be true is, simply, that snow should be white. One finds different versions of this account of truth in a number of philosophers, notably Ramsey (1927, 1991 [originally 1928]), Quine (e.g. 1986), and Prior (1971). But one of Dummett’s signal early contributions to philosophical discussion about truth was to demonstrate, as conclusively as such things can be demonstrated, that no version of minimalism about truth can cohere with a truth-conditional account of the metaphysics of content (Dummett 1959, 6–7). The precise form the demonstration takes depends on which species of minimalism is being considered, but we may take as representative the argument as it applies to the account of truth which A.N. Prior distilled from Ramsey’s writings (see Prior 1971). Prior put forward an explicit definition of truth whereby a statement is true if and only if there is a way things may be conceived to be such that the statement says that things are that way, and things are indeed that way (op. cit., 98). Symbolically, a statement s is true if and only if ∃P(s says that P ∧ P).4 Given that this is a definition of truth, the answer to our metaphysical question ‘What is it for a statement s to be true?’ must be this: ‘There is a way things may be conceived to be such that s says that things are that way, and they are that way’. The crucial point about this answer is that it takes as already understood the notion of a statement’s saying that things are thus-and-so. It then follows that we cannot return this answer to the question ‘What is it for a statement to be true?’ whilst also returning a truthconditional answer to the question ‘By virtue of what does a statement say what it says?’ For consider a truth-conditional theorist who answers the question ‘By virtue of what does the statement “La neige est blanche” say what it says’ by replying: ‘The statement says what it says by virtue of the fact that it is true if and only if snow is white’. Suppose that, when asked further what it is for a statement to be true, he replies: ‘It is for there to be a way things may be || 4 This definition of truth is not gainsaid by Tarski’s theorem that truth is indefinable, for Prior does not assume the truth of all Tarskian T-sentences. Indeed, he differs from Tarski in denying that truth is primarily a property of sentences at all; see Prior 1971, chap. 7. More generally, Prior regards Liar-like arguments as proofs that certain paradoxical utterances and inscriptions do not say anything, and so are not even candidates for evaluation as true; see op. cit., chap.6. (This view allows that inscriptions of, for example, ‘This statement is false’ in other, nonparadoxical contexts, might be true.) Much needs to be said about how this approach to the Liar can deal with revenge paradoxes. For a start on this task, see Rumfitt 2014.

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conceived to be such that the statement says that things are that way, and they are that way’. Now the way the present statement says that things are is the way they are when snow is white; so what it is for the statement to be true is for snow to be white. Accordingly, our theorist’s answer to the first question reduces to this: ‘The statement says that snow is white by virtue of the fact that snow is white if and only if snow is white’. It is then clear that his answer is inadequate: ‘snow is white if and only if snow is white’ is a tautology, and it cannot be right to say that a statement says what it says by virtue of a tautology. In rehearsing Dummett’s argument in the present argumentative context, I am claiming only that a minimalist theory of truth cannot combine with the truth-conditional theory of content to provide coherent answers to our two constitutive questions ‘What is it for a truth-bearer to be true?’ and ‘What is it for a statement to have the content that it has?’ When one tries to combine them, the apparent knot in the string is pulled straight, so that neither question is answered. I do not deny that there may be other philosophical projects to whose completion versions of these theories can jointly contribute. In particular, it may be that anyone who grasps the concept of truth knows (implicitly) that a truth-bearer is true if and only if, for some P, it has (or expresses) the content that P, and P; it also may well be that anyone who grasps the notion of having the content that P knows (implicitly) that any truth-bearer with that content is true just in case P. Whether true or false, these epistemological claims do not directly bear on the constitutive, or metaphysical, questions with which I am here concerned. The tension that Dummett identified between minimalist theories of truth on the one side, and truth-conditional accounts of what constitutes content on the other, is of enduring interest. Minimalist theories remain popular among philosophers who focus on the nature of truth: one thinks of the continuing discussion of the accounts proposed by Paul Horwich (1990) and by Robert Brandom (1994a). On the other hand, the most fully developed empirical semantic theories for natural languages specify the meanings of statements by assigning truth-conditions (of one form or another) to them, and many truthconditional semanticists are openly sceptical about the prospects for alternative forms of semantic theory.5 Now it is very far from clear that the truth-conditional semantic theories that now predominate in empirical linguistics require for their vindication a truth-conditional account of what constitutes linguistic content. Many empirical semanticists present themselves precisely as characterizing explicitly what a competent user of the relevant language implicitly knows, and || 5 For an attempt to counter one instance of such scepticism, see Horwich 2008.

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one may, it seems, engage in that task without taking a stand on the metaphysical question of what ultimately underpins assignments of content. I shall return to this point. For now, though, the crucial observation is this. Whether or not minimalism about truth commits one to a non-truth-conditional style of theorizing in empirical semantics, Dummett’s argument shows that it demands a nontruth-conditional account of the constitution of linguistic content. Accordingly, a whole class of philosophers – namely, minimalists about truth – need to find such an account to redeem the debt that their preferred theory of truth incurs within the theory of content. It does not follow from this, of course, that minimalists should adopt justificationism as the needed supplement to their account of truth: there are other non-truth-conditional answers to the question of what a statement’s content consists in. Ramsey, who saw very clearly the incompatibility between his redundancy theory of truth and a truth-conditional account of content, hoped to find the needed theory of content in a form of semantic pragmatism: ‘the meaning of a sentence is to be defined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects’ (Ramsey 1927, 51). So-called ‘success semantics’ (see e.g. Dokic and Engel 2002) is a modern attempt to execute his programme. Other neo-Ramseyans, anxious to give as much weight to the causes of assertions and beliefs as to their effects, have opted instead for functionalist theories of content (see e.g. Harman 1982 and Loar 1981). These answers deserve thorough development and careful assessment, but each of them faces problems.6 That is perhaps enough to show why minimalists about truth should take a close interest in the prospects for justificationism.

|| 6 Against success semantics, see Brandom 1994b and Blackburn 2005. On Ramseyan pragmatism more generally, see Rumfitt 2011. For problems with functionalist theories of content, see Loewer 1982. I set aside Grice’s account of meaning, for it takes for granted the assignment of content to mental states, and so cannot be parlayed into a general theory of content of the kind we are considering. On this limitation of Grice’s theory of meaning, see Dummett 1989, 171–174.

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2 Against the Heyting semantics As Williamson notes, Dummett’s own efforts to develop a justificationist theory of meaning have taken the Heyting semantics for the language of intuitionistic mathematics as a model.7 As Dummett explains this semantics, the meaning of each [logical] constant is to be given by specifying, for any sentence in which that constant is the main operator, what is to count as a proof of that sentence, it being assumed that we already know what is to count as a proof of any of the constituents. The explanation of each constant must be faithful to the principle that, for any construction that is presented to us, we shall always be able to recognize effectively whether or not it is a proof of any given statement. For simplicity of exposition, we shall assume that we are dealing with arithmetical statements… The logical constants fall into two groups. First are ∧, ∨ and ∃. A proof of A ∧ B is anything that is a proof of A and of B. A proof of A ∨ B is anything that is a proof either of A or of B. A proof of ∃xA(x) is anything that is a proof, for some n, of the statement A (⸢n⸣). The second group is composed of ∀, →, and ¬. A proof of ∀xA(x) is a construction of which we can recognize that, when applied to any number n, it yields a proof of A (⸢n⸣). Such a proof is therefore an operation that carries natural numbers into proofs. A proof of A → B is a construction of which we can recognize that, applied to any proof of A, it yields a proof of B. Such a proof is therefore an operation carrying proofs into proofs… A proof of ¬ A is usually characterized as a construction of which we can recognize that, applied to any proof of A, it will yield a proof of a contradiction (Dummett 2000, 8).

This semantic theory explains why certain classical logical laws are unacceptable to the intuitionist. A statement will count as intuitionistically valid if the semantic principles guarantee it to be provable no matter which atomic statements are provable. So a statement of the form ⸢A ∨ ¬ A⸣ will be valid only if either the first disjunct is provable or its negation is provable (i.e., it is provable that the first disjunct is unprovable). Since it cannot be assumed of an arbitrary statement that either it or its negation is provable, it cannot be assumed that an arbitrary instance of ⸢A ∨ ¬ A⸣ is intuitionistically valid: so excluded middle is not an intuitionist logical law. The reasoning just given could be formalized as a

|| 7 Many writers on intuitionism (although not Dummett) refer to the ‘Brouwer-HeytingKolmogorov (BHK) semantics’, but the triple-barrelled tag is misleading. Brouwer did not have a semantic theory at all – merely the idea that mathematical truth amounts to provability. As for Heyting and Kolmogorov, they had very different semantic ambitions. Heyting (1934) wanted to specify the meanings of the connectives using notions of construction and proof that intuitionists could accept, whereas Kolmogorov (1932) wanted to identify the intuitionistically acceptable sequents in terms of classical proofs. (For a recent contribution to Kolmogorov’s project, see Artemov 2001). Dummett’s work lies squarely in the Heyting tradition.

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valid argument in either classical or intuitionistic metalogic. So even if one starts out accepting all of classical logic, adopting the Heyting semantics will lead one to withdraw acceptance from certain classical laws. All the same, with the possible exception of the principle for conjunction, I think that the whole semantic theory adumbrated in this passage is deeply problematic. The basic problems may be brought out by considering the principle for ‘∨’. First, the Heyting axiom for ‘∨’ does not even succeed in giving its meaning within constructive mathematics. Let us say that a theory T has the disjunction property if a disjunctive statement is a theorem of T only if one of its disjuncts is also a theorem; and let us say that T has the existence property if an existentially quantified statement is a theorem only if a witness for it is also a theorem. (A witness for ∃x (x) is any statement in the form (c), where c is a singular term.) Intuitionistic sentential logic has the disjunction property; and first-order intuitionistic logic has both the disjunction property and the existence property. Moreover, certain first-order theories of particular interest to constructive mathematicians – notably Heyting arithmetic, HA – have both properties. The properties are indeed closely related. In any first-order intuitionistic theory, the existence property implies the disjunction property, since in first-order intuitionistic logic A ∨ B is equivalent to ∃x ((x = a → A) ∧ (x ≠ a → B)). A theorem of Harvey Friedman’s (1975) gives a partial but still substantial converse: in all recursively enumerable extensions of HA, the disjunction property implies the existence property. Since the Heyting semantic axioms purport to specify exhaustively the circumstances in which statements have proofs, the axiom for ‘∨’ implies: (P)

⸢A ∨ B⸣ is provable if and only if either A is provable or B is provable.

(P) will be acceptable to a constructivist only if the meaning of ‘proof’ is always determined in such a way that the class of provable statements has the disjunction property. As just noted, if ‘proof’ is understood to mean ‘proof in pure intuitionistic logic’, or ‘proof in Heyting arithmetic’, it will meet this condition. However, no constructivist will equate provability simpliciter with provability in either of these systems, or indeed in any other recursively axiomatizable formal theory. One of the morals the constructivist draws from Gödel’s first theorem is that any such equation is precluded: what is provable outruns what is provable in any particular formal system (see Dummett 1963). Rather, (P) will be acceptable only if, at any stage in the never-ending expansion of acceptable methods of proof, the class of statements that are provable by those methods has the

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disjunction property. An example of Kreisel’s (see Troelstra 1973, 91) shows, however, that this is not always so. Let Pr be the proof predicate of HA: that is, Pr (n, m) means that n is the Gödel number of a proof in HA of the formula whose Gödel number is m. Define (x) to mean Pr (x, ⸢0 = 1⸣) ∨ ∀y ¬Pr (y, ⸢0 = 1⸣). Assuming that HA is consistent on its intended interpretation, we have that ∀y ¬Pr (y, ⸢0 = 1⸣), from which it follows that ∃x (x). Accordingly, T = HA + ∃x (x) is an intuitionistically acceptable theory, and of course T (intuitionistically) entails ∃x (x). For no n, however, does T entail (n).8 Hence T lacks the existence property. The usual proof that the theorems of an axiomatizable theory are recursively enumerable goes through in intuitionistic logic. So, by Friedman’s result, T lacks the disjunction property too. T, however, represents a possible stage in the expansion of a constructivist’s methods of proof. The constructivist might start with Heyting arithmetic – a paradigm of a constructively acceptable mathematical theory. He might then consider the definition of (x), apply the argument above to conclude that ∃x (x), and hence expand his notion of provability from provability in HA to provability in T. In doing this, however, he will be moving from a theory which possesses the disjunction property to one which does not. So, at certain stages in a constructivist’s determination of what is provable, the theory comprising all provable statements will not possess the disjunction property, contrary to (P). It will not help to propose that (P) may yet be true if ‘provable’ is taken to mean ‘provable by absolutely any constructively acceptable method of proof’. For the idea that this gloss specifies a determinate notion of provability will be anathema to any constructivist: he will not take it to be a determinate matter what those methods are. On constructivist principles, then, (P) is unacceptable.9 Hence, the Heyting semantic axiom for ‘∨’, which implies it, does not satisfactorily specify the || 8 Sketch of proof (Kreisel): For each n, ¬Pr (n, ⸢0 = 1⸣) is true, and (since Pr is primitive recursive) provable in HA. Hence, for any n, HA|– (n) ↔ ∀y ¬Pr (y, ⸢0 = 1⸣). By logic, |– ∃x (x) ↔ [∃y Pr (y, ⸢0 = 1⸣) ∨ ∀y ¬Pr (y, ⸢0 = 1⸣)]. Now suppose for a contradiction that T|– (n). Then, by the definition of T, HA, ∃x (x)|– (n), whence HA|– ∃x (x) → (n). By the results above, this yields HA|– [∃y Pr (y, ⸢0 = 1⸣) ∨ ∀y ¬Pr (y, ⸢0 = 1⸣)] → ∀y ¬Pr (y, ⸢0 = 1⸣), hence HA|– ∃y Pr (y, ⸢0 = 1⸣) → ∀y ¬Pr (y, ⸢0 = 1⸣), whence HA|– ¬∃y Pr (y, ⸢0 = 1⸣). But then HA proves its own consistency, contrary to Gödel’s Second Incompleteness Theorem. This contradiction shows that for no n do we have T|– (n), as required. 9 In his book on intuitionism, Dummett in effect concedes this. Following Prawitz, he there distinguishes between ‘direct’ (or ‘canonical’) and ‘indirect’ (or non-canonical) proofs of statements (see e.g. Dummett 2000, 139ff.); only direct proofs of disjunctions are required to be proofs of one or other disjunct. This distinction involves a major departure from Heyting’s semantic theory, and in any case the distinction runs into notorious problems over the conditional; see e.g. Pagin 2009. See also the following footnote.

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meaning of ‘∨’ even within constructivist mathematics. Matters get worse when we try to generalize the axiom to cover other uses of ‘or’. A generalization is surely needed. The Heyting axiom tells us that a proof of a disjunction is anything that is a proof of either disjunct, but this tells us nothing about the contribution that the word ‘or’ makes to a disjunction neither of whose disjuncts admits of proof. Since neither scientific hypotheses, nor ordinary empirical claims about the world, admit of proof, this is a crippling limitation, and a respect in which the Heyting semantics contrasts unfavourably with a truth-conditional semantic theory. For scientific claims, ordinary empirical claims about the world, and mathematical statements all seem to admit of a univocal classification as true or as false, so that the truth-theoretic semantic principle ‘A disjunction is true when one of its disjuncts is true’ tells us something about the meaning of disjunctions composed out of any of these statements. As it stands, the Heyting axiom does not. Dummett sees that a generalization is needed. Since a mathematical proof justifies its conclusion, he argue, the natural generalization is (J)

A justification of ⸢A or B⸣ is anything that is a justification either of A or of B.

There is one immediately striking difference between (J) and (P). Whether a given piece of deductive argumentation constitutes a proof of a statement is an all-or-nothing matter: a deduction either proves its conclusion or it does not, tertium non datur. By contrast, whether a given piece of evidence, say, justifies an empirical claim is not usually an all-or-nothing matter. We sometimes find evidence that conclusively justifies an empirical claim, but in general our evidence provides greater or lesser support for it. This observation does not refute (J) or render it meaningless. We may suppose that there are features of any context of utterance which determine the standards of justification for assertions made in that context. We may then understand (J) as a schema, in which each occurrence of the term ‘justification’ is understood as contextually determined in the same way. Even when understood in this way, though, (J) departs so radically from our ordinary understanding of disjunction that what it specifies is scarcely recognizable even as a revision of that notion. Dummett himself concedes that (J) is untenable if it is understood to concern an individual’s justification for his assertions, at a particular time. ‘I may be entitled to assert [sc., may be justified in asserting] ⸢A or B⸣ because I was reliably so informed by someone in a position to know, but if he did not choose to tell me which alternative held good, I could not [assert either disjunct]’ (1991, 266). For this reason, Dummett asks us to read

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(J) as concerning the existence of justifications, not a given thinker’s apprehension of them. All the same, if a justification is to exist, he requires that a suitably placed thinker could have apprehended it, even if none in fact did: If I pass from saying, ‘A child ran out of the house’, to saying ‘Either a boy or a girl ran out of the house’, it may be said that I could have arrived at the latter statement by orintroduction – not on the basis that I was in fact in a position to assert either of the two disjoined statements ‘A boy ran out of the house’ and ‘A girl ran out of the house’, but on the strength of the fact that, given that I was entitled to make the statement I did make, I had an effective means available to me for putting myself in a position to make one or other of the disjoined statements (Dummett 1991, 268).

Perhaps the ‘effective means’ Dummett has in mind is to chase after the child seen running out of the house and find out whether it is a boy or a girl. But will such a means be available whenever we are entitled to assert a disjunction? Another of Dummett’s examples points to a whole range of cases where it will not. ‘Hardy may simply not have been able to hear whether Nelson said, “Kismet, Hardy” or “Kiss me, Hardy”, though he heard him say one or the other: once we have the concept of disjunction, our perceptions themselves may assume an irremediably disjunctive form’ (1991, 267). On its face, this is a clear counterexample to (J). If Hardy heard Nelson say one thing or the other, then there is – and he has – very strong justification for asserting the disjunction. But in the circumstances of the Battle of Trafalgar, there may have been no justification for asserting either disjunct that any observer could have apprehended. Hardy was as well placed to hear Nelson’s last words as anyone could have been, and all he could hear was that Nelson said either one thing or the other. Certainly, it would be implausible to claim that Hardy had an ‘effective means’ whereby he could have attained a justification of the requisite strength for one or other of the disjuncts. What was that means? To put his ear yet closer to Nelson’s mouth? To use an ear trumpet? Dummett’s last point is the crucial one: once we have the concept of disjunction, the contents of our perceptions may themselves be disjunctive. Since they may do so, the only way to protect (J) from this sort of counterexample would be to deny that perceptions can constitute justifications for disjunctive assertions. But in that case (J) forces so radical a departure from our ordinary understanding that the notion it characterizes is unrecognizable as our notion of disjunction.10

|| 10 The case also shows that it will not help to invoke the distinction between ‘direct’ and ‘indirect’ or justifications for statements (see previous footnote). The idea would be that, whatever may hold of indirect justifications, a direct justification of a disjunctive statement will

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This sort of case refutes not only (J), but also the more direct intuitionistic specification of the meaning of ‘or’ in terms of Beth trees (see Beth 1959). A node of a Beth tree is said to ‘force’ a disjunction if and only if it is ‘barred’ by a species of nodes every one of which forces one of the disjuncts. On the usual interpretation of Beth trees, this amounts to saying: a state of information verifies a disjunction if and only if there is a finite series of investigative steps the taking of which will put one in a state of information that verifies one of the disjuncts. Hardy’s state of information constitutes a counterexample to the ‘only if’ part of the claim. That state of information verifies the disjunction ‘Nelson said either “Kismet, Hardy” or “Kiss me, Hardy”’. But there may be no investigation which would put him, or anybody else, into a state of information that either verifies ‘Nelson said “Kismet, Hardy”’ or verifies ‘Nelson said “Kiss me, Hardy”’. The case also refutes the specification of the meaning of ‘or’ in terms of Kripke trees rather than Beth trees (Kripke 1965). Under that interpretation, the nodes of a Kripke tree are also taken to be states of information, and a state of information is taken to force a proposition when a thinker in the state would be entitled to assert the proposition. The semantic principle for disjunction, however, is simpler than in the case of Beth trees: a node in a Kripke tree is said to force ⸢A or B⸣ if and only if it either forces A or forces B. The cases that refute (J) directly refute this also.

3 Evidentialism Neither the Heyting semantics, then, nor the two most familiar alternative semantic theories for the language of intuitionistic mathematics, generalize to yield a satisfactory justificationist account of the meaning of ‘or’. How, if at all, can a justificationist do better? Let us begin by making the notion of justification more determinate. According to the justificationist, a statement’s content is given by the grounds for asserting it. But what grounds does one need to make an assertion? At least to a first approximation, a plausible answer11 is that one may assert that P only when one knows that P. Let us then take a possible ground of a statement to be something, apprehension of which would put a thinker in a position to know the || always justify one disjunct or the other. But what could be a more direct justification of ‘Nelson said either “Kismet, Hardy” or “Kiss me, Hardy”’ than the evidence Hardy acquired by listening to him speak? 11 See Williamson 2000, chapter 11.

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statement. For a justificationist, a statement’s possible grounds constitute its content. Where A is a statement, I shall use the notation |A|+ to signify A’s possible grounds. (The significance of the plus sign will emerge.) In the explanation of ‘possible ground’, the notion of ‘apprehension’ is perforce variegated. Grounds for mathematical statements are proofs, and apprehending a proof will be a matter of understanding it. At the other extreme, one may be in a position to know the truth of an observational statement simply by virtue of being in a certain perceptual state: if so, that state will qualify as a ground of the statement, and one will apprehend this ground by being in the state. If a statement’s content is to be determined by epistemic factors, then those factors need to stand some distance above the vagaries of what people actually know (which depends in part on what they actually believe). The notion of what a thinker is in a position to know does part of the necessary work of abstracting away from epistemic contingencies. Although the notion is now common currency in epistemology, it could do with more explanation than I can give it here, but one aspect is noteworthy for the arguments to come. I shall say that some premisses entail a conclusion when the conclusion follows from the premisses and it is in principle possible for someone to deduce the conclusion from the premisses. Then, as I shall use the notion, if someone is in a position to know that p, and p entails q, he is also in a position to know that q. Thus the notion of being in a position to know abstracts from deductive incompetence or deficiency. Any account of what determines content must say how the content of whole statements relates to that of their parts. According to the justificationist, a statement’s content is given by its possible grounds. So he needs to tell us how the possible grounds of complex statements relate to those of their components. For conjunctive statements, the story is straightforward. Anything, apprehension of which puts a thinker in a position to know ⸢A and B⸣, puts him in a position to know A and also puts him in a position to know B. Conversely, if someone is in a position to know A and to know B, he is in a position to apply the rule of conjunction-introduction to those pieces of knowledge and infer ⸢A and B⸣. So we have (C+)

For any possible ground x, x is a ground of ⸢A and B⸣ if and only if x is a ground of A and x is a ground of B.

Thus the possible grounds of a conjunction are simply the intersection of the possible grounds of each conjunct: (C+)

|A and B|+ = |A|+ ∩ |B|+.

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However, the corresponding claim for disjunction would be wrong. The corresponding claim would say that something is a possible ground of a disjunctive statement if and only if it is a possible ground of at least one disjunct. The ‘if’ part is correct: if someone is in a position to know A, he is also in a position to apply the rule of disjunction-introduction and infer ⸢A or B⸣ for arbitrary B. But, as the last section in effect showed, the ‘only if’ part is wrong. Inspector Morse might have evidence that puts him in a position to know the truth of the statement ‘Either Smith or Jones committed the murder’. All the same, that evidence may not put him in a position to know ‘Smith committed the murder’, and it may not put him a position to know ‘Jones committed the murder’. We have yet to find necessary conditions for being a ground of a disjunctive statement. How can we identify these conditions? Where U is any set of possible grounds of statements, let us define the closure of U, Cl (U), by the condition x ∈ Cl (U) if and only if x is a ground of every statement of which all the members of U are grounds. That is to say: x ∈ Cl (U) if and only if, for every statement C, x is a ground of C if every member of U is. I shall argue that x is a ground of ⸢A or B⸣ if and only if x ∈ Cl (|A|+ ∪ |B|+). To establish the ‘only if’ half of this biconditional, let us suppose that x is a ground of ⸢A or B⸣. What we need to show is that x is a ground of every statement of which all the members of |A|+ ∪ |B|+ are grounds. Let C, then, be an arbitrary statement meeting this condition. We have, in particular, that absolutely any possible ground that belongs to |A|+ is a ground of C. That is: absolutely any possible ground, apprehension of which puts one in a position to know A, puts one in a position to know C. This will be so only if A entails C. Similarly, absolutely any possible ground that belongs to |B|+ is a ground of C, which implies that B entails C. We have, then, that A entails C and that B entails C. So, by the logical law of dilemma, the disjunctive statement ⸢A or B⸣ also entails C.12 Now x is a ground of ⸢A or B⸣, so apprehension of x puts one in a position to know ⸢A or B⸣. And since ⸢A or B⸣ entails C, apprehension of x also puts one in a position know C. That is, x is a ground of C. We have shown, then, that

|| 12 For, given our definition of entailment, if A entails C and B entails C, then C certainly follows from both A and B; so (by dilemma) C also follows from ⸢A or B⸣. Also, since A entails C and B entails C, it is in principle possible to know that C follows from both A and B, and by going through the reasoning of the previous sentence we may in principle come to know, in addition, that C follows from ⸢A or B⸣. Hence ⸢A or B⸣ entails C, as required.

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when x is a ground of ⸢A or B⸣, it is also a ground of every statement of which all the members of |A|+ ∪ |B|+ are grounds. By the definition of closure, then, whenever x is a ground of ⸢A or B⸣ it belongs to Cl (|A|+ ∪ |B|+). It may be noted that this argument requires only the weak form of the law of dilemma, the form without side premisses, so the logical principle on which it rests is acceptable to classical, intuitionist, and even quantum logicians. To establish the ‘if’ half of our biconditional, suppose that x ∈ Cl (|A|+ ∪ |B|+). By the definition of closure, this implies that, for any statement C, if all the members of (|A|+ ∪ |B|+) are grounds of C, x is a ground of C. Now A entails ⸢A or B⸣, and B entails ⸢A or B⸣, so all the members of (|A|+ ∪ |B|+) are grounds of ⸢A or B⸣. Hence x is a ground of ⸢A or B⸣. Accordingly, we reach the justificationist semantic principle for ‘or’ that I propose: (D+)

x is a ground of ⸢A or B⸣ if and only if x ∈ Cl (|A|+ ∪ |B|+),

or, more briefly, (D+)

|A or B|+ = Cl (|A|+ ∪ |B|+).

The argument just given for (D+) may seem to cheat. The definition of closure quantifies over statements, and the argument for the ‘if’ half of (D+) presumes that the domain of quantification already contains the disjunctive statement ⸢A or B⸣ (or a statement that shares its content). It might then be objected that the proposed semantic axiom for ‘or’ offers no insight into how our understanding of the connective ‘or’ combines with our antecedent understanding of A and B so as to yield an understanding of the disjunctive statement of ⸢A or B⸣. It offers no insight, in other words, into how a competent speaker’s understanding of a complex statement’s parts combines to yield an understanding of the whole. And yet – in the eyes of many philosophers of language – providing such insight is the semantical task par excellence. So (D+) is not really a starter as a semantic axiom for ‘or’. In reply, we need to distinguish between different tasks that a semantic theorist might be set. One task is indeed that of accounting for the productivity of understanding, and (D+) is no help there. But this was not the task articulated in § 1. The project of the present paper is to see whether a justificationist answer can be developed to the constitutive question ‘By virtue of what does a statement say what it says?’ A satisfactory justificationist answer to that question can comprise axioms that specify the relations between the grounds of complex statements and those of their components, even when grounds are not compositionally determined upwards from atoms to molecules (as they usually will not

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be for disjunctive statements). Indeed, excessive attention to the problem of accounting for linguistic productivity may have led justificationists to overlook our comparatively simple solution to the constitutive problem as it arises for disjunction. Of course, the problem of accounting for productivity remains, but there is a solution to it that is fully consistent with the present form of justificationism. Any competent French speaker knows that ‘La neige est blanche’ says that snow is white. Any such speaker also knows that ‘La terre se déplace’ says that the earth moves. Suppose that such a speaker also knows that, whenever the French statement A says that P, and B says that Q, the disjunctive statement ⸢A ou B⸣ says that either P or Q. We can then account for his knowing that ‘La neige est blanche ou la terre se déplace’ says that either snow is white or the earth moves by reference to his knowledge of what the parts say.13 This approach to the problem of productivity casts no light on what makes it the case that statements say what they say, but that might well be a merit. On this approach, we cleanly separate the epistemological problem of accounting for our knowledge of what statements say from the metaphysical problem of explaining in what their saying what they say consists. At any rate, because it takes no stand on what determines what statements say, the minimalist approach to the problem of productivity just sketched is available to our sort of justificationist, who offers an answer to the metaphysical question but does not try to parlay that answer into an explanation of how we are able to understand statements that we have not heard before. An entire theory of the determination of linguistic content is latent in the proposed axiom for ‘or’. I call the theory that is so latent evidentialism. To see its shape, let us remark first that closure, in the present sense, is a closure operation in the sense favoured by lattice theorists. That is to say, the operation is INCREASING

U ⊆ Cl (U)

IDEMPOTENT

Cl Cl (U) = Cl (U)

and

|| 13 For this approach to the problem of productivity, see Davies 1981, 42ff. Davies finds theories of meaning of this kind wanting, but his argument against them presupposes a non-minimal notion of truth.

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MONOTONE

If U ⊆ V then Cl (U) ⊆ Cl (V).14

Let us call a set closed when it is identical with its own closure. By idempotence, the closure of any set is closed, and by monotonicity the closure of U is the smallest closed set containing U. We then have the following general axiom of evidentialism: (R+)

The possible grounds of a statement form a closed set.

The argument for (R+) is straightforward. Let U be the set of all possible grounds of the statement A, and consider an arbitrary member, x, of the closure of U. By the definition of closure, x is a ground of any statement of which all the members of U are grounds. Since every member of U is a ground of A, A is such a statement, so x is a ground of A. But then, since U comprises all the grounds of A, x must be a member of U. That is to say, any member of the closure of U must belong to U itself, so U is closed, as required. An evidentialist theory of content, then, will associate with each statement a closed set of possible grounds. The theory’s compositional principles will say how the grounds of a complex statement relate to the grounds of its components. The intersection of any two closed sets will be closed,15 so postulate (C+) respects our general principle that the grounds of any statement should form a closed set. So too does our postulate (D+) for disjunctions, since the closure of any set is closed. Although I cannot argue for the claim here, I believe that natural generalizations of these postulates can serve as semantic principles

|| 14 Proofs. INCREASING: immediate from the definition of closure. IDEMPOTENT: since closure is INCREASING, it suffices to show that Cl Cl (U) ⊆ Cl (U). Suppose then x ∈ Cl Cl (U). Then x is a ground of every statement of which every member of Cl (U) is a ground. Consider an arbitrary statement A of which every member of U is a ground. By definition, every member of Cl (U) will be a ground of A. Hence x is a ground of A. But that shows that x is a ground of every statement of which every member of U is a ground, so that x ∈ Cl (U), as required. MONOTONE: suppose that x ∈ Cl (U) and that U ⊆ V. Since x ∈ Cl (U), x is a ground of every statement of which all the members of U are grounds. Since U ⊆ V, it follows that x is also a ground of every statement of which all the members of V are grounds. That is, x ∈ Cl (V), as required. 15 Proof. Suppose that U = Cl (U) and that V = Cl (V). We need to show that U ∩ V = Cl (U ∩ V). Since INCREASING already yields U ∩ V ⊆ Cl (U ∩ V), it suffices to show that Cl (U ∩ V) ⊆ U ∩ V. Now U ∩ V ⊆ U, whence by MONOTONE Cl (U ∩ V) ⊆ Cl (U) = U. Similarly, Cl (U ∩ V) ⊆ Cl (V) = V. Together, these inclusions yield Cl (U ∩ V) ⊆ U ∩ V, as required.

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saying what the grounds are of universally and existentially quantified statements.16 Under evidentialism, the logic of disjunction need not be classical: because the possible grounds of a disjunction include things that do not ground either disjunct, our semantic principle for ‘or’ can accommodate logics (such as quantum logic) that invalidate distribution – i.e., the law that ⸢A and (B or C)⸣ always has as a logical consequence ⸢(A and B) or (A and C)⸣. But that, as it seems to me, is how things should be. The validity of the distributive law is not guaranteed by the meanings of ‘or’ and ‘and’ alone. Rather, it is sustained by those meanings in tandem with principles concerning logical consequence. A plausible principle is that genuine logical consequence should be absolute in the sense that, whenever we have an instance of logical consequence, it should remain so even if we accept additional premisses, or make additional suppositions (see especially McFetridge 1990). In the present framework, this amounts to the following requirement of stability: whenever a possible ground x belongs to the closure of a set of possible grounds U, the combination of x with an arbitrary possible ground y (if such a combination exists) will belong to the closure of the set formed by combining each member of U with y.17 If the space of possible grounds is stable in this sense, then the natural definition of consequence will validate distribution (see Sambin 1995, especially the remarks on distribution on p. 864, lemma 2 on p. 865 and theorem 4 on p. 868). So distribution is validated by the meanings of ‘or’ and ‘and’ along with the stability of the space of possible grounds – i.e., along with the thesis that logical consequence is absolute. In this way, evidentialism can accommodate the classical logic of ‘and’ and ‘or’.

|| 16 For the generalizations that I envisage, see Mares 2010. Mares works with a notion of ‘objective information’ rather than grounds, but modulo differences consequential upon that, his treatment of disjunction is equivalent to that proposed here. 17 The combination of two grounds may be defined as follows. The closure operation Cl induces a partial order ≤ on the space of possible grounds: x ≤ y if and only if y ∈ Cl ({x}). A possible ground z is then said to combine x and y when it is a minimal upper bound of x and y under the partial order ≤. That is, z combines x and y if and only if x ≤ z ∧ y ≤ z ∧ ∀w((x ≤ w ∧ y ≤ w) → z ≤ w).

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4 A snag and a way out: bilateral evidentialism Let us say that a statement is correctly assertible when one of its grounds obtains. Then evidentialism is committed to the following thesis: ( )

The conditions under which a statement is correctly assertible determine its content.

However, ( ) carries with it a problematic consequence. I defined a statement’s ground as something whose apprehension puts a thinker in a position to know the statement. Accordingly, a statement is correctly assertible only if it is true. This means that the theory of correct assertibility will include every instance of the schema of factivity: ⸢If it is correctly assertible that A, then A⸣. That is, using the symbol ‘□’ to mean ‘it is correctly assertible that’, it will include every instance of the schema □A → A. The theory of correct assertibility will also include every instance of the schema of distribution (over conjunction): □ (A ∧ B) → (□A ∧ □B). This also follows from the explanation of correct assertibility: if a thinker is in a position to know the truth of a conjunction, he is thereby in a position to know the truth of each conjunct. Given factivity and distribution, however, we can prove that truth and correct assertibility are equivalent: in addition to factivity, our theory must include every instance of the corresponding (material) biconditional schema A ↔ □A. The proof is an example of the sort of argument that used to be called ‘Fitchean’ (after Fitch 1963) but might more aptly be called ‘Churchy’ (see Salerno 2009). Let A be any statement, and consider the pair of statements A ∧ ¬□A (‘A and it is not assertible that A’) and A ∧ ¬A (‘A and not A’). By factivity, we have □(A ∧ ¬A) → (A ∧ ¬A), so that it is a consequence of the theory of assertibility that ¬□(A ∧ ¬A). The second of our pair of statements, then, is not assertible under any condition whatever. By distribution, however, □(A ∧ ¬□A) → (□A ∧ □¬□A), and by factivity (□A ∧ □¬□A) → (□A ∧ ¬□A), so that it is also a consequence of the theory of assertibility that ¬(□A ∧ □¬□A). Thus the first of our pair of statements is also not assertible under any conditions whatever. The statements A ∧ ¬□A and A ∧ ¬A, then, may be correctly asserted under exactly the same conditions – namely: never. Accordingly, by thesis ( ), they must share their content. Since the latter statement is invariably false, the same must go for the former, so we have ¬(A ∧ ¬□A), i.e. A → □A. This combines with factivity to yield A ↔ □A. On the assumptions, then, that assertibility is factive and that it distributes over conjunction, thesis ( ) entails that a statement’s correct assertibility is materially equivalent to its truth.

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While some anti-realist philosophers may welcome this consequence of ( ), I do not think that it is acceptable. For one thing, it renders evidentialism vulnerable to a version of the Paradox of Knowability (see again Fitch 1963 and, for the version of the argument that follows, Künne 2003, § 7.3). Even though we cannot specify an instance which we know to be true, most of us that believe that there are some true instances of the schema (*)

P and it is never known that P.

However, very weak assumptions render this plausible belief inconsistent with the thesis that a statement’s correct assertibility is materially equivalent to its truth. At least, this is so given our principle that a statement may be asserted only when it is known. For let P be a statement which makes (*) true. Since the relevant instance of (*) is true, the touted equivalence would yield It is possible that someone knows that (P and it is never known that P). It is surely necessary that when a thinker knows that (P and Q), he knows that P and knows that Q. So we may further deduce that It is possible that (someone knows that P and someone knows that it is never known that P). Now it is certainly necessary that when a thinker knows that P then P, so we have that It is possible that (someone knows that P and it is never known that P). This, however, is impossible. The reductio shows that the universal equivalence of truth and correct assertibility is inconsistent with the existence of unknown truths. Since the existence of unknown truths is so plausible, it is surely the equivalence that has to be rejected.18 But we have shown that that equivalence follows from thesis ( ), and that our evidentialist theory of meaning entails ( ). So evidentialism will not do as it stands. The deduction of A ↔ □A from ( ) shows that, if we want a theory of content in which truth does not collapse into correct assertibility, we must reject

|| 18 Anti-realist philosophers (e.g. Dummett 2001) have explored ways of avoiding the inconsistency. But see Künne, op. cit., 446–449.

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any theory on which a statement’s content is determined by its conditions of correct assertibility alone. This may seem to pose an insuperable problem for a justificationist; but whilst the difficulty latent in ( ) may rule out justificationism au pied de la lettre, a closely related theory of meaning avoids the collapse whilst being recognizably justificationist in spirit. Huw Price (1983 and 1990), Timothy Smiley (1996), Lloyd Humberstone (2000) and I (Rumfitt 2000) have pressed the merits of bilateral theories of the determination of content whose characteristic thesis is not ( ), but rather ( )

The conditions under which a statement may correctly be asserted, together with the conditions under which it may correctly be denied, jointly determine its content.

Let us postulate a theory of correct assertibility whose axioms are all instances of the schema of factivity (□A → A), all instances of the schema of normality (□(A → B) → (□A → □B)), and in which all classical tautologies are correctly assertible. (By virtue of normality, assertibility will distribute over conjunction.) In this theory, which lacks ( ), A will not in general imply □A. If we further assume that a statement may be correctly denied in precisely those circumstances in which its negation may be correctly asserted, then a theorem of Williamson’s (1990) shows that ( ) is already implicit in this theory of assertibility. So a bilateral theory of content, which conforms to ( ), differs from unilateral theories, which conform to ( ), in making room for a distinction between truth and correct assertibility. A way out of the difficulty, then, is to move from the unilateral evidentialism of the previous section to bilateral evidentialism. A bilateral evidentialist will accept the justificationist semantic principles articulated in § 3. For him, though, those principles tell only half the story. In formulating them, I drew upon our common understanding of what it is to be in a position to know a statement’s truth. But we also understand what it is to be in a position to know a statement’s falsehood. Let us call something, apprehension of which puts a thinker in a position to know the falsehood of statement A, an anti-ground of A. I use the notation|A|– to signify A’s possible anti-grounds. The bilateral evidentialist gives equal weight to grounds and anti-grounds: for him, a statement’s content is determined jointly by its possible grounds and its possible antigrounds. Of course, grounds and anti-grounds have to be appropriately related, and philosophers who are not dialetheists will postulate an Axiom of Consistency: a statement’s grounds and anti-grounds never overlap, |A|+ ∩ |A|– = ∅. The

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semantic principles to be given ensure that complex statements respect the Axiom so long as the atomic statements do.19 What are the principles that relate a complex statement’s anti-grounds to the anti-grounds of its parts? Where U is a set of possible anti-grounds for statements, let us define the closure of U, Cl (U), by the condition x ∈ Cl (U) if and only if x is an anti-ground of every statement of which all the members of U are anti-grounds. An argument parallel to that given in n.14 shows that the closure operation on anti-grounds is again INCREASING, IDEMPOTENT and MONOTONE; as before we may call a set of anti-grounds closed when it is identical with its own closure. Then, parallel to the principle (R+) of the last section, we have the following general axiom of bilateral evidentialism: (R–)

The possible anti-grounds of a statement form a closed set.

The argument for (R–) runs parallel to that given above for (R+). Let U be the set of all possible anti-grounds of the statement A, and consider an arbitrary member, x, of the closure of U. By the definition of closure, x is an anti-ground of any statement of which all the members of U are anti-grounds. Since every member of U is an anti-ground of A, A is such a statement, so x is an anti-ground of A. But then, since U comprises all the anti-grounds of A, x must be a member of U. That is, any member of the closure of U must belong to U itself, so U is closed, as required. We need to supplement the semantic axioms given in § 3 with further principles that say how the anti-grounds of statements built up using ‘and’ and ‘or’ relate to the anti-grounds of their parts. Adherents of classical logic will propose principles that are the duals of the axioms concerning grounds. We are in a position to know the falsehood of the disjunction ⸢A or B⸣ just when we are in a position to know both the falsehood of A and that of B, so we have (D–)

|A or B|– = |A|– ∩ |B|–.

|| 19 The fact that paradoxical utterances, such as ‘This very statement is false’, are prima facie violations of Consistency provides a bilateralist with a ground for the claim that they say nothing, as Prior held (see n.4 above).

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When it comes to anti-grounds, conjunctions are more problematic than disjunctions. A thinker who is in a position to know a statement’s falsehood is in a position to know the falsehood of any conjunction of which the statement is a conjunct. However, the converse does not hold. If I know nothing about a ball’s colour, I am not in a position to know the falsehood of ‘The ball is red all over’, nor the falsehood of ‘The ball is green all over’, but I am in a position to know the falsehood of the conjunction ‘The ball is both red all over and green all over’. However, an argument parallel to that given in § 3 for (D+) yields the principle: (C–)

|A and B|– = Cl (|A|– ∪ |B|–).

As with their positive counterparts, both (C–) and (D–) ensure that, so long as the atomic statements respect the master principle (R–), so will all the molecular statements built up from them using ‘and’ and ‘or’. With this apparatus in place, we can characterize negation as a logical switch that ‘toggles’ between grounds and anti-grounds, between being in a position to know a truth and being in a position to know a falsehood. For what are the grounds, apprehension of which puts us in a position to know the truth of ⸢Not A⸣? A plausible answer is that they are precisely the grounds, apprehension of which puts us in a position to know the falsehood of A. That is, we have: (N+)

x is a ground of ⸢Not A⸣ if and only if x is an anti-ground of A,

(N+)

|Not A|+ = |A|–.

i.e.,

Similarly, we need to ask what are the grounds, apprehension of which puts us in a position to know the falsehood of ⸢Not A⸣. A classical logician will answer that they are precisely the grounds, apprehension of which puts us in a position to know the truth of A. That is, we have: (N–)

x is an anti-ground of ⸢Not A⸣ if and only if x is a ground of A,

(N–)

|Not A|– = |A|+.

i.e.

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Together, (N+) and (N–) ensure that ⸢Not not A⸣ has the same grounds, and the same anti-grounds, as A itself. Given bilateral evidentialism, then, a statement and its double negation have the same content. We might call the conception of negation that (N+) and (N–) specify the Ramseyan conception. According to Ramsey, an object’s being red and its not being not-red ‘are simply the same fact expressed by other words’. He suggests that negation might most perspicuously be expressed ‘not by inserting a word “not”, but by writing what we negate upside down…[If] we adopted [such a symbolism] we should be rid of the redundant ‘not-not’, for the result of negating the sentence “p” twice would be simply the sentence “p” itself’ (Ramsey 1927, 42–43). Our account vindicates the semantic thesis that inspires this unusual form of syntax. The Ramseyan conception ensures the classical equivalence between A and ⸢Not not A⸣. In fact, in tandem with the assumption of stability, our bilateral semantic axioms validate the full classical logic of ‘and’, ‘or’, and ‘not’.20

5 Next steps Our bilateral evidentialist semantic axioms, then, determine which sequents involving ‘and’, ‘or’, and ‘not’ are valid. But there is more to the notion of a statement’s content than its strictly deductive behaviour. Do our axioms determine other aspects of what a statement says? Evidence often provides some degree of support for a statement even though it falls short of putting someone in a position to know it. It is natural to think of think of a statement’s Fregean sense as constituted precisely by its place in such a network of partial evidential support. It is metaphysically necessary that the statement ‘The atmosphere of the planet Phosphorus contains carbon dioxide’ is correctly assertible (and correctly deniable) just when ‘The atmosphere of the planet Hesperus contains carbon dioxide’ is. Yet these statements differ in sense (or ‘cognitive value’) because (for example) the evidence acquired by turning an astronomical spectrometer to the morning sky, and observing appropriately located signs of carbon dioxide, supports the first statement but does not – in and of itself – support the second. How tightly do our postulates about the conditions in which we are in a position to know a statement constrain those degrees of support? The answer is

|| 20 There are alternative bilateral axioms for negation that validate weaker logics for ‘not’ – notably, intuitionistic logic. See Humberstone 2000, especially 364–366.

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that they constrain it reasonably tightly, once they have been supplemented with some quite weak principles about the general structure of evidential support. Following Williamson (2000, chap. 10), let use the familiar dyadic or conditional probability operator P(A/B) as a measure of evidential probability: that is, let us understand it as a measure of the degree to which B supports the truth of A. Thus, where B is evidence, apprehension of which enables us to know the truth of A, we take P(A/B) to be unity; and where B is evidence, apprehension of which enables us to know the falsehood of A, we take P(A/B) to be zero. In our terminology, then, B is a ground for A just in case P(A/B) = 1, and B is an antiground of A just in case P(A/B) = 0.21 On this way of understanding P(A/B), the following postulates are highly plausible: I

0 ≤ P(A/B) ≤ P(A/A ∧ B) = P(t/B) = 1 P(f/C) = 0 unless P(D/C) = 1 for all D

II

P(A ∧ B/C) = P(B ∧ A/C)

III

P(A ∧ B/C) = P(A/C) P(B/A ∧ C)

In these postulates, t is a known logical truth of the relevant logic, and f is a known logical falsehood. Where it is knowable that C cannot obtain, we take it to be a ground of any statement; thus the second clause of postulate I says that any evidence that might obtain is an anti-ground of a known logical falsehood. Given these postulates, a theorem of Bas van Fraassen’s shows that our semantic axioms entail further plausible postulates that specify the relationship between the degrees to which evidence supports atomic statements and the degrees to which it supports complex statements.22 There is a strong case, then, for saying that our semantic axioms specify the contribution that ‘and’, ‘or’ and ‘not’ make to the Fregean sense of statements in which they occur. A great deal of work remains to be done. So far, the only connectives that our theory covers are ‘and’, ‘or’, and ‘not’. Only after we have extended it to cover a reasonably large fragment of a natural language will we be in a position to compare bilateral evidentialism with other accounts of the determinants of

|| 21 So, on this way of understanding it, P(A/B) can be 1 even when apprehension of B does not render A subjectively certain. See Williamson 2000, 213ff., for elaboration of this point. 22 See propositions (2-7) and (3-1) of van Fraassen 1981b (503, 505). Van Fraassen has a rather different way of understanding P(A/B) (see his 1981a), but the difference in interpretation does not affect his formal proofs.

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linguistic content. It would be foolish to anticipate the result of that comparison but, pace Williamson, the research programme of investigating how far the approach can run is very far from degenerate. Emancipated from the unhelpful intuitionist model, our bilateral form of justificationism is a promising place in which to seek the non-truth-conditional theory of content which minimalism about truth demands, but which is not at all easy to find.

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Williamson, T. 2000. Knowledge and Its Limits. Oxford: Clarendon Press. Williamson, T. 2007. The Philosophy of Philosophy. Oxford: Blackwell.

Timothy Williamson

Dummett on the Relation between Logics and Metalogics Abstract: The paper takes issue with a claim by Dummett that, in order to aid understanding between proponents and opponents of logical principles, a semantic theory should make the logic of the object-language maximally insensitive to the logic of the metalanguage. The general advantages of something closer to a homophonic semantic theory are sketched. A case study is then made of modal logic, with special reference to disputes over the Brouwerian formula (B) in propositional modal logic and the Barcan formula in quantified modal logic. Semantic theories for modal logic within a possible worlds framework satisfy Dummett’s desideratum, since the non-modal nature of the semantics makes the modal logic of the object-language trivially insensitive to the modal logic of the metalanguage. However, that does not help proponents and opponents of the modal principles at issue understand each other. Rather, it makes the semantic theory virtually irrelevant to the dispute, which is best conducted mainly in the object-language; this applies even to Dummett’s own objection to the B principle. Other forms of semantics for modal languages are shown not to alter the picture radically. It is argued that the semantic and more generally metalinguistic aspect of disputes in logic is much less significant than Dummett takes it to be. The role of (non-causal) abductive considerations in logic and philosophy is emphasized, contrary to Dummett’s view that inference to the best explanation is not a legitimate method of argument in these areas.

1 Introduction Philosophically, I grew up in the Oxford of the period 1973–80, where Michael Dummett’s thought was under more or less continuous debate, day and night. Familiarity with his work made us underestimate the difficulty of reading it cold: I remember a now-distinguished contemporary taken aback at the suggestion that some Americans found Dummett’s writing obscure. There was no presumption that Dummett was actually right: most Oxford philosophers hoped to avoid his anti-realism. Indeed, many of them spent their time struggling des-

|| University of Oxford

DOI 10/1515-9783110459135-009

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perately to do so. Yet they felt it always there, ready to engulf them if they made one false move. They tried to beat Dummett at his own game, perhaps with one or two changes in the rules. At a broader methodological level, Dummett represented the generation that had rejected ordinary language philosophy in favour of a more abstract and systematic mode of theorizing, still concerned to understand the use of natural languages but willing to model them by first-order formal languages and insistent on an explicit compositional theory of meaning. Less congenial were the behaviourist tendencies we suspected to be lurking in his arguments. But his originality, fertility, versatility, philosophical resourcefulness, technical power, and sheer idiosyncrasy defied such labels. Dummett was my supervisor for the year 1979–80, in which I completed my doctoral dissertation on the concept of approximation to the truth. He remains for me a paradigm of intellectual seriousness. He was also remarkably tolerant of my aggressively realist arguments, ready to discuss them on their own terms rather than seeking to impose his own starting-point. At least my thesis was different from the relentless concentration on his own writings in much of what he had to supervise: in that respect it may have come as something of a relief to him, despite its blithe tacit assumption of the futility of his life’s work. One incident may be of interest. We were discussing an argument I had given for regarding a four-termed relation of comparative similarity as more basic than a threetermed one, the only idea in my dissertation I subsequently developed for publication (Williamson 1988). Dummett thought it utterly fallacious. He reflected for a while, then said: ‘The difference between us in philosophical method is that you think that inference to the best explanation is a legitimate method of argument in philosophy, and I don’t’. On his view, the deep philosophical action came in determining whether a supposed explanation was so much as intelligible. After that, little was left for inference to the best explanation to do. On my view, philosophy does not fall into unintelligibility as easily as Dummett thought — plain falsity is a far more urgent danger — and in any case the way to test an explanation for intelligibility is by trying to work within its framework, not by seeking a guarantee of intelligibility in advance. I have recently defended such an abductive methodology for logic and metaphysics (Williamson 2013, pp. 423–429). How else Dummett expected to argue for a theory of meaning I do not fully understand. But, holding the Wykeham Chair of Logic that Dummett held when he taught me, I wonder whether I am as tolerant of disagreement in my students as he was of mine. Despite my socialization into a philosophical culture steeped in Dummett’s work, I find much of it deeply puzzling to read. I try to put my finger on a crucial premise of his arguments that I reject, but the difference between us is more

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global than that. I have to imagine my way into an alien but intriguing and powerful mindset, and find the experience surprisingly reminiscent of reading a philosophical masterpiece from a distant era. Although, sentence by sentence, the hermeneutic effort is less, putting the pieces together in one’s own terms can be almost equally hard. Latching onto a clear point of disagreement is therefore not simply a matter of philosophical point-scoring, even if it is that too. It is a clue to deeper differences of theoretical outlook. One such clear point of disagreement emerges in The Logical Basis of Metaphysics, when Dummett says ‘a difference over fundamental laws of logic must reflect a difference over the meanings of the logical constants’ (1991, p. 54). Elsewhere I have argued at length for the opposite view (2007, pp. 85–133). Rather than confront that issue again head-on, I will explore some characteristically thought-provoking remarks that Dummett then makes about the best way for proponents and opponents of a fundamental law of logic to make progress in their dispute by means of semantic theory. For Dummett, a semantic theory states in outline how the semantic values of whole sentences are determined by the semantic values of their constituents, explicitly articulating the contributions of the logical constants in a manner that would enable a soundness or completeness theorem to be stated and proved (if true) for a corresponding deductive system. The semantic theory does not attempt to show how its principles are correct in virtue of the use of the object-language by the community of its speakers. For Dummett, that essential task is to be carried out by a deeper meaning-theory for the language. Nevertheless, he envisages the semantic theory playing a role in enabling the parties to a dispute over a fundamental law of logic to understand their disagreement.1

|| 1 With permission, the remaining sections of this paper use material, significantly revised, from Williamson 2011. That paper was presented, in different forms, to both the 2010 Lauener Symposium in honour of Michael Dummett and a 2010 conference on Logic and Knowledge at the Sapienza University of Rome, whose proceedings form the kernel of Cellucci, Grosholz, and Ippoliti 2011. Cesare Cozzo gave a helpful commentary on the latter occasion (Cozzo 2011). I thank audiences at both events for discussion.

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2 Dummett on the virtue of insensitivity According to Dummett (1991, pp. 54–55): A thoroughly pernicious principle has gained considerable popularity in recent years. It is that in formulating a semantic theory the metalanguage must have the same underlying logic as the object-language. When this principle is followed, the proponent of a nonclassical logic has a perfect counter to an argument in favour of a classical law that he rejects, namely, that the argument assumes the validity of the law in the metalanguage.

Dummett illustrates such a stalemate with the dispute over the distributive law between classical and so-called quantum logic, as to whether A & (B ∨ C) entails (A & B) ∨ (A & C). He then continues in general terms: What is needed, if the two participants to the discussion are to achieve an understanding of each other, is a semantic theory as insensitive as possible to the logic of the metalanguage. Some forms of inference must be agreed to hold in the metalanguage, or no form of inference can be shown to be valid or to be invalid in the object-language; but they had better be ones that both disputants recognise as valid. Furthermore, the admission or rejection in the metalanguage of the laws in dispute between them ought, if possible, to make no difference to which laws come out valid and which invalid in the objectlanguage. […] If both disputants propose semantic theories of this kind, there will be some hope that each can come to understand the other; there is even a possibility that they may find a common basis on which to conduct a discussion of which of them is right.

As an illustration of such convenient insensitivity, he gives the semantics of intuitionistic sentential logic based on Beth or Kripke trees. Whether the metalogic is classical or intuitionistic, that semantics can be shown to validate exactly intuitionistic logic for the object-language. The aim of the rest of this paper is to assess Dummett’s claims in the quoted passages.

3 Some advantages of sensitivity Say that a semantic theory S projects a metalogic ML onto a logic L when L is the strongest logic whose validity is derivable from S, qua semantics for the language of L (the object-language), in ML, qua logic for the language of S (the metalanguage). In the first displayed passage, Dummett denies that a semantic theory should project each metalogic onto itself. The logic need not preserve the metalogic. In the second passage, he asserts that a semantic theory should pro-

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ject different metalogics onto the same logic (over as wide as possible a range of metalogics). The logic must be robust with respect to the metalogic. Dummett’s second claim implies the first, for trivially if the logic always preserves the metalogic, it is maximally sensitive to the metalogic. The converse does not hold, for a semantic theory might in principle constitute a one-one projection with no fixed points: then the logic would be maximally sensitive to the metalogic without ever preserving it. One obvious target for Dummett’s argument is the preference, often associated with Davidsonian philosophy of language, for homophonic semantics. The association was especially salient in Oxford when Dummett wrote. At least to a first approximation, a homophonic semantic theory projects each metalogic onto itself. By contrast, the Beth or Kripke semantics for intuitionistic logic is non-homophonic in structure, and projects even a classical metalogic onto intuitionistic logic for the object-language. Dummett’s argument applies to all semantic theories translatable into a homophonic theory, not just to the homophonic theory itself, since they will all project intertranslatable metalogics onto the same logic for the given objectlanguage. Yet elementary considerations suggest that a semantic theory should be translatable into a homophonic theory. The key fact about the sentence “Snow is white” which a semantic theory for English in any metalanguage should capture is that it means that snow is white. More generally, for any sentence s of the object-language, a semantic theory for that language should entail some truth like this: (M)

s means that p

In this schema, “s” gets replaced by a name of an object-language sentence and “p” by a sentence of the metalanguage. For the instance of (M) to be true, the former object-language sentence must indeed mean what the latter metalinguistic sentence means, or express the same proposition. In general, therefore, the object-language must be translatable into the metalanguage. Davidsonians try to achieve the same effect, despite substituting the extensional construction “is true in the object-language if and only if” for the non-extensional “means that”, by imposing complex constraints on the semantic theory. That elementary argument must be qualified in various ways. First, purely homophonic semantics is inadequate for languages with context-dependence. For example, although the sentence “I was born in Sweden” as uttered by me means that I was born in Sweden, as uttered by you it does not

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mean that I was born in Sweden; it means that you were born in Sweden. For present purposes we can ignore the complexities of context-dependence. Second, a more pertinent qualification is that a semantic theory, as Dummett envisages it, is primarily a theory of logical consequence, and so typically abstracts from the meanings of non-logical expressions, by generalizing over all interpretations or models of the object-language which fix the intended interpretations of the purely logical expressions. Thus not all sentences of the object-language need be translatable into a metalanguage adequate to express the metalogic of the object-language. For example, the metalanguage may lack synonyms of “snow” and “white”, even though the object-language has them. Nevertheless, since the model theory is supposed to abstract away only the logically irrelevant aspects of the semantics of the object-language, we may expect its semantic clauses for the logical constants to translate more or less homophonic semantic clauses for them, relativized to a model and usually other parameters too, such as an assignment of values to variables. That is just what we find in standard first-order model theory. It has clauses like the following, where M, a |= A means that the formula A is true in the model M under the assignment a of values to variables, dom(M) is the domain of quantification for M, and a(x/d) is the assignment like a except that a(x/d)(x) = d: M, a |= ¬A if and only if not M, a |= A M, a |= A & B if and only if M, a |= A and M, a |= B M, a |= ∃ x A if and only if for some d ∈ dom(M): M, a(x/d) |= A These are not homophonic clauses, since what correspond to the objectlanguage symbols ¬, & and ∃ in English as the informal metalanguage are “not”, “and” and “some”. Even if the clauses were formalized in an extension of the object-language, some non-homophonic features would remain, such as the reference to the assignment a(x/d) and, more significantly, to the domain of the model (Williamson 2003). Nevertheless, since the main connective in the righthand side of the biconditional for each logical constant is a rough translation of the logical constant into the metalanguage, and little more than the technically necessary minimum of extra structure has been introduced, we may still loosely call such model-theoretic semantics quasi-homophonic. Dummett’s main objection to quasi-homophonic semantics is that it blocks rational debate about questions of validity, because each side uses its preferred logic as a metalogic to vindicate that very logic for the object-language, and accuses the other side of begging the question when it does the same thing.

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More generally, on his view, quasi-homophonic semantics lacks explanatory power. Imagine a dispute over the validity of a fundamental logical law [!], which is in fact valid, although some philosophers argue otherwise. Friends of [!] correctly explain why it is valid. Their explanation invokes both a semantic theory and a metalogic. May it invoke [!] itself in the metalanguage? If it does, it will not persuade critics of [!]. But the purpose of explaining why something obtains is often not to persuade anyone that it does obtain. In explaining why there is life on Earth, a scientist is not trying to persuade anyone that there is life on Earth.2 It may be quite unreasonable to demand that friends of [!] explain why it is valid in the object-language without invoking [!] in the meta-language. If [!] is a fundamental logical law, a metalogic without [!] cannot be expected to project onto a logic with [!]. On this view, the metalogic should contain all the fundamental logical laws of the object-language. If the fundamental laws generate the non-fundamental ones, it follows that the metalogic should be at least as strong as the logic. It may need to be stronger: for instance, the metalanguage for a quantifier-free object-language must itself contain quantifiers, to express metalinguistic generalizations (such as soundness and completeness), so the metalogic must include quantificational logic. In quasi-homophonic semantics, the metalanguage may contain metalinguistic vocabulary beyond the expressive resources of the object-language. For example, the theory of logical consequence for a first-order language with unrestricted quantification may require a second-order metalanguage (Williamson 2003). This leaves the logic some room for insensitivity to the metalogic, since metalogics which coincide over the fragment of the metalanguage translatable into the object-language may differ elsewhere. But that is not the room Dummett wanted, since the metalogics of the parties to a dispute over validity for the object-language will thereby differ over the fragment of the metalanguage translatable into the object-language, not just elsewhere. Dummett may still complain that quasi-homophonic semantics lacks explanatory power. But the problem is not that its explanations of validity are strictly circular. For example, if we assume ∀ x x=x in explaining why that very formula is valid, the explanans is distinct from the explanandum; to suppose otherwise is to confuse use and mention. Nevertheless, the recipient may be left with an uneasy feeling that the explanation was too cheap: it did not cost

|| 2 Dummett 1975 emphasizes the distinction between explaining validity and persuading someone of it.

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enough in hard labour. If we assume ∀ x x≠x, we can “explain why” ∀ x x≠x is valid in a formally parallel way. However, the task was not to explain why ∀ x x=x, that is, why everything is self-identical. No explanation of that non-metalinguistic fact is on offer. We may well doubt that any is possible for something so simple and fundamental. The task was to explain why the formula ∀ x x=x is valid. The quasi-homophonic semantics enables us to explain that semantic fact, taking the non-semantic facts for granted. To expect more than that from a semantic theory is to look in the wrong place. In general, the task for a semantic theory is to explain the semantic facts, given the non-semantic facts. It is not to explain the non-semantic facts, even when they are logical facts. The same point applies to explaining why a rule of inference is valid, but is easier to miss there because the use-mention distinction is harder to apply to such rules than to single sentences. But the distinction still does apply, for simply to employ a logical rule such as modus ponens is not yet to think metalinguistically. None of this yields a methodology for resolving disputes about fundamental logical laws. Why should it? We reasonably expect such disputes to be hard to resolve. But they have not been put beyond the reach of reason. Once we explore and compare in detail the consequences of adopting different systems of logic, we have plenty of evidence on which to base a reasoned choice. In Dummett’s example, if rejecting one distributive law was all it took to resolve the puzzles of quantum mechanics in all other respects, with no other resolution in sight, we might indeed have bluntly rejected that law for both objectlanguage and metalanguage, without constructing any elaborate semantic theory to project a classical metalogic onto a quantum logic. The real trouble for quantum logic may be just that rejecting the distributive law does much less than advertised towards resolving the physical puzzles. The discussion so far has proceeded in highly schematic terms. We can test it by considering a case study. In this paper, the case is modal logic.

4 Case study I: propositional modal logic The standard metatheory of modal logic seems to meet Dummett’s constraint that the logic should be insensitive with respect to the metalogic. The semantic framework is possible worlds model theory, which is used to characterize the relevant consequence relations. The metalanguage is simply an extensional language with enough non-logical primitives to express set theory and the

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syntax of the modal object-language. It contains no modal operators. It does not even contain modal predicates such as “is possible” or “is a possible world”. In short, it is the language of mathematics and syntax. The logic onto which such a non-modal semantic theory projects the metalogic depends on no specifically modal principle of the metalogic. Let us check the point in detail for propositional modal logic. Syntactically, the object-language is standard: it contains countably many atomic formulas p, q, r, …, the one-place sentential operators ¬ and □, and the two-place sentential operator &. Other symbols are introduced as metalinguistic abbreviations in the usual way. For example, ◊ is ¬□¬. A model is any triple , where W is any nonempty set, R is any set of ordered pairs of members of W (a binary relation on W), and V is any function from atomic formulas to subsets of W. We define a three-place |= between a model M = , an element w ∈W and a formula A, by recursion on the complexity of A, thus (where “wRx” abbreviates “ ∈ R”): If A is atomic, M, w |= A if and only if w ∈ V(A) M, w |= ¬A if and only if not M, w |= A M, w |= A & B if and only if M, w |= A and M, w |= B M, w |= □A if and only if M, x |= A for every x ∈ W such that wRx Those definitions are stated purely in terms of mathematics and syntax. Although the clauses for ¬ and & are quasi-homophonic, the clause for □ is not, since the modal operator in the object-language is handled by non-modal quantification in the metalanguage. The model theory of propositional modal logic is developed simply as a piece of mathematics. For example, we can prove mathematically that for any binary relation R on a set W: M, w |= A → □◊A (the ‘Brouwerian’ schema) for every model M = , w ∈W and formula A if and only if R is symmetric. Similarly, we can prove that M, w |= □A → □□A (the 4 schema) for every model M = , w ∈ W and formula A if and only if R is transitive. The proofs involve no modal considerations whatsoever: they do not mention possibility or necessity. To work through an even simpler example, let us prove that M, w |= □A → A for every model M = , w ∈ W and formula A if and only if R is reflexive. First, suppose that R is reflexive on W. Consider any model M = , w ∈ W and formula A. If M, w |= □A, then, by the clause for □, M, x |= A for

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every x ∈ W such that wRx; but wRw because R is reflexive, so M, w |= A. Thus M, w |= □A → A by the clause for →, as required. For the converse, suppose that R is not reflexive on W. Hence for some w ∈W, not wRw. Consider a model M = such that V(p) = W-{w}. Thus by the relevant clauses M, w |= □p but not M, w |= p, so not M, w |= □p → p, which completes the proof. In motivating the semantics informally, we may describe W as a set of worlds, and R as a relation of relative possibility between worlds, where x is possible relative to w if and only if x would be possible if w obtained. We may call a formula A true at a world w in a model M if and only if M, w |= A. However, such ideas play no official role in the formal definitions or proofs. For example, in providing a model M = and w ∈W such that not M, w |= □p → p, we made no attempt to provide a possible case in which something obtains necessarily without obtaining; there is no such possible case. The set W may simply be {0}, and R simply {}. In effect, the technical study of modal logic has made such dramatic progress over the past fifty years by eliminating all modal considerations from its reasoning. Questions in the model theory of modal logic are answered by purely mathematical proofs. Philosophical disputes about possibility or necessity are irrelevant to this process. For that very reason, however, the model theory does not resolve those disputes. Let us explore the relation between the model theory of modal logic and philosophical questions about modality in more detail. For definiteness, we fix on an informal interpretation of the modal operators □ and ◊: as symbolizing metaphysical necessity and possibility respectively, say, rather than physical or epistemic modalities. Call a formula A metaphysically universal if and only if A is true on every interpretation of the atomic formulas with the intended interpretation of the operators. For present purposes we may assume that it is uncontroversial for the given object-language that (i) every truth-functional tautology is metaphysically universal; (ii) whenever A and A → B are metaphysically universal, so is B; (iii) every instance of the schema □(A → B) → (□A → □B) is metaphysically universal; (iv) whenever A is metaphysically universal, so is □A. Since the analogues of (i)–(iv) for provability axiomatize the weakest “normal” modal logic K, (i)–(iv) make every theorem of K metaphysically universal. This implies that there is a class of models (in the above sense) such that any formula is metaphysically universal if and only if it is true at every world in every model in the class. For let C be the class of all models M such that every metaphysically universal formula is true at every world in M. We need to show that every formula true at every world in every model in C is metaphysically universal. Suppose that A is not metaphysically universal. Thus ¬A is true on

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some interpretation Int of the atomic formulas and the intended interpretation of the operators (we assume bivalence: if A is not true on Int then ¬A is true on Int). The set of all formulas which are true on Int is K-consistent, in the sense that for no such formulas B1, …, Bn is ¬(B1 & … & Bn) a theorem of K: for if B1, …, Bn are all true on Int, so is B1 & … & Bn, so ¬(B1 & … & Bn) is not true on Int, so ¬(B1 & … & Bn) is not metaphysically universal, so by the above ¬(B1 & … & Bn) is not a theorem of K. It follows that the canonical model M = of K contains a world w at which all formulas true on Int are true. Let M* = be the submodel of generated by w; thus W* is the set of all x ∈ W to which w has the reflexive ancestral of R, R* = R ∩ W*2 and V*(p) = V(p) ∩ W* for every atomic formula p. Generated submodels preserve the truth-values of formulas at any point: for every x ∈ W* and formula B, M*, x |= B if and only if M, x |= B.3 Hence if B is true on Int then M*, w |= B because M, w |= B. In particular, since ¬A is true on Int, M*, w |= ¬A, so not M*, w |= A. But M* ∈ C: for if x ∈ W* then x is n steps of the relation R* from w; thus if a formula B is metaphysically universal, so is □nB by (iv) above (where □n is a sequence of n occurrences of □), so in particular □nB is true on Int, so M*, w |= □nB, so M*, x |= B by the clause for □. Thus A is not true at every world in every model in C. This proves that a formula is metaphysically universal if and only if it is true at every world in every model in C. We seem to have reduced questions of metaphysical universality to questions of model theory, given the rather modest assumptions (i)–(iv) about metaphysical universality. The catch is that the class C of models was itself defined in terms of metaphysical universality. Suppose, for example, that two philosophers disagree over the principle that what obtains necessarily possibly obtains. They both understand the modal operators as symbolizing metaphysical modalities; they are not talking past one another. In effect, they disagree on whether the B axiom p → □◊p is metaphysically universal. They both accept constraints (i)–(iv) on metaphysical universality, and so agree that the axiom is metaphysically universal if and only if it is true at every world in every model in the class C. By the fact noted above, if C contains only models for which R is symmetric, then the B axiom is true at every world in every model in C, so the axiom is metaphysically universal; in the other direction, if for some non-symmetric relation R on a set W, C contains all models of the form , then the B axiom is false at some world in some model in C, so the axiom is informally invalid. But to ask whether C contains a model with a non-symmetric

|| 3 See Hughes and Cresswell (1984, pp. 22–25 and 78–81 respectively) for more on the relevant properties of canonical models and of generated submodels.

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relation R boils down to asking whether every metaphysically universal formula is true at every w ∈ W in . If p → □◊p is false at some w ∈ W in then the question arises whether p → □◊p is metaphysically universal. We are back where we started. We may therefore expect arguments for or against the B principle to bypass the mathematical model theory, and address the modal issues directly. That is just what we find in practice, in particular when we look at Dummett’s own critique of the principle. He argues, against Kripke (1980, p. 157), that there could have been unicorns (Dummett 1993a, p. 346): They might, for instance, be of the order Artiodactyla, like deer, or of the order Perissodactyla, like horses. In the language of possible worlds, there are no unicorns in the actual world w, but there is a possible world u in which there are unicorns, which belong to the order Artiodactyla, and another possible world v in which there are also unicorns, which in that world belong to the order Perissodactyla. […] In world u, any animal, to be a unicorn, must have the same anatomical structure as the unicorns in u, and hence, in particular must belong to the order Artiodactyla. It follows that the world v is not possible relatively to u, and, conversely, that u is not possible relatively to v. How about the actual world w—is that possible relatively to either u or v? It would at first seem so, since the principal difference we have stipulated is that there are no unicorns at all in w. But u is a world in which it holds good that unicorns are necessarily of the order Artiodactyla, whereas in w it is possible for unicorns to be of the order Perissodactyla. Since a proposition necessarily true in u is possibly false in w, w cannot be possible relatively to u, although u is possible relatively to w. The relation of relative possibility (accessibility) is therefore not symmetrical.

In effect, Dummett is arguing that the principle p → □◊p is invalid when p is interpreted as expressing the proposition that it is possible for unicorns to be of the order Perissodactyla.4 Although Dummett puts his argument in terms of accessibility relations between worlds, he is not attempting to use model theory to avoid metaphysical controversy. Rather, he is adverting to an intended model. In speaking of possibility and necessity, he means the genuine articles, not whatever happens to play similar structural roles in an arbitrary model. The talk of worlds and their relative possibility serves to present the argument more perspicuously, not to replace metaphysics by semantics. One could articulate an equivalent argument

|| 4 Dummett’s argument requires the transitivity of the accessibility relation to argue from the accessibility of v from w and the inaccessibility of v from u to the inaccessibility of w from u, or alternatively the 4 schema (amongst other principles) to derive ¬□◊◊q from ¬□◊q, where q expresses the proposition that unicorns are of the order Perissodactyla (so p corresponds to ◊q). Dummett endorses the 4 schema in the next paragraph (1993, p. 347).

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using modal operators, without reference to worlds. Indeed, the crucial first sentence of the quoted passage is already so phrased: in effect, “There could have been unicorns of the order Artiodactyla and there could have been unicorns of the order Perissodactyla”. The switch to talk of possible worlds is presented as terminological: “in the language of possible worlds”. Critics of Dummett’s argument have accused it of confusing what could have been truly said with the word “unicorn” in counterfactual circumstances with what can be truly said with it in the actual circumstances about the counterfactual circumstances.5 If they are right, as they surely are, the argument rests on a semantic confusion, but that does not make it a semantic argument. It does not make the argument’s modal dimension redundant. Similar considerations apply to controversy over the 4 axiom □p → □□p. Suppose that two philosophers disagree over the principle that what necessarily obtains necessarily necessarily obtains. They both understand the modal operators as symbolizing metaphysical modalities; they are not talking past one another. In effect, they disagree on whether the 4 axiom is metaphysically universal. They both accept constraints (i)–(iv) on metaphysical universality, and so agree that the axiom is metaphysically universal if and only if it is true at every world in every model in the class C. If C contains only models for which R is transitive, then the 4 axiom is true at every world in every model in C, so the axiom is metaphysically universal; in the other direction, if for some nontransitive relation R on a set W, C contains all models of the form , then the 4 axiom is false at some world in some model in C, so the axiom is informally invalid. But to ask whether C contains a model with a nontransitive relation R boils down to asking whether every metaphysically universal formula is true at every w ∈ W in . If □p → □□p is false at some w ∈ W in then the question arises whether the model is really in C, which in turns partly depends on whether □p → □□p is metaphysically universal. Again, we are back where we started. Whether what necessarily obtains necessarily necessarily obtains is a metaphysical question, not a semantic one. It cannot be settled by purely modeltheoretic means. Not surprisingly, the most salient criticism of the principle has been overtly metaphysical, and couched in explicitly modal terms. Friends of the principle have responded at the same level. Both sides use semantic considerations in an auxiliary capacity, not as the core of their argument.6, 7

|| 5 See Reimer 1997 and Rumfitt 2010. For a related dispute over B, involving a proper name rather than a natural kind term, see Stephanou 2000 and Gregory 2001. 6 See Salmon 1989 and 1993 and Williamson 1990, pp. 126–143.

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5 Case study II: the Barcan formula The case of quantified modal logic is similar. The central controversy concerns the Barcan formula (Barcan 1946): BF

◊ ∃ x A → ∃ x ◊A

Informally, BF says that if there could have been something which met a certain condition, then there is something which could have met that condition. Many metaphysicians hold that there are actual counterexamples to BF. For instance, Queen Elizabeth I never had a child, but she could have done. By BF, it follows that there is something which could have been a child of Elizabeth I. But what is it? Given the essentiality of one’s actual origins, no actual person could have had Elizabeth I as a parent (Kripke 1980). Although some actual collection of atoms could have constituted a child of Elizabeth I, the collection would not have been identical with the child. According to those metaphysicians, there is nothing which could have been a child of Elizabeth I. Thus BF is false. Again, given the necessity of identity, BF implies that there could not have been more things than there actually are; but many metaphysicians regard the numerosity of the universe as contingent. There are two natural ways of extending models of propositional modal logic to interpret quantified modal logic: constant domain semantics and variable domain semantics. For both, the function V now maps each n-place atomic predicate F to an intension V(F), which maps each w ∈ W to an extension V(F)(w) for F. On constant domain semantics, each model has a single component set D to serve as the domain of the quantifiers. The semantic clause for a quantifier takes a form like this: M, w, a |= ∃ x A if and only if for some d ∈ D: M, w, a(x/d) |= A As an easily proved non-modal mathematical fact, on constant domain semantics every instance of BF is true at every world in every model under every assignment.

|| 7 For a more extensive discussion of metaphysical universality in propositional modal logic see Williamson 2013, pp. 92–118.

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On variable domain semantics (Kripke 1963), each model has a component function which maps each w ∈ W to a set D(w) to serve as the domain of the quantifiers when evaluated at w. The semantic clause for a quantifier takes a form like this: M, w, a |= ∃ x A if and only if for some d ∈ D(w): M, w, a(x/d) |= A As an easily proved non-modal mathematical fact, on variable domain semantics some instance of BF is false at some w ∈ W in some model under every assignment. For consider any model with w, x ∈ W such that wRx but not D(x) ⊆ D(w). In some such model M, V(F)(y) = D(y)-D(w) for all y. Thus M, x, a |= ∃ x Fx, so M, w, a |= ◊ ∃ x Fx. But if M, w, a(x/d) |= ◊Fx then d ∈ V(F)(y) for some y ∈ W, so not d ∈ D(w); hence not M, w, a |= ∃ x ◊Fx. Thus for A = Fx, BF is false at w in M. Informally, the set D in a constant domain model is imagined to contain whatever there is, with the background assumption that being (unlike being concrete) is non-contingent. Similarly, the set D(w) in a variable domain model is imagined to contain whatever there is in the world w, with the background assumption that being (like being concrete) is contingent. But these informal understandings play no role whatsoever in the model theory itself. As characterizations of metaphysical universality, constant domain semantics and variable domain semantics give incompatible results. We cannot accept both. The choice between them returns us to the metaphysical question of the contingency or otherwise of being. We have been given no way to bypass the modal controversy and settle the metaphysical universality of BF on non-modal grounds. Although variable domain models falsify BF, opponents of BF have a specific reason not to regard them as more than convenient representational devices. They typically hold that BF has actually false instances (read A as “x was a child of Elizabeth I”). Moreover, they typically hold that some such instances in no way depend on any tacit contextual restriction of the quantifiers. For example, although there could have been something which was a child of Elizabeth I, there is absolutely nothing, however widely the quantifier ranges, which could have been a child of Elizabeth I. Try capturing this idea in an intended variable domain model M. M should contain the actual world @ as one of its worlds. If BF has false instances at @ in M, for some world w D(@) does not include D(w), so some d in D(w) is not in D(@), so D(@) does not contain everything there is. Thus D(@) does not contain everything over which the quantifier ranges on its intended interpretation. Consequently, such opponents of BF

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should deny that there is an intended model.8 They should regard models as more like mere representational devices, which may picture a failure of BF but cannot instantiate one. Such a model may contain a world at which the true closed formulas coincide with those true on a given intended interpretation, but the explanation of their truth at the world in the model will be quite different from the explanation of their truth on the intended interpretation. We can explain on purely mathematical grounds why there are counterexamples in the variable domain semantics to BF. Indeed, in some of those counter-models the domains of all worlds are numbers, whose being is presumably non-contingent. The explanation ignores modal matters. Any argument for the coincidence of the sentences true at the world in the model with the sentences true on the intended interpretation would itself have to proceed partly in modal terms.

6 Other semantic theories for modal languages Given the results of section 5, opponents of BF may therefore seek a semantic theory more faithful to the intended meanings of the modal operators. The natural idea is to use a quasi-homophonic semantics in a modal metalanguage. A little work has indeed been done along such lines. It is a laborious business, by comparison with possible worlds semantics in a non-modal metalanguage. Even very simple results are very hard to prove; various obstacles remain to be overcome. Moreover, if we use such a metatheory, we cannot expect to achieve very much by semantic ascent. Our assessments of modal principles in the objectlanguage simply reflect our assessments of the same principles in the metalanguage.9 In general, we cannot expect to settle modal questions by non-modal reasoning. Of course, some philosophers try to reduce the modal to the non-modal. Perhaps the best example is David Lewis (1986). On his modal realism, quantification over possible worlds in a non-modal language represents the underlying metaphysical reality more perspicuously than does the use of modal operators. The possible worlds themselves are explained in non-modal terms, as mutually

|| 8 See Williamson 1998 for this point and a defence of BF. See also Linsky and Zalta 1994 and 1998 and Parsons 1995 for similar defences of BF. The point is argued in more depth in Williamson 2013, pp. 130–139. For the failure of the criterion of representational significance in Stalnaker 2010 to handle the example of BF see Williamson 2013, pp. 188–194. 9 For modal truth theory see Fine 1977, Davies 1978, Peacocke 1978, Gupta 1978 and 1980, and Rumfitt 2001; for modal model theory, see Humberstone 1996.

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isolated spatiotemporal systems. The actual world is merely one such system amongst many, just as here is merely one place amongst many, privileged only from its own perspective. Claims about how given objects could have been different describe or misdescribe how counterparts of those objects in other systems are different. Philosophers who agree on modal realism can use it to settle some modal questions by non-modal reasoning. But most philosophers reject modal realism as hopelessly implausible. They insist that this actual world contingently but objectively has a privileged metaphysical position. For them, the use of modal operators gives a more perspicuous representation of the underlying metaphysical reality in that respect than does quantification over worlds in a non-modal language. If modal realism is false, of what philosophical use is possible worlds model theory for modal logic? It is a powerful instrument for consistency proofs. We can show a formula to be underivable from a set of axioms and rules of inference by constructing a model with a world at which all the axioms are true and all the rules preserve truth but the formula is false. Moreover, if we have proved an axiomatic system sound and complete for a given class of models, we can sometimes derive consequences from it more efficiently by reasoning about the models than by reasoning within the axiomatic system itself. Alternatively, one might avoid the process of axiomatization altogether by specifying a modal theory as comprising just those formulas true at all worlds in all models in a formally specified class, and deriving its consequences by reasoning about the models. But in such applications the model theory plays no more than a useful auxiliary role. It does not enable us to bypass modal reasoning. For example, it provides no way of settling the disputes over the B, 4 and BF schemas by semantic means. Might some other form of semantics for modal languages do better? Possible worlds semantics is by far the best developed approach, and so constitutes the most authoritative test in this area of Dummett’s argument quoted at the start of this paper. It turned out to provide no independent criterion for determining the metaphysical universality of philosophically contentious modal schemas. A fortiori, nor does quasi-homophonic semantics. In the spirit of Dummett (1991), we might also consider more proof-theoretic semantic theories for modal languages. So far, proof theory has contributed comparatively little to the development of modal logic. Robert Brandom has recently attempted to apply his inferentialist approach to the semantics of modal logic. However, since his strategy is to show how to construct an inferentialist semantics for any normal modal logic, it is of no great help in deciding between normal modal logics (Brandom 2008, pp. 170–171).

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On current evidence, semantics is no royal road to the resolution of disputes in modal logic. It can clarify and discipline them, but it cannot transmute questions of logic into questions of semantics. Of course, confusion in semantics can cause error in logic. For unnoticed ambiguities can cause error in any inquiry, and false assumptions of synonymy may have the same effect. Confusion as to whether names are rigid designators can cause error in the modal logic of identity. Sorting out the confusion may be a precondition for correcting the error. But if getting the semantics wrong usually results in getting the logic wrong, it does not follow that getting the semantics right usually results in getting the logic right. In this respect, logic may differ less from the natural sciences than is often supposed. A bad semantic theory for the language of physics may mislead us into accepting false conclusions in physics. For example, on the basis of crude verificationism in semantics, a philosopher may assert that all past events have left causal traces in the present, which may turn out to be physically false. But that is no reason for physicists to expect much help from a good semantic theory for the language of physics. We should not be too quick to assume that the case of logic is radically different.10

7 Understanding and disagreement in logic In a passage from Dummett quoted at the beginning of the paper, he emphasizes the need for the two parties in a dispute over a logical principle to understand each other. He envisages this mutual understanding as produced by the insensitivity of the logic to the metalogic, given the sort of semantic theory he recommends, because the two sides can agree on what logic a given semantic theory of that sort would generate, assuming only uncontentious metalogical principles. For Dummett, the dispute may thereby evolve into one over the choice of semantic theory for the object-language. “Understanding someone’s utterance” can mean different things. If someone says “The Moon is larger than the Earth”, we may understand his utterance in one sense, because we know on the basis of our competence with the English language and awareness of the conversational context what proposition he strictly and literally expressed, but not understand his utterance in another sense, because we have no idea why he felt warranted in making it, thereby asserting what is generally known to be a falsehood. In a fundamental dispute

|| 10 For related discussion see Williamson 2007, pp. 10–47.

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over principles of logic, the initial understanding we lack of our opponent’s utterances is typically of the latter rather than the former kind. Whatever its formal ramifications, the dispute applies at least to principles formulated in a natural language, whose expressions both sides use with their standard public meanings, not with artificially concocted ones.11 For the natural language as object-language, the role of the rival semantic theories is descriptive, not stipulative. But they are supposed to give each side understanding in the latter sense of the other’s utterances, by explaining the considerations on which they were based. In the case studies of modal logic above, neither side has much difficulty in understanding what the other is strictly and literally saying. The challenge is to understand why they are saying it. In that task, however, the semantic theories are virtually useless. Those who accept symmetry as a constraint on the accessibility relation of a model must accept the B principle for the object-language, but that only raises the further question: why do they accept the symmetry constraint? An analogous point applies to those who reject both the symmetry constraint and the B principle. If you are puzzled by your opponents’ stance on the B principle, you will be equally puzzled by their stance on the symmetry constraint. If they explain their putative counterexample to symmetry, you may start to understand why they say what they do. But the example could just as well have been presented as a direct counterexample to the B principle itself, without the digression through model theory. One can often come to understand why people take the stance they do in a logical dispute when they articulate the considerations which move them. That does not depend on whether the considerations are semantic, logical or metaphysical. When we are challenged to explain why we accept what we take as our most fundamental logical principles, we may find ourselves with embarrassingly little to say. We can try to rebut objections, provide corroboration in the form of our logic’s strong track record, simplicity, elegance and integration with mathematics and science, and emphasize the problematic features of rival logics, while still feeling those considerations to be secondary by comparison with the utter obviousness of its principles. However, there are several reasons for not reworking the semantic theory with a view to deriving the contested principles of the logic from uncontested principles of the metalogic. First, the strategy may be impossible to implement. If the principle is fundamental enough, it may be derivable from less contested principles of the

|| 11 For more on logical disagreement without diversity of meaning see Williamson 2007, pp. 73–133.

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metalogic only given a contested semantic theory. The extra semantic structure invoked to avoid appeal to the contested principle in the metalanguage may simply support doubts about the faithfulness of the semantics. For instance, it is uncontentious that constant domain semantics validates BF; sceptics about BF simply focus their doubts on whether constant domain semantics is faithful to the intended meanings of the formulas. Second, contentiousness depends on one’s opponents. Virtually every putative principle of logic has been contested by some philosopher or other. The reworking of semantic theory needed to validate a principle of logic from principles of metalogic uncontested by one opponent may be quite different from the reworking needed to validate that principle from principles of metalogic uncontested by a quite different opponent. Semantics is a theoretical inquiry into meaning, not a debating technique. We should not tailor our semantic theories to suit the accident of which opponent currently exercises us the most, thereby distorting our semantic theories to gain a short-term dialectical advantage in logic. Of course, we cannot assume without further investigation that the case of modal logic is typical of the interrelations between logic, metalogic and semantics. In any difficult choice between logics, we must reflect explicitly on whether the argument forms they endorse coincide with the valid ones, which we can do systematically only if we consider semantic theories for the logical constants and syntactic constructions which characterize those forms. But it does not follow that the initiative lies with semantics in such an inquiry; as we have seen, its role may still be a subsidiary, clarificatory one. One simple line of thought, adumbrated earlier, suggests that the role of semantics will indeed be secondary. The starting point is that logical theorems are normally not metalinguistic in content. To return to the case of modal logic, against the background of a quantified modal system as strong as S5, the dispute over BF can resolve itself into a dispute over the single formula □∀x □∃y x=y (NNE), which says that necessarily everything is necessarily something, or equivalently: there could not have been something which could have been nothing. NNE and ¬NNE are not metalinguistic claims in any interesting sense. To try to explain why there could not have been something which could have been nothing, or why there could have been something which could have been nothing, by invoking semantic considerations would be to go off on a digression through the irrelevant. The same goes for any other theorem of any standard logical system. But if individual axioms and theorems are not metalinguistic in content, axiom and theorem schemas too do not introduce metalinguistic content, for a schema is only a convenient way of collecting its

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instances. Furthermore, if axiom and theorem schemas do not introduce metalinguistic content, nor do derived and underived inference rules. For an axiom or theorem schema is just the special case of a derived or underived inference rule with no premises, and leaving room for premises as well as a conclusion does not by itself introduce metalinguistic content where none was before. Thus logical principles in general are not metalinguistic in content, and so should not be explained (in the relevant sense) in semantic terms. Although we may need semantics to clear away confusions which block us from accepting valid logical principles, it is not semantics which explains the principles themselves. The same applies to rejecting invalid logical principles, once they are put in the form of universal generalizations, perhaps in higher-order logic. We should not expect semantics to exceed its proper task. This paper has concentrated on the relative roles of metalogic and of semantic theory in logical disputes, the topic of the passages originally quoted from The Logical Basis of Metaphysics. In the case studies, the metalogic and the semantic theory conforms to Dummett’s requirements, since which modal logic for the object-language the semantic theory projects the metalogic onto is trivially independent of any specifically modal feature of the metalanguage. Nevertheless, the semantic theory does not help the parties to the disputes in modal logic to understand each other, let alone to resolve their disputes. Indeed, we found no evidence for Dummett’s view that such disputes are fundamentally metalinguistic. Of course, according to Dummett, the ultimately decisive considerations should come from a deeper level of the theory of meaning, at which competing claims are supposed to be cashed out in terms of the hard currency of speakers’ observable use of the language. But we have virtually no idea what such a process might look like for modal logic, or why we should expect the semantic theorizing it would involve to be less contentious than the nonmetalinguistic logico-metaphysical theorizing involved in a more direct approach to the modal issues. My own view is that in such matters we have no promising alternative to abductive theorizing in a modal object-language.12 Thus my disagreement with Dummett can be traced back to the methodological difference he diagnosed between us more than a third of a century ago.

|| 12 Williamson 2013 undertakes such an approach.

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References Barcan, Ruth [Marcus, Ruth Barcan] 1946: ‘A Functional Calculus of First Order Based on Strict Implication’, The Journal of Symbolic Logic, 11, pp. 1–16. Brandom, Robert 2008: Between Saying and Doing: Towards an Analytic Pragmatism. Oxford: Oxford University Press. Cellucci, Carlo, Grosholz, Emily, and Ippoliti, Emiliano (eds.) 2011: Logic and Knowledge. Newcastle: Cambridge Scholars Publishing. Copeland, Jack (ed.) 1996: Logic and Reality: Essays on the Legacy of Arthur Prior. Oxford: Clarendon Press. Cozzo, Cesare 2011: ‘Discussion’, in Cellucci, Grosholz, and Ippoliti 2011, pp. 101–197. Davies, Martin 1978: ‘Weak necessity and truth theories’, Journal of Philosophical Logic, 7, pp. 415–439. Dummett, Michael 1975: ‘The Justification of Deduction’, Proceedings of the British Academy, 59, pp. 201–232. Reprinted in Dummett 1978, pp. 290–318. Dummett, Michael 1978: Truth and Other Enigmas. London: Duckworth. Dummett, Michael 1991: The Logical Basis of Metaphysics. London: Duckworth. Dummett, Michael 1993a: ‘Could there be unicorns?’, in Dummett 1993b, pp. 328–348. Revised version of ‘Könnte es Einhörner geben?’, Conceptus, 17, pp. 5–10. Dummett, Michael 1993b: The Seas of Language. Oxford: Clarendon Press. Fine, Kit 1977: ‘Prior on the construction of possible worlds and instants’, in Prior and Fine 1977, pp. 116–168. Gregory, Dominic 2001: ‘B is innocent’, Analysis, 61, pp. 225–229. Gupta, Anil 1978: ‘Modal logic and truth’, Journal of Philosophical Logic, 7, pp. 441–472. Gupta, Anil 1980: The Logic of Common Nouns. New Haven: Yale University Press. Hale, Bob, and Hoffmann, Aviv (eds.) 2010: Modality: Metaphysics, Logic, and Epistemology. Oxford: Oxford University Press. Hughes, George, and Cresswell, Max 1984: A Companion to Modal Logic. London: Methuen. Humberstone, Lloyd 1996: ‘Homophony, validity, modality’, in Copeland 1996, pp. 215–236. Kripke, Saul 1963: ‘Semantical considerations on modal logic’, Acta Philosophica Fennica, 16, pp. 83–94. Kripke, Saul 1980: Naming and Necessity. Oxford: Blackwell. Lewis, David 1986: On the Plurality of Worlds. Oxford: Blackwell. Linsky, Bernard and Zalta, Edward 1994: ‘In defense of the simplest quantified modal logic’, Philosophical Perspectives, 8, pp. 431–458. Linsky, Bernard and Zalta, Edward 1996: ‘In defense of the contingently nonconcrete’, Philosophical Studies, 84, pp. 283–294. Parsons, Terence 1994: ‘Ruth Barcan Marcus and the Barcan formula’, in Sinnott-Armstrong, Raffman, and Asher 1994, pp. 3–11. Peacocke, Christopher 1978: ‘Necessity and truth theories’, Journal of Philosophical Logic, 7, pp. 473–500. Prior, Arthur, and Fine, Kit 1977: Worlds, Times and Selves. London: Duckworth. Reimer, Marga 1997: ‘Could there have been unicorns?’, International Journal of Philosophical Studies, 5, pp. 35–51. Rumfitt, Ian 2001: ‘Semantic theory and necessary truth’, Synthese, 126, pp. 283–324. Rumfitt, Ian 2010: ‘Logical necessity’, in Hale and Hoffmann 2010, pp. 35–64.

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Salmon, Nathan 1989: ‘The logic of what might have been’, Philosophical Review, 98, pp. 3–34. Salmon, Nathan 1993: ‘This side of paradox’, Philosophical Topics, 21, pp. 187–197. Sinnott-Armstrong, Walter, with Raffman, Diana and Asher, Nicholas (eds.) 1994: Modality, Morality and Belief: Essays in Honor of Ruth Barcan Marcus. Cambridge: Cambridge University Press. Stalnaker, Robert 2010: ‘Merely possible propositions’, in Hale and Hoffman 2010, pp. 21–34. Stephanou, Yannis 2000: ‘Necessary beings’, Analysis, 60, pp. 188–191. Williamson, Timothy 1988: ‘First-order logics for comparative similarity’, Notre Dame Journal of Formal Logic, 29, pp. 457–481. Williamson, Timothy 1990: Identity and Discrimination. Oxford: Blackwell. Williamson, Timothy 1998: ‘Bare possibilia’, Erkenntnis, 48, pp. 257–273. Williamson, Timothy 2003: ‘Everything’, Philosophical Perspectives, 17, pp. 415–465. Williamson, Timothy 2007: The Philosophy of Philosophy. Oxford: Blackwell. Williamson, Timothy 2011: ‘Logics and metalogics’, in Cellucci, Grosholz, and Ippoliti 2011, pp. 81–100. Williamson, Timothy 2013: Modal Logic as Metaphysics. Oxford: Oxford University Press.

Alex Burri

Residues of Realism I The pursuit of truth and knowledge proceeds in what seems to be two different fields of human epistemic endeavor: nature and thought. We theorize not only about the world surrounding us but also about our ways of theorizing: about methods, standards of truth and knowledge, cognition, learning, the creations of the human mind, laws of probability and many other things constituting what one might call the realm of reason. The difference, if there is any, between these two subject matters does not lie in the inexhaustibility of the first as opposed to the exhaustibility of the second. As we know from the history of philosophy itself, hopes of establishing a complete or singular theory of theorizing are nil. The realm of reason, though self-reflexive, is not “ideal” in the sense of being fully transparent or comprehensible to us. Nevertheless, something akin to exhaustibility might well distinguish those two fields of investigation, namely, the ontological attitude with which we do, or should, approach them: realism in the case of nature, antirealism in the case of thought. Sir Michael Dummett, having an outstanding record of defending antirealism in more than just the philosophy of mathematics, will not agree with the last remark. Still, in the few pages to follow, I shall do two things. First, I shall present a brief argument in favor of scientific realism vis-à-vis nature, an argument that might also appeal to Dummettians. Second, and more importantly, I shall turn to mathematics as one of the sciences in the realm of reason in order to confront Sir Michael’s case for an intuitionistic, hence anti-realistic, treatment of number theory with what I take to be an ineliminable residue of realism in mathematics.

II Scientific realism is the view that the theoretical, i.e. non-observational, sentences of an empirical theory (a) must be understood literally, i.e. taken at face

|| Universität Erfurt

DOI 10/1515-9783110459135-010

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value, and (b) are either true or false. The first condition distinguishes realism from both instrumentalism and operationalism. According to the former, the non-observational sentences of a scientific theory are mere intellectual tools “for predicting future experience in the light of past experience”.1 They are, that is to say, “meaningless devices for the systematization of data”2 that can be regarded neither as literal descriptions of a hidden reality underlying observable phenomena nor as truth-apt. According to operationalism, those sentences must be reinterpreted “as covert, complex reports on observation”3 and cannot, therefore, be taken at face value although, as disguised observational statements, they are either true or false. The second condition distinguishes realism from so-called constructive empiricism according to which theoretical sentences do not, ipso facto, have a truth value but should nevertheless be understood literally.4 To lend substance to scientific realism one needs to bolster our belief in theoretical sentences being justified or true. The facile argument to that effect is purely inductive. Sir Michael has rendered it as follows: “these theories have ‘worked’ in the past, in the sense of having for the most part yielded true observation statements, and so we have confidence that they will continue to work in the future”.5 This smacks of circularity, however, since the principles of induction, whatever they may be, are themselves theoretical.6 The more ambitious, even though related, argument is due to Richard Boyd and Hilary Putnam: that the theoretical terms of a theory belonging to a mature science typically refer and that the laws of such a theory are typically approximately true are the only assumptions capable of explaining why the predictions of the theory in question turn out to be true.7

|| 1 Quine (1961), p. 44. – Whether Quine is an instrumentalist is of no importance here. At least some of his characterizations appear to fit the bill: “Science is a conceptual bridge of our own making, linking sensory stimulation to sensory stimulation” (Quine (1981), p. 2). 2 Horwich (1982), p. 182. 3 Ibid. 4 See van Fraassen (1980), pp. 10 f. 5 Dummett (1978), p. 220. 6 Hilary Putnam once bit the bullet: “true ideas are the ones that succeed – how do we know this? This statement too is a statement about the world; and we believe in the practice to which this idea corresponds […] on the basis that we believe in any good idea – it has proved successful! In this sense ‘induction is circular’. But of course it is!” (Putnam (1974), p. 239). 7 “The positive argument for realism is that it is the only philosophy of science that doesn’t make the success of science a miracle” (Putnam (1975), p. 73); “it seems that miracles are the only alternative to a realist explanation of the success of scientific practice” (Boyd (1983), p. 54).

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This, I take it, will not convince the skeptic. And ironically, realism, which is doomed to accept a clear-cut distinction between appearance and reality, is, to use Thomas Nagel’s apt phrase, “inescapably subject to skepticism and cannot refute it but must proceed under its shadow”.8 So how can we proceed under the shadow of skepticism without losing hold of the insight contained in the realist’s explanation for the success of science? If we were brains in a vat or the victims of an evil demon, both our scientific endeavor and our scientific hypotheses would be indistinguishable from how they present themselves to us now – and so would the realistic explanation of their success. Our observational stimuli would occur in the same order and display the same phenomenal regularities. In order to produce such an outcome, the computer feeding our sensory receptors or the evil demon tricking us into believing there is a material world outside of our minds would, however, have to go through a sequence of program steps or a sequence of considerations that simulates the workings of the causal mechanisms postulated by our predictively successful hypotheses taken at face value. The computer program or the mental deliberation of the demon would not have to be isomorphic to the postulated causal order but it would, roughly speaking, have to be functionally equivalent to it. Computers and demons are representational, if not intentional, systems doing what they do by processing symbol sequences or other thought-like entities. Hence, if they were the unobservable cause of our observational stimuli, what we take to be the external world would be far more Berkeley-like than, say, Locke-like. In a barely expressible sense, we would be wrong about the world’s metaphysical essence. But what difference would that make? Sir Michael himself has called our attention to the following: What would be the difference between God’s creating a material universe, in the whole of which there never was any creature able to experience it, and His creating nothing at all? Or, rather, what would be the difference between His creating such a universe and His merely conceiving of it? What difference would its existing make? There would surely be no difference.9

As far as the unobserved is concerned, be it a universe devoid of sentient beings, be it the deep cause of our observational stimuli, there is no tangible difference between the merely conceived-of, the thought-like or the representational on the one hand and the material, the substance-like or the thing in itself

|| 8 Nagel (1986), p. 71. 9 Dummett (2006), p. 97, his emphasis.

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on the other hand. What really matters is its causal arrangement, i.e. its functional structure that mediates between our preceding actions (turning our “head”, running a Michelson-Morley experiment or whatever) and our subsequent observational stimuli. As Quine said in a similar connection: “Save the structure and you save all”.10 In a skeptical scenario as well as in an ordinary one, whether the theoretical terms of a mature science should be said to refer or whether predictively successful hypotheses should be said to be approximately true is almost beside the point. What counts is whether the latter – think of them as differential equations – display, in Wittgenstein’s sense of “showing”, a mind-independent causal or functional structure that effectively links “bodily” actions to observational inputs or sensory stimulations. Such structures are all that is called for. They constitute both the real ontological commitment of theoretical sentences and the true core of scientific realism.

III In mathematics, the relationship between “theoretical” and “observational” sentences that is crucial to the realism/anti-realism issue boils down to the relationship between the axioms and certain theorems, let us call them “basic theorems”, that can also be established independently of the axioms themselves. At least it is a remarkable fact about elementary arithmetic that countless theorems such as “2 + 2 = 4” were known to be true way before Guiseppe Peano proposed the appropriate axioms. What is more, if it had not been possible to deduce every basic theorem from these axioms and some definitions, or if these axioms had even allowed to disprove one of the basic theorems, Peano’s proposal would simply have been considered inadequate. Rationalist philosophy, long impressed by Euclid’s seminal work and the more geometrico method read out from it, would have it the other way round: self-evident axioms whose truth can clearly and distinctly be apprehended establish the truth of the theorems derived by self-evident steps from them. Therefore, any conflict between the axioms and a putative theorem must lead to a refusal of the latter. But this is an epistemological fancy. As Quine has remarked:

|| 10 Quine (1992), p. 8.

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As for the end truths, the axioms of set theory, these have less obviousness and certainty to recommend them than do most of the mathematical theorems that we would derive from them. Moreover, we know from Gödel’s work that no consistent axiom system can cover mathematics even when we renounce self-evidence. Reduction in the foundations of mathematics remains mathematically and philosophically fascinating, but it does not do what the epistemologist would like of it: it does not reveal the ground of mathematical knowledge, it does not show how mathematical certainty is possible.11

The basic theorems, that is to say, are fundamental, indeed. They must be accounted for, as must the observational sentences in empirical science. Sir Michael is with us on this. And he has drawn anti-realistic conclusions from it. Let us see how that works. Focusing on arithmetic, the basic theorems he has in mind form “a very narrow range of statements of elementary number theory”,12 namely of those simple quantifier-free arithmetical sentences whose truth can be determined by “some computation procedure”,13 say, “598017 + 246532 = 844549” or “17 is prime”. As before, every viable axiomatization, every viable arithmetic theory must bring these fundamental statements about. So if, in the light of a counterexample, the Goldbach conjecture turns out to be false, all admissible axiom systems must be able to confirm this. And as before, if a proposed axiom leads to a consequence that contradicts one of the basic theorems, it has thereby been falsified and will be withdrawn from circulation. Besides providing us with implicit definitions of the terms occurring in them, one of the main purposes of the axioms is to systematize the data, i.e. the basic theorems. But in doing so, the axioms not only create a uniform and unified domain of discourse but also enable us to deduce, with the help of logical (and mathematical) inference rules, any number of theorems, both basic and non-basic ones. And this, in turn, has two consequences. First, many theorems become demonstrable in two different ways, namely by means of a direct or bottom-up method consisting of computation procedures plus logical weakening14 and by means of an indirect or top-down method consisting of deduction from axioms (and previously demonstrated theorems), i.e. of logical strengthen-

|| 11 Quine (1969), p. 70. 12 Dummett (1978), p. 219. 13 Ibid., p. 225. 14 “Logical weakening” is here to mean making inferences with the help of introduction rules, i.e. inferences leading from logically simpler premises such as “17 is prime” to logically more complex conclusions such as “∃x.x is prime” or “15 is prime ∨ 17 is prime”. Not every introduction rule weakens the conclusion in the face of the premises, however. “And”-introduction, for example, does not.

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ing.15 Second, both logical weakening and top-down theorem deduction add to the stock of available mathematical statements: they transcend the realm of basic theorems. But how do we have to deal with this additional stuff? Is it kosher? Can we accept it as justified or even true? On the one hand, some of it is clearly unproblematic. For example, to infer a non-basic theorem of the form “∃x.Fx” from a basic theorem of the form “Fa” amounts to a (degenerated) constructive existence proof that is, as such, beyond any doubt. Proving “∀n[1 + 2 + ... + n = n(n + 1)/2]” or the binomial theorem of elementary algebra by mathematical induction remains beyond dispute as well. For the principle of mathematical induction – used either as an axiom schema or as an inference rule, i.e. as an introduction rule for the universal quantifier in the realm of natural numbers – is fully acceptable even from the most scrupulous point of view in the philosophy of mathematics. To renounce that principle would be to confuse arithmetic with calculation. On the other hand, because the axioms are not to be regarded as selfevident statements that “force themselves upon us as being true”,16 some of the extra stuff might, on closer examination, turn out to be dubious. After all, the axioms, not being self-sustained truths, get their credibility from their capability of systematizing and implying the basic theorems (and the non-basic ones gained thereof by logical weakening). So why should a theorem that is deducible by means of the top-down method but not demonstrable (or, for that matter, not yet demonstrated) by means of the bottom-up method be counted as justified or even true? Think, to mention extreme cases, of the results of nonconstructive existence proofs requiring the axiom of choice. Epistemologically speaking, the situation might not be hopeless. One could, for example, fall back on conditionalization whenever a result obtained by topdown deduction lacks the property of being demonstrable by bottom-up means. Instead of asserting, say, the existence of a well-ordering of the real numbers, one could content oneself with the conditional “If the axiom of choice is true, then there is a well-ordering of the real numbers” that is provable in ZermeloFraenkel set theory without the axiom of choice. And such a retreat might be deemed satisfactory enough.

|| 15 “Logical strengthening” is here to mean making inferences with the help of elimination rules, i.e. inferences leading from logically more complex premises to logically simpler conclusions as, for example, in modus ponens. 16 Gödel (1990), p. 268.

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We can leave the epistemic issue aside, however, since Sir Michael has voiced a deeper concern with the extra stuff in question. His concern is semantic rather than epistemic: What could those theorems whose epistemic status is, as we might say, asymmetrically axiom-dependent possibly mean? What content do they have? What does the grasp of their meaning amount to? In order to escape from a holistic view of (diffused and intangible) meaning, we must be able to provide them with a determinate individual content, a content not lying “wholly in the rôle which they play within the mathematical theories to which they belong, and which are themselves significant precisely because they enable us to establish the correctness of finitistic statements [i.e. basic theorems]”.17 And this, Sir Michael insists, can only be accomplished if we provide for a congruence or “harmony” between the two different ways of demonstrating non-basic theorems: [A] loose way of putting the requirement is to say that there must be a harmony between the introduction and elimination rules […]. So far as a logically complex statement is concerned, the introduction rules governing the logical constants occurring in the statement display the most direct means of establishing the statement, step by step in accordance with its logical structure; but the statement may be accepted on the basis of a complicated deduction which relies also on elimination rules, and we require a harmony which obtains only if a statement that has been indirectly established always could (in some sense of ‘could’) have been established directly.18

Consider a statement indecidable within an axiomatic system, say, the continuum hypothesis or its negation in regard to Zermelo-Fraenkel set theory with the axiom of choice, or consider a statement, such as the Goldbach conjecture, not known to be provable or disprovable within Peano arithmetic; call it “p”. Then, by assuming classical logic, we can deduce the admittedly somewhat boring theorem “p ∨ ¬p” from the axiom system in question – or, to be more precise, from the formal language implicitly defined by those axioms. That theorem, however, cannot be established by bottom-up means, i.e. by “or”introduction, since both its disjuncts are either indecidable or unproved. Hence, the assertion of “p ∨ ¬p” violates the required harmony between indirect and direct methods of proof. But why, exactly, is that problematic? Here is one way of putting Sir Michael’s point: Because meaning determines use and vice versa, a sentence having a determinate content of its own must differ in use from a sentence hav-

|| 17 Dummett (1978), p. 219. 18 Ibid., pp. 221 f.

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ing a different content. The two sentences must, that is to say, be assertable under different circumstances. Now consider once again the continuum hypothesis (call it “p”) and the Goldbach conjecture (call it “q”). The former is about the size of the continuum while the latter deals with an alleged property of even numbers. Hence, intuitively speaking, the sentence “p ∨ ¬p” is about the size of the continuum as well; it can be taken to express the assumption that Platonic heaven is fully determinate with respect to the size of the continuum. In contrast, “q ∨ ¬q” can be taken to express the completely different assumption that Platonic heaven is fully determinate with respect to the question whether all even numbers greater than 2 have the property of being the sum of two primes. However, this difference in meaning between the two sentences “p ∨ ¬p” and “q ∨ ¬q” does not manifest itself in there being distinguishable conditions of assertability for each of them. Both are assertable for the very same reason: classical logic. Therefore, if we do not require a harmony between topdown and bottom-up means of demonstrating, i.e. justifiably asserting, theorems, then the difference in meaning between the two theorems “p ∨ ¬p” and “q ∨ ¬q” simply vanishes. Neither of them can be provided with a determinate individual content, and they become synonymous. In order to save meaning difference and meaning determinateness in such cases, we must opt for logical inference rules that keep top-down and bottomup methods of proof in balance. In other words, we have to relinquish classical logic in favor of intuitionistic logic. For the latter guarantees that a sentence of the form “p ∨ ¬p” can be justifiably asserted only if one of its disjuncts is justifiably assertable. Equivalently, intuitionistic logic does not legitimize reductio ad absurdum proofs, i.e. proofs that have a potentially non-constructive character: if, in trying to prove “p” by reductio, the assumption “¬p” leads to a contradiction, “¬¬p” can indeed be inferred from it (by negation introduction); but since “¬¬p → p” is equivalent to “p ∨ ¬p”, “p” itself cannot be inferred from “¬¬p” on an intutionistic basis. Once intuitionistic logic is in place, however, mathematical realism loses its cornerstone: since sentences of the form “p ∨ ¬p” whose disjuncts are either indecidable or unproved stop being justifiably assertable, the verificationtranscendent and mind-independent content they express no longer has to be taken as a fact. Mathematical reality, whatever it is, no longer has to be regarded as determinate ahead of our “discovery” that it is thus and so. Quite the contrary, it clearly seems that we make it determinate by proving one of the disjuncts in the first place.

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IV This anti-realist story has much to recommend it. Moreover, Platonism, the view that there (really!) are mind-independent and eternal mathematical objects such as numbers or pure sets, is very odd to begin with. But even in the realm of reason, realism is hard to get rid of. First, we should note that meaning determinateness cannot be ubiquitous according to anti-realist story just told: it stops short of the axioms. In contrast to the basic and the intuitionistically acceptable non-basic theorems to which we have double-sided epistemic access via bottom-up and top-down procedures, the axioms themselves remain epistemically inaccessible – at least once the rationalist recourse to their self-evidence has been rejected. They are neither deducible from higher principles nor demonstrable by recourse to basic theorems. And each of them is (unconditionally) assertable for the very same reason, namely because of its status as an axiom. Hence, in regard to the criteria mentioned above they lack determinate individual content. Second, axioms are mathematically indispensible. Although adopting an intuitionistic logic deprives them of their initial power to impose justification or truth on otherwise free-floating theorems, such a restriction cannot turn them into mere conveniences. For their role has never been confined to being epistemic engines. After all, mathematics, albeit an a priori venture, is a substantial, seemingly descriptive enterprise dealing with a content of its own. In contrast to logic, that is to say, mathematics has a genuine subject matter: it is about certain kinds of entities peculiar to its proper domain of discourse. This manifests itself in the fact that the language in which a mathematical theory is couched requires a particular non-logical vocabulary, a vocabulary containing predicates and singular terms as, for instance, “natural number”, “successor”, “ordered pair”, “function”, “zero” or “the empty set”. In order to have in the first place an outlined domain of discourse, able to serve as a more or less definite range for the variables, and in order to have applicable predicates at all, we need axioms that provide us with both a specification of the subject area and implicit definitions for our fundamental terms. Now these two findings stand in marked contrast to each other. How could the axioms fulfill the task of specifying the range of the variables or of implicitly defining distinct fundamental predicates if at the same time they lack determinate individual content? Since their descriptive-cum-definitional role cannot be dispensed with, we have to find a way of getting hold of their meaning. This, we just saw, cannot be done by verificational or constructive means. The approach to choose must therefore be pragmatic in character and consist in adopting an

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altered attitude towards them: at the minimum, we have to take them at face value in the sense mentioned above in connection with scientific realism because taking them non-literally would simply deprive them of their ability to play the intended role. Of course, such an attitude does not explain how the axioms get their content – if not by containing words already familiar to us from ordinary language (after all, the language games of mathematics may very well have their roots in proto-mathematical language games such as counting objects). But at least it opens a route to a performative approach to how knowledge of the axioms’ meaning manifests itself in use: taking the axioms literally allows us to declare them to be specifications or implicit definitions in the first place. It is precisely by such performative acts that we reveal our understanding of them. Moreover, nothing bars us from treating the axioms as true. For once we have opted for intuitionistic logic, we no longer have to fear that the use of the axioms in top-down deduction leads to a non-conservative extension “of that fragment of the language containing only observation statements”,19 i.e. basic theorems. In other words, handed-down truth is prevented from transcending bottom-up verification. But not only are we allowed to assume the truth of the axioms, we are also obliged to do so if those specifications and implicit definitions are to get off the ground. And in consequence, axiomatic realism is fully vindicated without thereby giving rise to Platonism. Mind-independence is expendable but literality and truth are not. Beyond that, a second and more surprising residue of mathematical realism comes to light in connection with the basic theorems themselves. Mathematics, we saw, has a genuine subject matter, making the use of axioms imperative. By contrast, logic is deemed to be topic-neutral, i.e. not to be about any particular kinds of entities. This topic neutrality seems to find expression in the fact that logic can be done with inference rules alone, in the absence of any axioms. Witness Gentzen-style systems of natural deduction. And it also seems that the basic theorems of mathematics can be established by innocent computation procedures not requiring any axiomatic foundation (threatened to be tainted with a realistic undercurrent). Both assumptions, however, are shaky. Predicate logic without identity indeed does not presuppose any axioms. But the same in not true of predicate logic with identity. The latter must rely on both the axiom “∀x.x = x” and the axiom schema “∀x∀y[(Fx ∧ x = y) → Fy]”. Now it is not just that the identity sign is required for expressing many basic theorems, but that its proper use is ultimately anchored in these axioms of identity. || 19 Ibid., p. 221.

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To see why, consider a basic theorem as, for instance, “598017 + 246532 = 844549”. In practice, it is demonstrated by the help of what Sir Michael calls a “computation procedure”, i.e. by applying an algorithm that has been designed to calculate the values of the addition function. But how do we know that the result obtained in this way is correct or that the used algorithm is indeed calculating the values of the addition function and not something else? When the going gets rough, when such questions start to be asked, we can, at least in principle, make recourse to a more elementary way of demonstrating the theorem: we can transform the initial equation, by a series of consecutive substitution steps, into a different but equivalent equation that seems to display its own truth. For example, in order to demonstrate the truth of “2 + 2 = 4” we draw on a number of definitions – “4 =df S3” (“4 is, by definition, the successor of 3”), “3 =df S2”, “2 =df S1”, “1 =df S0”, and “x + 1 =df Sx” – to generate the following transformation:20 2+2=4 S1 + S1 = S3 SS0 + 1 + 1 = SS2 SSS0 + 1 = SSS1 SSSS0 = SSSS0 The last line is an equation of the form “a = a” and as such it seems to be obviously true. But what, we must now ask ourselves, makes it true if not the fact that, for all x, x = x or, alternatively, the fact that the identity sign is, in part, implicitly defined by the axiom “∀x.x = x”? After all, sheer obviousness should not act as a guide to truth. Witness the obviousness of the comprehension axiom of naïve set theory which led to Russell’s antinomy. Moreover, the substitution rule employed in the foregoing transformation is cast in the same mold as the axiom schema “∀x∀y[(Fx ∧ x = y) → Fy]”. So the upshot is that both the direct verification and the truth of the basic theorems are axiom-dependent as well. Even bottom-up means of demonstrating a basic theorem, that is to say, contain a residual top-down element. Or, to put it otherwise, there is no clear-cut distinction between “observational” and “theoretical” statements in the realm of reason. This is bad news for anti-realism. For anti-realism pivots on the idea that any theoretical system has to be cut down to

|| 20 In an approximate way, this proof can already be found in Leibniz (1962), pp. 413 f. (Book Four, Chapter VII, § 10).

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observational size. And if there is no purely observational area to begin with, no such down-cutting can be fully accomplished. To be sure, the axioms in question are logical, not mathematical. But this does not mean that mathematics is off the hook – it is just to say that the realm of reason is not divided into distinct parts.

References Boyd, Richard (1983), “On the Current Status of the Issue of Scientific Realism”, Erkenntnis 19, 45–90. Dummett, Michael (1978), “The Philosophical Basis of Intuitionistic Logic”, in his Truth and Other Enigmas, Cambridge, Mass.: Harvard University Press, 215–247. Dummett, Michael (2006), Thought and Reality, Oxford: Clarendon Press. Gödel, Kurt (1990), “What Is Cantor’s Continuum Problem?”, in Solomon Feferman et al. (eds.), Kurt Gödel. Collected Works, Volume II, New York: Oxford University Press, 245–269. Horwich, Paul (1982), “Three Forms of Realism”, Synthese 51, 181–201. Leibniz, Gottfried Wilhelm (1962), Nouveaux essais sur l’entendement humain, in Deutsche Akademie der Wissenschaften zu Berlin (ed.), Gottfried Wilhelm Leibniz. Sämtliche Schriften und Briefe, Sechste Reihe (Philosophische Schriften), Sechster Band, Berlin: Akademie-Verlag. Nagel, Thomas (1986), The View from Nowhere, New York: Oxford University Press. Putnam, Hilary (1974), “The ‘Corroboration’ of Theories”, in Paul A. Schilpp (ed.), The Philosophy of Karl Popper, La Salle: Open Court, 221–240. Putnam, Hilary (1975), “What is Mathematical Truth?”, in his Mathematics, Matter and Method, Cambridge: Cambridge University Press, 60–78. Quine, W. V. (1961), “Two Dogmas of Empiricism”, in his From a Logical Point of View, Second Edition, Cambridge, Mass.: Harvard University Press, 20–46. Quine, W. V. (1969), “Epistemology Naturalized”, in his Ontological Relativity and Other Essays, New York: Columbia University Press, 69–90. Quine, W. V. (1981), “Things and Their Place in Theories”, in his Theories and Things, Cambridge, Mass.: Belknap, 1–23. Quine, W. V. (1992), “Structure and Nature”, The Journal of Philosophy 89, 5–9. van Fraassen, Bas (1980), The Scientific Image, Oxford: Clarendon Press.

Michael Frauchiger

Putting Reality into Perspective by Understanding Theoretic Propositions, Theistic Thoughts and Political Dealings Introduction If visualized from Dummett’s perspective, the concept of meaning—or the notion of content—and the concept of truth form the two pillars on which the bridge that connects semantics to metaphysics rests.1 Such a bridge metaphor can be drawn upon not only to set off a nexus between Dummett’s core fields of expertise within accurate philosophy, but also to bring out connections between Dummett’s unusually wide spheres of action that clearly exceed philosophy. In particular, deep understanding may be viewed as the key capability which forms the bridge between the activities of Dummett the professional analytic philosopher and those of Dummett the humanist, religious practical man who energetically campaigned against racism. For Dummett, academic writing on philosophical topics was clearly not the be-all and end-all of his animated, energetic and deeply committed life. Out of his sense of initiative, he wanted to act and to prompt himself and anybody willing to join his endeavours and campaigns to truly realize their objectives and values and to meet the challenges of orientations and norms which are different from their own. In short, and put emphatically, the following can not in any way be said of Michael Dummett: “he—with his specialized knowledge— more closely resembles a well-trained dog than a harmoniously developed person.”2 In contrast, the following consideration approximates Dummett’s vivid spirit of investigation: “It is not the result of scientific research that ennobles humans and enriches their nature, but the struggle to understand while performing creative and open-minded intellectual work.”3 This latter one of the

|| 1 Cp. Dummett (2006), 30. 2 See Calaprice, A., ed., (2011) The Ultimate Quotable Einstein, Princeton, Oxford: Princeton University Press, p. 105. 3 See Calaprice, op. cit., p. 386. || Lauener-Stiftung / Lauener Foundation for Analytical Philosophy

DOI 10.1515/9783110459135-011

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two just quoted statements from Albert Einstein needs though to be amplified in order to fit more adequately Dummett’s philosophically as well as morally and socially involved character: it is Michael Dummett’s struggle to understand while, at once, performing creative, responsible philosophical work and letting deeds, not only words speak of the ethical, broadly rational thoughts he grasps which has enriched his lucid, frank and spirited, empathic personality throughout his life. Dummett’s strong will and ability to understand the purport of diverse distinctively human matters—and in particular to explore our human modes of communication and the role of language within it, the structures of our thought and the relation between language and thought, the ways of our dealings with one another and our conceptions of reality—may be regarded as the main link between Dummett’s heterogeneous fields of work and dedication. Dummett indicates that the actual province of philosophy is understanding, which can be attained by reason aided by experience, through a deepening of our thought. It might be added that the province of understanding is the study of our human form(s) of life.4 We should note, though, that understanding, for Dummett, boils down to the knowledge of meaning and may be said to encompass the following: what can be understood is our common use of words and sentences or statements in a language as well as the conceptual contents, thoughts, propositions and facts— i.e. true propositions—which are expressed in different languages by intelligible, meaningful and (as is additionally required in the case of facts) consistent statements.5 Michael Dummett is best known for his philosophy of language—and the metaphysics, in the narrower sense, contingent upon it—as well as for his philosophy of logic and mathematics. The original contributions to the present

|| 4 Cp. e.g. Dummett (2001/2010), 31–42, 148 and Dummett (2006), 92f. 5 Cp. e.g. Dummett (2001/2010), 13–19, 40f., 89–99, 115–124, 127f., 137–142, Dummett (2006), 3–11, 14–18, 29f., Dummett (1978), 216f., 420, 423, 426 and (re the question of truth as attaching both to sentences—or rather statements—in a language and to the propositions or thoughts expressed by them) Dummett (2000), 9–13. Actually, for Dummett, understanding is tied to the knowledge of a language: to know some word or expression in a language amounts to understanding the meaning it has. The precise nature of understanding the meaning of expressions in a language needs in turn to be elucidated by a theory of meaning (which on its part gets justified by enabling such an elucidation as required). Dummett correspondingly writes: “A theory of meaning must incorporate a theory of understanding, since understanding is integral to the practice of using a language: two people can converse in a language they both know only because each understands what the other says.” (See Dummett (2001/2010), 133.)

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volume tackle in depth many different substantive issues and problems which are central or relevant to these main areas of Dummett’s rich philosophical oeuvre. I shall at first epitomize Dummett’s justificationist approach to semantics by touching on these central areas of his work as a basis for moving on to successively discuss in detail the following two important aspects of Dummett’s late work which have not been singled out for special recognition in the other contributions to this volume: for one thing Dummett’s remarkable theistic, broad metaphysics of the world as a whole, as it is in itself, and for another thing his tried, well-wrought egalitarian approach (with a thought-provoking religious cast) to improve the political principles and practical policy of asylum and immigration in Europe and elsewhere.

A Rough Outline of the Dummettian Justificationist Conception of Truth and Meaning Dummett’s philosophy of language and his philosophy of mathematics are systematically connected to each other. In particular Dummett’s widely and controversially discussed questioning and challenge of strong types of realism, based on his novel conception of semantic antirealism, draws and bears upon both his philosophy of language and his philosophy of mathematics.6 Dummett advocates that whether sentences (or propositions, which are expressed by such statements) are determinately either true or false depends on whether it is feasible in practice, or at least in principle, to recognize them as such, i.e. to assess them as true or as false. This antirealist view of truth runs contrary to platonism and other forms of mathematical realism, which implicate that all mathematical propositions are objectively either true or false, irrespective of whether a method can be devised for deciding which is the case. The notorious Goldbach Conjecture for instance has not been demonstrated to be true or false, despite numerous great efforts. Nevertheless, most mathematicians, being platonists or, in any case, strong realists, firmly assume that the Goldbach Conjecture must be determinately either true or false, which is to say that they presuppose a notion of potentially evidence-/verificationtranscendent, undecidable truth. In contrast, from the viewpoint of a Dummettian antirealist about truth, there are no good reasons which “objective-

|| 6 For lucid critical clarification of the Dummettian antirealist programme compare Wright (1987/1993), i. a. the Introduction.

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ly” speak in favour of making a firm assumption of that kind, since for an antirealist, an ascription of truth to a proposition entails the availability, in principle, of grounds for believing that it is true, and it is certain that there is no assurance that every single mathematical proposition can in principle be demonstrated to be true or false. Consequently, semantic antirealists contest the universal validity of the classical logical law of excluded middle, which states that for any proposition, either that proposition or its negation is true. That way, the antirealist view of truth upholds the intuitionist constructivist approach to mathematics and logic. The antirealists’ rejection of the semantic principle of bivalence, which states that every well-formed sentence or statement is determinately either true or false is closely linked with their denial that the logical law of excluded middle can be justified.7 Dummett’s semantic antirealism undermines, as well, metaphysical realism. This is because the view that the objects contained in “the” world (standing in certain relations to one another) exist independently of our human capability to recognize that they do so is, according to Dummett, equivalent to the view that statements are determinately either true or false independently of whether it is feasible for us in principle to recognize them as such. This semantic construction of metaphysical realism is of course diametrically opposed to Dummett’s semantic antirealism, which connects the truth (or the falsity) of a proposition with the availability, in principle at least, of evidence that provides the justificatory support for the assessment of its truth (or falsity); or, put in another way, the putative truth of a statement presupposes its assessability as true, which in turn involves necessarily the availability, in principle, of conclusive grounds for believing that it is true. Dummettian antirealists correspondingly reject the strong realist’s conception that the truth of our descriptions can intelligibly transcend the availability of evidence for it and that, consequently, our linguistic, conceptual abilities to depict determinate sectors of reality can intelligibly transcend our knowledge-acquiring capacities.8 Dummett’s antirealist view of truth has a significant bearing on his theory of meaning9. Dummett insists that the meaning of a statement both determines and is fully determined by its competent use; this is because a statement’s || 7 Cp. e.g. Dummett (1978), 215f., 229–233, Dummett (2001/2010), 134–136 as well as Wright (1987/1993), Introduction. 8 Cp. e.g. Dummett (1978), 215f., Dummett (2001/2010), 126–128, 130f. as well as Wright (1987/1993), Introduction. 9 For Dummett’s original presentation, in the 1970s, of his theory of meaning, see Dummett (1993), essays 1 and 2. That collection also includes some later essays that revise or refine some aspects of his original account.

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meaning boils down to the role that statement plays as an instrument of communication between individuals who are corporately engaged in exchange of ideas. Dummett thus maintains that language acquisition inevitably proceeds with a clear, focused orientation towards intersubjectively recognizable aspects of the competent employment of words in statements which are publicly, jointly and severally assessable as true or false. Clearly, theoretical explications along these lines of how we get to understand statements and how such understanding is fully manifested in linguistic practice finds accordance with Dummettian semantic antirealism. Evidently, the acquisition of a knowledge of the meaning (i.e. the acquisition of an understanding) of statements involves the learning of proper correlations between the making of confident, positive statements and the recognition of jointly, publicly accessible conditions as obtaining. Dummett consequently advocates that the meanings of sentences cannot be explicated with reference to truth-conditions, for truth-conditions of sentences (e.g. of statements concerning the remote past) potentially transcend our ability to recognize them as obtaining, i.e. truth-conditions can in principle transcend all of the evidence we could find for their obtaining. Instead, Dummett pleads for an explication of meaning based on conditions that are not potentially evidence-transcendent, but can in principle be intersubjectively recognized to obtain if they do: in particular, Dummett proposes to explicate the meanings of sentences with regard to assertibility-conditions, that is with respect to those conditions under which a sentence counts as correctly asserted, thus relating to conditions which are in principle publicly accessible. So an implicit (usually not verbalizable) knowledge of the meaning of a sentence amounts to being capable and observably in a position to arrive, in a publicly accountable way, at the recognition of certain conditions which, when they obtain, justify an assertion of that sentence. Dummett emphasizes that only the capacity to use the statements of a language in a correct way can be a hallmark of a grasp of their meanings and, for that matter, of those of the expressions they contain. The meaning of a word or more specifically of a predicate is accordingly explicated with reference to publicly recognizable conditions under which certain sentences containing that predicate are justifiably viewed as correctly asserted. Dummett makes clear that to possess the concept expressed by a predicate consists in the ability to apply that predicate in appropriate circumstances. The meanings of predicates, which therefore are given on the basis of (e.g. more or less instructed perceptual) experience, determine the criteria for recognizing various statements as true or not true.10 || 10 Cp. e.g. Dummett (1978), 215–218, Dummett (1988/1993), 133, Dummett (2001/2010), 133–

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Once the general criteria for the correct use of certain predicates (i.e. for the possession of the respective concepts) become apparent, sentences containing them that are consistent for use with these criteria can be intelligibly asserted, even if their truth-conditions transcend in practice or in principle our ability to corporately recognize them as obtaining. A statement in mathematics like the Goldbach Conjecture is intelligible, meaningful, although it cannot be demonstrated in practice whether it is true or false, and there may not even be any assurance that it can in principle be demonstrated to be true or false—according to Dummettian justificationist semantics, there may simply not be an answer to the question of its truth or falsity. Moreover connected sentences specifying some attributes of the theistic God (such as His11 being omniscient, creator, having an omnipotent as well as unlimitedly good will, etc.) may well be intelligibly asserted, i.e. consistently with the original integrated system of theistic precepts and tenets that are manifestly part of revelation, even if there is (in practice or in principle) not sufficient public evidence to be found for the obtaining or not of the truth-conditions of the relevant sentences. Following this, there may be truth gaps and factual gaps (facts being true propositions), respectively, that cannot be closed in our systematic descriptions, conceptions, and models, respectively, of “the” world.12 Dummett is occasionally perceived as a late advocate of logical positivist verificationism. However, Dummett’s semantic antirealism or justificationism is not a variety of the verificationism of the logical positivists.13 Dummett’s justificationist semantics and antirealist view of truth indeed entail that we have no right to assume that a statement which in principle cannot be recognized as true or false is nevertheless determinately either true or false; but Dummett does not advocate that for a statement to be (cognitively) meaningful, and thus intelligible, that statement must be in effect, in practice, verifiable. In fact (as discussed in detail below) Dummett himself does not hesitate to make statements about crucial features of the world as a whole and of the theistic God standing over against it, which are two concepts that are correlative to each other according to him. Dummett makes these intelligible and consistent || 136, 142–144, Dummett (2006), 41, 59, 100f. as well as Wright (1987/1993), Introduction. 11 In what follows, the theistic God will often be referred to as “He”—as is also the case in Dummett’s relevant writings. I see it as very important, though, to stress that with this usage of ‘He’ or ‘His’ or ‘Him’ (written with capital letter) when referring to God, no undertones of sexism whatsoever are intended! 12 Cp. e.g. Dummett (1988/1993), 133, Dummett (2001/2010), 49, 136 as well as Dummett (2006), 100f., 107. 13 Cp. Wright (1987/1993), viii–ix.

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statements in dealing with issues concerning metaphysics in the broadest sense, and he would scarcely claim that all such statements can in practice or in principle be verified or recognized as true or false and are so either true or false. Still, these statements of theistic-metaphysical explication—not, in general, of definition—are intelligible,14 since the explicantia they contain are concepts that can be acquired to be possessed by speakers in public situations, with an eye toward intersubjectively recognizable aspects of their use. So the terms which are employed to give an explication of key concepts in very broad metaphysics have meanings which are publicly acquirable and can, accordingly, be correctly applied in meaningful, intelligible explicative statements that serve to fathom out salient basic features of the world as it is in itself—as will become clearer below, Dummett’s notion of the world, not as we find it to be but as it is in itself, is a useful concept to inspire matters of broadly metaphysical and epistemological debate which quite likely cannot be settled by any empirical scientific theory.

The Theistic God and the World As It Is In Itself Standing Over Against One Another In the preface to his edited volume in honour of Dummett, Richard Heck recalls an episode at the 1987 New College graduate dinner which vividly illustrates Dummett’s sincere and fierce commitment to the moral advancement and humanization of the broader community he lives in: Michael rose to speak about the ivory tower, urging us not to forget our obligations to the community outside the College and University (a topic of particular importance, historically, in Oxford). As he spoke, at one point discussing the need for continued vehemence in battling against racism, someone laughed. I expect they were not laughing because of what Michael had said, but at something else, because they were not paying attention. The thought that someone would be so rude as not to be listening probably would not have occurred to Michael, however: in any event, his demeanour changed immediately. I do not recall exactly what he said, but it was something like, ‘If you are going to laugh at that, I need to give a different address.’ He launched directly into a completely impromptu, fierce, animated sermon on racism that would have been worthy of the Revd Jesse Jackson. I personally found it deeply moving, and sat through the end of it in tears. A

|| 14 Cp. e.g. Dummett (1978), 217 and Dummett (2006), 92.

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number of us confessed later that we had a strange desire to thank whoever had laughed, for giving us the opportunity to hear that particular speech.15

I might be off the mark but this and other similar testimony to Dummett’s religiously and morally motivated social and political commitment put me in mind of Pauline ardour of preparation for communities with human love and justice as the goal and measure of interactions between ethnically, culturally, socially and genderly diverse individuals. Several pertinent passages in Pauline epistles connected to this broad and long-range aspiration may spring to mind, e.g. in the First Corinthians and the Galatians. On the occasion of a conversation at the dinner table with Michael Dummett (during the 2010 Lauener Symposium in honour of him), I rather rashly mentioned his supposed contributions to philosophy of religion, but Dummett made it clear, in a considerate way, that he had not actually contributed to philosophy of religion, but to theology. With the benefit of hindsight, it has become clear to me that Dummett was able to relate to theology in a philosophical way similar to Paul the Apostle, who initiated Christian theology in a philosophical way by using arguments and revelatory evidence to make believers from diverse ethnic, cultural, and social backgrounds understand better in which respects the message of Christ ought to have repercussions on their everyday individual and social lives. Following Paulinian very early Christian theology, there is, of course, a whole string of theologically interested philosophers and philosophically minded theologians from late antiquity, the post-classical era and the early modern period who might be regarded as precursors of Michael Dummett the analytical philosopher who late in his life wrote about issues which are equally fundamental to philosophical metaphysics as to theology; but it would go clearly beyond the scope of this epilogic piece to endeavour to trace the historical lines of development which have been combined in an original way by Dummett in his philosophically and theologically substantiated metaphysics in the broadest sense. Our fellow sentient beings on earth are deemed by some of us humans to share our memory for emotional experiences, but to be deprived of a sufficiently sharp intellect; on the other hand, many of us believe that certain animals must have concepts and thoughts of some sort—thoughts without language. Dummett notes that different animal species have very different sensory faculties and intellectual capacities and thus experience “the” world in a variety of ways: the

|| 15 Heck (1997), vi.

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worlds they inhabit intersect our human ones but are markedly different from them. So Dummett consents that animals must have some building blocks to structure their apprehension, they must have some kind of concepts and thoughts (or rather “proto-concepts” and “proto-thoughts”) that are not accurately expressible in language and out of which their judgements about “the” world are built. Yet Dummett does not consider it plausible that interdisciplinary research will be able to form more than just a hazy conception of different animals’ perspectives and worlds, since for one thing we humans can describe animals’ perceptual experience, or the world as they find it to be, merely externally; and for another thing we can gain only a confused idea of different animals’ concepts and conceptual systems and thus of their communication of thoughts and of their factual judgements about “the” world.16 It might seem, though, that what really matters is that we humans possess more extensive analytical abilities and capacities for abstraction than animals: our increased intellectual capacities can greatly broaden the scope for progressively factoring out our particular human constitution and spatio-temporal location, with the objective to arrive at a purified presentation of the objective world that we experience and, thus, putatively arrive at a presentation of the world as it is in itself. Remarkable scientific efforts are heading in the direction of increasingly disentangling “the” external reality that impacts on us from our intersubjective, internal way of perceptually experiencing it, which is determined in part by the nature of our sense organs and our contingent size and location. There are manifold mathematical-theoretical efforts in science to abstract from the particularities of our diverse modes of apperception and conceptualization of various phenomena at different levels. Strongly realist, scientistic philosophers take it that such progressive abstracting and cleansing efforts in

|| 16 Cp. Dummett (2006), 38f., 93, 96f., 101, Dummett (2001/2010), 88f., 117–119, 122–124 and Dummett (1988/1993), 121–126. It needs to be noted at this point that according to Dummett’s “priority thesis”, conceptual contents—i.e. thoughts, propositions, and their constituents: concepts individuated by their possession conditions—can only be analyzed and understood through the analysis and knowledge of language. (Cp. Dummett (1988/1993), 128, 133.) Peacocke makes clear, though, that Dummett’s priority thesis does not at all exclude the possibility of thought without language. Dummett is merely committed to the point of view that animal conceptual thoughts without language can only be analyzed and hazily, approximately understood by making reference to language at some point. That is, Dummett consents that the conceptual contents certain animals must have to factually judge about—and to structure their apprehension of—“the” world are available independently of the linguistic expressions for them that enable us humans to form a hazy conception of these animals’ perspectives on “the” world. (Cp. Peacocke (1997), 2–5.)

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science will eventually result in a genuinely scientific characterization— independently of our specifically human modes of (direct or indirect) observation—of the real, ultimate truth of fact regarding the structure of the world as it is in itself. However, from a Dummettian justificationist point of view17, the entirely abstract, skeletal, barren mathematical models that would be described by completely cleansed, purely structural physical theories, and which would be connected with human experience just at several removes, would still remain abstractions of aspects in which we, humans, apprehend the world. A skeletal, barren structure can definitely not be claimed to be all there really is. The scientistic strongly realist quest for a highly abstract scientific-theoretical description of the world as it is in itself—seeking in effect a description of things as they really are, in themselves, and hence a description that would be no mere description which employs a particular conceptual vocabulary—turns out to be simply based on a contradictory objective.18 So Dummett understands the ontologies of scientific theories as the varied domains of facts that are expressed by theoretical descriptions—at different abstraction levels—of the world as we humans apprehend it. Dummett’s philosophy of language or more specifically his justificationist theory of meaning and truth are at the root of what I’d suggest to call his metaphysics in the narrow sense. Our basic human ambition and urge to explore the manifold phenomena we experience as well as the ever widening expansion of scientific measuring techniques and so of what we are in a position to observe corporately, intersubjectively, inevitably result in a wealth of theoretical descriptions and explanations of varying ontological aspects of the world as it can be represented dependently of us humans. These scientific ontologies are the subject of meta-

|| 17 Dummett summarizes that particular point of view as follows: “Although facts indeed impose themselves upon us, however, we cannot infer from this that they were there waiting to be discovered before we discovered them, still less that they would have been there even if we had not discovered them. The correct image, on a justificationist view, is that of blind explorers encountering objects that spring into existence only as they feel around for them. Our world is thus constituted by what we know of it or could have known of it. A realist might echo this saying, on the ground that, if a proposition is true, it might be known or have been known to be true. On a justificationist view, however, what we could have known extends only so far as the effective means we had to find out: the entailment is not from its being true to the possiblity of knowing it, but in the opposite direction. It would be wrong to say that we construct the world, since we have no control over what we find it to be like; but the world is, so to speak, formed from our exploration of it. The world of which I am speaking is our world, the world as we apprehend it. Our capacity to apprehend how the world is depends, of course, upon the concepts we possess—that is, upon our ability to describe it.” (See Dummett (2006), 92.) 18 Cp. Dummett (2006), 92–101.

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physics in the narrower sense that follows from Dummettian justificationist semantics and concerns “our” world, the world as we humans apprehend it and which thus neither transcends our human knowledge-acquiring capacities nor is simply a human construct. Each level of description—containing intelligible statements that express consistent propositions which are judged to be true according to the criteria supplied by the meanings conferred upon the relevant words and statements—matches a sector of the world as we apprehend it, that is of “our” world, which is real on account of our knowledge of it to be real. Notabene, the expression ‘sector of our world’ may not be appropriate here, because, for one, there may be gaps in the apprehended world and its sections, for according to the antirealist view of truth associated with Dummett’s justificationist semantics, there are meaningful, consistent statements we neither know to be true nor know to be false and which we thus have no right to assume to be determinately either true or false. More importantly, Dummett points out that the different levels of description of “our” world (being descriptive of the same or very similar occurrences, yet at different levels, e.g. as colours or as light of different wave-lengths) compete as equally true and cannot be harmonized into a single unified conception of “our” world.19 So a conception of “the” world as a whole, and in fact even of “our” world as a whole, cannot directly be attained through any particular scientific, theoretic perspective, which in principle can do no more than describe just one coherent factual ontology. Metaphysics in the broadest sense—in marked contrast to metaphysics in the narrower sense—cannot be committed to any particular theoretical perspectives devised by humans. Dummett thus, in fact, draws a

|| 19 Cp. Dummett (2006), 100f. – Dummett makes a distinction between descriptions of different levels according to whether they are formulated in theoretical terms whose meanings are given to lesser or greater extent by reference to our uninstructed perceptual experience (as contrasted with our intricate measuring methods). For instance, two valid consistent descriptions referring to light of different wave-lengths and colours respectively are both composed of statements recognized as true according to the criteria determined by the meanings of the words they contain. Meaningful, intelligible, consistent statements (or rather the propositions they express) are justly recognized as true when the criteria for so judging them are satisfied. There are different levels of description but not different standards of truth according to justificationism, and Dummett thus contests any scientistic, reductionist claim to be able to winnow ultimate, real truth from ordinary, mere truth. On the other hand, Dummett elucidates that some of these equally valid descriptions characterize the same phenomena at different levels and consequently compete, but cannot be harmonized into a unified description. He therefore concludes that we do not even have any single conception of the world how we humans apprehend it as being.

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dividing line between the narrow metaphysics of human aspects of reality, explorable from different scientific points of view, and metaphysics in the broadest sense of reality as such, as a whole, of which we may try to conceive an idea with the aid of theistic explications but not of scientistic ones, for reality as a whole cannot be cast in the mould of a scientific theoretical system. That way Dummett in some measure reconciles Kantian minimally realist epistemology with the philosophico-theological view that the divine, noumenal, transcendent realm lies outside the boundaries of any scientific approach whatsoever. Dummett’s minimal realism in his metaphysics in the broadest sense—which is consistent with his antirealism in his metaphysics in the narrower sense—is thus certainly not prone to frequently expressed criticisms that Dummettian metaphysical thinking per se cannot accommodate the commonsense truism that “reality” transcends (human) truth: in accordance with Dummett’s justificationist semantics, reality as a whole, as it is in itself, transcends both our human knowledge-acquiring capacities and our linguistic, conceptual capabilities to depict determinate sectors of reality. So it appears there are good reasons to presuppose beyond “our” human world (and the diverse worlds of the other sentient and rational species and possibly also extraterrestrial beings) a more fundamental, transcendent realm, to which Kantians use to refer as the noumenal sphere. That transcendent ultimate reality is often characterized “negatively”, e.g. as a non-mental, nonmaterial, non-spatial, neither finite nor infinite (timeless, “eternal”) and thus empirically, scientifically unexplorable realm. “Our” human world—which actually corresponds to a conglomeration of different intricate, subtly differentiated rational descriptions, which partly enable comparably feasible explanations of the same (scientifically or else commonsensically individuated) phenomena—is the subject of metaphysics in the narrow sense, which deals above all with the ontologies or models of the diverse theoretical conceptual systems which we develop in order to be applied in scientific descriptions and explanations of different aspects of “our” world. On the other hand, the noumenal sphere or world as it is in itself is the subject of metaphysics in the broadest sense. Strictly speaking, that noumenal realm appears to be unfathomable by human means of any kind whatsoever, even the most abstract logical and mathematical tools appear to avail us nothing with regard to a study of the ultimate reality as it is in itself. However, the critical question to be raised at this point is, rather, whether abstract logical and conceptual means are of little or of no avail when it comes to gain a basic understanding of the structure of the world as a whole. Evidently, Dummett is confident that little is much when we turn our attention to metaphysics in the broadest sense and so he can cheerfully

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endeavour to outline the basic structure of the noumenal or, respectively, the theistically divine realm. It is of interest at this point to briefly confront Dummett’s with Lauener’s minimal realism regarding the existence of reality as it is in itself. Lauener advocates a most minimal realism in epistemology (though not in ontology), which is pragmatically motivated. Lauener was atheist and would not have been ready to get involved in the venture of a theistically inspired metaphysics of things as they are in themselves. Still, he understood that the independent world as it is in itself needs, for sheer pragmatic reasons, to be simply postulated. Lauener puts this in a nutshell as follows: the notion of external truth is meaningless. This does not mean, however, that I do not believe in an outer world. My point is rather that no philosophy can do without some assumption or other – from nil nothing will follow – and that, seen from a pragmatic perspective, the most reasonable decision must be to accept as a presupposition the existence of a material environment even though its existence is not strictly provable within the system.20

Elsewhere Lauener clarifies this point further: we are unable to speak about “reality as such” or Kant’s “things-in-themselves” without using some language and thus imposing on “the world” a structure, we have ourselves devised. Each attempt of this kind results in a self-defeating enterprise. However, an empirically minded philosopher will not resign himself to give way to a form of linguistic idealism. In spite of the fact that “reality as such” remains unattainable he must insist that we talk about physical objects but only about those of a specific reality sector whose structure is determined by the forms of the linguistic system we have settled on. In other words, empirical facts have to take the shape impressed on them by the use of a language. My concession to realism is limited to the minimal claim that there must be an “outer world” which we have not created by our will. Yet, at the same time, I frankly admit that we have no logical demonstration or empirical evidence for it; such a general presupposition formulated in an ordinary prescientific language does not express more than my intuitive conviction that what I perceive is not merely imagined but caused by material processes originating in “the world”. Though not being able, for want of a Theory about the Universe – even cosmology deals only with very large reality sectors – to present a proof, I still can give practical reasons for believing in its existence. Without its presupposition our scientific activities would not make sense and all our endeavours would be in vain. Facing the utter absurdity of any other assumption with regard to human life, we do not have a choice: if we want to progress by devising successful theories we must remain confident that they do not rest on mere fancy. Let me finally remark that the adoption of such a postulate, expressed in an unregimented and therefore semantically unstable language, is not

|| 20 Lauener (1990), 219.

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vacuous with regard to epistemology as some have suspected, for it guarantees that we are quantifying over physical objects, not mental entities, when doing natural science and thus reveals our intent to remain within the scientific tradition.21

In contrast to Dummett, Lauener thus appears to hold that any effort to pursue metaphysics in the broadest sense remains aimless, and so he confines himself to give pragmatic reasons for postulating a thoroughly independent, external “world as such” or “thing-in-itself”. Lauener advocates a contemporary version of Kantian metascientific epistemology and concurs, on the whole, with Kant’s view that we can’t know but need to be able to conceive of a “thing” (or “world”) “in itself”, because there can’t be appearance without anything (hidden, transcendent) that appears.22 As regards metaphysics—wherever possible, he avoids this label—Lauener aligns ontology with his relativistic pragmatics of practical contexts (that are created for the development of separate scientific theories), which embeds model-theoretic semantics for stabilized theoretical conceptual systems. According to Lauener’s pragmatic or open transcendentalism, truth is internal and ontology varies from one scientific context to another. Lauener thus endeavours to confine metaphysical assertions to predications about models of specific conceptual systems; in fact he tries to narrow ontological questions as much as possible down to problems of formal semantic interpretation of methodologically far advanced theoretical systems in contemporary science. However, with regard to his postulate of a wholly independent world in itself, even Lauener cannot in any way avoid getting involved in some minimized metaphysics in the broadest sense; for in order to ensure ourselves that science describes nature and not just the fruits of the ingenious human mind, we need to postulate “something” wholly independent of us humans which “affects” us in ways we cannot explain and will in principle never be able to know (this recalls du Bois-Reymond’s notorious slogan “ignoramus et ignorabimus”). The case, though, that we cannot and won’t be able to explain why and how we humans (and the other sentient species for that matter) are able to apprehend “the” world, is not a good reason for Kantian philosophers to deny the existence of reality as it is in itself. Evidently, to postulate something unknowable is the metaphysical price we must pay in order not to fall prey to a thoroughgoing idealism. Yet if we agree with Lauener (and with Kant for that matter—at least according to Lauener’s interpretation of Kant) that the thing-initself is beyond experience and knowledge, do we moreover have to admit that

|| 21 Lauener (2002), 305f. 22 Cp. Immanuel Kant’s Critique of Pure Reason, e.g. B XXVIf.

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the “thing” or “world” as it is in itself is an indefinable, ineffable, unthinkable “something”? It is certain that Lauener would not be prepared to specify e.g. whether he presupposes one single thing-in-itself or rather a plurality of thingsin-themselves, because there can be no empirical evidence for settling such a question. Yet, in the fashion of Kant, Lauener readily concedes that the “thingin-itself” or rather “the outer world” is unattainable, i.e. beyond empirical knowledge, and very likely he would additionally need to admit that the outer world is inaccessible to all description and thought. Nevertheless, Lauener clarifies further his intuitive conviction that the outer world has not been created by our human will and that it someway originates material processes that cause the phenomena and objects of experience. Lauener thus comes to loosely define the concept of the outer world, which he needs to postulate for practical reasons deriving from the epistemology of science. Dummett appears to hold a minimal realism (in his metaphysics in the broadest sense) too, somewhat similar to Henri Lauener’s minimal realism23, though the reasons Dummett sets forth for believing in the existence of a transcendent world as such are not practical reasons deriving from the epistemological need to give sense to our scientific activities and endeavours, as in Lauener’s case, but pragmatic reasons informed by justificationist semantics as well as theistic-theological considerations. Yet Dummett does not revert to British (Hegelian) absolute idealism. He does not claim that reality as a whole, as it is in itself, is virtually a content of the mind of God. While at some point he concedes that God someway creates “the” world in conceiving of it, he consistently makes it clear that how the world is in itself must be the way in which God apprehends “the” world as being.24 That is, God does not downright construct “the” world by apprehending it but, instead, the way He finds it to be like—as it is in itself, comprehended in its entirety—is formed from His absolutely unique apprehension of it from all actual rational, truthful perspectives at once. The way in which God apprehends “the” world as being is very distinct from the way we humans apprehend it as being: whereas God apprehends the world in its totality, as a single reality, we humans apprehend it in an irreducibly perspectival, or indeed multiperspectival, kaleidoscopic mode. || 23 On the edges of the 2010 Lauener Symposium on themes from Dummett, I mentioned to Michael Dummett in passing that Henri Lauener advocated some kind of minimal realism, noticing that such comment made Michael Dummett sit up and listen attentively. I would have very much liked to raise the issue of minimal realism in an interview with Dummett that we had envisaged but which unfortunately was not possible in practice to be carried through prior to his death in 2011. 24 Cp. Dummett (2006), 96–109, and Dummett (2001/2010), 44.

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Dummett obviously goes much further than Lauener in the direction of explicating the concept of the world as it is in itself by proceeding with it in close coordination and consistent with his partial explication of the concept of the theistic God. At this point it should be noted, though, that Dummett’s original conception of the world as a whole turns out quite different from any Kantian conception (such as Lauener’s) of the noumenal world. Dummett’s discussion of the fundamental metaphysical structure of the world as it is in itself cannot be directly related to the ongoing controversies about the distinct metaphysical status of Kant’s things-in-themselves. In fact, Dummett is not concerned with Kant’s noumenal world consisting of things-in-themselves, and he therefore does not enter into the recent discussion about the correct interpretation of Kant’s distinction between the noumenal world and the phenomenal world—a discussion which can be outlined thus: while some analytic Kant interpreters (such as Lauener) take the view that we merely have knowledge of the phenomenal world of appearances (i.e. objects of empirical experience), and no knowledge at all of things-in-themselves in the independent noumenal world, others (such as Henry Allison) advocate that Kant presupposes just one type of things, which can be considered in two ways, either as they appear, or as they are in themselves; and according to still another, more recent Kant interpreter (Rae Langton), we can have knowledge only of relational (e.g. causal) properties of things in themselves, but not of their intrinsic attributes. – In marked contrast to these different Kant interpreters, Dummett is not concerned in his broad metaphysics with any Kantian noumenal realm of individual things-inthemselves but instead with the world as a whole, which combines and is made up of the effective, truthful perspectives of all different types of sentient, rational intramundane creatures and which is comprehensible as it is in itself only by a personal, unembodied, extramundane divine mind. The heterogeneous world as a whole, which Dummett attempts to fathom, is thus constituted of manifold structures of true propositions (or true thoughts, conceptual contents), which are expressed by sound theoretic language systems or, alternatively, by conceptual systems of coherent beliefs and judgements about “the” world which the various nonhuman sentient and reasonable beings are apt at forming. Although both reality as it is in itself and God are beyond observation, experimentation, scientific explanation and empirical knowledge in general, they are not only objects of revelation and religious convictions. They are also required a priori as transcendent guarantors for the plausibility and feasibility of a minimal realism in broadest metaphysics, which counteracts both idealism and the presumptuous absolutization of theoretical-empirical knowledge in strong

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scientistic, materialistic metaphysical realism. Dummett certainly would advocate, however, that it cannot suffice to make consistent a priori statements (statements capable of being a priori true) about the basic structure of the world as it is in itself. Instead, we can frame intelligible, consistent and theologically sound statements about the fundamental structure of “the” world and the most basic features of the theistic God, who apprehends it as a whole, by drawing on combined resources from revelation, contingent general knowledge, and their logico-semantic analysis and philosophico-theological reflection, which resolve apparent inner contradictions and eventually lead up to a clear understanding and well-founded acceptance—attained by rational thought—of the propositions that there is a theistic God and a world as a whole. So by giving consistent explications that assist a clearer understanding of the correlative concepts of the theistic God and of the world as it is in itself, good (sufficient) grounds for believing in their existence can be provided. A Dummettian approach like that opens up good prospects for getting an intelligible as well as plausible, though hazy and rough idea of the world as a whole, as it is in itself, which gives helpful, enlightening guidance to better understand in what way we humans and other sentient, rational creatures inhabit different but intersecting worlds, which can in principle be coherently related to one another.25 And yet Dummett does anything but endeavour to give an ontological argument (even less a proof) for the existence of God. On the contrary, he challenges in principle the plausibility of ontological arguments. Dummett diagnoses at one point that a main reason for the failure of ontological arguments is that they presuppose a Meinongian conception of ontology. According to Dummett, the proponents of the ontological argument must maintain (contrary to the Kantian and Fregean tradition) that the logical concept of existence is not to be interpreted as being true of everything and that accordingly we cannot quantify only over what exists but need to presuppose objects which do not exist (as Meinong does). By contrast, Dummett firmly repudiates the conception that there are objects which do not exist.26

|| 25 Cp. Dummett (2001/2010), 41f., and Dummett (2006), 101. – Dummett’s theistic reflections on the basic structure of the world as a whole show that there are still good reasons nowadays to venture on arguing for a substantive fundamental structure of the world as it is in itself. Dummett’s relevant argumentation (which is further clarified below) confirms that contemporary efforts in contributing to broad metaphysics are not at all condemned to end up as just some flight of metaphysical imagination. 26 Cp. Dummett (1993), 277–280. – Dummett stresses, incidentally, his conviction that some suitable means of handling statements that explicitly involve modal expressions or ascriptions

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Elsewhere Dummett indicates another good (or even conclusive) reason for his refusal of ontological arguments by hinting that he repudiates them and generally any arguments for the existence of God from premises deriving from unaided reason. He thus challenges the persuasiveness of any arguments from a priori, analytic premises alone (viz from premises that are in no way derived from observation of “the” world) to the conclusion that God exists. Consequently, Dummett advocates that contingent facts about “the” world which are evident to universal observation—without any specialized knowledge required— need to be invoked in order to provide good reasons for regarding the assertion that there is a God as true. More specifically, Dummett offers pragmatic reasons for accepting the existence of a theistic God, concentrating first of all on the circumstance that a sound, sensible explication of our conviction that there is such a thing as the single world as a whole, which is comprehended differently by different kinds of sentient beings within it, is only attainable by invoking an extramundane mind that apprehends it as it is in itself. The assumption of a theistic God’s existence is rational and constitutes a basic requirement for the development of metaphysics, with metaphysics not taken in the narrow sense as concerning our multifarious scientific-theoretical ontologies of facts but taken in the broadest sense as concerning the multiperspectival world as a whole. Thus, the following good (or, arguably, conclusive) pragmatic reason for believing in God’s existence can be given: there can be a meaningful, consistent conception of a unitary reality that we share with various other sentient beings only under the presupposition of the existence of an unembodied mind which knows it completely as it is, in its totality.27 In line with Dummett’s moral and political Christian egalitarianism (which will be clarified below in connection with his views on immigration and asylum), some further pragmatic reasons for rationally believing that God exists may be given, which relate to the social utility and moral and political value of the assumption of God’s existence for individuals and communities. Unlike Pascal’s notorious wager argument, the pragmatic reasons in question do not refer to the advantage of the belief in God for the happiness of the individual’s possible life after death but instead to the full social and moral benefits the assumption of God’s existence provides for this life. A good pragmatic reason for believing in God relates to the benefit individuals derive from their genuine belief in the existence of an omnibenevolent God, as regards the development of

|| of propositional attitudes will be found which does not require to interpret them as involving quantification over non-existent objects. 27 Cp. Dummett (2001/2010), 42, and Dummett (2006), VI, 102.

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their moral, prosocial traits, conscience and behaviour: genuine awe of God and life is characteristically linked to a sincere, unselfish, empathic way of living with humility, self-respect, ability to cope with crises as well as renunciation of contempt for humanity and a strong regard for morality, solidarity and social justice. At the end of his book The Nature and Future of Philosophy, Dummett writes: Finally, can philosophy settle what is surely the most important question of all, whether there are rational grounds for believing in the existence of God? There seems to me every reason to think that it can, and will even do so in the lifetimes of our great-grandchildren. My own belief is that it can be resolved positively; indeed, as a Catholic, I am committed to that belief.28

Dummett qua both a philosopher and a theist who has substantiated his view that the concepts of the theistic God and of the world as a whole stand over against one another has good or even conclusive reason for his belief in the proposition that the theistic God exists, even if that proposition—which rests on general revelation, accessible by purely rational thought aided by universal observation—may basically lack sufficient evidential support. Moreover, Dummett qua believing philosopher whose reflections on the content of his firm, clear intuition that there is an omnipotent, omnibenevolent God with a will as to the righteousness of human actions as well as on the associated egalitarian propositions regarding i. a. race, immigration and asylum bring him to the conclusion that these doctrines indeed are all intelligible and consistent has good reason to regard the assumption that a theistic God exists as true.29 Dummett does not try to reason with a philosopher who like Lauener is both an atheist and a strong Kantian agnostic on the issue of the structure of the noumenal outer world that he postulates. On the other hand Dummett does indeed set out to convince a theist who believes that we cannot in any way conceive an idea of the world as it is in itself—or for that matter an atheist who holds that our scientific-theoretical concepts can be successfully applied to

|| 28 Dummett (2001/2010), 151. 29 Actually, Dummett has good reason to regard the assertion that God exists as true, or rather to consider this assertion to be not false, without being able, though, to rule out the possibility that it is neither false nor true. It should be borne in mind here that according to the view of truth associated with Dummett’s justificationist semantics, there are consistent intelligible assertions we neither know to be true nor know to be false, any more than we can know them to be neither true nor false (and which we therefore do not actually have a right either to accept as being either true or false or to accept as being neither true nor false).

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characterize the world as a whole—of the broadly metaphysical overall picture which accrues from his justificationism and which he encapsulates as follows: God and the world as a whole stand over against one another.30 Through developing and vindicating that bracing basic idea, Dummett attempts to make the rationality of theism clear to agnosticists, pantheists, and atheists altogether as well as to bring home to strong scientific, materialistic metaphysical realists that a more weakly realist metaphysical presupposition of the world as a whole, which both transcends and comprises the largest possible explanatory domain of science, stands to reason. So Dummett’s basic idea concerning broad metaphysics is the following: “the only way in which we can make sense of our conviction that there is such a thing as the world as it is in itself, which we apprehend in certain ways and other beings apprehend in other ways” is to conceive “of a mind that apprehends it as it is in itself”—“how God apprehends things as being must be how they are in themselves (...) how things are in themselves consists in the way that God apprehends them.” That is to say, “our concepts of God and of the world as a whole stand over against one another.”31 In the above it has become clear that Dummett rejects the customary view that the diverse phenomenal worlds of us humans, and of the other sentient and rudimentarily reasonable terrestrials, are the more or less distorted and impure projections of the one alleged world-in-itself upon the consciousness of us earthlings of different types. Dummett argues that the world as it is in itself cannot be conceived wholly independently of how it is apprehended by some mind or other. There is nothing that would constitute the existence of the worldin-itself if there were no creatures to apprehend any of it; for unless there are sentient, rational minds, it would not be possible for either perceptual experience or inference to occur. Of course we can in principle conceive of the worldin-itself as an immensely detailed complex of mathematical structures, as it were without any reference to human experiential capacities, but such a purely abstract mathematical structure could not be specified as actually existing in a more robust sense. Hence Dummett advocates that the world as a whole must be conceived as encompassing and combining the particular truthful perspectives of all kinds of sentient, rational beings, that is, all the particular effective ways in which “the” world is apprehended by all sorts of different sentient creatures such as humans and different species of animals. The world as it is in itself is

|| 30 Cp. Dummett (2006), 96, 102. 31 Dummett (2006), 102.

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therefore constituted by all truths, i.e. by the totality of true propositions, which are built out of the concepts by means of which the different rational beings’ judgements about “the” world are built.32 The sphere of reality as it is in itself thus clearly transcends those cumulative sectors of reality which are in principle accessible to us, humans, and which increasingly open up new perspectives and possibilities for us. Dummett clearly grants that all ontologies of facts (i.e. all domains of linguistically expressible, intelligible, consistent as well as true propositions) which humans may in principle describe are but a limited selection from innumerable kaleidoscopic angles and configurations that are contained in the enormously complex unified system of the world as a whole. While the overall structure of the world as it really is, in itself, is thus simply not expressible, describable by human linguistic means, it is altogether graspable by the theistic God’s view from everywhere at once. Humans, animals and presumptive extraterrestrial sentient and rational beings are included in the world as it is in itself within those particular circumstances in which they are being apprehended from each other’s perspectives; and of course the contexts and specific structures within which we, humans, experience ourselves and everything around us are likewise part of the world as a whole. But what about the theistic God Himself, who apprehends the world from all actual perspectives at once? Can He also be part of the world as a whole? It would seem that we need to distinguish between the world as a whole and what I would call the “all-embracing reality as such”, which consists of the world as a whole and the theistic God qua spiritual entity apart from the world, which apprehends the world from (I quote) “no particular point of view, no location in the world, no perspective contrasted with other perspectives”.33 The world as it is in itself, is “created” and “designed” in the theistic God’s act of intervening by apprehending the world from everywhere at once (and thus allowing and substantiating particular perspective worlds such as ours). In addition to it, however, some exceptional kind of pantheistic, all-embracing reality as such needs to be presupposed (viz postulated), which is identical with the pair of the theistic God and the world apprehended by Him as it really is, in itself, as a whole.34

|| 32 Cp. Dummett (2006), 96–99. 33 See Dummett (2006), 96 (and compare pp. 96–102). 34 At this point, it may appear that the all-embracing, pantheistic reality as such can be gnostically interpreted as the all-containing, transcendent, hidden, ineffable highest divinity. Dummett’s theistic Creator God would then appear to be gnostically interpretable as the

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It is worth noting in this connection that the theistic God qua unembodied mind can therefore not be deemed to be maximally great in every respect, since the all-embracing, ultimate reality as such comprises the theistic God (along with the world as a whole apprehended by Him). Nevertheless, as will be clarified below, the theistic God is, in an important sense, conceived as a maximally great spirit, being omniscient, omnipotent, omnibenevolent, etc. Accordingly, it appears that an entity having such (compatible) properties that nothing greater could exist—as famously and controversially presupposed by Anselm—is not at all conceivable within Dummett’s outline of a broad metaphysics.35 Thus, according to Dummett’s understanding, God cannot be the Anselmian theistic God, viz the being than which no greater can be thought (“id/aliquid quo maius nihil cogitari possit/potest”). For, clearly, Dummett’s theistic God cannot be the creator of the all-embracing reality as such, which comprises God Himself and His creation. To rephrase this point in Augustinian terms of divine perfection in all (or some) respects: according to Dummettian broad metaphysics, the theistic God cannot, as Augustine holds, be absolutely perfect in all possible respects, since in respect of being creator He does not have the perfection that He has in respect of his other main attributes (such as omniscience, omnipotence, omnibenevolence). So, to recapitulate, over and above the different levels of our human descriptions of various aspects of “the” world (comprising domains of true propositions, viz ontologies of facts), Dummett broaches the more broadly metaphysical subject of the all-embracing reality as such, consisting of the extra-worldly God and the world apprehended by Him as it is in itself. Dummett makes wellfounded affirmative statements about the most basic features of the theistic God

|| spontaneously emanated demiurge, who in turn creates the material universe consisting of the physical aspects of all entities in the universe, including us humans. However, Dummett’s theistic God is not to be conceived of as an imperfect creator of “the” material world, including its physical inhabitants. Instead, Dummett’s omniscient God apprehends the world completely as it is in itself, that is, He comprehends it as the non-material totality of all truths from all differing perspectives on what appears to be “the” material world to the various sentient, rational beings, who are located within “that” world and thus appear to each other, and to themselves, as being material as well (in part at least). 35 Clearly the pantheistic all-embracing, ultimate reality as such is not a spirit (not an unembodied mind) and can therefore not be considered omniscient, omnipotent, omnibenevolent, etc. So if we do accept Dummett’s premise in broad metaphysics that the concepts of the theistic God and of the world as a whole stand over against one another, there can be no entity which is maximally great in every respect.

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and the fundamental structure of “the” world by characterizing God as the divine mind that is not located in “the” world and correspondingly apprehends “the” world from no particular point of view: God does not have perceptual experience of external things or events that impact on something that might be called His “sense organs”. His knowledge consists in His comprehension of all truth, i.e. of the world as it is constituted as a whole by the totality of true propositions or true thoughts of humans and animals as well as aliens (if there should in fact be any aliens). For Dummett, the world as it is in itself is thus conceivable only under the presupposition of a sole divine unembodied mind that comprehends it in its totality, as a single, unitary world, apprehended in different ways by different sentient creatures within it.36 As concerns the relation between faith and reason with regard to broad metaphysics, Dummett thus revives Augustine’s and Anselm’s approaches by starting from some religious propositions he takes himself to have good (sufficient) reason to believe so as to know anything about God and “the” world, whilst at the same time making sure that he does not uncritically accept these doctrines as true but seeks to better understand them through philosophical reflection on them. In order to now substantiate the basic features of his view of broad metaphysics, Dummett needs to be more specific about what properties are required of God and reach more detailed clarification of the primary attributes of the one theistic (and not pantheistic) God.37 In order to give a clearer sense to his suggestion of the way the world is in itself, in its totality, Dummett needs to conceive of God in terms of an omniscient spirit, and this in turn requires a further

|| 36 Cp. Dummett (2006), 96f., 102. 37 In this context, it is instructive to take a quick look back at Paulinian early Christian theology: in the Areopagus sermon (Acts 17:16–34), Paul makes affirmative statements about certain attributes of God, but only about a few contextually relevant attributes at a time. In order to make intelligible statements about God, Paul anthropomorphizes God in attributing recognizable human properties to Him such as being a creator of things—in late antique and medieval philosophy, this first step of giving a partial characterization of God is famously referred to as “via affirmativa”. Yet, Paul immediately hastens to limit his affirmative statement about God being creator by indicating that God as the creator of things should not be associated with things made with hands, nor should He be thought to need anything to back Him up—this second step is notoriously referred to as “via negativa”. – It is odd, though, that Paul then appears to further suggest that God and the world coincide, when he says, “For in him [God] we live and move and have our being.” (Acts 17:28 NIV) It may be that Paul endeavours here to align his Christian speech of God with the pantheism of his Stoic audience at the Areopagus in Athens. However, the Christian theistic God, the creator, can definitely not be equated with His own creation, “the world and everything in it” (Acts 17:24 NIV).

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elucidative analysis of the theistic nature of God. (It should be pointed out here that Dummett’s analysis of the theistic God’s primary attributes—which he reduces to the minimum necessary to give plausible elucidation of God’s creating and sustaining the world as a whole in the act of apprehending it—is principally for judging how the theistic God and the world as a whole can—in a conceptually consistent way—be comprehended as standing over against one another.) Dummett thus proceeds to make further explicative statements about central attributes of God commonly ascribed by the Christian or, more generally, the monotheistic religious tradition.38 Clearly monotheists do not fully agree on quite a few attributes of God and their further clarification, respectively; so in what follows, when referring to “the” theistic God I actually mean Dummett’s conception of the sole divine unembodied mind, that is, the divine spirit. As regards the compatibility of central characteristics of the theistic God, Dummett focuses on revealing the consistency primarily of God’s omniscience, His being Creator, His (omnipotent) will and His omnibenevolence. Moreover it becomes clear that Dummett’s justificationist semantics underlies not only his narrower metaphysics of our various human, scientific conceptions of “the” world but as well his theistic broad metaphysics of the world as it is in itself.39 Dummett ventures on the substantiation of the anthropomorphistic theistic conception of the personal God’s attributes by first comparing and contrasting God’s knowledge with human knowledge. God’s omniscience, in Dummett’s terms, is His comprehension of everything completely as it is in itself, which corresponds with Dummett’s broadly metaphysical basic assumption that the world as it is in itself is the world as only God apprehends it as being. Evidently the mode of God’s understanding and knowledge cannot be perception, observation and inference, nor can it be direct grasp of matter as being disposed, in

|| 38 From the point of view of philosophical metaphysics, the doctrinal intricacies of (e.g.) Christian theology cannot be decisive in analysing the concept of God and evaluating the proposition of the existence of God. As Dummett puts it: “such specifically Christian doctrines as the real presence of Christ in the Eucharist or the two natures, divine and human, possessed by the one person of Christ (...) help the delivery of revelation, not accessible by reason alone; hence, although theologians have the task of explaining their intelligibility by reasoning of a characteristically philosophical kind, the doctrines cannot figure as components of any strictly philosophical system. The existence of God, however, is held by most believers to be a truth attainable by purely rational thought, without appeal to any extraneous source of knowledge.” (See Dummett (2001/2010), 42.) 39 In relation to the following few paragraphs that look at the primary properties which are ascribed to the theistic God by Dummett, compare especially Dummett (2006), VI, 85–89, 96f., 102–109 and Dummett (2001/2010), 39–49, 151f.

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all its infinite complexity, in ways existing independently of being perceived by any sentient beings contained in this material universe. Instead, God’s complete, universal knowledge of “the” universe amounts to the comprehension of the extremely complex structure determining what truths the manifold kinds of sentient and rational beings contained within it will observe and infer according to their location in it and the kind of their sense organs and their powers of judgement. God is justly called Creator according to Dummett, since only by conceiving of a sole extra-worldly, divine unembodied mind that apprehends the world completely as it is in itself, any sense at all (and, for that matter, any coherent metaphysical interpretation) can be given to speaking of the world as it is in itself. So God creates “the” world in His act of apprehending it as it is, thus knowing it in its totality and correspondingly knowing absolutely all truths. That is, the world’s being as a whole can only be constituted in virtue of God’s grasp of the enormously complex structure which determines all thoughts or propositions which the various kinds of sentient, rational dwellers in “the” world can frame and can recognize as correct, and the truth of all these thoughts and propositions is constituted by God’s knowledge of them as He grasps the complex structure amounting to the world as it is in itself. What is more, Dummett may be interpreted to hold that only by God’s refining, structuring universal grasp of the utter chaos, the habitable unitary cosmos is created – that is, God’s apprehension of the world as forming an intrinsic whole is the prerequisite for us humans and all other sentient creatures within it to being able to live in valuable cohesion in a unified, single world. When interpreted in this way, Dummett stays remarkably close to biblical revelation while clarifying the metaphysical framework within which the intelligibility of the revealed theistic idea of creation can be explained by purely rational thought. More specifically, Dummett does not follow the received Christian doctrine that God created the world out of nothing (a theological tenet which did not gain acceptance until late antiquity, when creation out of some pre-existent, eternal matter came to be seen as limiting God’s omnipotence, an argument which Dummett would certainly not accept). By way of contrast, Dummett remains closer to the biblically revealed idea of creation to be found in the opening of the book Genesis, which states that when in the beginning God created the world, the condition of the earth was void and without form and God differentiated and formed this primordial, waste and shapeless matter (which had snarled up in the pre-existent Tohu wa-bohu) into a well-organized cosmos. So in concert with Genesis, Dummett advocates that God in the act of comprehending the world as it is in its totality does not originate “matter” but structures it

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into an intelligible whole, thus creating a cosmos hospitable to life. Such biblically revealed idea of creation qua world-formation is evidently at odds with the received materialistic theological view of creation out of absolutely nothing: the opening of Genesis suggests that God created the world from something (formless earth, deep, chaos water, etc.).40 However, beyond this, Dummett specifies that God ought not to be thought of as creating in a human way, by simply manipulating pre-existing matter, thus making physical objects which subsequently exist independently of Him. Instead, God creates through His word or inner speech, that is, He creates by His thought. So in God’s light the totally chaotic Tohu wa-bohu gets differentiated and transformed into the well-organized, unified, single cosmos, which exists exactly as conceived by God and is inhabited by different sentient, rational creatures, which apprehend it differently in their particular yet (in the upshot) coherently related ways. Thus, God as Creator of the world as a balanced whole is the giver of all manifold truths, indeed of all laws that govern the supremely complex world as a whole. Now the imposition of truths and laws is clearly an act, and God’s capability of action is in its turn a necessary condition of His having a will. Moreover Dummett specifies that necessary and sufficient of a being’s having a will is his capability of choosing an action to perform out of many possible alternatives. And he clarifies that the choices God makes, in acting as Creator, between alternative possible actions must be guided by His apprehension of their often very different consequences. That is, God must have motives for His actions and select an action out of a range of alternative possibilities for a reason. Dummett therefore concludes that God has a will in the fullest sense of the concept. Further, Dummett highlights that a will needs to be ascribed to God particularly concerning our human actions. At any rate, this applies if the principles of morality are identified with the way God wants us humans to act. Correspondingly, Dummett points out that according to all world religions, living an ethically upright life is considered to be essential for salvation (however each of these religions conceives of salvation). Dummett recognizes that religions differ in the principles of morality they take to be the correct ones but he considers it misleading to presuppose that there is a universal morality shared among all human individuals and communities (though he grants that it is reasonable to expect at least a partial moral consensus about the rightness of a few kinds of

|| 40 Cp. Genesis 1:1–4 NIV: “In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. And God said, ‘Let there be light,’ and there was light. God saw that the light was good, and he separated the light from the darkness.”

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human actions, e.g. as regards behaviour in daily life toward other people regarded as equals). So Dummett’s central point here appears to be that God’s will concerning human actions must be that we all should live an ethically upright life based on a consistent deontological moral code consisting of principles that are grounded in divine revelation and a significant part of which is accessible by the use of reason, partly by drawing out what is only implicit in the revelation. Such philosophical-theological development of religious moral doctrines allows, according to Dummett, for the moral development of religious institutions and communities, leading them to acknowledge something as in truth ethically wrong that had previously been considered morally allowable. Above and beyond that, it would appear that God’s will allows for different consistent codes or systems of correct moral principles. This moral pluralism is clearly bound up with the fact that we humans are evidently unable to harmonize our various sound moral codes into a single universal, strongly complete system of moral principles. (The question of Dummett’s stance on the relationship between religion and morality—particularly in so far as it concerns his moral and political philosophy of immigration and asylum—will be revisited further below.) And then Dummett addresses himself to certain notoriously puzzling questions related to God’s omnipotence (His omnipotent will) and to His unlimitedly good will (His omnibenevolence or perfect goodness), particularly as concerns the intricate, perennial problem of evil. Indisputably, there have been at all times grave evils of diverse sorts and immense, horrendous sufferings associated with them, of which the omniscient Creator God is undeniably aware of, as He comprehends the world as it really is, in its totality, and thus knows absolutely all truths and facts, which indeed are constituted by His knowledge of them. So God is familiar with each of the innumerable actual evils found in the world, which comprise the following kinds of ills: Physical evils (natural disasters, serious diseases, unbearable pains, natural death of loved ones, etc.), manmade evils (wrong policy decisions, economic crises, environmental damage, famines, severe accidents, breakup of close relationships, etc.) and moral evils (humiliation, defamation, treacherousness, heartlessness, narcissism, manipulations, (sexist, racial, political, religious) oppression, contempt for life, rape, torture, (offensive) warfare, dehumanization, genocide (such as the Holocaust or the untold democides of indigenous peoples in the Americas), etc.). The existence and extent of these various kinds of evils are prima facie incompatible with either God’s omnipotent will or His perfectly good will. So if God does have the power to prevent the manifold, apparently senseless evils, and if His will for us humans is that we live an ethically upright life, why does He allow scores of

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people to suffer intensively or permits them to unintentionally or willfully cause unnecessary and outsized suffering? Dummett neither denies that God’s will is omnipotent nor that God is morally perfect and accordingly has good will toward us humans. A good will according to Kant is the only thing that is good without qualification; in this vein, it may basically be said—in the present context—that to possess a good will is to possess a will that makes its decisions on the basis of a consistent system of correct moral principles or, in the case of God’s possession of a perfectly good will, to possess a will that makes its decisions on the basis of “the” strongly complete system of all true moral principles. So God’s unlimitedly good will essentially fulfills moral requirements, that is, exclusively moral considerations can be conclusive reasons for the omnibenevolent God’s decisions with regard to the shaping of human character and conduct. But then, how is it to be understood that we humans so often come to wrong and inconsistent decisions, and, what is more, that we often decide immorally, thereby incurring a heavy burden of moral guilt and, indeed, disclosing a serious lack of good will? It appears, paradoxically, that God’s perfectly good will in regard to the extent of goodness of our human will is constantly thwarted, although evidently He would have the omnipotence to prevent that. However, Dummett comes up with a viable clarification of those morally problematic aspects of God’s will that are not apparently compatible with His unlimited benevolence, i.e. His perfect goodness. Dummett proposes to distinguish between God’s immediate will and His overall will: it evidently must be the overall will of God that sentient, rational beings contained in the world as a whole, such as humans, should be free not to conform to His immediate will perforce. Dummett makes clear that there must be some overriding necessity for this serious anomaly, or seeming imperfection, which might eventually be divined by some rational being or become the subject of revelation. On the other hand, Dummett acknowledges that unlike e.g. a neo-Darwinian materialist, a theist must suppose that the riddle of evil might remain forever a mystery to us humans, or, as one might also say, a true “ignorabimus”. – Dummett’s response to the problem of evil cannot be called a theodicy, as Dummett merely states some mysterious overriding necessity, but he does not put forward any actual morally sufficient reasons for which the omnipotent, omnibenevolent God may allow us humans to flout His immediate will. Still, Dummett implies that God must have conclusive reason to allow humans to conduct themselves irresponsibly and malevolently: there must be an unfathomable compelling necessity for every actual man-made evil found in the world, compatible with the existence of the omniscient, omnipotent and omnibenevolent God. Also, it is important to note that Dummett points out that deliberate human individual decisions,

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including wrong decisions, evidently do conform to God’s overall will. And, what is more, God’s overall will apparently allows physical evils such as natural disasters, major diseases and severe pain, distress and suffering, which obviously cause disorientation and great stress, under which human misjudgement and wrong decisions inevitably affect morally reliable actions. It appears therefore that God gives priority to human deliberate decisions despite the endless man-made and moral evils that come as a result. In this respect, Dummett might e.g. argue in an Augustinian vein that God does not directly allow moral evil per se but allows us, humans, to deprive ourselves of the moral good: if we act autonomously on our own responsibility, we can deliberately select an action which is more or less good out of a range of alternative possibilities for a reason. So our actions are evil only in so far as they lack goodness, since there is no such thing as a moral evil in itself, but there are standards of the moral good which may not really be taken into account or may even be deliberately negated. Though God is the power of perfect goodness, His will concerning our conduct is not merely to work determining within us, but instead He wants us as reasonable, responsible free willing creatures who deliberately allow divine benevolence to work within them, notwithstanding the fact that they would be free to negate the moral good, which, in turn, necessarily opens up the possibility of our being morally evil. – Dummett does not get himself into speculations of that sort, though. Still, it’s worth noting at this point that the authentic, deliberate decisions of free willing individuals necessitate courage, i.e. a capacity for doing what one considers to be right in the face of serious difficulties and likely disadvantages for oneself. So authentic, thoughtful, deliberate decisions and autonomous, self-governed, self-responsible actions will most likely, at some point, involve failure and a lot of suffering. Yet while free willing, original persons—who are acting to the best of their judgement—cannot evade suffering evils nor inflicting certain man-made evils, they do learn to cope with suffering and to minimize its destructive impact by acquiring, for instance, humility, gratitude, open-mindedness, empathy, cooperativeness and perseverance. What is more, it is their suffering from setbacks and crises, i.e. their suffering the pains of maturation, which enable them to improve themselves further emotionally, intellectually and volitionally by developing and strengthening their character. The idea that to be mentally shaken up by suffering evils makes us become less defensive and more open—and thus functions as an engine of psychological maturation and moral growth—can be found in the Bible as well as in ancient Greek literature. Notably Aeschylus (in his Agamemnon, the first drama in his trilogy Oresteia) formulated the idea in an impressive manner:

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God, whose law it is that he who learns must suffer. And even in our sleep, pain that cannot forget falls drop by drop upon the heart, and in our own despite, against our will, comes wisdom to us by the awful grace of God.41

Such ideas of the inevitability of suffering and of the potentially positive value of trying to cope with the ills of human existence are by no means exclusively found in Christianity and other world religions such as Islam or Buddhism, but can as well be found in the various oral, non-institutional, non-proselyting religions (or systems of spiritual beliefs and ways of life) of indigenous smallscale societies. An apt example is traditional Apache religion, which takes as a starting point that this life is imperfect and all human beings are naturally evil in parts. The youth is accordingly taught how to both accept and manage their nature in order to become prepared to live a meaningful life in balance with the physical and spiritual realms: learning to cope with suffering, learning to be truthful, observant and useful, holding out in conscious prayer as well as practicing gratitude and appreciation of life and of the whole of creation, inter alia, play central roles in their maturation process.42

|| 41 Translation by E. Hamilton—see Edith Hamilton (1930) The Greek Way, New York, London: W. W. Norton, pp. 61 and 194. On pp. 61f., Hamilton clarifies that this citation from Aeschylus expresses the “thought that wisdom’s price is suffering and that it is always paid unwillingly although sent in truth as a gift from God”. It is interesting to note in this connection that following his brother’s assassination in 1963, Robert F. Kennedy received (from Jacqueline Kennedy) a copy of Edith Hamilton’s above-mentioned book, which left a lasting impression on him and from which he thereupon often quoted. On 4 April 1968, the day of the assassination of Martin Luther King (for whom Dummett, incidentally, felt great respect and admiration), Robert Kennedy gave a committed impromptu speech in front of a shocked and despairing (largely African-American) audience, in the course of which he cited the above-mentioned lines from Aeschylus’ drama Agamemnon and appealed for racial reconciliation and for a feeling of justice towards those who suffer within society. 42 Cp. Ball, E. (1970) In the Days of Victorio: Recollections of a Warm Springs Apache, Tucson: The University of Arizona Press as well as Ball, E. (1980/1988) Indeh, an Apache Odyssey, Norman: University of Oklahoma Press. – Pioneering oral history in the 1940s and 1950s, Eve Ball took down (in shorthand) verbatim narrations and accounts of almost seventy indigenous witnesses, many of them surviving witnesses of the Apache wars of the 1880s. Ball knew how to learn from the remarkable indigenous oral tradition, obtaining information from various valuable perspectives, and she succeeded in documenting crucial hidden aspects of Apache history and religion. In his Foreword to Ball (1970), p. viii, Father Albert Braun—who in 1916 had been assigned as a Franciscan missionary to the Mescalero Apache Reservation and had gradually won the respect and friendship of the Apache elders, who, incidentally, saw no fundamental conflict between Christianity and traditional Apache religion—wrote: “The Indians have a philosophy of life in many ways better than what we have. How, for instance, can we compare our value of suffering with that of the Apaches? In the old days they trained their

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The single, unitary world as a whole is constituted through God’s complete, allembracing knowledge of how it is in its totality and is thus conceived in being apprehended by God as it is in itself (as has been elucidated above). With this in mind, it is rather remarkable that Dummett is able to bring into play his justificationist stance against the common philosophico-theological view that “the” world created by God needs to adhere to the principle of bivalence. Dummett makes clear that God does not need to know the precise magnitudes for all quantities. More generally, God does not need to constitute the truth or falsity of all propositions (which we can intelligibly frame) by His knowledge of them or alternatively of their negations. God knows for every proposition framed and recognized as true by us humans (or any other sentient, rational inhabiters of “the” world) that it is true. But God in no way needs to know for every intelligibly framed, consistent proposition that it is true or else that it is false. When there apparently is an answer that we do not know, God for His part may not know any answer either, since there may simply not be any answer at all. For if God, being the Creator of all things by His thought, would be constrained to render every detail of “the” world determinate, then this would limit His omnipotence unjustifiably. All details crowding the world as a whole exist dependently of God, as conceived and thereby created by His thought; and so if God conceives of some parts of “the” world as partly undetermined, they simply are not fully determinate. And if there indeed are gaps in “the” world, then some questions will have no answer at all and the divine logic will, thus, not be classical. However, by no means does this imply that the divine logic is an intuitionistic one according to Dummett’s justificationist semantics. On a justificationist view, there may not be a determinate answer to an intelligible question for which we humans know no answer; that is, there may be gaps in “the” world, but we can never know that there are and, therefore, we humans

|| children to suffer because they knew suffering would come into every life.” In Ball (1980/1988), pp. 59f.—i.e. Ball’s second book from Apache memories, which is based on the whole wealth of her transcribed oral history primary sources—Ball relates what an Apache witness told her about certain aspects of the Apache religion: “The Apaches emphasized the life on earth through rites and ceremonies. (...) Nothing was ever said of reward or punishment. They emphasized good behavior. There should be no stealing. One should not tell lies. It was not a matter of future punishment, but of doing what was right because it was right. All Apaches can go to each other in distress, and if one asks help from another the help must be given. (...) Even small children were taught the use of medicinal herbs. Only in extreme cases did they seek the aid of the medicine man (...) His treatment was both physical and psychological. To the Apache it was also religious. Without the faith of the patient and his family and friends the herbs would be less effective, or perhaps of no aid.”

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need an intuitionistic logic. On the other hand, if there are gaps, i.e. certain intelligible questions that have no determinate answers, then God must know that there are, because for every determinate (true or false) proposition, God knows that it is true and false, respectively. Thus, it can be argued in accordance with Dummettian justificationist semantics that the divine logic is neither classical nor intuitionistic, but must be three-valued.43 We have seen that Dummett’s special conception of the sphere of reality as a whole stems from his presupposition of a sole omniscient Creator God, of “an omnipotent God who cares about mankind”44. Dummett’s Christian faith is apparently the main source not only of his ideas for a plausible metaphysics in the broadest sense but also of his moral outlook and political views. Yet obviously, Dummett does not take theistic principles, or alternatively Christian deontological principles, at face value. Quite the contrary, Dummett ties the uncritical acceptance of religious doctrines (which has repeatedly brought about revolting, atrocious crimes committed by religious believers against probable disbelievers) to an obsolete conception of faith: on this conception, the believer is obliged not merely to trust his beliefs, but invest his faith with certainty, that is to be literally certain that his faith is the absolutely, uniquely true one. In comparison, some Christian and non-Christian believers have in our times come to a more reasonable, revised conception of what is demanded by the virtue of faith, sincerely reaffirming the importance of respecting the principle of religious freedom. Dummett has always assumed the tolerant, open-minded attitude which at long last has come over a considerable number of religious believers and which he sums up as follows: They have come to understand how intimately anyone’s religious faith serves to determine his identity, whether that faith is, in their own eyes, mistaken or sound. They have learned how precious to each person his own faith is, and how deeply wounding to him is any manifestation of contempt for it or any insult directed to it. They have realized how unjust it is to compel anyone to abandon his religion or to adopt another.45

Dummett sees it as a matter of epistemic responsibility to endeavour to make sense of the religious propositions he accepts as a Christian. He goes over Christian doctrines with a fine-tooth comb, analysing and reexamining them through reasonable argumentation. Dummett draws a parallel between the philosophi-

|| 43 Cp. Dummett (2006), ix, 88f., 102f., 107–109. 44 Dummett (2001/2010), 45. 45 Dummett (2001/2010), 54.

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cal problems about the interpretation of scientific theories and those about the interpretation of religious doctrines: a philosopher who accepts a certain scientific theory as true needs to attempt to solve the interpretational problems that theory throws up; analogously, a philosopher who is positive about having good reason to accept a certain religious proposition as true feels compelled to critically reflect upon it in order to attain an adequate understanding of it and to show it to be consistent by resolving its apparent inner contradictions. Dummett considers such a task to be the object of philosophical theology, which evidently assumes that the contents of faith are accessible to reasonable reflection upon them. Dummett is careful to point out that a believing philosopher working e.g. in metaphysics or moral and political philosophy inevitably has to deal with tensions or even full incompatibility which may result between his philosophical conclusions and his religious faith; and he makes it clear that in such cases a believing philosopher needs to have the intellectual honesty to keep apart his philosophical and religious orientations. Dummett gets to the heart of the matter: A believer who devotes himself to philosophy must not do so with the intention of devising arguments to defend the tenets of his religion, or he ceases to act as a philosopher rather than an advocate or apologist. As a philosopher, he must follow where the argument leads. If it appears to lead to a conclusion contrary to his religious belief, he must still present it; he may acknowledge that he is convinced that the conclusion cannot be correct, even though he cannot see how to avoid it. Very few philosophical arguments survive unscathed by criticism; it is only by putting them forward to be honed by ensuing discussion that philosophy can progress.46

In his philosophy in general—and this clearly includes his philosophicaltheological reflections on broad metaphysics and on moral, political philosophy—Dummett rigorously follows the famous advice of Socrates as personated in Plato’s Republic: “we must follow the argument wherever, like a wind, it may lead us.”47

|| 46 Dummett (2001/2010), 45. – Cp. as well Dummett (2001/2010), 41. 47 Plato (1974), 394d (p. 65).

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An Anti-Racist Philosopher’s Christian Egalitarian View of Asylum and Immigration Dummett’s moral and political philosophy is to a certain extent embedded in his theistic broadly metaphysical view of “the” reality that consists in the way the omniscient, omnipotent and omnibenevolent God, who cares about human character and conduct, comprehends it. So Dummett’s egalitarian moral and political views and dedication have their origins in religious revelation but being a philosopher, Dummett endeavours to fathom them through reason: they are not self-evident and thus need to be continuously checked by argument and evidence. Dummett advocates a firm Christian egalitarianism, as will become evident from the following discussion of his ethico-political views on immigration and asylum. Basically, Dummett holds the view that human beings all have full, equal moral status as well as equal basic moral responsibilities without regard to social and economic class, gender, or race—or ethnicity, nationality, native language, physical appearance, etc. (Incidentally, it is interesting to note that with his firm egalitarian stance Dummett shows a radically different attitude than those influential sections of the Roman Catholic Church which are formally opposed to thoroughgoing egalitarianism, particularly in relation to gender equality.) What Dummett accordingly considers distinctive of Christian ethics is that racial and social differences are morally of no consequence—Dummett writes: Christian morality (...) condemns suicide, in opposition to the ancient Hindu practice of widows’ burning themselves on their husbands’ funeral pyres, and to the demands made by personal honor in Japan, in ancient Rome, and among British gentlemen; the ethic of honor is opposed by Christian humility and forgiveness. Above all, Christ taught that there is no one who does not count: those of a different race, those of lower caste or none at all, slaves—all are our neighbors.48

Paul in particular conceived of early Christianity as an inclusive moral community wherein all have responsibility to all their neighbours: There is neither Jew nor Gentile, neither slave nor free, nor is there male and female, for you are all one in Christ Jesus.49

|| 48 Dummett (2001/2010), 48. 49 Galatians 3:28 NIV.

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It can be argued, however, that, despite this moral precept (or ideal), women have often counted for less than men in the actual practice of Christian morality. Such a hostile, repressive attitude towards women appears to have been exemplified even by Paul himself: Women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the law says. If they want to inquire about something, they should ask their own husbands at home; for it is disgraceful for a woman to speak in the church.50

It must be noted, though, that many biblical scholars have argued that this antiwomen passage in First Corinthians cannot actually have been written by Paul, since at various points in his letters (probably most clearly in Romans 16:1-16) Paul does not at all wish to reduce women to silence but in fact makes personal references to several women he pays tribute to, calling them his equal, very hard-working co-workers in Christ (one of them, Junia, he even calls outstanding among the apostles) and characterising them as courageous and recognized leaders, deacons of house churches, benefactors of many people including himself, etc. What is more, beyond the New Testament, the “Passio Sanctarum Perpetuae et Felicitatis” (“The Passion of Saint Perpetua and Saint Felicitas”), reputedly one of the most reliable among the earliest of Christian texts, dating from the early third century, points in several related respects to the inclusive character of early Christian morality with respect to women. It is widely accepted by scholars that Perpetua was in fact the lead author of this text, so that the diary (intended for readers) of a female martyr was evidently enough to be highly appreciated and brought out sufficiently to survive—at Augustine’s time, incidentally, the passion of Perpetua and Felicitas had apparently become so popular that Augustine cautioned against attaching to it the importance of the Scriptures. Moreover in deliberately becoming a martyr after repeatedly refusing her own father’s (actually a nobleman’s) call for her to renounce her Christian faith, Perpetua rejects loyalties to her father, thus consciously failing to conform to Roman society’s norms, bringing disgrace to the name of her gentry family, in order to self-assuredly acknowledge a higher loyalty to Christ. Furthermore Perpetua autonomously asks for—and is consequently independently responsible for—the exceptional dream visions she receives in prison according to her own account. There might be a drawback, though: it has been argued that the coeval Christian editor of the passion of Perpetua and Felicitas disguised the

|| 50 1 Corinthians 14:34-35 NIV.

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(alleged) fact that it was actually Felicitas—a slave who worked in Perpetua’s house and was her friend and fellow convert—who received one or more of the visions of Perpetua and Saturus (who had introduced both of them to Christian faith) that are rendered in the narrative of the passion; the idea that a female slave of probable Berber origin (if not of Punic origin like Perpetua) can receive important visions might have overstretched the inclusivity even of the early Christian moral community. On the other hand, the execution of an aristocratic woman along with a slave—who are thus equal with each other in their martyrdom—clearly shows that social distinctions usually insurmountable in Roman society could be transcended by members of the significantly more inclusive Christian moral community. A philosophically (and juridically) minded theologian from the early modern period, Bartolomé de Las Casas, who developed pioneering egalitarian views, particularly in the form of an equal, universal natural right to liberty (more specifically to freedom from slavery) inhering in every human person, i.e. in human beings of all races and religions—and who courageously and steadfastly stood up in defence of the corresponding right of indigenous people— must be regarded as an important spiritual and ethical forebear of Michael Dummett the religiously inspired moral and political philosopher who has both intensely campaigned for more racial justice and set forth deontological principles specifying a state’s or citizenry’s responsibilities towards ethnically and culturally unfamiliar and underprivileged segments of society as a whole. In philosophy of language, philosophy of mathematics and logic, metaphysics, and in areas within epistemology and philosophy of mind which are connected with them, Dummett frequently starts from philosophical problems inherited from Frege, Wittgenstein or coeval philosophers, which he searchingly nuances and differentiates, offering clear, detailed arguments, illustrated with apt examples from both everyday life and recent history of science. By contrast, in his late and comparatively much rarer writings on topics in moral and political philosophy, Dummett argues to a large extent independently of reference to recent technical philosophical literature. Instead of this, he develops arguments for, and supplies evidence of, his ethico-political principles in common language, occasionally refined by some notes of definition or clarification. When one opens and starts to read Michael Dummett’s 2001 book On Immigration and Refugees it soon becomes evident that Dummett’s systematic reflections on a just treatment of refugees and migrants, devoid of ungrounded prejudice, rely on a wealth of commonsense firsthand observations over the decades of his and his wife’s, Ann Dummett’s, persistent struggle and campaign against racism. On

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the basis of this pertinent extensive experience, Dummett reviews and refines his pretheoretical, religiously inspired moral commitments and values. In doing so he chooses a deontological approach to specifying good or conclusive reasons for a state to reconsider its immigration policy, i.e. normative reasons that favour a more equitable treatment of migrants by an immigration state, which is to say, by the collective agents that make up a society comprised within a state: the citizens at large (the citizenry), the civic associations and especially the institutions and authorities of the structured community framework that (seriously) claims to be a democratic, constitutional, and pluralistic state. Dummett’s development of deontological principles for responsible treatment of helpless refugees and economically severely disadvantaged migrants by states of immigration is characterized, in a general sense, by his Christian faith. Dummett himself makes this religious kind of inspiration comprehensible as follows: Each religion presents its adherents not with a comprehensive conception of reality, but with salient features of such a conception that enable them to do two things: to decide how they should live their lives and to make some sense of what happens by attaching a significance and purpose to it (...) religio[us] belief endows our lives with a point, different for different religions, that for an unbeliever it is meaningless even to seek.51

As a matter of fact, the Gospel of Luke e.g. provides several salient indications of the high importance of a responsible treatment of the underprivileged who live on the fringes of a society, being at the mercy of its entrenched members. Christian, particularly Catholic, doctrine highlights the sanctity and dignity of human life, which implies due respect for the individual human life and supports the compassionate solidarity, through both words and deeds, with the most vulnerable and poorest members of any society. This moral-political aspect has been particularly accentuated in Catholic liberation theology in Latin America, which has been campaigning for social justice since the 1950s and which might be considered to be a very late continuation of Las Casas’ struggle against the merciless oppression of the indigenous population and the African slaves immediately following the discovery of the “New World”. It is interesting || 51 Dummett (2001/2010), 45. – To be sure, it needs to be added that official, sacred religious texts such as the New Testament or even a part of it, say, the Pauline epistles, hardly ever disclose in an entirely unequivocal fashion salient features of a religious (in the present case Christian) comprehensive conception of reality. There are unresolved tensions e.g. in the Pauline epistles and their readers need to decide which precepts and guidances they make a salient feature of. After all, Paul himself recommends, “test them all; hold on to what is good” (1 Thessalonians 5:21 NIV).

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to note that Jorge Bergoglio, the current pope, Francis, though not being adherent to (dogmatically Marxist versions of) liberation theology, clearly avows for a firm defense of society’s most vulnerable, disadvantaged members, tracing this option for the underprivileged back to the very beginnings of Christianity as well as to St. Francis of Assisi, who threw into relief a groundbreaking idea of poverty. In particular, Pope Francis keeps on warning the leading politicians and citizens of wealthy European countries to take a much firmer stand for severely disadvantaged migrants’ and refugees’ well-being. Dummett died one and a quarter years before Bergoglio became Pope Francis and so he may not have heard about him at all. However, the two undoubtedly share their equally keen and firm option for marooned refugees and migrants.52 Yet, whereas

|| 52 On some other matters of moral and political theology Dummett and Francis take very different views, actually. What needs particularly to be mentioned here is that while Francis appears to take the orthodox stance against contraception, Dummett argues consistently against the Catholic Church’s condemnation of the use of such contraceptives as condoms or the pill. I give in the following some connected extracts from Dummett’s argument in question, which is to be found in Dummett (2001/2010), pp. 49–53: “Paul VI’s encyclical Humanae Vitae reiterating the prohibition on contraception (...) greatly damaged the respect of the faithful for the Catholic Church’s moral teaching in general, since many of them do not accept the ban on contraceptives (...) But it has also damaged the integrity of Catholic moral theology. The encyclical did not merely reaffirm a long-standing tradition: it also dealt with something quite new— the Pill. The condemnation of its use for a contraceptive purpose accorded with the manner in which it had been usual to say why it was wrong to use other devices for that purpose, namely precisely because of the purpose. But the Church’s recognition that the use by a husband and wife of the “natural” or rhythm method, whereby they confine sexual intercourse to infertile periods, is morally legitimate denied it the right to hold the purpose of reducing the frequency or number of pregnancies to be in itself wrong. A condemnation of the use of contraceptive devices such as condoms could not therefore be consistently based upon their intended purpose, only on a claim that an act involving such a device was intrinsically wrong, regardless of the purpose (...) Moral philosophy cannot accommodate such a prohibition. A certain type of act, defined by a given form of description, may be intrinsically wrong. If so, it can never be morally justified by an ulterior purpose, however commendable; this is what is meant by saying that the end does not justify the means. For instance, to give someone a fatal dose of poison must in all circumstances be wrong: even if the purpose is to frustrate the known plan of the victim to massacre an entire family, it will still be wrong. It would be a misuse of the principle of double effect to appeal to it in justification of such a murder. (...) Double effect can be invoked only when the act is in itself morally legitimate, even though in the particular circumstances it will have foreseeable evil side effects. Nothing can be a side effect if it is the means by which the objective of the act is realized. (...) No one supposes that it is intrinsically wrong for a woman to take the Pill, for example for its original purpose of regularizing irregular periods. (...) Equally, the intention, on the part of a married couple, of reducing the frequency or number of the wife’s pregnancies is, as already noted, recognized by the Church as legitimate

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Francis appears to be motivated by what he considers God’s or Christ’s surprising mercy upon us, which in turn triggers our correspondent mercy on our fellow human beings, Dummett considers it a reasonable duty, which notably intellectuals have, to endeavour to contribute to more racial and generally social justice given the opportunity. But Dummett also maintains that religious moral teachings nonetheless have their own authority: It is wrong to claim (...) that to treat a moral precept as part of natural law is to base it solely upon fallible human reason; it may be part of revelation even if accessible without it. Traditional Christian morality stems in great part from the teaching of Christ, of the Epistles and of ancient texts like the Didache, and is too integrated a system for a Christian to regard it as mere opinion, but precepts not manifestly part of the original integrated system cannot be taught as with divine authority.53

It is important to note at this point that irrespective of his own Catholic background, Dummett vehemently opposes it if, in today’s world, a state founds its identity on a particular religion such as Christianity. Nor, Dummett claims, should a specific language or a particular ethnos be construed as part of a state’s identity: whatever one may choose to call it—nationality, race, ethnicity, or speech community—no racial, ethnic, national or language affiliation ought to be conceived of as an integral part of the identity of a state. As a first step on the way to the establishment of a socially more just society Dummett proposes

|| and, in appropriate circumstances, praiseworthy. In the ruling of Humanae Vitae, we have therefore a condemnation as morally wrong of an act not intrinsically wrong but held to become wrong when it is done for a particular end, even though that end is likewise not in itself wrong. It is incomprehensible how this could be so (...) Pope Paul VI did many good things, yet his most important action, the issuing of Humanae Vitae, has engendered unresolved moral chaos. This chaos is most sharply seen in the confusion about condoms in countries where AIDS has tragically become pandemic. Some Church authorities oppose the use of condoms in an absolute and doctrinaire spirit (...) Some Church authorities with a more balanced sense of moral priorities have sought to justify the use of condoms by appeal to the principle of double effect. If the use of a condom is wrong only when it is done to prevent conception, like the use of the Pill according to Paul VI, then no justification is needed when it is done for a different purpose, such as to protect against infection. If, on the other hand, the use of a condom is intrinsically wrong, as violating the integrity of the marriage act, it follows from what was said above about double effect that to try to justify it by appeal to that principle is an abuse of the principle. But the existence of the horrifying disease of AIDS should surely prompt some rethinking about the blanket condemnation of contraceptives, including condoms.” 53 Dummett (2001/2010), 49.

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accordingly, by way of a basic general principle, that “no state ought to take race, religion or language as essential to its identity”.54 Accordingly, a state is not legitimized to discriminate against any of its citizens (and any other inhabitants, for that matter) on account of their ethnic origin, religion or native language. Any kind of persecution, contempt or disadvantage of ethnic, religious or linguistic minorities either by state laws, regulations, or actions of dominant groups should not be tolerated so that the members of minorities feel able to fully identify themselves with the state they inhabit. They must not be given to understand that they are merely tolerated. Dummett makes it clear that what ultimately matters is that each citizen has a chance to feel that he fully belongs, that is, to feel that he has what Dummett calls “the right to be a first-class citizen”. This fundamental right in turn is incompatible with imperialism, since the fact to be governed by a class of persons from somewhere else makes the indigenous inhabitants feel themselves to be second-class citizens. The consequent right to “self-government” entails that authority must be based on the will of the governed people, vital points being that a suitable voting system is adopted55 and, in particular, that due weight is given to the values and views of all citizens so that minorities are duly represented. Only in a just democratic system each citizen (or indeed inhabitant) would be a first-class citizen, but Dummett points out that none of the existing constitutions recognized as democratic reaches the required standard, even the best being just very rough approximations to such an ideal, although very serious efforts to realize this ideal are a requirement of justice. Dummett brings out that the identity of a state can only be grounded in a mutual ideal, a shared vision of the just society the citizens are determined to create—a collective ideal, as needs to be added, which constitutes the priority objective of a state’s citizenry and which they therefore do not tacitly consider to remain always and evermore beyond their search.56

|| 54 Cp. Dummett (2001), 4–6. 55 Dummett’s 1997 book Principles of Electoral Reform presents in an accessible way his significant, enlightening ideas about voting procedures. (Dummett understands the theory of voting to be the hitherto most salient application of social choice theory; and his simplified exposition in his Principles of Electoral Reform is, inter alia, addressed to politicians and other people responsible in practice for devising voting procedures—cp. Fara/Salles 2006, 351f.). This popular book of Dummett’s on voting systems could actually be very valuable for elevating the quality of today’s growing public debate surrounding conventional forms of elections by popular vote that turn out to be an impediment to the development of more truly participative forms of functioning democracy. 56 Cp. Dummett (2001), 4–14.

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Such a conception of the identity of a state clearly does not permit any nationalistic or rather racial, ethnic, linguistic or religious bias to be connected with a state’s immigration policy. But Dummett’s elaboration (based on contemporary historical evidence) of deontological principles for what states and their citizenry owe to refugees and migrants from disproportionately disadvantaged world regions goes far beyond this. That becomes clear when we take note of Dummett’s criticism that political philosophers such as Rawls, who enquire into the nature and foundations of justice with regard to the institutional arrangements of a society and the actions of its individual citizens, have rarely overstepped the boundaries of a citizenry determined by the authority of a single state, and have thus rarely asked what duties a state has towards foreign persons who are not its citizens. Dummett, on the other hand, proceeds on the assumption that it’s not sufficient for social justice to focus primarily on fair relations between a state and its citizens. It’s not the case that a state has duties merely to its own citizens. Marooned external newcomers such as asylum seekers or migrants from devastated or poor countries need as well be duly considered when it comes to just relations between a state and the society governed by its laws; for society needs to function justly as a whole, justice being a feature of a state, its laws, institutions, authorities and its whole society involving its citizenry, civic associations and all of its actual foreign inhabitants as well as emerging helpless, destitute or persecuted people being at the mercy of that one state to which they’re applying for an authorization to stay. Non-citizens, whether they are present in the community for some time already or have just arrived, need to be taken into consideration as part of society as a whole. This aspect of social justice is deepened and substantiated by another general, fundamental principle Dummett proposes: everyone, i.e. each of the actual or expectant members of a society, has to be given their due. To ensure that everyone gets their due is more basic and vital an obligation of a state than to make certain that everyone gets his or her desert. According to egalitarianism every individual within a just society as a whole must be accorded absolutely equal treatment, which is considerably beyond the scope of equality of opportunity.57 || 57 Dummett clarifies the point that equality of opportunity is not sufficient for equal treatment as follows (see Dummett (2001), 23–24): “God deals out very unequal hands: some suffer continual illness, some enjoy robust health; some have ten talents, some five, some only one. Even complete equality of opportunity can guarantee no more than that the most gifted secure more for themselves than others do; it can still result in great disparity of wealth and power. (...) For the egalitarian, it is the duty of the state to correct for inherited inequalities as much as can be done (...) It is easy to explain how accidental advantages can lead over time to grotesque disparities of wealth and power, to the divisions of a society by class and status. Almost all

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Dummett insists that in any matter, deviation from exactly equal treatment for individuals demands justification. And he indicates that Rawls’ proposal that inequalities within a civil society are legitimate only if they benefit the least well-off among its citizens will not suffice for the required justification, notably with regard to the justifiability of a state’s treatment of refugees and migrants, since justice cannot be regarded as bearing exclusively on the functioning of single societies defined by single sovereign nation states. What is certain is that a state must give individuals who seek to enter its land their due. The elaboration and application of his strictly egalitarian principle might be seen as the pivotal point of Dummett’s philosophical analysis of immigration and refugee policy, which he backs up with relevant well-researched and detailed contemporary historical evidence: he discusses a range of recent developments related to refugees worldwide, which constitute a suitable frame for the appraisal of the egalitarian quality of a state’s asylum and immigration policy. For instance Dummett criticizes that there is no international convention which has laid any obligation on a state to admit stateless people expelled from the country in which they were living, emphasizing that it is unjust for a state to deny someone the right to live somewhere, i.e. not to grant stateless refugees asylum. Moreover Dummett draws attention to the principle that states are obliged not to send refugees anywhere from which they may be returned to the country they have fled, since to deny human beings the right to refuge from persecution is to deny them their due, which is a manifest injustice. But nevertheless many refugees whose immigration applications are rejected by European states are actually returned by them to imprisonment and torture; that is, these wealthy European states refuse help to people threatened by injustice and thereby collaborate with that injustice, incurring part of the responsibility for it. Furthermore Dummett complains about the fact that the 1951 Geneva Convention on the Status of Refugees recognizes as refugees only those seeking escape from persecution (for reasons of race, religion, political opinion, etc.) but not refugees from civil war, starvation or the complete impossibility of supporting oneself or one’s family. Dummett accordingly queries the prominent employment by European governments of the depreciative term ‘economic migrant’: he argues that all conditions that deny somebody’s ability to live a minimally decent human life in his country of origin, disabling him e.g. from preventing his childrens’ death from malnutrition, ought to be grounds for claiming refuge somewhere else. In addition Dummett points out that Western states in effect created the infamous, criminal

|| societies are disfigured by such inequities: only hardness of heart or ideological dogma can blind an observer to their flagrant injustice.”

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traffic in illegal immigrants by erecting barriers to make it impossible for refugees to reach their borders, notwithstanding that these same states have an internationally recognized duty not to prevent refugees from reaching their territories. Typically enough, none of the wealthy, “developed” states have accepted similarly large numbers of refugees fleeing from severe crisis situations as some of the poorest countries in the world, which have repeatedly accepted refugees by the million. In sum, Dummett is highly critical of the exclusionist policy of the vast majority of the very wealthy states in the world, especially considering that the horrifying inequalities and ever widening gulf between rich states and poor countries has the most powerful effect on migration between them. Dummett warns that this disparity is a gross injustice and presents the gravest problem in the world today, a problem which is likely to prove fatal, since it threatens the stability of the world.58 The strict egalitarianism which is at the basis of Dummett’s proposals as to what moral considerations based on sound arguments and principles should inform a state’s asylum and immigration policy is, as stated above, characterized by his Christian faith. Dummett’s struggle against racism and his later moral-political philosophy are both deeply connected with his sense of duty to assist the underprivileged and oppressed, which arises out of his religious view of life.59 In the first instance, Dummett illustrates his basic egalitarian principle of justice, that everyone has to be given their due, by the parable of Jesus in the Gospel of Matthew which is called the “Parable of the Labourers in the Vineyard”. Briefly, it relates that each of the workers who had been hired by the owner of a vineyard at successive times of the day received the same amount at the end of the day, a full day’s pay, regardless of how late in the day they had been hired, i.e. how long they had actually been working. The content of this parable clearly goes counter to any right-libertarian idea of unchecked pursuit of self-interest, and Dummett plausibly comments on it as follows:

|| 58 Cp. Dummett (2001), 4–11, 23–45, and Fara/Salles (2006), 360. 59 At one point, being interviewed, Dummett tries to dispel the prejudgement, common among analytic philosophers, that there is a tension between doing analytical philosophy and having a religious belief. Dummett mentions some renowned fellow accurate philosophers (Peter Geach, Elizabeth Anscombe, Dagfinn Føllesdal) being Catholics as well, but concedes that [I quote] “particularly in America, it’s not just atheism, it’s straightforward materialism that has become almost axiomatic among analytic philosophers”. And shortly afterwards he continues: “I mean if you declare yourself as an atheist or a materialist, you’re just as much giving the conclusion in advance, in fact rather more than if you declare your adherence to a religious faith.” (See Fara/Salles (2006), 361.)

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The story is strikingly egalitarian: each was paid the same. Justice does not consist in giving each what he deserves – Hamlet had a sharp word for that idea: it consists in giving each his due. There are some things which are everybody’s due. The basic conditions that enable someone to live a fully human life are the due of every human being, just in virtue of being human: these are what are nowadays called ‘human rights’.60

There is no doubt that Christian charity is of fundamental importance for Dummett’s strictly egalitarian conception of justice, especially considering that it has been recalled (ever since Kierkegaard) that the obligation to love your neighbour as yourself overcomes the touch of subjectivism inherent in other forms of the Golden Rule, since as integral part of the Christian Great Commandment it is integrated with the obligating love of God; and in keeping with Dummett’s metaphysics in the broadest sense, we can go on to say that the piety to God turns into the committing awe of reality as a whole, the being of which is constituted by God’s knowledge of it, and which thus transcends the domains of any human subjective as well as intersubjective (or indeed objective) rational, sound perspective. As quoted above, Dummett points out that a religion presents its adherents with salient features of a comprehensive conception of reality that enable them to decide how they should deal with what happens by attaching a significance and purpose to it—the human obligation of charity is definitely the most salient of all those features. Dummett cogently connects the idea of charity with the story of the Good Samaritan (contained in the Gospel of Luke): When Christ reiterated the Old Testament commandment to love your neighbour as yourself, his listeners asked him, ‘Who is my neighbour?’. He responded by telling the story of the good Samaritan. Is it not time for both politicians and public to ask the same question?61

One of the reasons for the importance of the didactic story of the good Samaritan is that it reminds us about the duty between all individual humans to assist each other in need (irrespective of their particular religious, cultural, ethnic, political identities). In addition to this, however, the story appeals to politicians

|| 60 Dummett (2001), 26. 61 Dummett (2001), 45. – In referring here to the Old Testament commandment to love your neighbour as yourself, Dummett is likely to be thinking of the following passage from the book of Leviticus, which is particularly relevant when it comes to the duties of a state’s citizenry to refugees and immigrants: “When a foreigner resides among you in your land, do not mistreat them. The foreigner residing among you must be treated as your native-born. Love them as yourself, for you were foreigners in Egypt.” (Leviticus 19,33f. NIV).

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and public to construct a more solidary and just society. This second reading of the story might be traced back to different comments on it by liberation theologians and by Martin Luther King, Jr., respectively. (Michael Dummett, incidentally, met Dr. Martin Luther King in 1956 during the bus boycott in Montgomery, Alabama.) Reflecting on the relationship between the duties of a state and the duties of its individual citizens, Dummett comments: Duties of states derive from duties of the citizens, and citizens have duties not only to their fellow countrymen but to anyone else in the world whose condition of life they can affect.62

On the other hand, Dummett makes it clear that there is a greater distinction between justice as an individual attribute and as an attribute of a state and its society than is the case with any other of the virtues: whereas the courage of a society is the sum of the courage of its members, a just society is one that functions justly as a whole; even if the vast majority of its members act justly, it is not thereby guaranteed that every individual within that society is accorded equal treatment.63 Dummett’s Christian egalitarian view of a state and society that functions justly involves a far-reaching conception of the state’s duties to helpless, destitute migrants who are at its mercy, asking for an authorization to stay, seeking refuge from oppression, persecution or other catastrophic conditions in their country of origin. The Catholic Church, in which Dummett was rooted from his early adulthood (following several years of being an atheist) up to his death, is often perceived as an ultraconservative institution, which is, inter alia, related to its denial of equal treatment for men and women with regard to ordination to priesthood or its condemnation of contraceptives (to which reference has been made above). But sometimes the same Catholic Church appears almost like an avant-garde movement campaigning vigorously for eliminating persistent injustices in those Western democratic states which consider their societies to be the morally most advanced, politically most developed and culturally most highly civilized societies in the entire world. A today still topical and enlightening example of such commitment to reinforce social solidarity and justice in states and their societies is the famed encyclical Pacem in Terris of 1963, issued by Pope John XXIII64, which emphasizes the need for individuals as well as states

|| 62 Fara/Salles (2006), 360. 63 Cp. Dummett (2001), 23–27, 44–45. 64 Angelo Giuseppe Roncalli, John XXIII, affectionately called “il Papa buono”, not only advocated human rights and social equality in theory; long before he became pope, during the

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to be subject to rights and duties, urging all humans to assist each other in social-political aspects by fully respecting human rights such as rights to bodily integrity, to education and to suitable means for the proper development of one’s life. Dummett highlights that in this encyclical, the moral imperative was stated which demands that all states should recognize the principle of open borders, and he goes on clarifying that in it Pope John: said that, ‘when there are just reasons in favour of it’, every human being ‘must be permitted to emigrate to other countries and take up residence there’. The reason he gave for this was that the fact that someone is a citizen of a particular state ‘does not debar him from membership of the human family, or from citizenship of that universal society, the common, worldwide fellowship of men’. (...) Pope John even went some way in the same encyclical towards declaring economic migration a basic human right, saying that ‘among man’s personal rights we must include his right to enter a country in which he hopes to be able to provide more fittingly for himself and his dependants’.65

Dummett makes clear that if the principle of open borders is accepted as the norm, the right of residence will normally not remain the privilege of citizenship. And furthermore he points out that immigrants, once residents in a state, pay taxes and may have some special needs which must not be overlooked so that it appears to be unjust to deny them the right to vote. However, despite arguing convincingly that a denial of the vote to non-citizen residents is indeed unjust, Dummett holds that it would be imprudent to give them this right because that would fuel the existing or potential public hostility to immigrants. (Dummett believes, though, that to give residents at large the right to vote for merely local elections might be practicable.) Yet for Dummett there is no doubt that the state’s duty first of all is to guarantee just living conditions for all its residents, whether citizens or not, by administering transparent, fair laws.66 It needs to be noted at this point that Dummett does not regard it as the duty of a state to assist people who are in need but who neither are its citizens nor are

|| Holocaust period, Roncalli showed strong commitment as apostolic nuncio to save Jewish refugees persecuted by Nazi Germany and its collaborators. His encyclical Pacem in Terris has made an impact, inter alia, on the development of religious freedom, dialogue with other religions, women’s rights as well as international relations. In 2014 Roncalli, John XXIII, was canonized by Pope Francis, not on the basis of a miracle, as would be traditionally required, but based on his exemplary, virtuous life and his merits for the Second Vatican Council, which he conceived and convened and which led to a partial renewal of the Catholic Church. 65 Dummett (2001), 81. 66 Cp. Dummett (2001), 80–84.

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located in its territory (or waiting at its borders, as needs to be added). A state has the duty to protect its citizens when, living or travelling abroad, they get into major difficulties; on the other hand, no state has such a duty towards people who are neither its citizens nor its residents (unless, as should be added here, they have been its residents before and are stateless). Somewhat hesitantly, Dummett adds the reservation that if the human rights of major groups of people in another country, i.e. outside its jurisdiction, are grossly violated, a state is morally still justified in endeavouring to protect them (notwithstanding that they are not its citizens); but he adds that all such interventions by single states, the NATO or even the United Nations have so far proved inefficient and ineffective. Dummett further admits that a state has a limited right to deny entry to intending immigrants. For instance, a state has a mandate for what today is ambiguously called “national security”. This includes a duty to protect its citizens from terrorist assaults and to curb organized crime. Dummett accordingly admits the right of a state to refuse applications for immigration or asylum to persons reasonably suspected of terrorist or criminal activities, whose presence would be dangerous. It might well be asked, though, what a state really is to do about prospective immigrants who are reasonably suspected of having committed e.g. serious violations of human rights in their country of origin but who do not constitute any clearly identifiable threat to the society of the immigration state itself: should they be admitted and be brought before the courts forthwith or might they be rightfully denied the right of residence? In any case, according to Dummett, refused immigrants need to be given the opportunity to appeal against this decision so that the grounds offered for the refusal can be tested by evidence and argument before a tribunal. Moreover Dummett acknowledges that states can legitimately depart from the normal principle of open borders in a few types of exceptional circumstances. For one thing a state has a legitimate reason for restricting immigration in case immigration is reasonably feared to lead to over-population (with the population density in the concerned region of the world in mind). According to Dummett, however, this does not apply to any of the wealthy immigration states at the present time. And for another thing the native population of a state, united by culture, religion or language, has the right not to be submerged (or “swamped”) by immigrants who do not share its culture and thus has the right to ensure that such immigration does not swell to a disproportionately high size. Dummett, however, warns that the concept of submergence is for the most part illegitimately invoked, since it is almost always the case that even a relatively high level of immigration does not threaten the native culture at all: an

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indigenous population which is not oppressed by imperialist invaders can easily and fruitfully assimilate new features.67 Dummett’s motivation to formulate moral-political principles concerning the duties of a state to refugees, to intending immigrants and to its non-citizen residents was grounded in experiences he had during his military service as a young man in British colonies in Southeast Asia as well as, most importantly, before and during the campaign against racism he waged in Britain from the mid-1960s on, in collaboration with his wife. In his “Closing Speech to Lauener Symposium”, Dummett mentions that he loved Malaya when he was based there right after the Second World War.68 This was because when he lived there, different peoples with different cultures and religions (Malays, Chinese, Indian Tamils) appeared to live together in great harmony. Elsewhere, however, Dummett clarifies that the immigration of Chinese and Indians to Malaya had been strongly encouraged by the British colonial authorities because they required additional labour from outside to promote commerce and to work the rubber plantations, respectively. During colonial rule Malays, Chinese and Indians neither merged nor influenced each other, but after independence was declared, conflict between the communities erupted due to their different social and economic needs. Dummett makes clear that this fateful development traces back to the scandalous indifference with which the colonial regime ignored the wishes of the native inhabitants of the territory they governed.69 Dummett always felt a loathing for racial prejudice and it grew steadily when he personally witnessed its social manifestations in Britain and the United States during the 1950s and 1960s. Dummett has been highly critical of the xenophobic, blatantly racialist and anti-immigrant atmosphere which he has encountered since his return to Britain a few years after the Second World War. He states that already then there were extreme racist groups in Britain which demanded to halt immigration of “coloured” people from poor regions of the world. During the 1950s and 1960s Britain largely disengaged from its colonies: decolonization proceeded rapidly and this led to an increase in immigration to Britain from its former colonies, particularly from the Indian sub-continent and the Caribbean region. The successive Conservative and Labour governments reacted to the growing public hostility towards “coloured immigration” by

|| 67 Cp. Dummett (2001), e.g. 14–21, 80–85, and Fara/Salles (2006), 359. 68 See this volume. 69 Cp. Dummett (2001), 15–16, 20.

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exerting an ever-tightening control on such immigration. As Dummett puts it, these measures of the government meant, in turn, that the racist groups got increasingly reassured so that racist feelings in the public were in a way officially encouraged. Dummett is particularly incensed by the manipulative role which most politicians (Conservative and Labour alike) as well as the majority of the media have played in fueling hostility and hatred towards immigrants and refugees in order to gain votes and readers, respectively. Dummett examines this manipulative process, as he has experienced it, pointing out that when the control on immigration could finally not be tightened any further, politicians consciously directed the attention of the public to asylum seekers, deliberately arousing public hostility towards refugees. Thereby the popular demand to reduce immigration of “coloured” economical migrants got focused on a demand to reduce the number of asylum seekers admitted. By identifying the arrival of “coloured” migrants and refugees as a terrible threat to British society while promising to endeavour to halt this kind of influx, politicians have gotten votes—they have been skilled at unrighteously making putrid political capital out of fueling public xenophobia and racism. At the same time the readers of populist media derived pleasure from being given just any reasons for hating and scapegoating poor immigrants and asylum seekers, which also resulted in an increased incidence of violent racist attacks on “coloured” migrants. So politicians and media joined forces in actively encouraging the xenophobic and racist attitudes and violence in the public. Dummett’s searing indictment states: The manipulation of minds for unworthy ends may well be the sin against the Holy Ghost, which shall not be forgiven. Poisoning the mind of the public at large against a group of people who ask our help and deserve our pity – or, indeed, against any whole group – is a worse crime than simply treating its members unjustly. Yet the two main British political parties have colluded in it for years.70

Similar trends have been evident in many other European countries: Dummett sees most West European countries caught in the grip of hysteria about immigration, at the latest from the 1990s on, with many politicians in the centre and on the left succumbing to it and the whole situation being exploited by rapidly emerging right-wing populist parties. A premise of the rhetoric of many western European politicians is that immigration of poor people of unfamiliar ethnic and cultural backgrounds is an intolerable threat for European states which must be resisted as much as possible. A distorted picture is presented in a manipulative way that hordes of “economic” migrants and bogus asylum-seekers are swamp|| 70 Dummett (2001), 45.

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ing the developed states of the Western world, which have to protect their righteous (more probably self-righteous) citizens against such social parasites and cheap labour whatever the cost. Moreover, as discussed above, any fears that the especially robust cultures of wealthy Western states could be submerged by desperate asylum seekers and helpless migrants are altogether without foundation according to Dummett.71 Dummett indicates that [I quote] “by 1968, Britain had become irretrievably identified by the black people living here as a racist society”72, and in 1980 he states: Black people and white now inhabit two different Britains. Most white people are completely unaware of what is common experience for black people; they are oblivious of the conditions we as a nation have created. They know little or nothing of the racial murders and the assaults on black people’s property occurring with ever-increasing frequency; of the cynical dilatoriness of the police when telephoned for help, or their indifference when attacks are reported to them; of the brutality practised by the police themselves against black people, to the extent that it is now hazardous for any black youngster to visit the West End of London; (...) of the effect on young people born in this country of being asked, at hospitals or employment exchanges, to produce their passports to prove their ‘immigration status’ (...) Such things are due to eighteen years of indoctrination in the belief that no calamity could be greater than that one black person who can be got rid of should be allowed to remain. Those, numerous among civil servants, magistrates and the police, who have succumbed to this indoctrination are now, by any objective standards, literally mad; but madness of this sort is very usual amongst us, and so it is not noticed. (...) The feelings of black people do not count; they have to be dealt with, but they are not cared about. (...) Black people feel deprived both of respect and of any ground for hope. They have been very patient. In a society indifferent to and oblivious of their sufferings, they have suffered much injustice and much violence, and they have remained almost wholly law-abiding. No one can be expected to endure for ever, without hope of improvement. If their endurance snaps, things will then either very quickly get much worse, or at last begin to get better.73

At the sight of the fatal trends in the society he lived in, Dummett felt bound to engage in a political struggle, prioritizing his political and humanitarian activities over his creative work in philosophy for most of the second half of the 1960s: “I conceived it my duty to involve myself actively in opposition to the racism which was becoming more and more manifest in English life.”74 The ethos which has determined Dummett’s sense of responsibility for the better|| 71 Cp. Dummett (2001), e.g. ix, xif., 3f., 20f., 39–45, 83f., 152f., and Fara/Salles (2006), 358f. 72 Dummett (1973/1981), x. 73 Dummett (1973/1981), xxiii–xxv. 74 Dummett (1973/1981), x.

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ment of the society to which he belongs is expressed by him thus: “I’ve always thought that intellectuals, if they see a possibility – which usually there isn’t – of making a practical difference, they have the duty to try to do so.”75 Where possible, philosophers ought to undertake responsibility by giving the public the benefit of their analytic capabilities not just in identifying social grievances and rational solution approaches, but in making as well effective practical contributions that force attention upon workable ways of how to improve upon these grievances; this is probably the only convincing way to demonstrate the sincerity, feasibility and credibility of philosophical proposals which define a set of just principles that ought to govern policies on asylum and immigration. Consequently importance needs to be placed on the fact that Dummett (who was dedicated exclusively to theoretical philosophy before and subsequent to the peak of his struggle against racism) has elaborated on his moral and political ideas much later in his life, by way of processing his experiences as a campaigner for racial justice with a practical philosopher’s eye. Up to a certain point Dummett appears to share his pragmatic vision of the intellectuals’ responsibility and social activist role in society with Bertrand Russell. However, whereas Dummett acted out of his religiously inspired strictly egalitarian, deontological morality, Russell’s atheistic or, as the case may be, agnostic as well as partially emotivist incentive often has a streak of plain consequentialism. That clear divergence in Dummett’s and Russell’s reasons for their social-political involvement is well illustrated by Dummett’s following statement: Bertrand Russell, in a television interview given shortly before his death, was asked whether he thought that the political work on which he was engaged at the end of his life was of more importance than the philosophical and mathematical work he had done earlier. He replied, ‘It depends how successful the political work is: if it succeeds, it is of much more importance than the other; but, if it does not, it is just silly.’ One may, all the same, have to undertake something knowing there is only a small chance of success: if someone is faced by a great and manifest evil to the elimination of which he has some chance of making a contribution, the countervailing reasons must be strong to justify his refusing to make it.76

The objection Dummett brings up here to Russell’s slightly ironic statement concerning the attainment of intended effects relates to what is a distinctive feature of Dummett’s moral, humanitarian commitment: political courage, i.e.

|| 75 Fara/Salles (2006), 357. 76 Dummett (1973/1981), x.

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the general willingness he has demonstrated to do what he feels is right in the face of probable disadvantages for himself, which naturally has required a continuous self-critical examination of his values and beliefs. Dummett spent a lot of time on his struggle against racism and the underlying lack of tolerance and solidarity towards non-white immigrants in Britain, along with his wife Ann Dummett, whose [I quote] “involvement in the struggle was even more intense than my own”77. He played an important part in organizing campaigns against the injustice of the immigration laws through the national “Campaign Against Racial Discrimination” (CARD) and later through the “Joint Council for the Welfare of Immigrants”, the latter continuing to flourish today. Dummett was involved in the establishment of both these independent organizations and remained on their executive committees for many years. What is more, Dummett became deeply involved in conducting casework on behalf of intending immigrants threatened with being rejected on entering the country. Dummett describes his antiracialist commitment to social casework as follows: in 1964 I became involved, together with my wife Ann, in the struggle against racism in Britain. For four years I devoted every minute that I could spare to that struggle; I carried out my teaching duties, but abandoned all attempt at creative work in philosophy. (...) It was more or less accidental that I became particularly involved with immigration. At that time the entry clearance system was not in operation: people arrived at Heathrow from the Caribbean or the Indian subcontinent, and were summarily put back on the next returning plane if the immigration officer refused them entry. It was, however, possible to intervene to ‘make representations’ on behalf of anyone refused entry if one could do so before the person was put on the plane. (...) Local community groups from all over the country were sometimes able to intervene when their members were expecting relatives; but the system was very haphazard, and many people were sent home without having anyone to make representations for them. Acting in the name of CARD, I set up an unofficial network of informants at Heathrow who would telephone me, at any hour of the day or night, when they heard of someone’s being refused. I had then to telephone the Chief Immigration Officer, and tell him, when at last I got through, that I wished to make representations; next I had to dash to the airport, find out the background facts and make my representations to the immigration officer. Remarkably, these were often successful; but the system was still haphazard, and very disruptive of my teaching work.78

It is quite evident that Dummett paid a high price for his idealistic, indeed selfsacrificing behaviour in trying to obtain more justice for the victims of the unfair immigration policy of that society towards which he felt loyalty and with which

|| 77 Cp. Dummett (1973/1981), x. 78 Dummett (2001), ix–x.

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he threw in his lot and felt truly at home79, in spite of everything. The fact that the publication of Dummett’s pathbreaking first book about Frege’s philosophy80 was considerably deferred is not just due to his self-critical and creative methods of work, his continuous revisions to his manuscripts contingent on his worries that [I quote] “my ideas remain sufficiently static for all parts of the manuscript to appear satisfactory (...) how can anyone publish what he knows he can improve?”81 To a much greater degree, of course, the delay of almost a decade was due to Michael Dummett’s extraordinary personal commitment to combating racial injustice in Britain. When he finally thought himself justified in returning to completing his book about Frege’s philosophy during the late 1960s, Dummett was faced with a very difficult task:

|| 79 On the natural human sentiment and need of being at home somewhere—not only in some locality one feels familiar with, but in a society as a whole—as well as on dual identities, notably of immigrants and their descendants, who need to be left free and without pressure to find out how to fuse distinct cultural traditions within their own lives, compare Dummett (2001), 17–21. 80 At the end of the preface to the first edition of this book, Dummett notes in a pensive mood: “There is some irony for me in the fact that the man about whose philosophical views I have devoted, over years, a great deal of time to thinking, was, at least at the end of his life, a virulent racist, specifically an anti-semite. This fact is revealed by a fragment of a diary which survives among Frege’s Nachlass (...) When I first read that diary, many years ago, I was deeply shocked, because I had revered Frege as an absolutely rational man, if, perhaps, not a very likeable one. I regret that the editors of Frege’s Nachlass chose to suppress that particular item. From it I learned something about human beings which I should be sorry not to know; perhaps something about Europe, also.” See Dummett (1973/1981), xii. 81 Cp. Dummett (1973/1981), x. – Elsewhere, later on, Dummett makes a statement which stands in remarkable, enlightening tension with his afore-mentioned remark (“how can anyone publish what he knows he can improve?”): “I do not see that anyone is an authority on whether his present views are closer to the truth than his earlier views. Of course, he must think that they are, because otherwise he would not hold his present views; but he cannot maintain that there is any strong objective probability that later views will be an improvement on earlier ones. (...) Of course, there has to be some reasonable chance that one’s later views on a given topic will be an improvement on the earlier ones, or there would be no point in continuing to think about the topic having once expressed oneself upon it. The chance is, however, seldom so high that one should want to suppress the earlier writings; and the behaviour of philosophical authors confirms this. Wittgenstein, for example, did not wish the Tractatus to go out of print: on the contrary, he wanted it reprinted together with the Philosophical Investigations.” See Dummett (1993), xii.

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Returning to a book one had almost completed some years before is a frustrating experience. Some chapters I physically could not find; some seemed badly expressed, or, at least, not how I wanted any longer to express them.82

In addition, there were existential worries: Both my wife and I thought that I had ruined my career by involving myself in the struggle against racism, and therefore for some time not producing any work of my own, but just doing my teaching. I did think I’d thrown away my career, but luckily it didn’t work out that way.83

In the end, actually, Dummett was knighted (in 1999) for his contributions to philosophy as well as to racial justice. I’ve mentioned above that Dummett’s undogmatic Christianity highlights the sanctity and dignity of individual human life; this view of life demands basic respect and empathy for each individual person, manifested in words and deeds. Dummett, on numerous occasions, has shown these capabilities, in contact with particularly vulnerable people such as refugees and also in his capacity as an academic teacher who has been regarded by most (or maybe all) of his students to be an encouraging, challenging and broad-minded mentor. It is evident that Dummett, as a philosophical researcher and teacher, endeavoured to breathe new life into the beaten track of contemporary academic philosophy. There is a wealth of evidence to be found that Dummett’s students much appreciated his detailed attention to their projects and his encouragement of their developing independent, distinct views, which often ran counter to his own position. At this point, three selected testimonies out of numerous testimonies need to be enough to highlight Dummett’s epistemic authority and responsibility as a philosophy mentor as well as to illustrate the gratitude which he received from his students for having undertaken the task of resigning his Senior Research Fellowship at Oxford’s All Souls College in order to accept the appointment as Wykeham Professor of Logic and Philosophy in the University of Oxford—Christopher Peacocke’s, Timothy Williamson’s and Crispin Wright’s observations provide particularly valuable insights here. Peacocke writes: My personal and intellectual debt to Michael Dummett is greater than my debt to any other philosopher. It was my immense good fortune to have my graduate work supervised by Michael, and a formidable experience it was too. The full power of Michael’s abilities were brought to bear—always in a supportive fashion—on my early fumblings in the sub-

|| 82 Dummett (1973/1981), xi. 83 Fara/Salles (2006), 357.

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ject. Perhaps the most important thing I learned from these lengthy supervision sessions was that a properly argued philosophical position can come only from a deep understanding of, and feeling for, the sources of the opposing conception. (...) Michael’s selfless decision to resign a Senior Research Fellowship to shoulder the burdens of the Wykeham Chair is only the most notable event in a lifetime’s devotion to advancing philosophy in Oxford. In the 1970s we used to ask Michael to join our discussion groups when we were thinking about his work, and he always made himself available. (...) You could ask Michael anything, even something which questioned the whole basis of his approach to philosophy; and you would always receive a thoughtful answer.84

The following reminiscences of Michael Dummett’s virtues as a philosophy teacher are from Williamson: Several Chinese philosophers had told me the saying that your teacher is your teacher for life. So I talked about the difference between respecting one’s teachers and agreeing with them. My attitude had been encouraged by Michael Dummett himself. In the drafts of dissertation chapters I gave him, my methods of argument and conclusions were often radically wrong-headed from his perspective, as I well knew. He never said a word about that. He seemed to enjoy entering the alien world of my callow thoughts, mildly raising a question from within that looked minor at first, but as our conversation developed turned out to go to the heart of the matter. Neither of us persuaded the other; indeed, I realized that our disagreement over methodology went even deeper than I had thought. But simultaneously my respect for him became all the greater, for the breadth of his philosophical empathy, his capacity to discern underlying patterns at the most abstract level, and his intellectual seriousness — manifested not least when he burst into laughter on noticing another of the absurdities around us.85

Wright describes in a visual, vivid way the unusual qualities of Dummett as a lecturer who aroused enthusiasm for careful, responsible philosophical work in his students:

|| 84 Peacocke (1997), 1. – Wright gives some more detailed information on the discussion groups thinking about Dummett’s work in the 1970s which Peacocke mentions above: “An older and very special debt is to the other members of a small discussion group, composed of Gareth Evans, John McDowell, Christopher Peacocke and me which met regularly in Oxford in 1976–8. We talked about nothing but realism and the theory of meaning. Evans insisted that detailed minutes were taken at every discussion. The result was an unusual continuity of focus and a sustained debate from which, I think, each of us learned far more about these questions than would have been likely otherwise.” See Wright (1987/1993), xii. 85 See Timothy Williamson, “Teacher for life”, (online, in ‘The Stone’, under ‘Opinionator’), The New York Times, 4 January 2012, http://opinionator.blogs.nytimes.com/2012/01/04/ remembering-michael-dummett/

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Philosophers are chiefly remembered for what they wrote, but my personal memory of Michael Dummett is of a very challenging but very supportive supervisor and a superb lecturer. (...) Dummett liked to use the then newfangled white boards, on which he wrote using variously colored water-soluble pens, erasing by means of a contraption that combined the qualities of a water pistol with a square of blotting paper. He lectured with an extraordinary fluency, hardly ever referring to his notes, at the same time producing highly legible, multicolored text on the board almost as fast as he could speak it, spraying, smudging and erasing as he went along, and smoking incessantly using a cigarette holder which, along with the pens, he lodged between his fingers — we waited for him to put one of the pens in his mouth and take a drag, or inadvertently extinguish his cigarette with a spurt from the eraser, but it never happened. The lectures contained a wealth of detail, both technical and philosophical. Dummett’s erudition was remarkable — his undergraduate background had been in PPE (Philosophy, Politics and Economics), and his grasp of logic and mathematics had been almost entirely self-taught. But my abiding memory is of the passion of his delivery, the determination to get things right, and the sense he radiated of the deep interest of the issues and the huge importance of thinking about them well.86

It was undoubtedly not easy for Dummett to accept what he considered to be his duty as a salaried member of the academic milieu: to face up to very extensive efforts of teaching, supervision of student theses, and further academic duties, instead of being able to focus on his creative philosophical writing and on the other projects that were very important to him personally. The effects continued to be felt long after his retirement: Dummett continued to suffer (like most of his equally illustrious colleagues) from not having enough time and tranquillity for completing the projects closest to his heart.87 It says something about Dummett that at one point in an expert discussion he uttered, “I don’t see why one shouldn’t be interested in more than one thing. I

|| 86 See Crispin Wright, “Master Class”, (online, in ‘The Stone’, under ‘Opinionator’), The New York Times, 4 January 2012, http://opinionator.blogs.nytimes.com/2012/01/04/rememberingmichael-dummett/ 87 In 2005 Dummett writes: “I had in advance envisaged retirement as a period in which I should be my own master: free to work at what I chose, at whatever pace I chose. I found it quite unlike that. I have been retired for thirteen years now, and this is the first time I have felt myself free to do whatever work I like. As is shown by the craters on the Moon and on all other planetary satellites that have been inspected by space probes, those bodies have been bombarded by meteoroids and rocks flying around in space. In my experience, a retired academic is like a planetary satellite: he is bombarded by requests (...) There has not been a moment in those thirteen years when I have not been writing something under pressure.” See Dummett (2006), V.

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think it’s a pity in fact to concentrate just on one thing”88. The versatility and openness of mind, which is stressed here by Michael Dummett accentuates the great relevance of continually finding a balance between the technical refinement of one’s work on specific philosophical themes and the vital receptiveness to other kinds of issues from everyday life which deeply interest and animate us. A one-sided focus on technical aspects in philosophy often leads to stereotyped ideals and skilled yet rabulistic argumentation marked by habitual fear of making a mistake, which hampers courageous, independent choice of subject and thus spoils all the possibilities for development of an epistemically and morally responsible, authentic philosophical work such as Dummett’s. Dummett’s open-mindedness is further underlined by the fact that his selfcritical attitude, the versatility of his interests, his acquisition of varied experience, and his sensitivity to the variations in the cultures of different places and times mirror his continuing efforts to encourage and make use of the rationalitypromoting plurality of intelligible, consistent perspectives (or ways of thinking, arguing, valuation, etc.). It is notable that Dummett’s demand on himself to help vulnerable, disadvantaged people to enjoy the basic conditions for a fully human life has been focused on refugees and migrants from world regions, communities and walks of life which have been haughtily disregarded by selfproclaimed “exceptional”, “developed” Western societies (in the hazy prevailing socio-cultural-political-economic sense of the phrase).

References Dummett, M. (1973/1981) Frege: Philosophy of Language, second edition, London: Duckworth, 1981. First edition 1973. Dummett, M. (1978) Truth and Other Enigmas, Cambridge MA: Harvard University Press. Dummett, M. (1988/1993) Origins of Analytical Philosophy, London: Duckworth, 1993. (Revised version of a series of lectures originally published in Lingua e Stile XXIII (1988), 3–49, 171–210.) Dummett, M. (1993) The Seas of Language, Oxford: Oxford University Press. Dummett, M. (2000) “Sentences and propositions”, in: Teichmann, R., ed., Logic, Cause and Action: Essays in honour of Elizabeth Anscombe, Royal Institute of Philosophy Supplement 46, Cambridge: Cambridge University Press, 2000, 9–23. Dummett, M. (2001) On Immigration and Refugees, London, New York: Routledge.

|| 88 Fara/Salles (2006), 351.

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Dummett, M. (2001/2010) The Nature and Future of Philosophy, New York: Columbia University Press, 2010. (Original version appeared in an Italian translation by Eva Picardi, published by Il Nuovo Melangolo under the title La natura e il futuro della filosofia in 2001.) Dummett, M. (2006) Thought and Reality, Oxford: Oxford University Press. Fara, R./Salles, M. (2006) “An interview with Michael Dummett: from analytical philosophy to voting analysis and beyond”, in: Social Choice and Welfare 27, issue 2 (2006), 347–364. Heck, R., ed., (1997) Language, Thought, and Logic: Essays in Honour of Michael Dummett, Oxford: Oxford University Press. Lauener, H. (1990) “Holism and Naturalized Epistemology Confronted with the Problem of Truth”, in: Barrett, R./Gibson, R., eds., Perspectives on Quine, Oxford, Cambridge MA: Blackwell, 1990, 213–228. Lauener, H. (2002) “Epistemology from a relativistic point of view”, in: LinneweberLammerskitten, H./Mohr, G., eds., Interpretation und Argument, Würzburg: Königshausen & Neumann, 2002, 303–322. Peacocke, C. (1997) “Concepts Without Words”, in: Heck (1997), 1–33. Plato (1974): The Republic, translated by G.M.A. Grube, Indianapolis: Hackett. Wright, C. (1987/1993) Realism, Meaning and Truth, second edition, Oxford, Cambridge MA: Blackwell, 1993. First edition 1987.

Contributors Alex Burri is full professor for Theoretical Philosophy at the University of Erfurt (Germany). His areas of research include philosophy of language and mind, epistemology, and metaphysics. Michael Dummett (1925‒2011) was Emeritus Wykeham Professor of Logic at Oxford University and Emeritus Fellow of New College and of All Souls College, Oxford. Among his many influential books are Frege: Philosophy of Language (1973), Elements of Intuitionism (1977), Truth and Other Enigmas (1978), Voting Procedures (1984), The Logical Basis of Metaphysics (1991), Frege: Philosophy of Mathematics (1991), Origins of Analytical Philosophy (1993), The Seas of Language (1993), On Immigration and Refugees (2001), Truth and the Past (2004), Thought and Reality (2006), and The Nature and Future of Philosophy (2010). Dummett wrote numerous original, pioneering articles, mainly on the philosophy of language, the philosophy of mathematics and logics, and on metaphysics as well as on the history of philosophy, practical philosophy, the theory of voting, theology, and on the game of tarot. Michael Dummett was a leading campaigner for racial equality and justice. He was a member of the British Academy, of the Academia Europaea and of the American Academy of Arts and Sciences. Dummett was knighted in 1999 for services to philosophy and to racial justice and was a recipient of various prizes and awards, including the Lakatos Award, the Rolf Schock Prize, and the Lauener Prize for an Outstanding Oeuvre. Michael Frauchiger, Dr. phil., has been lecturing in the fields of philosophy and methodology of science and the humanities with the University of Applied Sciences of Zurich, the Open University and the University of Bern, and additionally has been an Advanced Research Fellow of the Swiss National Science Foundation. Since its establishment in 2003, he has been the Managing member of the board of trustees of the Lauener Foundation for Analytical Philosophy. His areas of research are epistemology, philosophy of language and logics, metaphysics as well as some interrelated topics in ethics, political philosophy, philosophy of religion and philosophy of psychology and action. Daniel Isaacson is University Lecturer in the Philosophy of Mathematics Emeritus at Oxford University. He arrived in Oxford as a graduate student in 1967. His doctoral thesis was, in part, a response to Dummett’s “Truth”. After positions at the University of Washington in Seattle and the Rockefeller University in New York, he returned to Oxford and was appointed University Lecturer in the Philosophy of Mathematics in 1974, when Dummett resigned his post in the philosophy of mathematics. Except for a year when he was a Visiting Professor of Philosophy at the University of California, Berkeley, he has remained in Oxford, since then. His work in the philosophy of mathematics develops a structuralist understanding of the nature and reality of mathematics. Dale Jacquette (1953‒2016) was ordentlicher Professor für Philosophie, Abteilung Logik und theoretische Philosophie, at Universität Bern, Switzerland, until his death in 2016. He received his AB in Philosophy from Oberlin College in 1975 and his MA and PhD in Philosophy from Brown University in 1981 and 1983. His research areas included philosophical logic, analytic metaphysics, especially ontology and philosophy of mind, and selected figures and periods in the history of philosophy. He is the author of numerous books and articles on logic, metaphys-

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ics, philosophy of mind, and aesthetics. Recently he published Logic and How it Gets That Way (Routledge 2010), Alexius Meinong, The Shepherd of Non-Being (Springer 2015), and Frege: A Philosophical Biography (Cambridge University Press 2018). Eva Picardi is Professor of Philosophy of Language at the University of Bologna. She is the author of the monographs La chimica dei concetti (Bologna, 1994) and Le Teorie del significato (Roma-Bari, 1999). She is also the author of many essays and articles on the philosophy of Dummett, Quine, Davidson, Rorty, and Wittgenstein. Her research interests focus on the history of logic and analytic philosophy, with special attention to the philosophy of Gottlob Frege. She has been visiting fellow at the universities of Erlangen, Helsinki, Bielefeld and All Souls College, Oxford. Dag Prawitz is professor emeritus of theoretical philosophy at Stockholm University. His main areas of research are logic, philosophy of mathematics, and philosophy of language. Within logic he has mostly worked in proof theory, in particular in what he has called general proof theory, concerned with issues such as the concept of proof, identity of proofs, and the structure of proofs; an early result was a normalization theorem for classical and intuitionistic natural deduction. Within philosophy of language he has been interested mainly in theories of meaning that relate the meaning of sentences to how they are established as true. Ian Rumfitt is a Senior Research Fellow of All Souls College, Oxford. He was Professor of Philosophy at the University of Birmingham (2013‒2016) and at Birkbeck College, London (2005‒ 2013), having previously taught at University College, Oxford (1998‒2005) and the University of Michigan at Ann Arbor (1993‒1998). His doctoral studies at Oxford were supervised by Sir Michael Dummett. He works mainly in the philosophy of language, philosophy of logic, and philosophy of mathematics, and has published many papers in those areas. His book The Boundary Stones of Thought: An Essay in the Philosophy of Logic appeared from the Clarendon Press in 2015. Timothy Williamson has been the Wykeham Professor of Logic at Oxford University since 2000. His publications include Identity and Discrimination, Vagueness, Knowledge and its Limits, The Philosophy of Philosophy, Modal Logic as Metaphysics, Tetralogue, and about 200 academic articles on logic, metaphysics, epistemology, and philosophy of language. He is a Fellow of the British Academy and of the Royal Society of Edinburgh, Foreign Honorary Member of the American Academy of Arts and Sciences, Member of the Academia Europaea, Foreign Member of the Norwegian Academy of Science and Letters, and Honorary Member of the Royal Irish Academy. He has held visiting positions at MIT, ANU, Canterbury University (NZ), Princeton, UNAM (Mexico), Chinese University of Hong Kong, University of Michigan, and Yale. Crispin Wright, FBA, FRSE, FAAAS, was a student of Michael Dummett’s and then his colleague at All Souls College, Oxford during the first part of his career. He has published widely on epistemology, the philosophy of mathematics, the metaphysics of truth and meaning, and the philosophies of Wittgenstein and Frege, and is founder and former director of both the Arché research centre at St Andrews and the Northern Institute of Philosophy at Aberdeen. He is currently Professor of Philosophy at New York University and Professor of Philosophical Research at the University of Stirling.

Index Aeschylus 217–218 Allison, H. 204 Anscombe, G. E. M. 3, 15, 25, 231, 245 Anselm (of Canterbury) 210–211 Artemov, S. N. 131, 150 Augustine (of Hippo) 210–211, 217, 223 Austin, J. L. 2, 6, 25 Ball, E. 218–219 Barcan Marcus, R. 153, 166, 174–175 Bergoglio, J. M. (Pope Francis) 226–227, 234 Berkeley, G. 9, 179 Berker, S. 126, 150 Beth, E. W. 136, 150, 156–157 Blackburn, S. W. 130, 150 Boolos, G. 30, 33–34, 58, 64–65, 79 Boyd, R. 178 Bradley, F. H. 89 Brandom, R. 55–56, 58, 129–130, 150, 169, 174 Braun, A. 218 Brouwer, L. E. J. 4–5, 8, 131 Burali-Forti, C. 66, 74–75 Burge, T. 94 Burgess, J. 33 Cantor, G. 30–31, 58, 66–67, 72, 74–77 Carnap, R. 30–31, 38–39, 47, 50, 54–55, 59, 61, 112 Church, A. 95 Copernicus, N. 27 Cozzo, C. 103, 112–113, 121 Cresswell, M. 163, 174 Crossley, J. 7 Darwin, C. R. 27 Davidson, D. 3–4, 19, 36, 53, 59–60, 85, 95– 96, 98–99 Davies, M. 140, 150, 168, 174 Dodgson, C. L. (Lewis Carroll) 26 Dokic, J. 130, 150 Drai, D. 84–85, 94–95 Du Bois-Reymond, E. 202

Dummett, A. 1, 3–4, 6–8, 10, 16, 19, 26, 224, 240 Dummett, M. A. E. 1–23, 29, 31–41, 46–47, 50–56, 59, 61, 63–68, 70–71, 74–76, 79–85, 87–88, 91, 93–94, 96–100, 103– 109, 111–112, 114–115, 121, 123–135, 144, 150–151, 153–160, 164–165, 169–170, 173–174, 177–179, 181, 183, 187–222, 224–246 Eardley-Wilmot, J. E. 7 Einstein, A. 27, 189–190 Engel, P. 130, 150 Euclid 180 Evans, G. 243 Farquharson, R. 20, 23 Field, H. 54, 59 Fine, K. 34, 45, 59, 64, 168, 174 Fitch, F. B. 143–144, 150 Føllesdal, D. 231 Frege, G. 2–3, 5–6, 8–11, 13, 19–21, 23, 25– 26, 29–61, 63–65, 78–79, 81–89, 91– 100, 107–108, 205, 224, 241, 245 Friedman, H. 132–133, 150 Geach, P. 231 Gentzen, G. 105, 114, 118, 122 Gödel, K. 8, 68, 70, 72, 79, 95, 101, 132–133, 150, 181–182 Goldbach, C. 181, 183–184, 191, 194 Goldfarb, W. 44 Gregory, D. 165 Grice, P. 130 Gupta, A. 168, 174 Hale, B. 32, 40, 54, 60 Halmos, P. 3 Hamilton, E. 218 Hammersley, J. 3 Hare, R. 13 Harman, G. H. 130, 150 Heck, R. 33, 44, 53–54, 60, 195–196, 246 Hegel, G. W. F. 203

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Henkin, L. 3 Heyting, A. 8, 116–117, 122, 124–125, 131– 134, 136, 150 Horace 22 Horwich, P. 129, 151, 178 Hughes, G. E. 163, 174 Humberstone, L. 145, 148, 151, 168, 174 Husserl, E. 87 Junia 223 Kant, I. 13, 55, 200–205, 207, 216 Kennedy. R. F. 218 Kierkegaard, S. 232 King, M. Luther 4, 218, 233 Kolmogorov, A. 131, 151 Kreisel, G. 8, 133 Kripke, S. 19, 136, 151, 156–157, 164, 166– 167, 174 Künne, W. 144, 151 Lakatos, I. 124 Langton, R. 204 Las Casas, B. de 224–225 Lauener, H. 189, 196, 201–204, 207, 236, 246 Leibniz, G. W. 187 Lemmon, J. 16, 23 Levine, J. 84–85, 97–98 Lewis, D. 168, 174 Linnebo, Ø. 54, 60 Linsky, B. 168, 174 Loar, B. F. 130, 151 Locke, J. 179 Loewer, B. 130, 151 Luke (the Evangelist) 225, 232 Mann, S. 19, 23 Mares, E. D. 142, 151 Martin-Löf, P. 111, 118, 121 McDowell, J. 243 McFetridge, I. G. 142, 151 McLeod, J. 19, 23 Meinong, A. 205 Milton, J. 19 Montini, G. B. (Pope Paul VI) 226–227 Myhill, J. 3

Nagel, T. 179 Newton, I. 27 Pagin, P. 133, 151 Parmenides 98 Parsons, T. 168, 174 Pascal, B. 206 Paul (the Apostle) 196, 211, 222–223, 225 Peacocke, C. 123–124, 151, 168, 174, 197, 242–243, 246 Peano, G. 30–31, 46, 54, 180, 183 Plato 221, 246 Plutarch 13–14 Prawitz, D. 133 Price, H. 145, 151 Prior, A. N. 128, 146, 151 Putnam, H. 34–37, 58, 61, 178 Quine, W. V. O. 16, 19, 21, 36, 38, 47, 50, 53, 61, 128, 151, 178, 180–181 Rahman, H. 6 Ramsey, F. P. 128, 130, 148, 150–151 Rawls, J. 229–230 Reimer, M. 165 Resnik, M. 33, 61 Ricketts, T. 42, 44, 60–61 Robinson, R. 3 Roncalli, A. G. (Pope John XXIII) 233–234 Ruffino, M. 54, 61 Rumfitt, I. 165, 168, 174 Russell, B. 30–32, 39, 44, 47–48, 50, 52, 64– 68, 70–71, 74, 77–80, 89, 101, 187, 239 Salerno, J. 143, 151 Salmon, N. 165, 175 Sambin, G. 142, 151 Schroeder-Heister, P. 115, 122 Scott, D. 7, 13 Screech, M. 1–2 Shapiro, S. 65, 80 Smiley, T. J. 145, 151 Socrates 221 St. Anthony of Padua 2 St. Felicitas 223–224 St. Francis of Assisi 226

Index | 251

St. Perpetua 223–224 Stalnaker, R. 168, 175 Stephanou, Y. 165 Strawson, P. 13–14 Sullivan, P. 40 Sundholm, G. 118 Tarski, A. 3, 53, 70, 128 Taylor, C. 13 Tennant, N. 56, 61 Thatcher, M. 7, 22 Thiel, C. 53, 59, 60–61 Troelstra, A. S. 133, 151 Van Fraassen, B. 149, 151, 178

Waismann, F. 6, 18 Wang, H. 6, 18 Weiner, J. 44 Weyl, H. 39 Williamson, T. 124–126, 131, 136, 145, 149– 152, 242–243 Wittgenstein, L. 3–5, 15–17, 23, 44, 55, 57, 58, 60–61, 89, 93–94, 180, 224, 241 Wright, C. 21, 30–34, 39–40, 54, 56, 59, 60– 61, 191–192, 194, 242–244, 246 Zalta, E. 168, 174