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Intuitionism and Logical Revision UNIVERSITY OF SHEFFIELD Intuitionism and Logical Revision Julien Murzi A thesis submitted in partial fulfillment for the degree of Doctor of Philosophy Faculty of Arts and Humanities Department of Philosophy Sheffield October 2010 To my parents. To Barbara. To Bob and Dam. Abstract The topic of this thesis is logical revision: should we revise the canons of classical reasoning in favour of a weaker logic, such as intuitionistic logic? In the first part of the thesis, I consider two metaphysical arguments against the classical Law of Excluded Middle-arguments whose main premise is the metaphysical claim that truth is knowable. I argue that the first argument, the Basic Revisionary Argument, validates a parallel argument for a conclusion that is unwelcome to classicists and intuitionists alike: that the dual of the Law of Excluded Middle, the Law of Non-Contradiction, is either unknown, or both known and not known to be true. As for the second argument, the Paradox of Knowability, I offer new reasons for thinking that adopting intuitionistic logic does not go to the heart of the matter. In the second part of the thesis, I motivate an inferentialist framework for assessing competing logics-one on which the meaning of the logical vocabulary is determined by the rules for its correct use. I defend the inferentialist account of understanding from the contention that it is inadequate in principle, and I offer reasons for thinking that the inferentialist approach to logic can help model- theorists and proof-theorists alike justify their logical choices. I then scrutinize the main meaning-theoretic principles on which the inferentialist approach to logic rests: the requirements of harmony and separability. I show that these principles are motivated by the assumption that inference rules are complete, and that the kind of completeness that is necessary for imposing separability is strictly stronger than the completeness needed for requiring harmony. This allows me to reconcile the inferentialist assumption that inference rules are complete with the inherent incompleteness of higher-order logics-an apparent tension that has sometimes been thought to undermine the entire inferentialist project. I finally turn to the question whether the inferentialist framework is inhos- pitable in principle to classical logical principles. I compare three different regi- mentations of classical logic: two old, the multiple-conclusions and the bilateralist ones, and one new. Each of them satisfies the requirements of harmony and sepa- rability, but each of them also invokes structural principles that are not accepted by the intuitionist logician. I offer reasons for dismissing multiple-conclusions and bilateralist formalizations of logic, and I argue that we can nevertheless be in harmony with classical logic, if we are prepared to adopt classical rules for disjunction, and if we are willing to treat absurdity as a logical punctuation sign. Acknowledgements I am greatly indebted to my supervisors in Sheffield: Bob Hale and Dominic Gregory. Their help and support has been invaluable. They have always provided extremely helpful feedback on my work, and they have always responded, some- times with a great deal of olympic patience, to my many queries. Bob's presence has been a constant encouragement for me. His constant and unfailing support, both philosophical and human, is behind every single bit of my work in Sheffield. Dom has acted as a second primary supervisor, always managing to be helpful, present, and sharp. I warmly thank them both. This thesis is for them. Many thanks, too, to my previous supervisor in Rome, Cesare Cozzo. His analytical skills, as well as his philosophical rigour, have always been an example for me. Many thanks to Carrie Jenkins, who supervised my work at the time of my first visit to Arche, St Andrews, in the Spring of 2006, to Martin Smith, under whose keen direction I worked during my second visit to Arche, in the Spring of 2008, and to Stephen Read, with whom I had the pleasure to work during the Spring of 2008 and during the academic year 2009/10. Very special thanks are also due to Rosanna Keefe, Ian Rumfitt, and Stewart Shapiro for their very helpful feedback on my work, and for their constant encouragement; to Tim Williamson, for valuable correspondence and discussion, and for the comments he gave on my joint paper "The Paradox of Idealisation" at the Oxford Graduate Conference in November 2007; to Crispin Wright, whose comments at a seminar in St Andrews inJuly 2006 very much contributed to what I can refer to as my realistic tum; to Gabriele Usberti, for his very helpful and generous feedback and support, and to Frederik Sljemberg, who has given so many helpful comments on my early work. Many of the subjects presented herein have been object of exceedingly stimu- lating discussions with many friends and colleagues: IC Beall, Colin Caret, Hartry Field, Salvatore Florio, Ole Thomassen Hjortland, Luca Incurvati, Marcus Ross- berg, and Florian Steinberger. I am most grateful to all of them. I am also very iv much indebted to Simona Aimar, Catarina Dutilh Novaes, Richard Dietz, Dick de Jongh, John Divers, Michael Gabbay, Joe Salerno, Stewart Shapiro, and Elia Zardini, who have been kind enough to read and comment on my work, and to Volker Halbach, Lloyd Humberstone, Dag Prawitz, Greg Restall, and Neil Tennant for very helpful email exchanges. I am very grateful for the feedback I received from audiences at conferences and talks in Rome, Lisbon, Nancy, St Andrews, Aix-en-Provence, Sheffield, Geneva, San Francisco, Leeds, Bristol, Norwich, St Andrews, Oxford, Baltimore, Cambridge, Canterbury, Hejnice, London, Amsterdam, Dubrovnik, Siena and Aberdeen. Roy Cook, Marcus Rossberg, Tim Williamson, Joe Salerno, Luca Incurvati and Peter Smith, and Dick de [ongh were the commentators in San Francisco, St Andrews, Oxford, Baltimore, Cambridge, and Amsterdam respectively. Many thanks to all of them. Very special thanks are due to the Department of Philosophy in Sheffield: for giving me confidence, for providing such a friendly and stimulating environment, and for letting me teach a course of my own on some of the topics of this thesis; a challenging, but truly amazing, experience. I should also thank my students in Sheffield, both graduate and undergraduates, from whom I have learned a great deal. Special thanks are also due to the University of St Andrews, and to its research center, Arche, which provided an exceedingly stimulating environment for my work, in a very friendly and hospitable atmosphere. I also wish to thank my philosopher friends and colleagues in Rome, Sheffield, St Andrews, and around the world. To mention but a very few: Aaron Bogart, Theodora Achourioti, Paolo Bonardi, Pablo Cobreiros, Colin Caret, Mike De, Roy Dyckhoff, Matteo Falomi, Graeme A. Forbes, Kate Harrington, Andrea Iacona, Frederique Janssen-Lauret, Peter Milne, Sebastiano Moruzzi, Sergi Oms, Francesco Orsi, Jonathan Payne, Kathy Puddifoot, Sara Protasi, Cristina Roadevin, Sam Roberts, Gil Sagi, Valerio Salvi, Andrea Sauchelli, Daniele Sgaravatti, Andreas Stokke, Margot Strohminger, and Richard Woodward. I very gratefully acknowledge the very generous financial support of the Uni- versity of Rome "La Sapienza", of the Royal Institute of Philosophy, of the Univer- sity of Sheffield, of the Analysis Trust, and of the Petrie Watson Exhibition Fund. A final very big thank to Barbara and to my parents, whose love, understanding, and support made this thesis possible. Julien Murzi Sheffield, October 2010 Contents Abstract ii Acknowledgements iii Contents v 1 Introduction 1 1.1 From metaphysics to logic . .. 2 1.1.1 The Basic Revisionary Argument. .. 3 1.1.2 The Paradox of Knowability. .. 4 1.2 Inferentialism and logical revision . .. 5 1.2.1 Logical inferentialism . .. 6 1.2.2 Proof-theoretic harmony. .. 7 1.2.3 Inferentialism and separability . .. 8 1.2.4 Classical inferentialism 9 1.2.5 Conclusions............................ 10 I From metaphysics to logic 11 2 The BasicRevisionary Argument 13 2.1 Dummett's challenges . .. 15 2.1.1 Some definitions . .. 15 2.1.2 Dummett's case against Semantic Realism 18 2.2 Tennant and Salerno on logical revision . .. 27 2.2.1 Tennant on manifestation and logical revision . " 27 2.2.2 Salerno on logical revision. .. 33 2.3 The Basic Revisionary Argument . .. 35 vi CONTENTS 2.3.1 Brouwer's line of argument . 35 2.3.2 From Brouwer to Wright. 37 2.3.3 Introducing the Basic Revisionary Argument . 41 2.4 How basic is the Basic Revisionary Argument? . 47 2.4.1 How Basic is the Basic Revisionary Argument? .. 48 2.4.2 Objections and replies . 49 2.4.3 Wright on epistemic modesty . 51 2.4.4 Intuitionism and Dialetheism . 55 2.4.5 How Basic is the Single Premise Argument? 56 2.5 Conclusions . 58 3 The Paradox of Knowability 61 3.1 The Church-Fitch argument 62 3.1.1 The proof . 63 3.1.2 Possible ways out: a shopper's guide 65 3.2 Victor's anti-realism . 67 3.2.1 Dummett on anti-realism and basic statements . 67 3.2.2 A weaker Manifestation Requirement 68 3.2.3 Williamson on basic statements . 68 3.2.4 Basic statements and logical revision. 70 3.2.5 Summing up . 71 3.3 A seemingly trivial way out . 71 3.3.1 Proof-types and-proof tokens 72 3.3.2 Proofs as Aristotelian types . 73 3.3.3 Truth and provability .... 74 3.4 The standard intuitionistic response 76 3.4.1 Dummett's new line ..... 76 3.4.2 The Standard Argument . 77 3.4.3 Knowability and bivalence . 79 3.5 The Paradox of Idealisation . 80 3.5.1 Hierarchical treatments ... 81 3.5.2 Strict Finitism and the Paradox of Idealisation 82 3.5.3 Objections and replies .
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