On Common Solutions to the Liar and the Sorites Sergi Oms Sardans Aquesta tesi doctoral està subjecta a la llicència Reconeixement 3.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento 3.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution 3.0. Spain License. On Common Solutions to the Liar and the Sorites Sergi Oms Sardans Phd in Cognitive Science and Language Supervisor: Jose´ Mart´ınez Fernandez´ Tutor: Manuel Garc´ıa-Carpintero Sanchez-Miguel´ Department of Philosophy Faculty of Philosophy University of Barcelona CONTENTS Abstract iv Resum v Acknowledgements vi 1 Introduction 1 1.1 Truth . 1 1.2 The Liar . 3 1.3 The Formal Liar . 6 1.4 The Sorites ............................ 10 1.5 Forms of the Sorites ....................... 12 1.6 Vagueness . 17 1.7 This Dissertation . 18 2 The notion of Paradox 20 2.1 A Minor Point . 20 2.2 The Traditional Definition . 22 2.3 Arguments and Premises . 25 2.4 The Logical Form . 28 2.5 A First Attempt . 29 2.6 The Notion of Paradox . 31 2.7 Two Consequences . 35 2.8 Two Objections . 36 3 Solving Paradoxes 39 3.1 Solving One Paradox . 39 3.2 Solving more than One Paradox . 45 3.3 Van McGee: Truth as a Vague Notion . 47 i CONTENTS ii 3.3.1 The Sorites in Partially Interpreted Languages . 47 3.3.2 The Liar and vagueness . 49 3.3.3 Solving the Paradoxes . 54 4 Why a Common Solution 56 4.1 Why a Common Solution? . 56 4.2 Graham Priest: Inclosures and Contradictions . 61 4.2.1 The Inclosure Schema . 61 4.2.2 PUS . 63 4.2.3 Paraconsistency . 67 4.2.4 Solving the Paradoxes . 70 5 Jamie Tappenden: Truth-functional and Penumbral Intu- itions 72 5.1 Kripke and SK .......................... 72 5.1.1 Weakness of the Logic . 78 5.1.2 Revenge . 80 5.2 Intuitions and Sharpenings . 81 5.3 Gaps and Supervaluations . 83 5.4 Some Unsuccessful Objections . 86 5.4.1 The Penumbral Intuition . 86 5.4.2 Articulation and Implicatures . 87 5.4.3 The Sorites ........................ 88 5.5 Some Objections . 89 5.6 The Liar . 95 5.7 Solving the paradoxes . 99 6 Paul Horwich: Semantic Epistemicism 101 6.1 Vagueness . 101 6.2 Minimalism and Semantic Epistemicism . 104 6.2.1 Deflationism . 104 6.2.2 Minimalism . 106 6.3 The Generalization Problem . 111 6.3.1 The Generalization Problem and Minimalism . 111 6.3.2 Minimalism and the Liar . 117 6.4 Horwich’s Proposal . 120 6.4.1 The Construction . 120 6.4.2 Kripke and Supervaluations . 123 6.4.3 The Minimal Theory of Truth . 131 6.5 Some Objections . 134 6.6 Solving the Paradoxes . 137 CONTENTS iii 7 Hartry Field: Towards a conditional for the Liar and the Sorites 138 7.1 The Liar, the Sorites and Indeterminacy . 138 7.1.1 Rejecting LEM . 140 7.1.2 Paracomplete Logics . 142 7.2 The Construction . 147 7.2.1 The First Conditional . 147 7.2.2 The Second Conditional . 156 7.2.3 Validity . 158 7.3 Vagueness . 163 7.4 Solving the Paradoxes . 167 8 Conclusions 168 References 172 ABSTRACT In this dissertation I examine some of the most relevant proposals of common solutions to the Liar and the Sorites paradoxes. In order to do that, I present first a definition of what a paradox is so that, with this at hand, I can characterize in detail what should we expect from a common solution to a given collection of paradoxes. Next, I look into the reasons we might have to endorse a common solution to a group of paradoxes and some consequences are drawn with respect to Vann McGee’s and Graham Priest’s proposals to cope with both the Liar and the Sorites paradoxes. In the next chapters, three authors are examined in some detail. First, Jamie Tappenden’s account is judged inappropriate, specially in the case of the Liar paradox. With respect to the Sorites, it is showed to be at least as problematic as Supervaluational approaches. Second, Paul Horwich’s epistemicist proposal is examined with a special focus on the treatment of the Liar paradox. Horwich’s account about how to construct his theory of truth is formalized and critically discussed with the use of a fixed-point construction. In the last chapter, I introduce and discuss some logics based on the work of Hartry Field that use two conditionals in a language with a truth predicate and vague predicates. RESUM En aquesta tesi examino algunes de les propostes m´esimportants de soluci´oco- muna a les paradoxes del Mentider i la Sorites. Per tal de fer-ho, introdueixo, primer, una definici´ode la noci´ode paradoxa i, amb ella, caracteritzo en detall qu`ecal esperar d’una soluci´ocomuna a un grup de paradoxes. A continuaci´o, considero quines s´onles raons que podem tenir per tal d’adoptar una soluci´o comuna a una colecci´ode paradoxes i extrec algunes conclusions respecte les pro- postes de Vann McGee i Graham Priest per fer front al Mentider i la Sorites. En els tres cap´ıtolsseg¨uents, examino tres autors en detall. Primer, rebutjo la proposta de Jamie Tappenden per inapropiada, especialment en el cas del Men- tider. Pel que fa a la Sorites, mostro que la teoria que Tappenden defensa ´es,al menys, tan problem`aticacom les propostes superavaluacionistes. En segon lloc, examino la teoria epistemicista de Paul Horwich, amb especial atenci´oa la seva aplicaci´oal mentider. A trav´esd’una construcci´ode punt fixe, formalitzo i dis- cuteixo cr´ıticament la proposta de Horwich sobre com construir la seva teoria de la veritat. En l’´ultimcap´ıtol,introdueixo i discuteixo algunes l`ogiques,basades en les propostes de Hartry Field, que usen dos condicionals en llenguatges amb un predicat de veritat i predicats vagues. ACKNOWLEDGEMENTS First of all, I would like to express my deepest gratitude to my supervisor, Jos´eMart´ınez,for his excellent guidance, support, patience and for providing me with the best atmosphere for doing research. I would also like to express my gratitude to the members of my thesis committee: Dan L´opez de Sa, Gabriel Uzquiano and Elia Zardini. I would like to thank them for accepting to critically read my thesis. I would also like to express my sincere gratitude to my tutor, Manuel Garc´ıa-Carpintero. I had the opportunity to begin my PhD thanks to a FPU scholarship from the Spanish Ministry of Education and Science. I also enjoyed two visiting re- search grants from the same institution; the first at the University of Sheffield and the second at the New York University. This dissertation has been pos- sible thanks to a number of research projects: Reference, Self-Reference and Empirical Data (FFI2011-25626) from the Spanish Ministry of Education; PERSP—Philosophy of Perspectival Thoughts and Facts (CSD2009-00056) from the Spanish Ministry of Education and Science; Localism and Global- ism in Logics and Semantics (FFI2015-70707-P) from the Spanish Ministry of Economy and Competitiveness. I would like to thank all the people that, at one moment or another, were at Logos during the time I was working on this dissertation, specially Dan L´opez de Sa, Manuel Mart´ınez,Manuel P´erezOtero, Sven Rosenkranz and Elia Zardini for helping me with invaluable comments to earlier drafts of this thesis. At the University of Barcelona, I would also like to express my gratitude to Calixto Badesa and Joost Joosten for their comments and support. During these years at Barcelona I have made many friends, who helped me in many ways: Marc Artiga, Joan Bertran, Mar´ıaEsteban, Marta Jorba, Paco Murcia, Chiara Panizza,Umberto Rivieccio, Gon¸caloSantos, Luz Sabina, Pilar Terr´esand, of course, Oscar Cabaco, whose friendliness and help has been invaluable. At Sheffield, I would like to thank, first, my supervisor, Rosanna Keefe, who helped me and provided me with extremely helpful discussions. Also, at Sheffield, I want to thank Kathy Puddifoot, Kate Harrington, Paniel Reyes, Bernardo Aguilera, Inga Vermeulen, Julien Murzi and Jonathan Payne. I would like to specially thank, at Barcelona, London Ontario, Sheffield and everywhere, Cristina Roadevin. At New York, I am grateful to my supervisor, Hartry Field, for sharing with me some of his time and providing me with extremely helpful discus- sions. I would also like to thank Paul Horwich, for his patience and help, and David Samsundar, for his friendliness and warm farewell. Per acabar, vull agrair a tota la meva fam´ılial’esfor¸cque, durant aquests anys, han fet per ajudar-me a acabar aquest treball. En especial, a l’Oriol Oms, per animar-me a acabar; al Josep Maria Sardans, per les seves preguntes i, sobretot, als meus pares, el Ramon i la Magda, per la seva paci`encia. Deixo, tamb´e,un record i tot el meu agra¨ıment pel Pepito i la Llu¨ısa. Per`oel meu agra¨ıment m´esprofund ´esper la Cristina, sense la qual aquesta tesi no hauria estat possible, i pel Guillem i l’Al´ıcia,sense els quals aquesta tesi s’hauria acabat tres anys abans. CHAPTER ONE INTRODUCTION 1.1 Truth The investigation on the notion of truth is one of the main problems in philosophy and also one of the oldest ones. At least since Pilate asked ‘what is truth?’ (John, 18:38), human beings have tried to look into the nature of truth. As Russell (1950) sensibly stated, looking at the question ‘What is truth?’ —so as to obtain a general definition of truth— is not the same as asking ‘Which beliefs, or sentences, are true?’ —so as to obtain a general criterion for truth.