Type-free Truth Inaugural-Dissertation zur Erlangung des Doktorgrades der Philosophie an der Ludwig-Maximilians-Universit¨atM¨unchen vorgelegt von Thomas Schindler aus M¨unchen 2015 Erstgutachter: Prof. Dr. Dr. Hannes Leitgeb Zweitgutachter: Prof. Dr. Volker Halbach Datum der m¨undlichen Pr¨ufung:29.01.2015 Abstract This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski's and Kripke's theories of truth, and then discuss the prospects of classical type-free truth. In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke's theory of truth. Strategies in these games can be interpreted as reference-graphs (or dependency-graphs) of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine's idea of stratification. This will yield classical disquotational theories that interpret full − second-order arithmetic without set parameters, Z2 (chapter 9). We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory. Contents Acknowledgements . viii Co- and single-authored publications . .x I. Truth as a logico-mathematical notion 1 1. Introduction 3 1.1. Deflationism . .4 1.2. Virtual classes . .9 1.3. Overview of the thesis . 12 2. Technical preliminaries 15 2.1. Peano arithmetic . 15 2.2. Weak and strong diagonalization . 18 3. Escaping the liar 23 3.1. Typing. Tarski's hierarchy . 23 3.2. Non-classical solutions. Kripke fixed points . 27 4. Classical untyped truth 35 4.1. Expressing infinite conjunctions . 35 4.1.1. What does it take to express an infinite conjunction? . 36 4.1.2. Some observations . 42 4.2. More on infinite conjunctions . 48 4.3. Reflecting on classical truth . 52 II. Grounded truth 57 5. A graph-theoretic analysis of the semantic paradoxes 59 5.1. Reference and paradox . 60 5.2. Semantic dependence. Leitgeb (2005) . 63 5.3. Sensitivity-graphs . 67 v Contents 5.4. Kripke-games and reference-graphs . 73 5.4.1. The grounding game . 73 5.4.2. The verification game . 76 5.4.3. Kripke-paradoxicality . 80 6. Axioms for grounded truth 83 6.1. KF and VF . 83 6.2. Simultaneous axiomatizations of groundedness and truth . 87 6.3. Proof-theoretic analysis . 92 6.4. Comparison . 97 III. Truth, Definability, and Comprehension 99 7. Truth-sets and second-order structures 101 7.1. The Translation Lemma . 101 7.2. Complexity of fixed-point theories . 105 7.3. Positive disquotation . 110 8. Hyperarithmetic sets and ramified truth 115 9. Stratified truth 121 9.1. Zigzag theories . 121 9.2. Some systems of stratified truth . 124 9.3. Consistency . 131 9.4. Comprehension with parameters . 134 IV. Appendix 137 10.Ordinal notations 139 10.1. Notation for ordinals below 0 ...................... 140 10.2. Notation for ordinals below Γ0 ...................... 141 10.3. Kleene's O ................................. 142 11.Recursion Theory 143 11.1. Indices . 143 11.2. The Arithmetical and the Analytical Hierarchy . 143 11.3. Hyperarithmetical sets . 144 11.4. The Ramified Analytical Hierarchy . 145 11.5. Subsystems of second-order arithmetic . 146 vi Contents 12.Graph theory 149 13.Bibliography 151 Index 158 vii Contents Acknowledgements Writing this thesis would have been impossible without the help and support of many people. First, I want to thank my supervisor Hannes Leitgeb for all his support and encouragement during the writing of this thesis. His many comments and advices on my papers and talks were an invaluable help to me. His paper \What truth depends on" [55] had a great impact on me, as witnessed by Part II of my thesis. A special thanks goes to Jeff Ketland, who supervised my work before I became a doctoral fellow at the MCMP. Before he left to Oxford, I had almost weekly meetings with him in which I could probe my ideas and would always get valuable feedback. Volker Halbach was kind enough to read some of my papers and provide helpful comments on it. My conversations with him in Amsterdam and Chiemsee sharpened my views on some of the topics that I deal with in this thesis. Almost everything I know about axiomatic truth theories derives from my reading of his book [38]. I am very happy that he agreed to be the external examiner of this thesis. I am deeply indebted to my friend Timo Beringer, with whom I had countless conversations about logic and philosophy in the last 5 years or so. He proof-read all of my papers and discussed all my ideas with me. He sharpened my views on many topics and was always very helpful when I had certain technical questions. My joint paper with him, \Reference-graphs, games for truth, and paradox" is one of the papers that I am most proud of. I am obliged to Lavinia Picollo, with whom I had many discussions about meta- mathematics and deflationism in the last year. She certainly influenced my thinking about these topics very much. I am very proud of our joint paper “Deflationism and the purpose of truth". Stanislav Speranski carefully read chapters 6-8 of this thesis and provided helpful suggestions. We had several meetings where we discussed the contents of chapter 7, which allowed me to improve this chapter significantly. I thank my colleagues at the MCMP for providing such a stimulating enviro- ment. A special thanks goes to Catrin Campbell-Moore, Johannes Stern and Mar- tin Fischer with whom I had several discussions about the topics in this thesis. A thanks goes also out to my office colleagues, Johannes Korbmacher, Hans-Christoph Kotzsch, Albert Anglberger and Thomas Meier. I thank Eduardo Barrio and the logic group of Buenos Aires, who made my stay in Buenos Aires a real pleasure. I thank the DAAD for supporting that trip financially. Thanks to Jonas Raab and Christian Meyer for listening to all of my stories, and thanks to Jonas for proof-reading so many of my papers. I also want to take the opportunity to thank my teachers in logic, Godehard Link, Marek Polanski and Karl-Georg Niebergall. I have presented my work at several workshops and conferences, including Vienna, viii Contents Chongching, Amsterdam, Bristol, Oslo, Canterbury, Chiemsee, Buenos Aires and, of course, Munich. I thank the attendees of these conferences for their feedback. In particular, I thank Leon Horsten, Graham Leigh, Toby Meadows, Julien Murzi, David Ripley, J¨onneSpeck, and Philip Welch. I am very grateful to Stephan Hartmann and Dieter Donder for having agreed to be on the committee of my defense. This work was generously supported by the Alexander von Humboldt foundation and the Munich Center for Mathematical Philosophy. I could not have written this thesis without their financial support. Last but certainly not least I want to thank my family, Mama, Armin, Tanja, Tina and Toni for all their support during the last years. ix Contents Co- and single-authored publications Sections 4.1-4.2 and parts of section 1.1 were jointly written with Lavinia Picollo from the University of Buenos Aires. The material is part of our paper [65], which is currently in preparation. Of course, I take full responsibility for the way the ma- terial is presented here. Sections 5.3-5.4.3 were jointly written with Timo Beringer from the LMU Munich. The material is part of our paper [7], which is currently in preparation. The paper contains a lot of additional material that has not been included in this thesis. Of course, I take full responsibility for the way the material is presented here. All other parts of this thesis were written solely by myself. All proofs in this book were carried out solely by myself, except those that resulted from joint work with Ms. Picollo or Mr. Beringer, as indicated above. A few theorems that I have proved are implicit in the work of others, and I have indicated the relevant paper in such cases. All theorems in this book that were not proved by myself are stated without proofs, and I have indicated the owner of the result in brackets.