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The Metaphysical Commitments of Logic Thomas Nicolaas Pieter Albert Brouwer University of Leeds School of Philosophy, Religion and History of Science Submitted in accordance with the requirements for the degree of Doctor of Philosophy June 2013 i ii The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. 2013 The University of Leeds and Thomas Nicolaas Pieter Albert Brouwer The right of Thomas Nicolaas Albert Brouwer to be identified as the Author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988. iii iv Acknowledgements First and foremost, I want to thank my supervisors, Robbie Williams, Jason Turner and Robin Le Poidevin. Without their guidance, this thesis would be nothing like it is, though all of its shortcomings are of course my fault. Their commitment, interest and insight have made my project a pleasure to complete. Apart from my supervisors, I am also indebted to the members of the Centre for Metaphysics and Mind for philosophical conversation both on topics related to my thesis and other subjects. Special mention goes to Ross Cameron, Andy McGonigal and Paolo Santorio in this regard. Appreciation goes to Aaron Meskin for his persistent efforts to get me to give a talk on aesthetics, which I finally did. I want to especially thank John Divers for his support. The postgraduate community in Leeds is incomparable. I want to thank all of my fellow postgrads for helping to make Leeds a great place to do a PhD, and my fellow philosophers in particular for endless discussion on all philosophical topics. For listening to me trying out arguments and views and helping me out on many occasions, I want to particularly thank Jon Banks, Jordan Bartol, Callum Sida-Murray and Carl Warom. Fond thanks also go to Sarah ‘plain old’ Adams, Dani Adams, Richard Caves, Dave Cornell, Fran Fairbairn, Mirja Holst, Wouter Kalf, Alice Kay, Kerry McKenzie, Henry Merivale, Robb Muckle, Alex Oldemeier, Andy Peet, Will Perry, Rob Pezet and Paul Ramshaw. Most of all, I want to thank Mike Bench-Capon, who is not only clever himself but the cause of cleverness in others. And special mention goes to Efram Sera-Shriar, who did much to make me feel at home when I first arrived in Leeds. I also want to express my gratitude to the administrative staff at the School, for always being ready to help out a confused postgrad, and always there for a friendly chat. In particular I want to thank Sarah Cason for always cheering me up. In a category of her own, I want to thank Akiko Frischhut for good philosophy, good food and good company, and for making Geneva one more home away from v home. Finally, I want to thank my family back home in the Netherlands, who have provided so much moral support and continue to do so. I could not have done it without them. I hope they will forgive me for being so far away for such a long time. vi Abstract This thesis is about the metaphysics of logic. I argue against a view I refer to as ‘logical realism’. This is the view that the logical constants represent a particular kind of metaphysical structure, which I dub ‘logico-metaphysical structure’. I argue instead for a more metaphysically lightweight view of logic which I dub ‘logical expressivism’. In the first part of this thesis (Chapters I and II) I argue against a number of arguments that Theodore Sider has given for logical realism. In Chapter I, I present an argument of his to the effect that logico-metaphysical structure provides the only good explanation of the semantic determinacy of the logical constants. I argue that other explanations are possible. In Chapter II, I present another argument of his to the effect that logico-metaphysical structure is something that comes along with ontological realism: the view that there is a non-language-relative fact of the matter about what exists. I argue that the connection between logical and ontological realism is not as close as Sider makes it out to be. In the second part of this thesis (Chapters III – V) I set out a positive view of the logical constants that can explain both why their meanings are semantically determinate, and why they form part of our vocabulary. On that view, the primary bearers of logical structure are propositional attitudes, and the logical constants are in our language as vehicles for the expression of logically complex propositional attitudes. In Chapter III, I set out an expressivist theory of propositional logic. In Chapter IV, I use this theory to explain how the logical connectives end up having determinate meanings. In Chapter V, I extend the expressivist theory to predicate logic. vii Table of Contents Introduction 1 I. The Metaphysics of Logic 1 II. The Metaphysical Commitments of Logic 2 III. Logical Realism 4 IV. Sider's Arguments for Logical Realism 6 V. The ‘Quinean’ Argument from Ideological Commitment 7 VI. The Structure of the Thesis 11 Chapter I: The Argument from Semantic Determinacy 15 0. Introduction 15 I. The Indeterminacy of Meaning 17 I.I. The Kripkenstein Problem 18 II. Interpretation and Meaning 24 II.I. Lewis’s Eligibility Response 25 III. A Kripkenstein Argument for Logical Constants 30 III.I. Kripkenstein Connectives 31 III.II. Kripkenstein Quantifiers 34 VI. Eligibility Responses 36 V. Conclusion 39 Chapter II: Quantifiers and Metaontology 41 0. Introduction 41 I. Quantifiers and the Realism-Deflationism Debate 43 viii II. The Meaning of “” 44 III. The Structure of the Arguments 47 IV. Hirsch's Argument: a Closer Look 50 V. Eligibility and Aprioricity 54 VI. The Sider-Hirsch Dialectic 60 VI.I. What Ought we Disagree About? 61 VI.II. Verbal Debates about Real Facts 63 VII. Semantics and Metasemantics 64 VII.I. Local and Global Quantifier Meanings 65 VII.II. Domain Assignments 69 VIII. Conclusion 70 IX. Appendix: Unrestricted Quantification 71 Chapter III: Logical Expressivism 77 0. Introduction 78 0.I. Some Preliminaries and Caveats 81 I. Propositional Attitudes, Simple and Complex 84 I.I. Rejection 84 I.I.II. A Spectrum of Acceptance and Rejection 87 I.II. Complex Propositional Attitudes 89 I.III. Why Complex Attitudes Are Propositional Attitudes 91 I.IV. Generating Greater Complexity 93 II. Sequentese 95 II.I. The Symbol “►” 96 II.II. Some Connectives Replaced 97 II.III. Embedded Sequents 100 III. Pinning Down Sequentese 102 III.I. Syntax of Sequentese 103 III.II. Translating Sequentese 104 ix III.II.I. Translating Sequentese: Sequentese to English 105 III.II.II. Translating Sequentese: from PL to Sequentese 108 III.II.III. Translating Sequentese: from Sequentese to PL 111 III.II.IV. Translating Sequentese: Recoverability 113 IV. Reasoning in Sequentese 115 IV.I. Motivating the Inference Rules 118 IV.II. Motivating the Structural Rules 124 IV.III. Embedding and Extraction Rules for Sequentese 129 IV.IV. Is Sequentese Classical? 132 IV.IV.I. Classical Derivability Implies Sequentese Derivability 133 IV.IV.II. Sequentese Derivability Implies Classical Derivability 136 V. Conclusion 141 Chapter IV: Radical indeterminacy revisited 145 0. Introduction 145 I. Chapter I Revisited 147 II. Metasemantics and Indeterminacy 149 II.I. A Lewisian Approach 149 II.II. Radical Interpretation 150 III. Lewis and Naturalness 155 IV. The Metasemantics of the Logical Connectives 158 IV.I. Psychological Assumptions 159 IV.II. Assumptions about Language 162 IV.III. Pinning Down the Constants 163 V. Reversing the Polarity on the Reference Magnets 165 VI. Conclusion 169 x Chapter V: Expressivism for Quantifiers 170 0.Introduction 170 I. Strategies 172 I.I. Tractarian Quantification 173 I.II. Ramsey-Mill Quantification 175 I.II.I. Predication 176 I.II.II. Quantification 179 I.III. Correctness 181 I.IV. On the Way to Predicate Sequentese 183 II. Sequentese with Predicates: Syntax 185 II.I. Predication 185 II.II. Open and Closed Sequents 186 III. Sequentese with Predicates: Translation Manual 188 IV.I Sequentese with Predicates: Inference Rules 190 IV.II. Justifying the Inference Rules 197 V. Sequentese with Predicates is Standard Predicate Logic 198 Conclusion 201 I. Explanatory Tasks 201 II. Logical Expressivism: Challenges and Applications 203 II.I. Challenges 204 II.II. Applications 205 Bibliography 211 xi xii Introduction I. The Metaphysics of Logic 1 II. The Metaphysical Commitments of Logic 2 III. Logical Realism 4 IV. Sider's Arguments for Logical Realism 6 V. The ‘Quinean’ Argument from Ideological Commitment 7 VI. The Structure of the Thesis 11 I. The Metaphysics of Logic This thesis is about the metaphysics of logic. It investigates what we can conclude from the fact that we have the logical constants in our language, that we use them all the time, and that they are so useful to us. Does it tell us anything about the world we live in? Does the world itself have some sort of logical structure to it? Are our theories of logic in some sense theories about the world? I defend a negative answer to these questions. Our logical terms do not reflect a special feature of reality. Logic is, in the end, all about us. We, of course, are part of the world. But there is also a sense in which we’re not part of it. We can stand apart and look at the world and consider it as a thing separate from us. Insofar as we are creatures that can do that, logic belongs with us and not with the world, in the grand scheme of things.

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