Topics in Philosophical Logic The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Litland, Jon. 2012. Topics in Philosophical Logic. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9527318 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA © Jon Litland All rights reserved. Warren Goldfarb Jon Litland Topics in Philosophical Logic Abstract In “Proof-Theoretic Justification of Logic”, building on work by Dummett and Prawitz, I show how to construct use-based meaning-theories for the logical constants. The assertability-conditional meaning-theory takes the meaning of the logical constants to be given by their introduction rules; the consequence-conditional meaning-theory takes the meaning of the log- ical constants to be given by their elimination rules. I then consider the question: given a set of introduction (elimination) rules , what are the R strongest elimination (introduction) rules that are validated by an assertabil- ity (consequence) conditional meaning-theory based on ? I prove that the R intuitionistic introduction (elimination) rules are the strongest rules that are validated by the intuitionistic elimination (introduction) rules. I then prove that intuitionistic logic is the strongest logic that can be given either an assertability-conditional or consequence-conditional meaning-theory. In “Grounding Grounding” I discuss the notion of grounding. My discus- sion revolves around the problem of iterated grounding-claims. Suppose that ∆ grounds φ; what grounds that ∆ grounds that φ? I argue that unless we can get a satisfactory answer to this question the notion of grounding will be useless. I discuss and reject some proposed accounts of iterated grounding claims. I then develop a new way of expressing grounding, propose an account of iterated grounding-claims and show how we can develop logics for grounding. In “Is the Vagueness Argument Valid?” I argue that the Vagueness Ar- iii Warren Goldfarb Jon Litland gument in favor of unrestricted composition isn’t valid. However, if the premisses of the argument are true and the conclusion false, mereological facts fail to supervene on non-mereological facts. I argue that this failure of supervenience is an artifact of the interplay between the necessity and determinacy operators and that it does not mean that mereological facts fail to depend on non-mereological facts. I sketch a deflationary view of ontology to establish this. iv Contents I Proof-theoretic Justification of Logic 1 1 Intuitionistic Rules 2 1.1 Introduction . 2 1.2 Philosophical Remarks . 6 1.3 Verificationist Meaning-Theory . 9 1.3.1 Boundary Rules . 16 1.3.2 Another Approach . 22 1.4 Pragmatist Meaning-Theories . 23 1.4.1 Examples . 24 1.4.2 Rules of inference . 25 1.4.3 Canonical Proofs . 26 1.4.4 Typographical conventions . 29 1.4.5 Basic facts about canonical arguments . 29 1.4.6 Justifying introduction rules . 31 1.5 Properties of Valid Sequents . 33 v 1.5.1 Structural properties . 33 1.5.2 Cut-elimination . 33 1.5.3 Introduction rules . 45 1.5.4 Completeness . 46 1.5.5 Doing without the conditional . 54 2 General Rules 56 2.1 General Introduction Rules . 58 2.1.1 Canonical Arguments . 61 2.1.2 Characterization of the general introduction rules . 62 2.1.3 Maximality of intuitionistic logic . 63 2.2 General Elimination Rules . 67 2.2.1 Canonical Arguments . 68 2.2.2 Cut-elimination . 72 2.2.3 Conservativity Results . 83 2.2.4 Maximality of intuitionistic logic . 86 2.3 Stability . 88 2.4 Summing Up . 92 II Metaphysics 94 3 Is the Vagueness Argument Valid? 95 3.1 Introduction . 95 3.1.1 Overview . 97 3.2 Technicalities . 98 3.3 The Argument . 100 vi 3.3.1 The premisses . 100 3.3.2 Sider’s argument for (P3)................. 102 3.3.3 Models . 103 3.4 The Contingency of Composition . 104 3.5 Existence Deflated . 106 3.5.1 Existence on the cheap . 106 3.5.2 ‘Just consists in’ . 107 3.5.3 Basic and non-basic worlds . 108 3.5.4 Non-basic worlds and determinacy . 109 3.5.5 Penumbral Counterfactuals . 110 3.5.6 Matters dialectical . 113 3.6 Arbitrariness . 114 3.6.1 Intraworld Arbitrariness . 114 3.6.2 Interworld arbitrariness and Sharp Cut-Offs . 115 3.6.3 Whence the lack of sharp cut-offs? . 118 3.6.4 In terms of possible worlds . 119 3.6.5 Another way of blocking the argument . 120 3.7 Loose Threads . 121 3.7.1 Counterfactual excluded middle . 121 3.7.2 “Drawn from different possible worlds” . 122 3.8 Supervenience . 123 3.9 Is the Actual World a World? . 125 3.10 Conclusions . 129 4 Grounding Grounding 131 4.1 Introduction . 131 vii 4.2 The Status Problem: a Special Case . 133 4.3 Clarifications . 135 4.3.1 The Form of Grounding-Claims . 135 4.3.2 Grounding as a primitive . 136 4.3.3 Higher-Order Quantification . 137 4.3.4 Explaining explanations . 140 4.3.5 Matters of Ideology . 140 4.4 The Status Problem in General . 141 4.4.1 No layering of objects? . 145 4.5 An Account of Iterated Ground . 147 4.6 The Essentialist View of Iterated Ground . 149 4.6.1 Canonical forms . 150 4.6.2 The problem of constituents . 152 4.6.3 Do essences enter into the grounds? . 153 4.6.4 The problem of factuality . 155 4.7 The Antinomy of Grounding . 159 4.8 Zero-Grounding . 161 4.8.1 Zero-grounding and factuality . 161 4.8.2 Zero-grounding and essence . 162 4.9 Grounding an Explanation: The Grammar of Ground . 163 4.10 Grounding and Explanation: Explanatory Arguments . 165 4.11 Expressive Limitations . 168 4.12 Grounding as an Operator . 169 4.13 Objections and Clarifications . 172 4.14 Conclusions . 177 viii A Appendix 179 A.1 (P1b), (P2b), (P3c) are jointly consistent . 179 Bibliography 181 ix For my father x List of Figures 1.3.1 Intuitionistic Introduction Rules . 12 1.4.1 Intuitionistic Elimination Rules . 26 2.1.1 A general introduction rule . 59 2.2.1 A General Elimination Rule . 68 A.1.1Model of (P1b)........................... 180 xi Acknowledgments While writing this thesis, the Harvard Philosophy Department was my intellectual home; it was here that I came of age as a philosopher. I owe a great debt to Warren Goldfarb for suggesting that I look into Dummett’s proof-theoretic justifications of logical laws. Untold false starts later, I’ve finally settled all the technical questions I started out with in my second year paper—and more. Many thanks also to Peter Koellner for forcing me to think hard about what the philosophical significance of this work is. Although none of the technical work in this dissertation directly relates to his own work, my debt to Koellner’s seminars in the philosophy of mathematics, mathematical logic and set theory is enormous. For the more metaphysical papers, my largest debt is to Ned Hall. The paper on the vagueness argument started out as a two-page note. Many patient discussions later the result is the chapter in the dissertation. For discussions about this paper I’m also very grateful to Bryan Pickel. I’ve been interested in grounding, and the logic of grounding, for many years, but my attempts were not successful. In spring 2011 I came up with the idea behind the logics of ground which I sketch in this paper. Apart from my committee, I here owe special thanks to Susanna Siegel and the other members of the job-market seminar at Harvard for reading and commenting on numerous drafts. For their patient and insightful comments on earlier versions of this paper I’m very grateful to Shamik Dasgupta and Michael Raven. Most of all, thanks to Louis deRosset for lengthy and insightful exchanges about the logic of ground. xii Needless to say, any errors remain my own. On a personal note, this dissertation would not have existed without my family—Na’ama, Omri and Ingunn. xiii Part I Proof-theoretic Justification of Logic 1 1 Intuitionistic Rules 1.1 Introduction According to Dummett (see e.g., his 1973, 1976, 1978, 1991), there are two points where our linguistic practice makes contact with the extralinguistic world. First, we verify statements on the basis of observation; second, we draw ultimate conclusions from our statements—conclusions which have consequences for action. For Dummett our practice of verifying statements and our practice of drawing ultimate conclusions from statements are fea- 2 tures which are manifest in our use of language. Dummett suggests that each of these aspects of use can be the central notion of a meaning-theory; the former leading to ‘verificationist’ meaning-theory, the latter to a ‘prag- matist’ meaning theory. Such meaning-theories would not be subject to the manifestation argument. In a verificationist meaning theory the meaning of a statement φ is spec- ified by laying down what counts as a canonical verification of φ. In a pragmatist meaning theory the meaning of a statement φ is specified by specifying which ultimate consequences can be canonically derived from φ.1 Central to Dummett’s philosophy is the concern that these two aspects of use have to be in harmony. Our actual usage can be criticized if they’re not. For instance, our actual usage can be criticized if we draw conclusions from our statements which aren’t licensed by the meaning conferred on the statements by their verification conditions.