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Frontiers of Conditional Logic City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2-2019 Frontiers of Conditional Logic Yale Weiss The Graduate Center, City University of New York How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/2964 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Frontiers of Conditional Logic by Yale Weiss A dissertation submitted to the Graduate Faculty in Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York 2019 ii c 2018 Yale Weiss All Rights Reserved iii This manuscript has been read and accepted by the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Professor Gary Ostertag Date Chair of Examining Committee Professor Nickolas Pappas Date Executive Officer Professor Graham Priest Professor Melvin Fitting Professor Edwin Mares Professor Gary Ostertag Supervisory Committee The City University of New York iv Abstract Frontiers of Conditional Logic by Yale Weiss Adviser: Professor Graham Priest Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional|especially counterfactual|expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that cat- alyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel v embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. My dissertation examines deontic and connexive conditional logic as well as the un- derappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take|in contrast to Lewis and Williamson|a non-vacuous line. Thus, in my view, a counterpossible like \If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, al- though it has an impossible antecedent. I exploit impossible (non-normal) worlds|originally developed to model non-normal modal logics|to provide non-vacuous semantics for coun- terpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. Acknowledgments and Permissions I would like to express my gratitude, first and foremost, to Professor Graham Priest. I first began thinking seriously about the philosophy and logic of conditionals in his course with Hartry Field on the subject in the fall semester of 2014|my first semester as a graduate student|and have yet to stop. At every stage of my work on the subject, from just learning it to producing original research, Graham Priest made himself available to provide feedback and advice. This dissertation, beginning with the subject to which it is devoted, owes much to his influence. I owe special thanks, as well, to the other members of my committee. I am grateful to Professor Melvin Fitting for his comments, not only on this dissertation, but on the various papers and presentations over the years that built up to it. I thank Professor Edwin Mares for agreeing to serve on my committee (from far off New Zealand!) and for his valuable remarks on matters technical and philosophical. Finally, I owe thanks to Professor Gary Ostertag, especially for critical comments which have helped hone and clarify the philosophical content of chapter 5. Thanks are also due to, among others: the anonymous referees and editors of various papers of mine on conditional logic (selections of some of which are included herein), the members of the \Priest Club" and the members of the Logic and Metaphysics Workshop (for listening to me present some of this material and providing helpful feedback), Professor Consuelo Preti (without whose influence I might never have ended up at the Graduate vi vii Center), Professor Branden Fitelson (for introducing me to an automated theorem proving tool used herein), and my parents, Veronica and Wayne Weiss. This dissertation incorporates material that I have previously published and I am pleased to acknowledge the publishers' permissions to reuse this material. Some of the material from chapter 4 has been adapted from: Weiss, Y. (2018). Connexive Extensions of Regular Conditional Logic. Logic and Logical Philosophy, forthcoming. The publisher of the paper, which can be obtained at http://doi.org/10.12775/LLP. 2018.012, is Nicolaus Copernicus University in Toru´n,Poland. The material is reused here with the permission of the editors of Logic and Logical Philosophy. A substantial amount of the material from chapter 6 is drawn from: Weiss, Y. (2018). Basic Intuitionistic Conditional Logic. Journal of Philosophical Logic, forthcoming. This paper is available from the publisher at https://doi.org/10.1007/s10992-018-9471-4. It is reprinted by permission of the publisher under license 4442590680916 with the following required acknowledgement: Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature, Journal of Philosophical Logic, \Basic Intuitionistic Conditional Logic," Yale Weiss, COPYRIGHT 2018, advance online publication, 23 July 2018 (doi: 10.1038/sj.jpl.) Contents Contents viii List of Tables xi 1 Introduction 1 1.1 The Scope of Conditional Logic . .1 1.2 Overview of the Dissertation . .4 2 Conditional Logic 9 2.1 Syntax . 10 2.1.1 Axiom Systems . 10 2.1.2 Modal Logic in Conditional Logic . 16 2.2 Uniform Semantics . 20 2.2.1 Proposition Indexed Interpretations . 21 2.2.2 Algebraic Semantics . 26 2.2.3 Determination Results . 31 2.3 Niche Semantics . 34 2.3.1 Half-Classical Semantics . 34 2.3.2 Preorder and Sphere Semantics . 38 viii CONTENTS ix 3 Tableaux for Lewis-Stalnaker Logics 44 3.1 Tableaux for Lewis Systems . 46 3.1.1 Tableaux Systems . 46 3.1.2 Determination Results . 55 3.2 Tableaux for Stalnaker's System . 63 4 Connexive Conditional Logic 71 4.1 Obligation: Absolute and Conditional . 73 4.1.1 Deontic Modal Logic . 74 4.1.2 Deontic Conditional Logic . 77 4.2 Axiom Systems . 82 4.2.1 Deontic Connexive Systems . 85 4.2.2 Non-Deontic Connexive Systems . 88 4.3 Semantics . 90 4.3.1 Proposition Indexed Interpretations . 90 4.3.2 Conditional Algebraic Interpretations . 93 5 Counterpossible Logic 97 5.1 Vacuousness and Counterpossibles . 98 5.1.1 The Case against Vacuousness . 100 5.1.2 Non-vacuous Semantics . 109 5.2 Axiom Systems . 112 5.2.1 Principles of Counterpossible Logic . 112 5.2.2 Systems of Counterpossible Logic . 115 5.3 Determination Results . 117 6 Intuitionistic Conditional Logic 123 CONTENTS x 6.1 Intuitionistic Logic . 124 6.1.1 Axiomatics and Semantics . 125 6.1.2 A Conditional Embedding of Intuitionistic Logic . 128 6.2 Intuitionistic Conditional Logic . 132 6.2.1 Axiom Systems . 132 6.2.2 Semantics . 138 6.3 Determination Results . 146 7 Concluding Remarks 152 Bibliography 155 List of Tables 2.1 Rules . 11 2.2 Axiom Schemata . 12 2.3 Major Systems of Conditional Logic . 15 2.4 Modal Axiom Schemata . 17 2.5 Function Constraints . 22 2.6 Algebraic Constraints . 27 2.7 Preorder Constraints . 40 4.1 Deontic Systems of Conditional Logic . 86 4.2 Non-Deontic Systems of Connexive Conditional Logic . 89 4.3 Deontic and Connexive Function Constraints . 91 4.4 Deontic and Connexive Algebraic Constraints . 94 5.1 Counterpossible Semantic Constraints . 111 5.2 Counterpossible Systems of Conditional Logic . 116 6.1 Intuitionistic Axiom Schemata and Rules . 126 6.2 Intuitionistic Systems of Conditional Logic . 133 6.3 Intuitionistic Relational Constraints . 140 6.4 Heyting Algebraic Constraints . 142 xi Chapter 1 Introduction In what follows, I introduce the subject of conditional logic and explain what this dissertation contributes to it. The history and scope of conditional logic is canvassed in section 1.1. An overview of the results and aims of the main chapters of this dissertation is given in section 1.2. 1.1 The Scope of Conditional Logic Debate over the logic of conditionals is as old as propositional logic.
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