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Metaphysical Dependence and Set Theory City University of New York (CUNY) CUNY Academic Works All Dissertations, Theses, and Capstone Projects Dissertations, Theses, and Capstone Projects 2013 Metaphysical Dependence and Set Theory John Wigglesworth 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/1697 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] METAPHYSICAL DEPENDENCE AND SET THEORY by JOHN WIGGLESWORTH 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 2013 ii c 2013 John Matthew Wigglesworth All Rights Reserved iii This manuscript has been read and accepted for the Graduate Faculty in Philosophy in satisfaction of the dissertation requirement for the degree of Doctor of Philosophy. Richard Mendelsohn Date Chair of Examining Committee Iakovos Vasiliou Date Executive Officer Graham Priest Arnold Koslow Melvin Fitting Kit Fine Supervisory Committee THE CITY UNIVERSITY OF NEW YORK Abstract METAPHYSICAL DEPENDENCE AND SET THEORY by JOHN WIGGLESWORTH Advisor: Professor Graham Priest In this dissertation, I articulate and defend a counterfactual analysis of metaphysical dependence. It is natural to think that one thing x depends on another thing y iff had y not existed, then x wouldn’t have existed either. But counterfactual analyses of metaphysical dependence are often rejected in the current literature. They are rejected because straightforward counterfactual analyses fail to accurately capture dependence relations between objects that exist necessarily, like mathematical objects. For example, it is taken as given that sets metaphysically depend on their members, while members do not metaphysically depend on the sets they belong to. The set /0 metaphysically { } depends on /0,while /0does not metaphysically depend on /0 . The dependence is asymmetric. { } But if counterfactuals are given a possible worlds analysis, as is standard, then the counterfactual approach to dependence will yield a symmetric dependence relation between these two sets. Be- cause the counterfactual analysis fails to accurately capture dependence relations between sets and their members, most reject this approach to metaphysical dependence. To generate the desired asymmetry, I argue that we should introduce impossible worlds into the framework for evaluating counterfactuals. I review independent reasons for admitting impos- sible worlds alongside possible worlds. Once we have impossible worlds at our disposal, we can consider worlds where, e.g., the empty set does not exist. I argue that in the worlds that are ceteris paribus like the actual world, where /0does not exist, /0 does not exist either. And so, according { } iv ABSTRACT v to the counterfactual analysis of dependence, /0 metaphysically depends on /0,as desired. Con- { } versely, however, there is no reason to think that every world that is ceteris paribus like the actual world, where /0 does not exist, is such that /0does not exist either. And so /0does not metaphys- { } ically depend on /0 . After applying this extended counterfactual analysis to several set-theoretic { } cases, I show that it can be applied to account for dependence relations between other mathemati- cal objects as well. I conclude by defending the counterfactual analysis, extended with impossible worlds, against several objections. Acknowledgements I am eternally grateful to my wife, Victoria. Her patience, encouragement, and understanding made the completion of this project possible. I also express my deepest thanks to Graham Priest, who challenged me to think in new ways about many ideas in this thesis and beyond. His guidance has made me a better philosopher. But I could not have accomplished anything without my parents. They have always believed in me, and they taught me to believe in myself. There are many others who deserve thanks. My committee members, Arnie Koslow and Mel Fitting, devoted much of their valuable time to reading many drafts. This thesis would not be what it is without their insightful comments. I am grateful to Daniel Nolan and Jonathan Schaffer for their challenges to the main arguments in this thesis. I appreciate the constructive feedback and critique from audiences in New York, Melbourne, Bristol, and Nottingham, where parts of this work have been presented. And I must also thank Ricki Bliss for keeping me interested in metaphysics when I had my doubts. Lastly, I am forever indebted to Gary Matthews and Kris McDaniel, my first philosophy teachers, who inspired me to follow this path. vi Contents Abstract iv Acknowledgements vi List of Figures ix Chapter 1. Introduction 1 1. The Plan 5 Chapter 2. Metaphysical Dependence, Then and Now 10 1. Then: a selective history of metaphysical dependence 14 2. Now: metaphysical dependence on the contemporary scene 20 Chapter 3. The Modal Analysis of Dependence 32 1. Set-Theoretic Dependence 32 2. The Modal Analysis and How it Fails 46 3. An Alternative Modal Analysis 51 Chapter 4. Impossible Worlds 58 1. Logical Laws 61 2. Metaphysical Laws 68 3. Worlds vs. Points 72 4. Realist Theories of Impossible Worlds 75 5. Anti-Realist Theories of Impossible Worlds 97 Chapter 5. In Favor of Realism 102 1. The Argument For 102 2. The Arguments Against 132 vii viii CONTENTS 3. Appendix 136 Chapter 6. The Counterfactual Analysis of Dependence 138 1. Conditional Logic 141 2. Set-Theoretic Dependence 144 3. An Account of Minimal Metaphysical Dependence 164 4. Mathematical Structuralism and Metaphysical Dependence 169 Chapter 7. Objections and Replies 179 1. High Level Objections 179 2. Why not this way? 183 3. Specific Objections 189 4. Conclusion 200 Bibliography 205 List of Figures 1 The set a = a 159 { } 2 The sets a = c,a and b = c,b . 160 { } { } 3 The sets /0and /0 . 160 { } 4 The set a = a 162 { } 5 The set b = b 162 { } 6 Alternative graphs of the set a = a . 163 { } ix CHAPTER 1 Introduction Dependence has become something of a “hot topic” in metaphysics. Some see it as a central concept in the study of reality. A concept as important, or arguably more important than the concept of existence. For once we know what exists, it is how those things relate to one another that tells us what reality is like. And it is thought that one of the fundamental ways in which things relate to one another is in virtue of metaphysical dependence. Metaphysical dependence gives structure to the world. One explanation as to why metaphysical dependence is so important is its ubiquity. If we look, we can find metaphysical dependence everywhere. Properties depend on objects; wholes depend on parts; holes depend on hosts; moral facts depend on non-moral facts; modal facts depend on non-modal facts; legal facts depend on non-legal facts, the world depends on mind; everything depends on God. God depends on Herself. And so on. I do not claim that these dependence relations actually hold. They are examples of dependence claims that can be made and have been made. One explanation as to why metaphysical dependence is ubiquitous is that it may just be an umbrella term, used to capture many different concepts. There is a plurality of relations, each of which could plausibly be labeled a dependence relation: supervenience, composition, constitution, determination, causation, truth-making, grounding, priority, holding “in virtue of”, explanation, entailment, and arguably set-membership. Of course, these are not all terms for the same relation. Rather, it may be that the phrase ‘metaphysical dependence’ is ambiguous between them, or between the elements of some subset, or some superset, of them. The goal of this project is to single out one plausible understanding of metaphysical dependence, and show that under this particular conception of dependence, sets metaphysically depend on their members. And not vice versa. At least, not usually. Under this 1 2 1. INTRODUCTION conception, in general, things that are members of sets do not metaphysically depend on the sets that they are members of. When it comes to sets and their members, metaphysical dependence is in most cases asymmetric. The claim that sets metaphysically depend on their members may strike you as odd. It is widely accepted that some concrete objects metaphysically depend on others. But why think the same holds for sets? Sets are mathematical objects. And mathematical objects are said to exist necessarily, if they exist at all. They could not have failed to exist. In what sense, then, could they depend on anything else? Their existence is, in effect, guaranteed. Nothing else is needed. Certainly, additional complications arise when one tries to articulate dependence relations between objects that exist necessarily. But the fact that it is complicated or odd does not justify ignoring the thought. It is the thought that we wish to explore. And what we show is that, at least in the set-theoretic case, the claim that some mathematical objects metaphysically depend on others is true. And we give a framework within which we can understand these dependence claims. That being said, on a general level, we wish to remain neutral with respect to the ontological status of mathematical objects. The claim we defend is to some degree a hypothetical one: If mathematical objects exist, then they enter into certain dependence relations. For example, if sets exist, then they depend on their members. We take as an assumption, then, that purely mathematical objects like numbers and sets exist, and that they exist necessarily. Indeed, it is for the most part standard to think that if mathematical objects exist, then they exist necessarily. But we do not try to defend these claims here; we simply try to articulate a conception of metaphysical dependence that would hold between certain mathematical objects, given that they exist.
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