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Numbers and Necessity NUMBERS AND NECESSITY Craig Warmke A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Philosophy. Chapel Hill 2015 Approved by: Robert M. Adams Thomas Hofweber Keith Simmons L.A. Paul William G. Lycan ©2015 Craig Warmke ALL RIGHTS RESERVED ii ABSTRACT Craig Warmke: Numbers and Necessity (Under the direction of Robert Merrihew Adams and Thomas Hofweber) I offer a new account of property parthood, and on its basis, novel semantic approaches to first- order logic, modal propositional logic, and quantified modal logic. I also use it in my metaphysical accounts of sets and the natural numbers. Along the way, I analyze property parthood in terms of the mental conjunction of ideas. What results: idealist accounts of necessity and number with certain explanatory advantages over standard accounts. iii For Laura. iv ACKNOWLEDGEMENTS I have been in grad school for nine years and accrued debts to people far and wide. Rather than list the entire ledger, I’ll tell a few stories instead. First. When I applied to PhD programs, I emailed three people about working with them: Thomas Hofweber at UNC, Laurie Paul at Arizona, and Stephen Yablo at MIT. Only Arizona accepted me, but Laurie had offers to teach at other schools, including UNC. She very graciously promised to try to take me with her if she decided to go elsewhere. Soon after I agreed to attend Arizona, Laurie took the UNC job. But she stuck around Tucson for my first year, worked closely with me, and then, as promised, took me to UNC. I arrived in Chapel Hill at the same time as Bob Adams. He taught a seminar on existence our first semester at UNC, and I wrote a paper for him which developed a new semantics for modal logic. Bob encouraged me to continue this project, which grew into the present dissertation. He co-supervised my dissertation for two years with Keith Simmons until they both left for the northeast. At that point, UNC rules required me to adopt another supervisor from UNC. Since I was working on logic and philosophy of mathematics at the time, Thomas Hofweber was the natural choice to step in for Keith. For all this moving and shuffling, I’ve had the enviable experience of having four world-class advisers. And I owe them each a unique debt. I’m grateful for the lessons Laurie taught me at both Arizona and UNC and for letting me tag along. I’m grateful for Bob’s encouragement and depth of philosophical and historical insight. For Keith’s patience, availability, philosophical clarity, and formal precision. And for Thomas’s approachability and philosophical and professional guidance. UNC has been so important for my philosophical development. So I sincerely thank Stephen Yablo for ignoring my email and rejecting me from MIT. Second. My brother Brandon and I went to the same small school in Chicago, but neither of us studied philosophy. After Brandon graduated, he began to claw his way into academic philosophy. He took some classes at some community colleges and eventually took a class with Daniel Sutherland v at UIC. Partly on the basis of Sutherland’s letter of recommendation, my brother gained admission into the MA program at Northern Illinois. Probably on the basis of Brandon’s performance, I also was admitted to NIU. This fall, I’ll return to NIU as an Assistant Professor. There are three people I’d like to thank here. First, my brother. He is my best friend and also an excellent philosopher. I wouldn’t be in philosophy without him. Being in the same profession as my brother has been one of my life’s greatest joys. Second, Tomis Kapitan. He taught me modal logic and possible worlds semantics at NIU. He retired recently, and, as I understand it, I’ll be filing the shoes he left behind—a fulfilling and challenging task. Finally, Daniel Sutherland. I might not be in philosophy without the letter he wrote for my brother. Amusingly, Sutherland and I are now also brothers of a sort, since Bob Adams has helped advise both of our dissertations. Third. Laura and I married soon after I graduated from college. We spent our savings on gas and tuition so I could commute over an hour to attend graduate school in a field in which I had no background. She gave up job opportunities and moved cross country twice so I could study where I felt I should. She agreed to have two children with me in graduate school even though job prospects for philosophers are grim. Besides my parents, no one has believed in me more or loved me as much in such a costly way. Like any dissertation, mine has a table of contents which captures the topics to come. But each entry in the TOC carries a special meaning for me beyond its normal one. I see a section and remember the life I had while I wrote it. I see ups and downs as I scan my TOC, and Laura’s love and support is a common thread through them all. So it’s only fitting for me to dedicate this whole thing to her. vi TABLE OF CONTENTS LIST OF FIGURES . x 1 Introduction . 1 2 Property Parthood . 7 2.1 Introduction. 7 2.2 Abundant Properties . 9 2.3 Candidate Analyses . 10 2.4 Competing Conceptions . 14 2.5 The Mereology. 20 2.6 Conclusion . 25 3 Modal Intensionalism . 27 3.1 Introduction. 27 3.2 Inclusion . 29 3.3 Frames . 30 3.4 Truth, Necessity, and Possibility . 32 3.5 Pure Semantics. 37 3.6 Applied Semantics . 39 3.7 Explanation and Finite Cognition . 41 3.8 Conclusion . 44 4 Logical Intensionalism . 45 4.1 Introduction. 45 4.2 Montague Semantics . 49 vii 4.3 The Containment Theory . 53 4.4 The Alternative . 61 4.5 Evaluation . 69 4.6 Conclusion . 77 4.7 Appendix 1: Examples . 78 4.8 Appendix 2: Non-existents . 80 5 Modal Idealism . 83 5.1 Introduction. 83 5.2 Ideas ............................................................................ 87 5.3 Parts of Ideas . 93 5.4 Worldhood. 97 5.5 Propositional Ideas . 100 5.6 Truth ............................................................................103 5.7 Necessity . 105 5.8 Names . 106 5.9 Predicates. 108 5.10 Negation . 109 5.11 Quantification . 112 5.12 The Formalism . 113 5.13 Parthood . ..
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