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Locations, Bodies and Sets Locations, Bodies and Sets J. Jeremy Meyers Locations, Bodies and Sets ILLC Dissertation Series DS-2012-10 For further information about ILLC-publications, please contact Institute for Logic, Language and Computation Universiteit van Amsterdam Science Park 107 1098 XG Amsterdam phone: +31-20-525 6051 e-mail: [email protected] homepage: http://www.illc.uva.nl/ Copyright c 2012 by J. Jeremy Meyers Locations, Bodies and Sets Academisch Proefschrift ter verkrijging van de graad van doctor aan de Leland Stanford Junior University op gezag van Dean Richard P. Saller ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Aula der Universiteit op maandag 23 juni 2012, te 12.00 uur door J. Jeremy Meyers geboren te Falls Church, Virginia, Verenigde Staten van Amerika. Promotor: Prof.dr. J. van Benthem Co-promotors: Prof. John Perry, Prof. Grigori Mints, Prof. Tom Ryckman “In terms of part-whole theory itself, the use or non-use of sets makes no substantial difference to expressive power. The sum axiom may be ex- pressed as an axiom using sets, while with predicates a schema must be used if second-order quantification is to be avoided... Many of the writings of mereology are concerned with bringing logical refinement to the existing system, by suggesting new primitives, shorter axioms, and the like. While they are interesting in themselves, these con- tributions take the classical theory for granted and do not question its presuppositions. For someone who questions the cogency of some of these presuppositions it is remarkable how little work has been done on systems of mereology weaker than the classical ones of Leśniewski and Leonard and Goodman.” Peter Simons Parts pages 54 and 65 5 Acknowledgement Without comparison my greatest debts are to my parents, my son Jason, and my sister Felicite. They were by far my biggest source of support over the years, and I could not have completed this dissertation without their patient devotion. My advisor Johan van Benthem has been a superior advisor and a source of in- spiration over the years. His approach to philosophy and logic is as innovative as it is motivational. My ideas concerning space, time, location, and nominalism all were inspired by his views. His work with Marco Aiello was crucial to many of the ideas in appearing in chapter VII. My understanding of nominalistic mereology as a fragment of first-order logic is just a variation on his own famous doctoral thesis. In addition, John Perry has been a great supporter and intellectual inspiration over the years. His work on indexicals and his theory of situations play a significant role in the dissertation. I owe special thanks to Lauri Hella and the Logic Group at Tam- pere, Finland for extraordinarily generous involvement with the project. Especially important technical contributions were made by my good friends Antti Kuusisto and Jonni Virtema. I also give special thanks to Grigori Mints for helping my professional development in a great many ways, not least of which has been the contribution of substantial improvement to the present project. I cannot forget Bob Tully for helping me as an assistant professor at West Point from 2007 to 2010. And I give a great thanks to Randall Dipert for inspiring me at a young age to continue with philosophy and helping me achieve my goals. As a cadet at the United States Military Academy I profited from continued discussions over many years with my very close friends Josh Carlisle and Andrew Pearce who were also philosophy majors. And I would not have made it to this point without the 6 7 support of my parents at that time. I have benefited a great deal from audiences and friends at Stanford, University of Amsterdam, University of Helsinki, and University of Groningen. Particular thanks are extended also to Jesse Alama, Alexei Angelides, Carlos Areces, Alexandru Baltag, Patrick Blackburn, Torben Braüner, Balder ten Cate, Lena Kurzen, Nina Gierasim- czuk, Tal Glezer, Tomohiro Hoshi, Alistair Isaac, Micah Lewin, Marc Pauly, Darko Sarenac, Sonja Smets, and Ben Wolfson. Contents 5 Acknowledgement 6 1 Introduction 1 1.1 Background and Problems . 1 1.2 My Argument . 6 1.3 Dissertation Abstract . 6 1.4 The Philosophical Assumptions I Make . 10 1.5 Chapter Overview . 13 1.5.1 Ontology . 13 1.5.2 Nominalistic Mereologic . 14 1.6 List of Items Published . 16 2 Mereological Structure and Physical Reality 17 2.1 Nominalism and Mereology . 17 2.1.1 Reality and Abstract Objects . 18 2.1.2 Fragmented and Tensed Versions of Reality . 20 2.1.3 Mathematical Entities and Sets . 23 2.1.4 Universals and Properties . 26 2.1.5 Is the Parthood Relation a Trope? . 28 2.1.6 Extensional and Intensional Objects . 30 2.2 “Mereologicality” . 31 2.2.1 Mereological Extensionality . 32 8 CONTENTS 9 2.2.2 Unrestricted Fusion and the Closure Problem . 41 2.2.3 What are closing substances? . 45 2.2.4 Interconnectedness . 47 2.2.5 Tentative Conclusions . 50 3 Mereological Situations and Locality 52 3.1 Interconnectedness and the Body . 52 3.2 First Personal Objects . 53 3.2.1 The Reality of Localized Situations . 55 3.3 Mereological Situations . 59 3.3.1 The Question of Parametrized Situations . 62 3.3.2 One Organism in Multiple Locations . 64 3.3.3 Multiple Individuals in the Same Location? . 67 3.3.4 Tense and Simultaneity . 69 3.3.5 Inanimate Objects and the Ship of Theseus . 71 3.3.6 Unrestricted Fusions of Locations . 72 3.3.7 Idealism? . 73 4 Inscriptional Nominalistic Mereologies 76 4.0.8 Leśniewski’s Conception of Formal Languages . 77 4.0.9 Interpretability and Equiformity . 78 4.1 Formal Languages as Physical Objects . 81 4.1.1 Formal Languages are Abstract . 86 4.1.2 The Uses of Logic and Mereologic . 89 5 A Formal Language for Mereology 93 5.1 The Question of Mereological Languages . 93 5.1.1 FO Formulas and Quantification . 95 5.1.2 Arithmeticality in First-Order Logic . 100 5.1.3 Upshot . 102 5.2 The Question of Modal Mereological Languages . 103 5.2.1 A Hybrid Language for Mereology . 106 CONTENTS 10 5.2.2 The Question of Infinitary Nominalistic Mereology . 110 5.2.3 Mereobisimulation and Nominalistic Mereology . 112 5.2.4 Empty Names, Quantifiers, and Variables . 115 6 Hm and Ho 118 6.1 Syntax and Semantics . 120 6.1.1 Truth Conditions over Models . 121 6.1.2 First-order Translation . 123 6.1.3 Logical Expressions . 124 6.1.4 Mereological and Boolean Relationships in Hm . 125 6.1.5 Ho Expressivity . 130 6.2 Some Definable Frame Classes . 132 6.2.1 Frame Definability and Validity over Fully Named Models . 132 6.2.2 The Definability of the BA and GEM Conditions . 133 6.2.3 Atomic and Atomless BA Classes . 136 6.2.4 Definability in Ho . 136 6.3 Logics . 138 6.3.1 Khm ................................ 138 6.3.2 Khm + BA and Khm + M ..................... 144 6.3.3 Remarks about a “Kho” . 147 6.4 Invariance and Characterization . 147 6.4.1 Mereobisimulations . 148 6.4.2 Expressivity in Hm . 149 6.4.3 The lack of counting in Hm . 150 6.4.4 Characterization of Hm and Ho . 153 6.5 Conclusion . 159 7 Capturing the Structure of Locations 162 7.1 Complete Extensional Models . 163 7.2 Is the Employment of Mathematical Models Nominalistically Acceptable?166 7.2.1 Regions and Spatial Logic . 168 7.3 Mereobisimulations between Models of Locations . 170 CONTENTS 11 7.4 A Pixellated Geometry of Solids . 173 7.4.1 Regular Open Sets . 173 7.4.2 Regular Serial and Chequered Open Sets . 176 7.4.3 The Atomless Mereobisimulation . 178 7.4.4 Corridors, Coverings, and Monotonicity . 180 7.4.5 The Mereobisimulation between CH and RO . 185 7.5 Complete Atomic BAs . 191 7.6 Finite Model Property . 207 7.7 Concluding Observations . 209 8 Conclusion 211 8.1 A Summary of the Argument . 212 8.2 Mereologic and Beyond . 213 8.3 Final Vision . 215 A Boolean Algebras 217 A.1 Topology and regular open algebras . 218 A.2 Boolean subalgebras . 219 A.3 Homomorphisms, Atoms, and Finite BAs . 220 A.4 Formal theories of BAs . 221 B Survey of Mereology 222 B.1 Types of mereologies . 222 B.2 Mereological Concepts . 222 B.3 Symbolism for Formal Systems . 225 B.4 Mereological Principles . 225 B.4.1 Parthood Axiomatizations and Concepts . 226 B.4.2 Composition Principles and Mereo-Boolean Variants . 228 B.5 PCIs . 232 B.5.1 GPCI . 232 B.5.2 EPCI . 235 B.6 SCIs . 235 CONTENTS 12 B.6.1 TSCI . 237 C Leśniewski’s Systems 238 C.0.2 Terms . 240 C.0.3 Quantification . 241 C.0.4 Formal Definitions . 242 C.0.5 Ontology .............................. 242 C.0.6 Mereology ............................. 246 D Free Logics 251 Bibliography 254 List of Tables 5.1 Definitions of mereological relations and operators. 107 6.1 Mereological operators expressed in Hm. 128 13 List of Figures 6.1 Two BA-models that have H([≤]; [≥]; α)-bisimilar points. In M0, w0 is the complement of the denotation of i. In M, w is not the complement 0 of the denotation of i. However, w !H([≤];[≥],α) w . Transitive edges and reflexive loops are omitted. 151 6.2 Two BA-models that are totally mereobisimilar. Transitive edges and reflexive loops are omitted. 152 7.1 Two objects that have the same corridor configuration. 180 7.2 8 corridors generated by 4 named nRCH open subsets of R2.
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