Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators). The connection with Tarski’s topo- logical semantics is developed throughout the text, and the frst substantive chapter is devoted to a new and simplifed proof of Tarski’s completeness result via well- known fractal curves. This work may be applied in the many formal areas of philosophy that exploit probability theory for philosophical purposes. One interesting application in meta- physics, or mereology, is developed in the introductory chapter. We argue, against orthodoxy, that on a ‘gunky’ conception of space—a conception of space accord- ing to which each region of space has a proper subregion—we can still introduce many of the usual topological notions that we have for ordinary, ‘pointy’space. 1 To Dana and Grisha, for turning a philosopher into a mathematician, and to Barry and Paolo, for turning her back again. i Acknowledgements This dissertation had its beginnings in the Colorado mountains. I went there in a break between Summer Session 2009 and the beginning of the fall semester to visit a friend, Darko Sarenac. At the time, I was having serious doubts about fnishing my degree, and was exploring the possibility of dropping out to become a photographer in the more remote parts of Southwestern New Mexico. Darko and I discussed the different things I might photograph, and even, as I remember, made planstotraveltogether with mycamerato Wyoming andthe South. Soon enough, though, we got to talking about logic. On a hike up a mountain as a storm set in, I learned that there was a deep connection between topology and modal logic—indeed, that there was a whole feld called topological modal logic that had been quite active at Stanford and elsewhere in the last several years. In those 48 hours, Darko taught me the basics. By the time he dropped me off at the airport in Denver, we had plans to write a paper together. (This paper forms the frst chapter of the dissertation.) Thank you, Darko. Without you, I would be somewhere in New Mexico. I want to thank, most of all, the people who have worked with me throughout my graduate career at Berkeley. I was incredibly fortunate to have Barry Stroud as an advisor from very early on. Our conversations shaped the way that I think about so many things, and his way of doing philosophy has been a great infuence on me. Although the topic I eventually chose for my dissertation was quite far afeld from Barry’s own interests, he was the frst to encourage it. More than anyone else, Barry was witness to the many ups and downs of my career at Berkeley, and I always felt his staunchsupport and confdence. I was also very fortunate to have Paolo Mancosu as an advisor and mentor. Paolo was the frst professor to take me on as a Graduate Student Instructor. I learned from him how logic could be taught in a way that was clear, engaging, and philosophically rich. When it came time to my writing a dissertation in modal logic, Paolo was aware of the many professional challenges that lay ahead and despite this, was fully supportive of the work I was doing and the project I had chosen. He provided me with invaluable advice and help at critical moments. In the frst few days of working together, Darko surreptitiously sent an e-mail to Grigori Mints at Stanford, encouraging him to be in contact with me. I still remember seeing Grisha for the frst timeaftermany years at a talkby Dana Scott in the Berkeley Logic Colloquium. At the talk, Dana introduced a new, probabilistic semantics for modal logic—a semantics about which very little was known at the time. Some days after the talk, Grisha approached me. “Tamar,” he said, “Vai you ii not prove completeness?” in his inimitable accent. Grisha became a mentor to me of the best kind, always pointing me in the way of interesting questions, and giving practical advice at every turn in the road. I am very grateful for his taking me in with such generosity and kindness. I am indebted on so many scores to Dana Scott. First, for introducing the prob- abilistic semantics: the semantics which this dissertation is about. What beautiful defnitions and ideas! Second, for taking me on as an informal student, once I had begun working on a completeness proof, as Grisha had suggested. But most of all, for his generosity and keen insights. I have enjoyed great conversations with Justin Bledin, Branden Fitelson, Peter Koellner, Mike Martin, Christos Papadimitriou, James Stazicker, Seth Yalcin, and many others. I want to thank my sister and parents, for the support—and tremen- dous endurance!—they showed throughout this long endeavor. iii Contents 1 Introduction 1 1.1 Introduction . 1 1.2 Modal beginnings . 4 1.2.1 Early motivations . 5 1.2.2 Relational semantics for modal languages . 6 1.3 Kripke semantics . 9 1.4 Space and topological semantics ........................................................... 12 1.4.1 A mathematical view of space ................................................. 12 1.4.2 Topological semantics ............................................................... 15 1.5 Measure and probabilistic semantics ................................................... 19 1.5.1 Measure ....................................................................................... 20 1.5.2 Probabilistic semantics .............................................................. 22 1.6 Gunk via the Lebesgue measure algebra ............................................. 26 1.6.1 Motivations ................................................................................. 27 1.6.2 The approach based on regular closed sets ............................. 28 1.6.3 The measure-theoretic approach ............................................. 31 1.7 Game plan ................................................................................................ 36 2 Topological Completeness of S4 via Fractal Curves 38 2.1 Introduction ............................................................................................. 39 2.2 Kripke semantics for S4 .................................................................. 40 2.2.1 Language, models, and truth ................................................... 40 2.2.2 Kripke’s classic completeness results ...................................... 41 2.3 Infnite binary tree ................................................................................... 42 2.3.1 The modal view of the infnite binary tree, T2 ......................... 42 2.3.2 Building a p� m orphism from T2 onto fnite Kripke frames 43 2.4 Topological semantics for S4 .......................................................... 46 2.4.1 Topological semantics ............................................................... 47 iv 2.4.2 Interior maps and truth preservation in the topological se- mantics ........................................................................................ 49 2.4.3 Topological completeness results for S4 ............................. 50 2.4.4 The infnite binary tree and the complete binary tree, viewed topologically ............................................................................... 51 2.5 Fractal curves and topological completeness ...................................... 54 2.5.1 The Koch curve .......................................................................... 55 2.5.2 Completeness via the Koch curve ............................................ 59 3 Completeness of S4 for the Lebesgue Measure Algebra 62 3.1 Introduction ............................................................................................. 63 3.2 Topological and algebraic semantics for S4 .................................... 64 3.3 The Lebesgue measure algebra ............................................................. 67 3.4 Invariance maps .....................................................................................