!"# #$%"" &'( ( )#"% * +, %- ( * %. ( %/* %0 * ( +, %. % +, % %0 ( 1 2 % ( %/ %+ ( ( %/ ( %/ ( ( 1 ( ( ( % "# 344%%4 253333 #6#787 /0.' 9'# 86' 8" /0.' 9'# 86' (#"8'# Analyticity, Necessity and Belief Aspects of two-dimensional semantics Eric Johannesson c Eric Johannesson, Stockholm 2017 ISBN print 978-91-7649-776-0 ISBN PDF 978-91-7649-777-7 Printed by Universitetsservice US-AB, Stockholm 2017 Distributor: Department of Philosophy, Stockholm University Cover photo: the water at Petite Terre, Guadeloupe 2016 Contents Acknowledgments v 1 Introduction 1 2 Modal logic 7 2.1Introduction.......................... 7 2.2Basicmodallogic....................... 13 2.3Non-denotingterms..................... 21 2.4Chaptersummary...................... 23 3 Two-dimensionalism 25 3.1Introduction.......................... 25 3.2Basictemporallogic..................... 27 3.3 Adding the now operator.................. 29 3.4Addingtheactualityoperator................ 32 3.5 Descriptivism ......................... 34 3.6Theanalytic/syntheticdistinction............. 40 3.7 Descriptivist 2D-semantics .................. 42 3.8 Causal descriptivism ..................... 49 3.9Meta-semantictwo-dimensionalism............. 50 3.10Epistemictwo-dimensionalism................ 54 3.11Chaptersummary...................... 56 4 One-dimensional alternatives 57 4.1Introduction.......................... 57 4.2Metaphysicallyimpossibleworlds.............. 58 4.3Switchersemantics...................... 61 4.4Chaptersummary...................... 69 iii 5 Belief ascriptions 71 5.1Introduction.......................... 71 5.2Simpledoxasticlogic.....................72 5.2.1 A posteriori necessities ................ 73 5.2.2 Beliefs de re ...................... 73 5.2.3 Indexicals....................... 74 5.2.4 Beliefs de se ..................... 75 5.2.5 Logical omniscience ................. 76 5.3Two-dimensionaldoxasticlogic............... 76 5.4Themeta-semanticstrategy................. 87 5.5Structuredpropositions................... 91 5.6Structured2D-semantics................... 94 5.7Richard’spuzzleaboutindexicals..............106 5.8Theactualityoperator....................108 5.9Theapriorioperator.....................110 5.10Thenand-problem......................113 5.11Chaptersummary......................116 6 Conclusion 119 Appendix A Partial semantics 123 A.1Varyingdomains.......................123 A.2Twofactsaboutexistenceclaims..............135 A.3Non-denotingterms.....................137 A.4 Three possible semantics for Lι ..............143 Appendix B The now- and the actuality operator 147 B.1Theactualityoperatorisuneliminable...........147 B.2 The now operatorisuneliminable..............157 B.3 No N-free sentence is universally equivalent to NPa....161 Appendix C Two-dimensional semantics 163 C.11D*-consequenceequals2D-consequence..........163 ∗ C.2 LB is more expressive than LB ...............166 C.3Structured2D-semantics...................170 C.4Anexample..........................179 Svensk sammanfattning 183 Bibliography 189 iv Acknowledgments Originally, I was going to write a thesis about the nature of deductive reasoning. I quickly realized, however, that it was going to have to be a thesis about the nature of deductive and mathematical reasoning. Still suspecting that my focus was too narrow, I decided to broaden the scope of my investigation even further. The question I eventually had in mind was: what is logic and mathematics? Needless to say, the project derailed. Having ventured for a while into the area of decision theory, prob- ability and formal epistemology, as part of my obligatory course work I was going to write a paper for Kathrin Gl¨uer-Pagin on proper names and indexicals. It was in connection with a course she had given on the topic. That’s when I first came in real contact with Kripke, Kaplan and two-dimensional semantics. I never finished that paper. Unbeknownst to me, it was instead going to be the starting point of this thesis. Then in 2015, I helped organize the second PLM master class with David Chalmers, where I got to present a paper on a posteriori necessities. That paper essentially became the fourth chapter of this thesis. So, thank you David and Kathrin! There are many other people I would like to thank. First, I would like to thank my supervisors, Peter Pagin and Dag Westerst˚ahl, for teaching me how to think rigorously. Thank you also for your patience! Secondly, I would like to thank John Cantwell and Sebastian Enqvist for reviewing the penultimate version of the thesis. Thirdly, I would like to thank friends and colleagues at the Department of Philosophy at Stockholm University, for commenting on my work at one point or other and for making these five and a half years a great experience: S¨oren H¨aggqvist, Hans Mathlein, Mattias H¨ogstr¨om, Karl Karlander, Jonas Olson, Frans Svensson, Torbj¨orn T¨annsj¨o, Lars Bergstr¨om, Kjell Svensson, Anna Petronella Foultier, Bj¨orn Eriksson, Staffan Carlshamre, Jonas Akerman,˚ Arvid B˚ave, Asa˚ Wikforss, Gunnar Svensson, Niklas Olsson Yaouzis, Asa˚ Carlsson, Paul Needham, Sofia Jeppson, Dugald v Murdoch, Mikael Janvid, Valentin Goranko, Anandi Hattiangadi, Kris- ter Bykvist, G¨osta Gr¨onroos, Mikael Pettersson, Dag Prawitz, Richard Dawid, Per Martin-L¨of, Lisa Furberg, Levi Spectre, Robert Callerg˚ard, Daniel R¨onnedal, Maria Svedberg, Henrik Ahlenius, Katharina Berndt Rasmussen, Conrad Bakka, Jonathan Egeland Harouny, Stefan Buijs- man, Lisa Hecht, Madeleine Hyde, Asa˚ Burman, Mats Ingelstr¨om, Ivan Kasa, Johan Lindberg, Daniel Ram¨oller, Karl Nygren, Hege Dypedokk Johnsen, Emma Wallin, Amanda Thorell, Nils S¨afstr¨om, Sama Agahi, Olle Torpman, Henning Strandin, and Anders Lundstedt. A special thanks to my dear friend and colleague Sara Packal´en, for her support through thick and thin. Finally, I would like to thank my family: my parents Elisabeth and Bertil, and my brothers and sisters Sofia, Marcus, Frida and Olof. Without you I couldn’t have done it, obviously. vi Chapter 1 Introduction A glass couldn’t contain water unless it contained H2O-molecules. In other words: whenever a glass contains water, it contains H2O-molecules. Likewise, a man couldn’t be a bachelor unless he was unmarried. In other words: whenever a man is a bachelor, he is unmarried. Now, the latter is what we would call a conceptual or analytical truth. It’s also what we would call a priori. But it’s hardly a conceptual or analytical truth that if a glass contains water, then it contains H2O-molecules. Neither is it a priori. The fact that water is composed of H2O-molecules was an empirical discovery made in the eighteenth century. The fact that all bachelors are unmarried was not. But neither is a logical truth, so how do we explain the difference? Understanding the nature of logical-, analytical-, necessary- and a priori entailment (and the relation between them) is, I believe, crucial for understanding and evaluating many philo- sophical arguments and theses. Take, for instance, Hume’s dictum that onecannotderiveanought from an is. Under what interpretation of derive is that an interesting, yet plausible, claim? The same goes for Moore’s open question argument against naturalism in meta-ethics, and the zombie argument against physicalism in the philosophy of mind. In Naming and necessity, Kripke convinced many of us that we need to distinguish between metaphysical and epistemic possibility. What’s necessarily true is not the same as what’s analytically or a priori true, and vice versa. The fact that a glass contains H2O-molecules if it con- tains water is a good example. It’s metaphysically necessary, but it’s not epistemically necessary. But what is then the relation between these two kinds of possibilities, if it’s not identity? Two-dimensional semantics is a framework that promises to shed light on these issues. The main idea, championed most notably by David Chalmers and Frank Jackson, is that when one evaluates an expression relative to a possible state of affairs (or 1 CHAPTER 1. INTRODUCTION possible world, for short), one can do it in two ways: by considering the world as actual, or by considering it as counterfactual. The distinction is supposed to be closely related to the distinction between indicative and subjunctive conditionals. For instance, 1. If Oswald didn’t shoot Kennedy, someone else did. (indicative, and true) 2. If Oswald hadn’t shot Kennedy, someone else would have. (subjunctive, and probably false) 3. If the stuff filling rivers and lakes always has been XYZ rather than H2O, then rivers and lakes still contain water. (indicative, and true) 4. If the stuff filling rivers and lakes always had been XYZ rather than H2O, then rivers and lakes would still have contained water. (subjunctive, and false) Moreover, the idea goes, in order to evaluate an expression, one needs to consider a pair of worlds: one considered as actual, and the other considered as counterfactual. Relative to our world (where the stuff fill- ing rivers and lakes is H2O) considered as actual and a world considered as counterfactual where the stuff filling rivers and lakes is XYZ, the ex- pression Rivers and lakes contain water will be false. But relative to the same world considered as both actual and counterfactual, the same expression will