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Taming the Impossible Matthias Jenny Taming the Impossible by Matthias Jenny B.A., University of Zurich (2010) M.A., Brandeis University (2012) Submitted to the Department of Linguistics and Philosophy in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2017 ○c Massachusetts Institute of Technology 2017. All rights reserved. Author........................................................................... Department of Linguistics and Philosophy September 8, 2017 Certified by . Vann McGee Professor of Philosophy Thesis Supervisor Accepted by . Roger White Professor of Philosophy Chair of the Committee on Graduate Students 2 Taming the Impossible by Matthias Jenny Submitted to the Department of Linguistics and Philosophy on September 8, 2017, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Philosophy Abstract The semantic paradoxes and other statements about impossibilities have proved to be obstacles to a satisfactory theory of conditionals. In my dissertation, which consists of two parts, I propose a new approach to the impossible that yields an improved theory of conditionals. A prominent response to the semantic paradoxes is glut theory. Glut theorists avoid paradox by giving up material modus ponens. But they argue that they can help themselves to this rule in areas where no paradoxes loom. In chapter 1, I argue that this does not work and that giving up modus ponens in paradoxical domains leaves glut theorists with a weak logic everywhere. One option that’s available to glut theorists involves pragmatic innovation. In chapter 2, I explore the consequences of giving glut theorists the pragmatic resources that are already available the proponents of gap theory, the dual of glut theory. The resulting hybrid theory, which makes use of two distinct speech acts of assertion, is glap theory. Surprisingly, the logic of glap theory is a quite strong logic that adds to the logics of glut and gap theory two hybrid forms of modus ponens. Turning to counterfactual conditionals, the second half of my dissertation concerns the vacuity thesis, which says that all counterpossible conditionals are vacuously true. In chapter 3, I argue that the strongest case against the vacuity thesis comes from counterpossibles as they appear in relative computability theory. I show that relative computability theorists crucially invoke counterpossibles when they define the central notions of their theory. I also provide a model theory for a quantified language that can express such counterpossibles. The logical properties of counterfactuals about relative computability deserve closer attention. In chapter 4, I provide an axiomatization of a propositional fragment of the model theory developed in chapter 3 and prove that the axiomatization is complete and that the resulting conditional logic is decidable. This logic display some surprising features. While validating modus ponens, it also contains a restricted form of the import-export law. Thesis Supervisor: Vann McGee Title: Professor of Philosophy 3 4 Acknowledgments I am incredibly lucky to have had Vann McGee as my thesis advisor. I got a first glimpse of his generosity and dedication when I submitted a draft of my final paper for the first class I took with him: to my amazement, Vann’s comments were longer than the paper itself! This set the tone for our working relationship for the years to come. Between regular meetings that spanned whole af- ternoons, long written comments on numerous drafts, and countless excellent lectures and seminar meetings, Vann has been my greatest intellectual influence. Without him, this thesis truly wouldn’t have been possible—trust me, I’m an expert on the impossible. Aside from the personal debt I owe him, Vann’s research is a constant source of inspiration. I will always try to emulate the combination of philosophical depth and technical skill displayed in his papers and books. I am also truly grateful to the other members of my thesis committee, all of whom have made me a better philosopher through their mentorship, teaching, and writing. Agustín Rayo has been so encouraging during the past five years and he intervened in crucial moments to prevent me from going astray. Bob Stalnaker’s theories were the initial spark for much of my dissertation and his mentorship helped me guide along the way. Steve Yablo often drew my attention to ideas that always changed the direction of my thinking for the better. My departmental colleagues have taught me so much and they listened patiently to my depart- mental talks and read my drafts. Thanks to Marion Boulicault, Sylvain Bromberger, Alex Byrne, Jennifer Carr, Nilanjan Das, Kevin Dorst, Lyndal Grant, Sally Haslanger, Sam Hesni, Richard Holton, Sophie Horowitz, Abby Jaques, Brendan de Kenessey, Justin Khoo, Rae Langton, Ju- lia Markovits, Rebecca Millsop, Daniel Munoz, Sofia Ortiz-Hinojosa, Kevin Richardson, Damien Rochford, Said Saillant, Bernhard Salow, Ginger Schultheis, Melissa Schumacher, Brad Skow, Jack Spencer, Judy Thomson, Ian Wells, Quinn White, and Roger White. And special thanks to my of- fice mates David Boylan, Cosmo Grant, Ari Koslow, Rose Lenehan, Matt Mandelkern, and Milo Phillips-Brown for making my daily life more entertaining. Philosophy at MIT is more than the sum of its parts, but these people have made my five years here particularly enjoyable. In addition to my committee and my departmental colleagues, I am grateful to the philosophers and logicans at the University of Connecticut. UConn always felt like an intellectual home away from my MIT home. Thanks to Jc Beall, Jared Henderson, Nathan Kellen, Suzy Killmister, Andrew Parisi, David Ripley, Marcus Rossberg, Lionel Shapiro, Stewart Shapiro, Andrew Tedder, and Sam Wheeler. And thanks to my teachers at Brandeis, especially Eli Hirsch, Sophia Malamud, Berislav Marušic,´ and Palle Yourgrau, for putting me on the right path and for their continued mentorship. Thanks also to Danielle Bromwich, Teresa Bruno, Noel Dominguez, Nykki Dular, Amy Ferrer, Mary Grenham, Sally Haslanger, Abby Jaques, Ari Koslow, Dana Miranda, Milo Phillips-Brown, Kevin Richardson, Lisa Rivera, Hayley Webster, and everyone else involved in the first two install- ments of PIKSI-Boston, especially our amazing Alain Locke Fellows, for giving me hope that our 5 discipline can be better than it has been in the past. The members of the administrative staff at MIT’s Department of Linguistics and Philosophy do an amazing job keeping everything running smoothly. Thanks to Chrissy Graham, Mary Grenham, Chris Naylor, Jen Purdy, and Matt Sikorski for immediately jumping into action whenever I had a problem. Thanks also to the staff at MIT’s International Students Office, especially David Elwell. Many thanks to my friends and family for their love, friendship, and unwavering support over the years: Abby, Blake, Can, Diego, Dustin, Kevin, Lukas, Marco, Martha, Matt, Migge, Milo, Nykki, and my parents, Ursula and Ulrich, and my sister, Mirjam. Finally, thanks to Vann for bringing his canine friend Cherry with him to his office and to class almost every day. Cherry, you’ve made Vann’s lectures on logic less stressful and my hours spent in his office even more enjoyable. Please don’t tell Vann you’re the real reason why I asked him to be my advisor. 6 Contents Introduction . .9 Part I: Taming the Paradoxes 13 1 The Flight from Gluts 15 1.1 Introduction . 15 1.2 Classical recapture for gap and glut theorists . 17 1.3 Perlocutionary recapture: shrieking and whistling . 19 1.3.1 Shrieking . 19 1.3.2 Whistling . 20 1.4 Illocutionary recapture: metalinguistic negation . 24 1.4.1 Expressive limitations . 26 1.4.2 Embedding metalinguistic negation . 28 1.5 Conclusion . 31 2 Glap Theory 35 2.1 Introduction . 35 2.2 Thesis: gluts . 37 2.3 Antithesis: gaps . 38 2.4 Synthesis: glaps . 41 2.5 Revenge? . 45 2.6 Taking stock . 48 2.7 Restrictive and permissive consequence . 49 2.8 Restrictive and permissive derivability . 51 2.9 Discussion . 53 2.9.1 Glut vs. glap theory . 53 2.9.2 Gap vs. glap theory . 54 2.9.3 One glap theory or two? . 56 2.10 Transparent truth . 57 7 2.10.1 Multiple-conclusion R and P ........................ 58 2.10.2 Adding a truth predicate . 60 2.10.3 Deriving the structural rules . 61 2.10.4 Comparisons . 63 2.11 Subervaluationism . 64 2.11.1 Supervaluationism and subvaluationism . 65 2.11.2 Subervaluationist consequence and derivability . 66 2.11.3 Discussion . 69 2.11.4 Truth . 70 2.12 Comparing glap theory and subervaluationism . 70 Part II: Taming the Undecidable 74 3 Counterpossibles in Science 77 3.1 Introduction . 77 3.2 Background . 79 3.3 Philosophical humility . 85 3.4 Understanding and misunderstanding reducibility . 87 3.5 Patching up the orthodoxy . 102 3.6 Interpreting the indices . 104 3.7 Conclusion . 108 3.8 Appendix: Model theory . 109 4 A Conditional Logic of Turing Reducibility 115 4.1 Introduction . 115 4.2 Syntax . 119 4.3 Models . 120 4.4 Axioms and rules . 123 4.5 Discussion . 125 4.6 Soundness . 129 4.7 Some facts about `PT ................................. 130 4.8 Completeness and decidability . 134 4.9 Intended models . 141 4.10 On the full propositional language . 143 8 Introduction Here are some claims about conditionals that, from a pretheoretical perspective, are hard to deny: Identity is a law of logic. This is the claim that every sentence of the form 휑 ! 휑 is a law of logic. If it’s raining, then it’s raining. If that’s not a law of logic, then it’s unclear what would be. Modus ponens is a rule of logic. This is the claim that every instance of 휑 ! 휑 is logically valid. If it’s raining, then the streets are wet. It is raining. So the streets are wet.
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