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Epistemic Modality, Mind, and Mathematics

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Epistemic Modality, Mind, and Mathematics Epistemic Modality, Mind, and Mathematics Hasen Khudairi June 20, 2017 c Hasen Khudairi 2017, 2020 All rights reserved. 1 Abstract This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as con- cerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the compu- tational theory of mind; metaphysical modality; deontic modality; the types of mathematical modality; to the epistemic status of undecidable proposi- tions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Each essay is informed by either epistemic logic, modal and cylindric algebra or coalgebra, intensional semantics or hyperin- tensional semantics. The book’s original contributions include theories of: (i) epistemic modal algebras and coalgebras; (ii) cognitivism about epistemic modality; (iii) two-dimensional truthmaker semantics, and interpretations thereof; (iv) the ground-theoretic ontology of consciousness; (v) fixed-points in vagueness; (vi) the modal foundations of mathematical platonism; (vii) a solution to the Julius Caesar problem based on metaphysical definitions availing of notions of ground and essence; (viii) the application of epistemic two-dimensional semantics to the epistemology of mathematics; and (ix) a modal logic for rational intuition. I develop, further, a novel approach to conditions of self-knowledge in the setting of the modal µ-calculus, as well as novel epistemicist solutions to Curry’s and the liar paradoxes. Solutions to previously intransigent issues concerning the first-person concept, the distinc- tion between fundamental and derivative truths, and the unity of intention and its role in decision theory, are developed along the way. 2 Acknowledgements From 2014 to 2017, I was a Ph.D. Student at the Arché Philosophical Re- search Centre for Logic, Language, Metaphysics and Epistemology at the University of St Andrews. St Andrews is an ideal place to live and work. At Arché, I was supported by a St Leonard’s College (SASP) Research Scholar- ship, for which I record my gratitude. This book is a revised version of my dissertation. The dissertation was written between the foregoing years, and revised in the years that followed. For comments which lead to revisions to individual chapters, I am grateful to Aaron Cotnoir, Josh Dever, Peter Milne, and Stephen Read. For productive conversations at Arché, I am grateful to Mark Bowker, Sarah Broadie, Federico Faroldi, Katherine Hawley, Patrik Hummel, Ryo Ito, Bruno Jacinto, Li Kang, Kris Kersa, Martin Lipman, Poppy Mankowitz, Matthew McKeever, Daniel Nolan, Laurie Paul, Andrew Peet, Simon Prosser, Justin Snedegar, Mark Thakkar, Jens Timmermann, Michael Traynor, Gabriel Uzquiano, Brian Weatherson, and Erik Wielenberg. For her administrative assistance, I am grateful to Lynn Hynd. For productive discussion at conferences, workshops, et al., I am grateful to Jody Azzouni, Ralf Bader, George Bealer, Jacob Berger, Corine Besson, Ned Block, Susanne Bobzien, Otavio Bueno, Fabrizio Cariani, David Chalmers, Roy Cook, Paul Linton Cowie, Brian Epstein, Delia Graff Fara, Kit Fine, Peter Fritz, Zachary Gartenberg, Nemira Gasiunas, Tamar Gendler, Sandy Goldberg, Sally Haslanger, Benj Hellie, Christopher Hill, Sean Kelly, Uriah Kriegel, Maria Lasonen-Aarnio, Anita Leirfall, Hannes Leitgeb, Mary Leng, Stephan Leuenberger, Øystein Linnebo, Julien Murzi, Myrto Mylopoulos, Bryan Pickel, Agustin Rayo, Mark Richard, David Ripley, Sonia Roca-Royes, Gideon Rosen, David Rosenthal, Marcus Rossberg, Ian Rumfitt, Sarah Sawyer, Claudia Schaer, Wolfgang Schwarz, Erica Shumener, Jonathan Simon, Robert Stalnaker, Ravi Tharakan, Achille Varzi, Katja Vogt, Robbie Williams, Tim- 3 othy Williamson, Jessica Wilson, Keith Wilson, Crispin Wright, Stephen Yablo, and Seth Yalcin. From 2005 to 2008, I was an honors undergraduate in philosophy at Johns Hopkins University. For their encouragement and example, I am grateful to Michael Williams, Hent de Vries, Meredith Williams, Dean Moyar, and Maura Tumulty. For his friendship and for visits at the beginning and end of the time this dissertation was written, I am grateful to Armand Leblois. For their unwavering support of my endeavors over the years, I am grateful to my parents. The book is dedicated to Alison Bowen. Chapter 5 is forthcoming as "Conceivability and Haecceitism" in Dusko Prelevic and Anand Vaidya (eds.), The Epistemology of Modality and Philo- sophical Methodology, Routledge. Chapter 6 has been published as "Ground- ing, Conceivability, and the Mind-Body Problem" in Synthese 195 (2):919- 926 (2018), doi:10.1007/s11229-016-1254-2. Chapter 11 has been published as "Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism" in Don Berkich and Matteo Vincenzo d’Alfonso (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence – Themes from IACAP 2016, Springer (2019). 4 Table of Contents 1. Methodological Forward (p. 11) Part I: A Framework for Epistemic Modality 2. Modal Cognitivism and Modal Expressivism (p. 25) 2.1 The Hybrid Proposal 2.1.1 Epistemic Modal Algebra 2.1.2 Modal Coalgebraic Automata 2.2 Material Adequacy 2.3 Modal Expressivism and the Philosophy of Mathematics 2.4 Concluding Remarks 3. Cognitivism about Epistemic Modality (p. 41) 3.1 Introduction 3.2 An Abstraction Principle for Cognitive Algorithms 3.3 Epistemic Modal Algebra 3.4 Examples in Philosophy, Cognitive Science, and Quantum Information Theory 3.5 Objections and Replies 3.6 Concluding Remarks Part II: Epistemic Modality and Conceivability 4. Conceivability and Haecceitism (p. 59) 4.1 Introduction 5 4.2 Super-Rigidity 4.3 Two Dogmas of Semantic Rationalism 4.3.1 The First Dogma 4.3.2 The Second Dogma 4.3.2.1 The Julius Caesar Problem 4.3.3 Mereological Parthood 4.3.4 Summary 4.4 Determinacy and Consistency 4.5 Concluding Remarks 5. Grounding, Conceivability, and the Mind-Body Problem (p. 76) Part III: Epistemic Modality, Knowledge, and Apriority 6. Attention, Phenomenal Content, and Inexact Knowledge (p. 86) 6.1 Introduction 6.2 Perceptual Representation 6.3 Phenomenal Representation 6.3.1 The Two-stage Theory of Phenomenal Content 6.3.2 Category-theoretic Semantics for Phenomenal Representation 6.4 Attention and Inexact Knowledge 6.5 Concluding Remarks 7. Non-Transitive Self-Knowledge: Luminosity via Modal µ-Automata (p. 98) 8. Bayesian Apriority (p. 105) 8.1 Introduction 8.2 Definitions 8.3 An Informal Characterization of the Theory 8.4 The Formal Regimentation 8.4.1 Epistemic Modality 6 8.4.2 Implicit Definition 8.4.3 Evidential Confirmation 8.4.4 Grounding Structure 8.4.5 Iterated Ground 8.4.6 Apriority and Abduction 8.5 The First-Person Concept 8.5.1 The Empirical and Metaphysical Selves 8.5.2 Super-rigidity and Immunity 8.5.3 Immunity and Indeterminacy 8.6 Epistemic Modality and Deontic Modality 8.7 Concluding Remarks Part IV: Epistemic Modality and the Philosophy of Mathemat- ics 9. Epistemic Modality, Necessitism, and Abstractionism (p. 130) 9.1 Introduction 9.2 The Abstractionist Foundations of Mathematics 9.3 Abstraction and Necessitism 9.3.1 Hale and Wright’s Arguments against Necessitism 9.3.2 Hale on the Necessary Being of Purely General Properties and Objects 9.3.2.1 Objections 9.3.3 Cardinality and Intensionality 9.4 Epistemic Modality, Metaphysical Modality, and Epistemic Utility and Entitlement 9.5 Concluding Remarks 10. Ω-Logicism: Automata, Neo-logicism, and Set-theoretic Realism (p. 153) 10.1. Introduction 10.2 Definitions 7 10.2.1 Axioms 10.2.2 Large Cardinals 10.2.3 Ω-Logic 10.3 Discussion 10.3.1 Neo-Logicism 10.3.2 Set-theoretic Realism 10.4 Concluding Remarks 11. Epistemic Modality and Absolute Decidability (p. 167) 11.1 Mathematical Modality 11.1.1 Metaphysical Mathematical Modality 11.1.2 Epistemic Mathematical Modality 11.1.3 Interaction 11.1.4 Modal Axioms 11.2 Departures from Precedent 11.3 Knowledge of Absolute Decidability 11.4 Concluding Remarks 12. Grothendieck Universes, and Indefinite Extensibility (p. 179) 12.1 Introduction 12.2 Indefinite Extensibility in Set Theory: Modal and Extensional Approaches 12.3 Grothendieck Universes 12.4 Modal Coalgebraic Automata and Indefinite Extensibility 12.5 Concluding Remarks 13. A Modal Logic for Gödelian Intuition (p. 193) 13.1 Introduction 13.2 Rational Intuition as Cognitive Phenomenology 13.3 Modalized Rational Intuition and Conceptual Elucidation 8 13.4 Concluding Remarks 14. An Epistemicist Solution to the Alethic Paradoxes (p. 204) 14.1 Introduction 14.2 Scharp’s Replacement Theory 14.2.1 Properties of Ascending and Descending Truth 14.2.2 Scharp’s Theory: ADT 14.2.3 Semantics for ADT 14.2.4 Further Montagovian Desiderata 14.2.5 Satisfying the Desiderata 14.3 Issue 1: Validity 14.4 Issue 2: Hybrid Principles and Compositionality 14.5 Issue 3: ADT and Indeterminacy 14.5.1 First Extension: The Preface Paradox 14.5.2 Second Extension: Absolute Generality 14.5.3 Third Extension: Probabilistic Self-reference 14.5.4 Fourth Extension: The Sorites Paradox 14.6 Issue 4: Descending Truth, Ascending Truth, and Objectivity 14.7 Issue 5: Paradox, Sense, and Signification 14.8 Epistemicism and Alethic Paradox Part V: Epistemic Modality, Intention, and Decision Theory 15. Two-Dimensional Truthmaker Semantics (p. 227) 15.1. Introduction 15.2 Two-Dimensional Truthmaker Semantics 15.2.1 Intensional Semantics 15.2.2 Truthmaker Semantics 15.2.3 Two-Dimensional Truthmaker Semantics 15.3 New Interpretations 9 15.3.1 Fundamental and Derivative Truths 15.3.2 Decision Theory 15.3.3 Intentional Action 15.4 Concluding Remarks 16. Epistemic Modality, Intention, and Decision Theory (p. 243) 16.1 Introduction 16.2 The Modes of Intention 16.2.1 Intention-in-Action 16.2.2 Intention-with-which 16.2.3 Intention-for-the-Future 16.3 Intention in Decision Theory 16.4 Concluding Remarks Bibliography (p.
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