Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic by Corey Mulvihill A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Philosophy Waterloo, Ontario, Canada, 2015 c Corey Mulvihill 2015 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract Hilbert's choice operators τ and ", when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert's operators in relation to his evolving program in the foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett's program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+" without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett's contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett's framework is improved by these results as they show that questions of realism and anti-realism are not an \all or nothing" matter, but that there are plausibly metaphysical stances between the poles of anti-realism (corresponding to acceptance just of intutionistic logic) and realism (corresponding to acceptance of classical logic), because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as \is smart," which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong (i.e., classical) logic. v Acknowledgements I would first like to thank my advisor David DeVidi, who both encouraged and, when necessary, restrained my efforts, and whose own work inspired so much of the subject matter of this thesis. I would also like to thank my committee, from the Waterloo philosophy department, Tim Kenyon, Doreen Frazer and especially Gerry Callaghan, who valiantly stepped in at the last minute; Steve Furino, from the Waterloo faculty of mathematics; and my external examiner Greg Lavers for his careful reading and helpful comments. In addition, I must thank all the Waterloo Philosophy Department Faculty and Staff. Especially Debbie Detrich who from my first arrival at UW has advised me, and helped me navigate the UW system, looking out for me as she does all the graduate students in the Philosophy Department, we are very lucky to have her. I should also thank my graduate colleagues at Waterloo, who created such a wonderful and welcoming environment. Especially my year's Ph.D. cohort: Mike McEwan and Eric Hochstein. They made living in Waterloo tolerable with their friendship, discussions in various uptown establishments, and the Saturday breakfasts that followed. In a similar vein I would be remiss if I did not thank the Univeristy of Western Ontario philosophers, and Ivey folk, who made living in London bearable. In addition to those mentioned above, I would also like to thank the people who I en- countered when I started my journey in philosophy, at the University of Ottawa|especially since I forgot to include an acknowledgement's section in my master's thesis. Thanks, first, to Mathieu Marion, my master's supervisor, who first introduced me to non-classical logics; and to Sophie Rietti and Patrice Philie, friends and now colleagues at Ottawa, for their friendship and insight. Thanks also are due to my former graduate student colleagues, in particular: Dan McArthur, who encouraged me to return to academia; Mark Young, whose friendship and loyalty can only be described as supererogatory; Shane Ralston, who I am proud to say made it through his Ph.D. at Ottawa without me getting him arrested once; and Dean Lauer whose wonderful companionship kept us out late so many nights, often until closing hours. Thanks to my wonderful gigantic family, both the Rowes and the Mulvihills, and now, since I made the best decision of my life and married my wonderful wife, the Mazutises and Trences. Please know that I am aware of how lucky I am to have so many wonderful and supportive people in my life. A special thank you is due to my amazing parents, Bob and Helen, who have supported me through the years, both morally and financially, and to my sister, Shauna Mulvihill, whom the fates have punished with me as a sibling. Finally, to my brilliant and beautiful wife Daina Mazutis and my daughter Emilija Mulvihill; I know I could not have finished this without you Daina. Thank you for your support and the most wonderful daughter in the world. Thank you all. vii Contents Introduction 1 1 Introduction 3 1.1 Logic and Metaphysics . .5 1.1.1 Problems with Dummett's views . .7 1.2 The Axiom of Choice and Hilbert's Programme . .7 1.3 Structure of Thesis . .9 I Historical Context 11 2 The Origins and Development of Hilbert's Programme 13 2.1 Introduction . 13 2.2 Early Constructivism . 17 2.3 The Road to Hilbert's Programme . 19 2.3.1 The Invariant Theory Papers . 20 2.3.2 From the Foundations of Geometry to the Foundations of Mathematics 22 2.3.3 Hilbert's Problems . 23 2.4 The Origin of the Ignorabimus ......................... 24 2.5 Axiomatizing Arithmetic . 27 2.5.1 Hilbert's \On the Concept of Number" . 28 2.5.2 Hilbert \On the Foundations of Logic and Arithmetic" . 30 2.5.3 Poincar´e'sResponse to Hilbert (1905) . 33 ix 2.5.4 Hilbert's 1905 Lecture Course `Logical Principles of Mathematical Thought' . 34 2.6 The Middle Period . 36 2.6.1 Hilbert and Logicism . 37 2.7 Hilbert's Programme . 41 2.7.1 Hilbert's \New Grounding of Mathematics" . 42 2.8 Choice Operators . 49 2.8.1 Hilbert and Bernay's Grundlagen der Arithmetik .......... 49 2.8.2 Russell's -Operator and Hilbert's "-Operator . 50 2.8.3 Hilbert's Epsilon Theorems . 55 3 Intuitionistism, Intuitionistic Logic and Anti-Realism 59 3.1 Introduction . 59 3.2 Defining Intuitionism . 63 3.2.1 Philosophical Intuitionism . 64 3.3 Brouwer's intuitionism . 65 3.3.1 Intuitionism and Formalism . 66 3.3.2 Hilbert and Brouwer . 68 3.4 Intuitionistic Logic . 69 3.4.1 Formalizations of intuitionistic logic . 70 3.4.2 Axiom Systems for Intuitionistic Logic . 71 3.4.3 Introduction and Elimination Rules . 72 3.4.4 Intermediate Logics . 74 3.5 Dummett and Intuitionistic Logic . 76 3.6 Dummett and Logical Laws . 80 3.6.1 Harmony and Introduction Rules . 81 3.6.2 Proof Theoretic Justification . 85 3.6.3 Canonical Proofs and Demonstrations . 92 3.7 The Introduction and Elimination Rules for Intuitionistic Logic +" .... 94 x II Formal Results 97 4 Deriving Logical Axioms from Choice Principles 99 4.1 Introduction: The Intuitionistic "-calculus . 99 4.2 Syntax . 101 4.2.1 Notational Conventions . 101 4.3 DeMorgan's Law and the Intuitionistic "-calculus . 103 4.3.1 Bell's Second Proof of DeMorgan's Laws from Epsilon . 104 4.4 Proof of DeMorgan's Laws in Intuitionisitic Logic + " without Identity . 106 4.5 Linearity Axiom and Intuitionistic Predicate Calculus + " ......... 109 4.5.1 Dummett's Scheme . 109 4.6 Extensional Epsilon and the Law of Excluded Middle . 111 4.6.1 The Proof of the Law of Excluded Middle from Intuitionisitic Logic+" and the Axiom of Epsilon Extensionality . 111 4.6.2 Proof of LEM from Intutionistic Logic+" and a Weakened Exten- sional Axiom . 113 5 Semantics for Intuitionistic Logic + " 115 5.1 Semantic Considerations . 115 5.2 Classical Semantics for " -Calculus . 117 5.3 Existing Semantics for Inutitionistic "-Calculus . 118 5.4 Bell's Semantics for Intuitionistic First Order Logic + " .................................... 119 5.5 Bell's Model and Completeness . 124 5.6 DeVidi Semantics for Intuitionistic "-Calculus . 125 5.7 Skeletons and Ground Terms . 126 5.7.1 Skeleton Terms . 127 5.7.2 The Alpha Axiom . 128 5.8 Identity .