From Language to Thought: On the Logical Foundations of Semantic Theory Dissertation Presented in Partial Fulllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Giorgio Sbardolini, MA Graduate Program in Philosophy The Ohio State University 2019 Dissertation Committee: Stewart Shapiro, Advisor Øystein Linnebo William Taschek Neil Tennant © Copyright by Giorgio Sbardolini 2019 Abstract Sentences have meanings: the things we say, and the things we believe. Semantics is the theory of meaning, and thoughts, i.e. the meanings of sentences, are among the objects of semantic theory. But what are meanings? What is the place of meaning in the natural world? In the discussion below, I shall motivate formal constraints on the logical and metaphysical foundations of semantic theory. Some philosophers have suggested that semantics is a piece of modal metaphysics. The modal approach to meaning covers a lot of empirical and conceptual ground, but imposes a sharp separation between epistemology and metaphysics. The modal approach ultimately fails, since the mechanism invoked to recombine the metaphysical with the epistemic leads to inconsistency. The lesson is that semantics is not modal metaphysics, and that sameness of meaning is a hyperintensional notion. Other intensional paradoxes follow more generally from assuming that thoughts can be individuated to a more or less precise degree, i.e. as the only thoughts having a certain property. These assumptions are often very plausible. Some contemporary accounts of the intensional paradoxes save consistency at the cost of rejecting these plausible assump- tions. This puzzling situation leads naturally to wonder about the conditions for referring to thoughts: how do we individuate them? Reference to abstract objects may be achieved by singular terms whose semantic prop- erties are established by abstraction. On this proposal, reference is explained by a criterion ii of identity for the referents, which is in turn established by an abstraction principle on which sameness of meaning is equated with hyperintensional equivalence. Such notion cannot be as ne-grained as contemporary accounts of structured propositions take it to be, on pain of ruling out a compelling pragmatic account of redundancy in the use of lan- guage: speakers naturally take certain pairs of sentences to have the same meaning, so that syntactic complexity does not inevitably amount to semantic dierence. Any plausible hyperintensional notion of equivalence faces, in higher-order logic, the Russell-Myhill paradox. However, consistency can be restored by a dynamic understand- ing of abstraction, on which the truth conditions of identity statements are dened in- crementally. From this perspective, there are dierences between what is absolutely true, i.e. for any renement of the identity relation, and what is potentially true, i.e. for some accessible renement. This distinction tames Russell-Myhill while still leaving room for hyperintensionality, and accommodates plausible assumptions about the individuation of thoughts. On the resulting picture, thoughts are “shadows of sentences”, to use an image of W. V. O. Quine, and quantication over thoughts is understood predicatively. This is the logic and metaphysics for the foundations of semantics. iii Acknowledgments I am grateful to many people who made my time as a graduate student an excellent expe- rience, both intellectually and personally. Special thanks to my adviser, Stewart Shapiro, who has always been insightful and supportive. I have never left Stewart’s oce after a philosophical discussion feeling that no progress had been made, and I have always had the encouraging awareness that the development of my research was my responsibility, whether the mistakes and blunders, or the achievements. It is a privilege to be Stewart’s student. Most of the research I have done has been provoked by reection on topics to which members of my dissertation committee have spent many years contributing. Although it is sometimes said that there is no progress in Philosophy, there is no denying that the present work builds on top of recent scholarly conversations in which Øystein Linnebo, William Taschek, and Neil Tennant have been primary participants. I strive to further develop this conversation—hopefully in directions that they are sympathetic to. I want to express my gratitude for your guidance and mentoring, and for the time and energies you have dedicated in helping me. I have beneted from conversation with many others, in many dierent ways: some with challenging questions, some with advice while I was writing, some for help on earlier drafts, all from whom I have learned something in the course of a few intense and excit- ing years. I would like to thank Francesco Berto, David Braun, Ben Caplan, Ben Lennertz, iv Tristram McPherson, Chris Pincock, Carl Pollard, Graham Priest, Craige Roberts, Richard Samuels, Kevin Scharp, Gabriel Uzquiano, Tim Williamson, the faculty members and the graduate students of the Ohio State Philosophy department, the faculty members and the graduate students of the Ohio State Linguistics department, and audiences at the univer- sities of Milan, Virginia Tech, Connecticut, Stockholm, Munich, Padua, Oslo, Amsterdam, St. Andrews, and Bualo, where parts of this work have been presented, invariably to receive terric feedback. I also would like to thank anonymous reviewers who have given helpful comments on parts of this dissertation. I gratefully acknowledge the generous support of the Ohio State Graduate School for a Presidential Fellowship for the year 2019, and the combined support of the Ohio State Graduate School for a Matching Tuition Award and of the ConceptLab at the University of Oslo for a Collaborative Fellowship for the year 2017. Finally and most importantly, heartfelt thanks to my friends, in America and in Eu- rope, and to my family, for help in life beyond Philosophy. v Vita April 21, 1988 . Born - Monza, Italy 2010 . B.A. Philosophy, Università degli Studi di Milano 2012 . M.A. Philosophy, Università degli Studi di Milano 2013-present . .Graduate Teaching Associate, Ohio State University Publications Research Publications Two-dimensional Paradox. Australasian Journal of Philosophy, 2018. with S. Negri: Proof Analysis for Lewis Counterfactuals. The Review of Symbolic Logic, 9, 1: 44–75, 2016. Fields of Study Major Field: Philosophy vi Table of Contents Page Abstract . ii Acknowledgments . iv Vita................................................. vi 1. Introduction . 1 1.1 Propositional variables . 5 1.2 Propositional attitudes . 15 2. Two-dimensional paradox . 20 2.1 Kaplan’s principle . 21 2.2 Two-dimensionalism . 22 2.3 Diagonals and Antidiagonals . 27 2.4 Beyond intensionality . 37 3. Prior’s gambit . 43 3.1 Prior’s paradox . 43 3.2 Prior’s theorem . 48 3.3 A most paradoxical argument . 54 3.4 Coherentism about thoughts? . 59 4. Reference to Thoughts . 63 4.1 Puzzles about quantication . 63 4.1.1 Background . 66 4.2 The dilemma about mathematical truth . 71 4.3 A Metasemantic Benacerraf . 73 vii 4.4 Descriptive Singular Terms . 77 4.4.1 A Fregean argument . 81 4.5 “Standard names” . 83 4.6 Begrisschrift §3.................................. 86 5. Wittgenstein’s fundamental thought . 95 5.1 Structured propositions . 95 5.2 On redundancy . 100 5.3 Principles of conversation . 102 5.4 A look ahead . 110 6. Metasemantic Predicativism . 112 6.1 Aboutness paradox . 113 6.1.1 Neither truth nor sets . 114 6.1.2 The structure of reality . 118 6.2 Predicative logic . 124 6.2.1 Dynamic abstraction . 128 6.2.2 Revised Prior’s theorem . 135 6.3 Addendum . 139 7. Conclusions and further work . 152 Bibliography . 158 viii Chapter 1: Introduction An utterance of the English sentence ‘Snow is white’, relative to a context, expresses the thought that snow is white. That thought is what I come to believe, if you tell me that snow is white. We can describe our agreement that snow is white, by saying that our mental states have that thought in common. Thoughts are universals of semantics, and they can be employed to account for the contents of sentences, various aspects of our mental lives, and epistemic notions like agreement and disagreement. For these purposes, thoughts are assumed to be true or false with respect to how things are in the world: there is thus a direct connection between the world on one side, and meaning and belief on the other. But if thoughts are true or false, it follows that at least some logical relations hold among them: material equivalence (being both true or both false), truth preservation (if these thoughts are true, so is this), and satisability (these thoughts can be true together). What else can be said about the logic of thoughts? And what are they, that their nature might support such logical structure? These two questions, about the logic and the nature of thoughts, are the topic of the present work. I shall address them from two sides. First, the problem of paradox. In- tensional paradoxes arise fairly quickly when we have the linguistic resources to discuss formal theories of thoughts, threatening to reduce them to triviality. The logic of thoughts 1 should at least avoid triviality, and yet, the situation here being no dierent than other do- mains of inquiry troubled by the paradoxes, it is controversial what measures to adopt in response. Secondly, the problem of propositional identity. Material equivalence does not suce for the identity of thoughts: the thought that snow is white is true if and only if the thought that the sky is blue is true. Indeed, they are both true. Certainly however, those are not the same thought. Moreover, the thought that snow is either white or not, and the thought that the sky is either blue or not, are necessarily such that the former is true if and only if the latter is. Indeed, they are both necessarily true.