Stand Over There, Please The Dynamics of Vagueness, the Origins of Vagueness, and How Pie-Cutting Relates to Ancient Heaps of Sand Seth Cable Master of Science Thesis Master of Science Programme in Logic Institute for Logic, Language and Computation University of Amsterdam August 2002 Although many are thanked in the Preface, this thesis is dedicated to only the following individuals: Michael and Jane Cable, who, at some point during the Carter administration, thought it wise to make a little boy. Mary Cable, who con’t spel, but has, concentrated at the tip of her pen, more insight into a person’s guts than I have in the whole of my fat, balding head. Suzanne Kalamar, who submitted herself to heroic personal sacrifice, just so she could spend a single harried week in a country without SUV’s. God in heaven, Zsuzsa, I hope you don’t drop off the face of my Earth. H. L. Mencken, whose timeless brilliance in the American idiom and genuine intellectual bravery made existence tolerable during those useless minutes before and after bed. ii Contents Preface v 1 Introduction 1 1.1 The Meaning of “Vague” . 1 1.1.1 Borderline Cases . 2 1.1.2 Susceptibility to Sorites Paradoxes . 3 1.1.3 Standard-Sensitive Context Dependency . 4 1.2 The Question . 7 1.3 The Problem . 8 1.4 Our Inscrutable Knowledge of Vagueness . 10 1.4.1 Lexical Semantics and the Study of Vagueness . 10 1.4.2 The Inscrutability of Vagueness . 12 1.4.3 Objections to the Argument and Replies . 13 1.5 Consequences for Our Study . 16 2 Context Change and Vague Predicates 19 2.1 Chapter Overview . 19 2.2 Dynamic Semantics: An Elementary Refresher . 20 2.3 A Classical Semantics for Vague Predicates . 22 2.4 Vague Predicates and Information Change . 24 2.5 A Dynamic Semantics for Vague Predicates . 25 2.6 Truth and Borderlines . 30 2.7 Vagueness and Comparison . 34 2.7.1 The Relationship Between Gradability and Vagueness . 34 2.7.2 The Effects of the Comparative upon the Context . 35 2.8 The Logic of Vague Predicates . 37 2.8.1 Sentence Connectives and the Projection of Indeterminacy . 38 2.8.2 Validity and Consequence . 41 2.8.3 Connections with Supervaluationism . 46 2.9 The Semantics of “Definitely” . 48 2.9.1 A First, Crude Pass . 48 2.9.2 The Preferred Approach . 52 iii 3 Vague Predicates, Vague Selections 57 3.1 Chapter Overview . 57 3.2 Why This is Not Enough . 58 3.2.1 Non-Existent Contexts . 58 3.2.2 The Strange Case of Higher Order Vagueness . 59 3.2.3 Broken Promises . 62 3.3 A Reductive Theory of Vagueness . 62 3.4 Vague Selections . 64 3.5 Primitive Degrees . 69 3.6 The Reduction, Informally Described . 72 3.6.1 Degree Orderings Are Dense . 72 3.6.2 The Non-Synonymy Principle . 74 3.6.3 Standard Sensitivity and Vague Selections . 75 3.7 The Reduction, Formal . 77 3.7.1 Formal Theory of Vague Selections . 77 3.7.2 Relationship Between Vague Selections and Contexts . 81 3.7.3 Downsizing Our Goals . 84 3.8 Overcoming Our Adversities . 87 3.8.1 Proving the Law of Higher Order Vagueness . 87 3.8.2 Borderlines Equivalent to Standards . 89 3.8.3 On the Nature of Vague Predicates . 91 4 The Heap 93 4.1 Chapter Overview . 93 4.2 Paradox, Fallacy, or Misunderstanding? . 94 4.2.1 What is at Issue . 94 4.2.2 The Sorites is Not Successful . 96 4.2.3 The Sorites Rests on a False Premise . 98 4.2.4 What Now Must be Done . 99 4.3 Approaching the Sorites . 100 4.4 Precise Boundaries Within Well Orderings . 103 4.5 The Sizes of Vague Selections . 104 4.5.1 The Minimality of Vague Selections . 104 4.5.2 The Minimality of the Uncertainty Orders . 105 4.5.3 The Law of Quasi-Tolerance . 107 4.6 Quasi-Tolerance and the Sorites Paradox . 108 4.6.1 The Epistemological Question . 109 4.6.2 The Psychological Question . 111 4.6.3 Interim Summary . 112 4.7 Quasi-Tolerance and the Higher-Order Sorites . 112 4.8 Sorites Susceptibility Equivalent to Standards . 115 iv Preface This thesis is an investigation into the semantics of vague expressions, although vague predicates are predominantly discussed. It proposes an analysis of vague predicates within the framework of dynamic semantics, and argues that this analysis predicts the observed relationship between the properties of having borderline cases, of being susceptible to sorites paradoxes, and of being context- dependent. The first, introductory chapter concerns the questions of what, precisely, a “theory of vagueness” is and what it can hope to achieve. The material here is intended to provide the lens through which the more technical work of the following chapters should be viewed, and it is hoped that the characterization offered here of a program for the study of vagueness clarifies and unifies the existing literature on this subject. In outline, the chapter explains first the con- cept of “vagueness” as it appears in the linguistic and philosophical literature, and then discusses the necessary limits to any scientific study of this property. In regards to those limits, it is argued that a complete explanation of the seman- tics and logic of vague language is logically impossible, and that the most we as theorists can hope for is a set of increasingly refined approximations. These necessarily limitations, moreover, affect how one should describe the study of vagueness as an intellectual activity, and so how the reader should understand the material of the next three chapters. In Chapter 2, the basic semantic analysis for vague predicates is proposed. This analysis uses the formal techniques of dynamic semantics to capture the unique information change potential of predications employing vague language. Moreover, the analysis is argued to correctly predict the relationship between the context dependency of a predicate and its having “borderline cases.” Some basic properties of the resulting logic of vague predicates are then discussed, including its wide retention of classical logic and its ability to grab hold of “penumbral connections.” Finally, the vagueness operator “definitely” is introduced, as well as its proposed information change potential, and some effects on our logic of adding the operator to our language are proven. Chapter 3 begins with a host of difficulties for the basic analysis of the previ- ous chapter. In particular, the analysis allows for the possibility of unnaturally precise uses of vague predicates and fails to predict the relationship between the presence of borderline cases and so-called “higher-order vagueness.” It is suggested that these problems may be overcome if a more sophisticated the- v ory of the origins of linguistic vagueness is adopted. The concept of a “vague selection” is introduced, as well as the hypothesis that all vagueness in lan- guage may be reduced to the “vagueness” of these selections. This hypothesis is spelled out formally within our dynamic semantics, and the resulting analysis is demonstrated to overcome the challenges raised at the chapter’s beginning. The fourth and final chapter concerns the sorites paradox. The basic ques- tions concerning the sorites are reviewed, as well as some reasons against ac- cepting it as a genuine paradox. It is then argued that the reductive theory of vagueness put forth in the previous chapter provides a unique perspective on what is occurring within the sorites argument, a perspective from which one spies a potential “resolution” to the paradox. Finally, it is argued that this analysis of the sorites, when combined with our reductive theory of vagueness, predicts that predicates are susceptible to sorites arguments if and only if they are vague. Acknowledgements Writing this thesis was, for scattered reasons, a great personal joy, my delight in these ideas perhaps revealing itself at times in the density of the text. Stoked by my acknowledged weakness for ostentation (al´aDon King), the furnaces of my obsequiousness rattle from an overheated desire to wax lyrical over the many people and institutions which were midwife to me during this difficult labor 1. Nevertheless, I keep it short. Highest thanks go to Professor Frank Veltman, who supervised my research on this project. His expertise in these subjects supplied direction when it was lost and levied criticism where it was most needed. He also read my myriad of meandering project reports, which, as the reader may already imagine, could become annoyingly hectic. I wish here to also extend thanks to the other mem- bers of my defense committee, Professors Jeroen Groenendijk and Johan van Benthem. Thanks are also due to the Fulbright Foundation, the Netherland-America Foundation, The Netherlands America Commission for Educational Exchange, the Institute for Logic, Language and Computation (h.f. “ILLC”) and the Spinoza Project, all of whom at some point in the past year gave me lots of money. The ILLC must also be singled out for individual thanks. The reasons for gratitude should be obvious, but let me state for the record that the ILLC is a singularly stimulating research environment, one in which I have grown and developed in ways I would not have before imagined, and which paid me the goodwill of letting me in its doors. The ILLC and the Universiteit van Amsterdam are the main characters of the heretofore most fascinating drama in my life, and they shall never be forgotten. The material making up Chapter 2 of this thesis was presented at the 2002 Szklarska Poreba Workshop on the Roots of Pragmasemantics.