Formal Representation s of Suspended Judgment Doctoral thesis for obtaining the academic degree Doctor of Philosophy (Dr.phil.) submitted by Ali Zolfagharian at the Faculty of Humanities Department of Philosophy iii Konstanz, 2019 Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-2-gpnx2i068ae09 Introduction iv The distinction between ignorance and suspension v Formal Representations of Suspended Judgment Doctoral thesis for obtaining the academic degree Doctor of Philosophy (Dr.phil.) submitted by Ali Zolfagharian at the Faculty of Humanities Department of Philosophy Konstanz, 2019 Introduction vi Date of the oral examination: 23.04.2020 1. Reviewer: Prof. Spohn 2. Reviewer: Prof. Müller 3. Reviewer: Jun.-Prof. Dr. Carolin Antos-Kuby The distinction between ignorance and suspension vii This page intentionally left blank Introduction viii Contents 0. Introduction 1 1. Suspension of Judgment 4 1 The distinction between ignorance and suspension 6 2 The necessity of non-belief for suspension 9 3 Believing and suspending at the same time 14 3.1 Two-minded 14 3.2 Suspending while having a high degree of belief 15 4 Is suspension a negation-invariant operator? 17 5 Degrees of suspended judgment 18 6 Qualitative and Quantitative epistemic states 20 7 Doxastic attitudes: definitions and relationships 22 2. Belief Revision 24 1 Traditional Belief Revision (TBR) 27 1.1 Logical consequence operator 28 1.2 Belief set 28 1.3 Belief base 30 1.4 Update rules 30 1.4.1 Expansion 31 1.4.2 Contraction 32 1.4.3 Revision 38 1.4.4 Two general rules 40 2 The representation of doxastic attitudes in TBR. 41 2.1 Disbelief set 41 2.2 Suspension and ignorance 42 3 System of Spheres and Indeterministic Belief Revision 44 3.1 Indeterministic Belief Revision: from spheres to ellipses 47 3.1.1 relational belief revision 47 3.1.2 Epistemic Entrenchment 50 4 The representation of suspended judgment in IndBR 52 4.1 Accepted proposition 52 The distinction between ignorance and suspension ix 4.2 Basic doxastic attitudes 53 4.3 Epistemic bi-lattice 53 4.4 Some unwanted results 55 5 Qualitative acceptance Revision 56 5.1 Acceptance base 58 5.2 Inferable sets 59 5.3 Accepted in 60 5.4 Assessable proposition 60 5.5 Accepted proposition 61 5.6 Basic doxastic attitudes 61 5.7 Conditionally accepted propositions 62 5.8 Epistemic change 62 5.8.1 Expansion 63 5.8.2 Contraction 63 3. Bayesian Epistemology 69 1 Preliminary 73 1.1 Probabilism 75 1.2 regularity 75 1.3 Principle of indifference 76 1.4 Principal principle 77 1.5 Unwanted result 80 1.6 Reflection principle 81 1.7 Conditionalization 81 1.8 Crucial questions 82 2 Representing the degree of suspension and ignorance 82 3 Dutch book argument for probabilism 84 3.1 Czech book argument 86 3.1.1 Ignorance 87 3.2 Package principle and betting on our actions 88 4 Quantitative notion of belief and basic doxastic attitudes 89 5 BE cannot distinguish suspension and ignorance 90 Introduction x 4. Dempster-Shafer theory 92 1 Preliminary 93 1.1 Set of possibilities 93 1.2 Mass Function 95 1.3 Belief Function 100 1.4 Plausibility function 103 2 The degrees of suspension, disbelief, and ignorance 104 3 Qualitative doxastic attitudes 107 3.1 From belief function to the quaternary epistemic states 109 3.2 Truth table (as a four-valued logic) 116 3.3 Qualitative Ignorance and suspended judgment 119 4 Suspension/Ignorance and the rule of combination 120 4.1 The Dempster rule of combination. 121 4.2 Yager’s rule 124 4.3 Inagaki’s unified combination rule 127 4.4 Dubois and Prade’s disjunctive consensus rule 128 5 Basic doxastic attitudes in DS 129 5. Ranking theory 131 1 Classic Ranking Theory 133 1.1 Negative Ranking function 133 1.2 Positive Ranking function 134 1.3 Two-sided ranking function 135 1.4 Conditional Rank 135 1.5 Spohn Conditionalization 136 1.6 Regularity 137 1.7 Qualitative belief and degree of unopinionatedness 137 1.8 Doxastic attitudes in RT 138 1.9 What are Ranks? 140 1.9.1 The quantitative nature of ranks 141 1.9.2 The qualitative nature of ranks 146 1.10 Grading of suspension 149 1.11 Qualitative Suspension and Ignorance 151 2 General Ranking Theory 152 The distinction between ignorance and suspension xi 2.1 Basic ranking function 153 2.2 Acceptance base 153 2.3 Assessable propositions 154 2.4 Genral ranking function 156 2.5 Accepted propositions 159 2.6 Belief, disbelief, suspension, and ignorance 160 2.7 General positive and negative ranking function. 161 3 Comparison between RT and GRT 162 6. Acceptance Revision 163 1 Definitions and relationships among doxastic attitudes 164 2 TBR, IndBR, and QAR 164 3 BE and DS 166 3.1 Definition of quantitative doxastic attitudes 168 3.2 Acceptance threshold 168 3.3 Unwanted results 169 4 RT and GRT 170 5 Acceptance revision 170 5.1 Mass function 171 5.2 Acceptance base 171 5.3 Acceptance function 172 5.3.1 Unknown propositions or non-assessable propositions 173 5.3.2 Assessable propositions 173 5.4 Accepted propositions 175 5.5 Qualitative doxastic attitudes 175 5.6 Quantitative doxastic attitudes 175 6 Comparison 176 Bibliography 178 Introduction xii To Sahar The distinction between ignorance and suspension xiii Acknowledgements I would like to express my sincere gratitude to Prof. Wolfgnag Spohn, for his continuous supervision, guidance, inspirations, criticism and his invaluable support. I would also like to express my sincere appreciation to Prof. Thomas Müller for his encouragements and guidance. I’ve participated in his colloquium for six semesters, it was an enjoyable and valuable experience. I would like to extend my sincere gratitude to the committee of LGFG for the scholarship. Their trust in me was heartwarming. Introduction xiv FORMAL REPRESENTATIONS OF SUSPENDED JUDGMENT Abstract This work investigates various theories in Formal Epistemology in order to observe their capacity of distinguishing suspended judgment from ignorance. Besides, it suggests how every theory in formal epistemology could be changed and improved to represent suspended judgment properly. After an inquiry about the nature of suspended judgment and introducing its characteristics, I observe and suggest some improvement in various theories in formal epistemology namely AGM belief revision, Indeterministic belief revision, Bayesian Epistemology, Dempster Shafer theory of evidence, and Ranking Theory. The text also suggests a new theory, the Acceptance Revision, which can properly represent all doxastic attitudes namely belief, disbelief, suspended judgment and ignorance. The distinction between ignorance and suspension xv FORMALE DARSTELLUNGEN VON URTEILSENTHALTUNG Zusammenfassung Diese Arbeit untersucht verschiedene Theorien der formalen Erkenntnistheorie, um ihre Leistungsfähigkeit dahingehend zu beurteilen, ob sie die doxastische Einstellung der Urteilsenthaltung von bloßem Nichtwissen unterscheiden können. Außerdem werden in der Arbeit Vorschläge gemacht, wie jede Theorie in der formalen Erkenntnistheorie geändert und verbessert werden kann, um die Urteilsenthaltung angemessen abzubilden. Nach einer Untersuchung des Phänomens der Urteilsenthaltung und einer Einführung in ihre Charakteristika, schlage ich eine Verbesserung verschiedener Theorien in der formalen Erkenntnistheorie vor, nämlich AGM Belief Revision, Indeterministic Belief Revision, Bayesian Epistemology, Dempster Shafer Theory of Evidence und Ranking Theory. In meiner Arbeit schlage ich auch eine neue Theorie vor, die ich Acceptance Revision nenne, die alle doxastischen Einstellungen richtig darstellen kann, nämlich Glaube, Ablehnung, Urteilsenthaltung und Nichtwissen. 0 Introduction Ideas and perspectives related to the suspension of judgment are scattered like an archipelago. This text gives a detailed map and insight into the meaning of suspension of judgment and its formal representation in various prevalent theories in formal epistemology. I found that reaching a unified account of suspended judgment is not achievable (it was not acheivable regarding belief as well). During the inquiry, I tried to keep the plurality of various possible approaches, written and unwritten; to help the reader to form her ideas if she does not agree with the conclusion. In the first chapter, the nature of suspended judgment is discussed. There are five key questions which shed light on the nature of the suspension of judgment (suspension). This inquiry leads us into ten formulae which hold in the entire Introduction 2 dissertation. This chapter contains the main assumptions, the adopted philosophical approaches, and the definitions of basic and primary concepts. From the second chapter, we play the game of mathematics-meet-epistemology to find the proper formal representation of suspension. The starting point of the chapter is the preliminary report of the belief revision theory which considers the qualitative notion of belief to represent our doxastic states. The inadequacy of the belief revision in distinguishing ignorance and suspension compels us to turn to indeterministic belief revision. This version can represent formally all four possible doxastic states namely; belief, disbelief, suspension, and ignorance. For a better interpretation of the indeterministic belief revision, the four-valued logic by Belnap is applied. The chapter plays an instructive role in understanding the issues related to the formal representation of suspended judgment. Chapter three addresses the quantitative notion of belief by discussing Bayesian epistemology. Bayesian probability and the development of probability theory by Kolmogorov helped epistemologists to construct a formal representation of degrees of belief. Bayesian epistemology is the received view in formal epistemology. One of its pillars is the Dutch book argument. I give some counterexample to invalidate the theory. Also, it fails to distinguish suspension and ignorance. The problem arises because of the principle of indifference. We learn about Dempster-Shafer theory of evidence as a generalization of Bayesian epistemology in chapter four. The principle of indifference would not be applied in the theory of evidence.