Hume, Induction, and Probability Peter J.R. Millican The University of Leeds Department of Philosophy Submitted in accordance with the requirements for the degree of PhD, May 1996. The candidate confirms that the work submitted is his own and that appropriate credit has been given where reference has been made to the work of others. ii iii Abstract The overall aim of this thesis is to understand Hume’s famous argument concerning induction, and to appraise its success in establishing its conclusion. The thesis accordingly falls into two main parts, the first being concerned with analysis and interpretation of the argument itself, and the second with investigation of possible responses to it. Naturally the argument’s interpretation strongly constrains the range of possible replies, and indeed the results of Part I indicate that the only kind of strategy which stands much prospect of defeating Hume’s argument is one based on a priori probabilistic reasoning – hence the overwhelming majority of Part II is devoted to a thorough investigation of this approach. Leaving aside the many incidental discussions of others’ work, the principal novel ideas in Part I of the thesis concern: (a) Clarification of Hume’s distinction between “demonstrative” and “probable” reasoning (§1.4); (b) Identification of arbitrariness as a key notion in the argument (§2.2); (c) Substantiation of the argument’s normative significance (§2.4); (d) Coherent interpretation of Humean “presupposition” (§2.5, §3.2); (e) Elucidation of Hume’s conclusion and its lack of dependence on his analysis of causation (§§4.1-2); (f) Construction of a complete structure diagram for the argument (§4.3); (g) Refutation of Stove’s well-known alternative diagram (§5.1); (h) Likewise of Stove’s analysis of Hume’s conclusion and his allegation of deductiv- ism (§5.2); and finally (i) Exposition of a three-way ambiguity in Hume’s notion of “reason”, which both clarifies its context in relation to predecessors such as Locke, and iv also explains how the argument can be genuinely sceptical while at the same time making sense of Hume’s unambiguously positive attitude towards science (§§6.1-2). The main discussion of Part II is prepared by (j) A demonstration of the impotence of the traditional non-probabilistic counters to Hume’s argument (§§7.2-5) and (k) A clarification of the available probabilistic perspectives from which a potential justification of induction could begin. The principles of probability theory (based on Jeffreys’ system) are briefly presented, and the presuppositions for what follows set out. Then the heart of Part II consists of four chapters providing a theoretical framework within which are situated and discussed attempted justifications by: (l) De Finetti (§§10.3-4); (m) Williams and Stove (§10.5); (n) Mackie (§§11.2-6); and (n) Harrod and Blackburn (§§12.1-7). All of these fail, but reflection on the “continuing uniformity” strategy of Mackie combined with the scale-invariance hinted at by Harrod surprisingly suggests: (o) A new method of argument based on Jaynes’ solution to the “paradoxes of geometrical probability” (§§13.4-6), which appears to represent the “last best hope” of the probabilistic approach to justifying induction. The thesis ends by asking whether the derivation of this method should be considered as an anti-Humean modus ponens or on the contrary as a sceptical modus tollens. The issues involved here enter formal terrain beyond the scope of the thesis, but nevertheless there are clear indications that the presuppositions of the new argument are themselves insecure and subject to Humean doubt. On the whole, then, and despite a small ray of hope for the inductive probabilist, Hume appears to be vindicated. ____________________ v Contents Preface ix Acknowledgements xiii PART I HUME’S ARGUMENT CONCERNING INDUCTION: STRUCTURE AND INTERPRETATION 1 Chapter 1 The Context and Topic of Hume’s Argument 3 1.1 Introduction 3 1.2 The Treatise and the Enquiry 6 1.3 The Topic of the Argument 10 1.4 “Demonstrative” and “Probable” 14 Chapter 2 Probable Arguments and the Uniformity Principle 19 2.1 The Overall Strategy of Hume’s Argument 19 2.2 “All Probable Arguments are Founded on Experience” 20 2.3 “All Probable Arguments Presuppose that Nature is Uniform” 27 2.4 “Foundation in the Understanding” and “Rational Justification” 30 2.5 The Uniformity Principle and its Presupposition 35 Chapter 3 Seeking a Foundation for the Uniformity Principle 45 3.1 “The Uniformity Principle can only be Justified by Argument” 45 3.2 “There is No Good Argument for the Uniformity Principle” 47 Chapter 4 The Conclusion of Hume’s Argument, and a Coda 55 4.1 Hume’s Conclusion: “Probable Inferences are Not Founded on Reason” 55 4.2 Coda: the Irrelevance of Causal Powers, and a Parting Shot 57 4.3 The Complete Structure of Hume’s Argument 64 vi Chapter 5 Stove’s Analysis and his Interpretation of Hume’s Conclusion 69 5.1 Stove’s Structural Analysis and Hume’s Alleged Deductivism 69 5.2 Stove’s Probabilistic Interpretation of Hume’s Conclusion 75 Chapter 6 Hume, Scepticism, and Reason 81 6.1 Is Hume an Inductive Sceptic? 81 6.2 Three Senses of “Reason” 86 6.3 “Reason” in the Argument Concerning Induction 95 PART II PROBABILISTIC REASONING AS AN ANSWER TO HUME101 Chapter 7 Introduction: Strategies for Refuting Hume 103 7.1 The Shape of Part II 103 7.2 The Inductive Justification 105 7.3 The Analytic Justification 107 7.4 The Pragmatic Justification 109 7.5 The Impotence of the Non-Probabilistic Attempts at Justification 112 7.6 Conclusion: Probability or Bust! 115 Chapter 8 Probability Theory (a) Interpretations and Options 117 8.1 Four Interpretations of Probability 117 8.2 Clarifying the Options 120 8.3 Conceivable Probabilistic Perspectives for the Justification of Induction 127 Chapter 9 Probability Theory (b) Basic Principles 129 9.1 Introduction: Fundamental Principles from Two Bayesian Perspectives 129 9.2 A Summary of Jeffreys’ Axiom System 130 9.3 The Basis of Personalism 139 9.4 Bayes’ Theorem 141 9.5 Prior Probabilities and Indifference 143 vii Chapter 10 Seeking the Individual Probability of Future Events 147 10.1 Introduction: Future Events versus Continuing Uniformity 147 10.1 Bernoullian Trials and the Law of Large Numbers 148 10.2 An Inverse Law of Large Numbers? 151 10.3 De Finetti’s Theorem on Exchangeable Events 154 10.4 What Has De Finetti Proved? 159 10.5 Williams and Stove 165 10.6 A Humean Punch Line: Why These Attempts Must Fail 169 Chapter 11 Continuing Uniformity (a) Mackie 171 11.1 The Probability of Continuing Uniformity: Introduction 171 11.2 A Criterion of Success 174 11.3 Mackie’s Defence of Induction 176 11.4 The Mathematics of Mackie’s Argument 178 11.5 Satisfying the Criterion 181 11.6 The Incompleteness of Mackie’s Argument 182 Chapter 12 Continuing Uniformity (b) Harrod and Blackburn 185 12.1 Harrod’s Initial Argument, and an Objection 185 12.2 Harrod’s Revised “Square Array” Argument 190 12.3 Blackburn’s Discussion of Harrod’s Argument 195 12.4 Two Decisive Objections to Harrod’s Argument 202 12.5 A Further Objection to Blackburn 205 12.6 Scale Invariance and Infinities 207 12.7 Conclusion 210 viii Chapter 13 Continuing Uniformity (c) An Objective Prior? 211 13.1 Introduction: The Need for an “Objective Prior” 211 13.2 Finite and Infinite Applications of the Principle of Indifference 212 13.3 The Paradoxes of Geometrical Probability 214 13.4 Indifference Amongst Problems: Jaynes’ Answer to Bertrand’s Paradox 220 13.5 Deriving an “Objective” Prior for the Extent of Temporal Uniformity 223 13.6 The Log Uniform Distribution 227 13.7 An Inductively Favourable Conclusion? 230 PART III CONCLUSION 233 Chapter 14 Induction Defended or Hume Victorious? 235 Modus Ponens, or Modus Tollens? 235 APPENDICES 239 APPENDIX 1: An authoritative text of Section IV of Hume’s First Enquiry 241 APPENDIX 2: Structure diagrams of the Treatise and Enquiry arguments, with Hume’s statement of their stages 257 APPENDIX 3: A Proof of Chebyshev’s Inequality 267 REFERENCES 271 (a) Works Cited in Part I 272 (b) Works Cited in Parts II and III 274 INDEXES 281 ix Preface This thesis falls into two main parts. In the first, I have attempted to provide a detailed and thorough analysis of Hume’s famous argument concerning induction, focusing in particular on the presentation of that argument in Section IV of his Enquiry Concerning Human Understanding (though Appendix 2 provides a corresponding outline analysis of the argument from Section I iii 6 of the Treatise of Human Nature). In the second part, I look at possible responses to Hume’s argument so interpreted, and in particular, examine the question of whether a consideration of mathematical probability theory could in principle yield an answer to it, by providing a route through the one major logical gap that Hume’s discussion leaves unclosed. My keen interest in Hume goes back to my undergraduate days at Oxford, where I found his clear no-nonsense style a breath of fresh air after an initially exciting but ultimately disillusioning flirtation with Kant. Hume’s incisive and penetrating arguments, particularly on the subjects of causal and religious knowledge, were in many respects a liberation, cutting a commonsense swathe through a jungle of wish- fulfilling natural theology and obscurantist a priori metaphysics. However these same arguments seemed to leave in their wake an unsettling epistemological instability: their sceptical thrust did not strike merely at the ambitious sophistry attacked so wittily in the Dialogues Concerning Natural Religion, but also threatened to undermine the very foundations of the solidly empirical science that Hume had sought to put in its place.