Philosophy and Probability

"Probability is increasingly important for our understanding of the world. What is probability? How do we model it, and how do we use it? Timothy Childers presents a lively introduction to the foundations of probability and to philosophical issues it raises. He keeps technicalities to a minimum, and assumes no prior knowledge of the subject. He explains the main interpretations of probability--frequentist, propensity, classical, Bayesian, and objective Bayesian--and uses stimulating examples to bring the subject to life. All students of philosophy will benefit from an understanding of probability, and this is the book to provide it."--Publisher's description Cover Contents Preface Acknowledgements List of Boxes and Figures 1. Probability and Relative Frequencies 1.1 Introduction 1.2 Von Mises’s Relative Frequency Interpretation 1.2.1 Probability and mass phenomena 1.2.2 Convergence of relative frequency 1.2.3 Randomness—the impossibility of a gambling system 1.2.4 Operations on collectives 1.2.5 Objections to von Mises’s interpretation 1.3 Kolmogorov and Relative Frequencies 1.3.1 Relative frequencies as probabilities—the Kolmogorov axioms 1.3.2 The measure-theoretic framework 1.3.3 Doob’s reinterpretation of von Mises 1.3.4 Van Fraassen’s modal frequency interpretation 1.3.5 Problems with Kolmogorovian interpretations 1.4 Finite Frequency Interpretations 1.5 Conclusion 2. Propensities and Other Physical Probabilities 2.1 Elements of a Propensity Interpretation 2.1.1 Probability as a disposition 2.1.2 Single-case probabilities 2.2 Problems with Propensity Interpretations 2.2.1 Indeterminism and the reference class 2.2.2 Empirical content 2.2.3 Humphreys’s paradox 2.2.4 Why are propensities probabilities? 2.2.5 Are propensities relative frequencies? 2.2.6 Is there a separate propensity interpretation? 2.3 Conclusion 3. Subjective Probability 3.1 Introduction 3.2 Dutch Book Arguments 3.2.1 Fair bets 3.2.2 The forms of bets 3.2.3 How not to bet 3.2.4 Adding up bets and probabilities 3.2.5 Conditional bets and probability 3.3 The Application of Subjective Probabilities 3.3.1 Bayes’s theorem and Bayesian epistemology 3.3.2 Example: Beer 3.3.3 Disconfirmation 3.3.4 Am I this good a brewer?—falsification 3.3.5 Am I this good a brewer?—The Duhem-Quine problem 3.3.6 The Bayesian account of the Duhem-Quine problem 3.3.7 Other Bayesian solutions 3.4 Problems with the Dutch Book Argument 3.4.1 The literal interpretation of the Dutch Book argument 3.4.2 The as-if interpretation 3.4.3 The ‘logical’ interpretation 3.5 Probability from Likelihood 3.5.1 Problems with probabilities from likelihood 3.6 Probabilities from Preferences 3.6.1 Problems with utility theory 3.7 Other Arguments Equating Degrees of Belief and Probability 3.8 Is Bayesianism Too Subjective? 3.8.1 Bayesian learning theory 3.8.2 Convergence of opinion 3.8.3 The problem of induction 3.8.4 Diachronic Dutch Books 3.9 Is Bayesianism Too Flexible? Or Not Flexible Enough? 3.10 Conclusion 4. Subjective and Objective Probabilities 4.1 The Principle of Direct Inference 4.2 Betting on Frequencies 4.3 The Principal Principle 4.3.1 Humean supervenience and best systems analyses of laws 4.3.2 The big bad bug and the New Principle 4.4 Exchangeability 4.5 Conclusion 5. The Classical and Logical Interpretations 5.1 The Origins of Probability—The Classical Theory 5.1.1 The Rule of Succession 5.1.2 The continuous case of the Principle of Indifference 5.2 Problems with the Principle of Indifference 5.2.1 Problems with the Rule of Succession 5.2.2 The paradoxes 5.3 Keynes’s Logical Interpretation 5.3.1 The discrete case and the justification of the Principle of Indifference 5.3.2 Keynes on the continuous case 5.3.3 Keynes on the Rule of Succession 5.4 Carnap 5.4.1 The logical foundations of probability 5.4.2 The continuum of inductive methods 5.5 Conclusion 6. The Maximum Entropy Principle 6.1 Bits and Information 6.2 The Principle of Maximum Entropy 6.2.1 The continuous version of the Principle of Maximum Entropy 6.2.2 Maximum entropy and the paradoxes of geometric probability 6.2.3 Determination of continuous probabilities 6.3 Maximum Entropy and the Wine-Water Paradox 6.3.1 Problems with the solution—dimensions or not? 6.4 Language Dependence 6.4.1 The statistical mechanics counterexample 6.4.2 Correctly applying the Principle? 6.4.3 Language dependence and ontological dependence 6.4.4 The scope of the Maximum Entropy Principle 6.5 Justifying the Maximum Entropy Principle as a Logical Constraint 6.5.1 Maximum entropy as imposing consistency 6.5.2 Problems with the Maximum Entropy Principle as consistency 6.6 Conclusion Appendices A.0 Some Basics A.0.1 Percentages A.0.2 Kinds of numbers A.0.3 Sizes of sets—countable and uncountable A.0.4 Functions, limits A.0.5 Logarithms A.1 The Axioms A.1.1 Conditional probability, independence A.2 Measures, Probability Measures A.2.1 Fields A.2.2 Fields, σ-fields A.2.3 Measures A.2.4 Probability measures A.2.5 Some useful theorems A.3 Random Variables A.3.1 Sums of random variables A.3.2 Expectation A.3.3 Continuous random variables A.4 Combinatorics A.4.1 Permutations A.4.2 Combinations A.5 Laws of Large Numbers A.5.1 Bernoulli random variables and the binomial distribution A.5.2 Laws of Large Numbers A.5.3 Behaviour of the binomial distribution for large numbers of trials A.6 Topics in Subjective Probability A.6.1 Strict coherence A.6.2 Scoring rules A.6.3 Axioms for qualitative and quantitative probabilities A.7 The Duhem-Quine Problem, Language, Metaphysics A.7.1 A probabilistic translation of Quine’s programme References Index A B C D E F G H I J K L M O P Q R S T U V W Z Probabilities And Relative Frequencies -- Propensities And Other Physical Probabilities -- Subjective Probabilities -- Subjective And Objective Probabilities -- The Classical And Logical Interpretations -- The Maximum Entropy Principle. Timothy Childers. Includes Bibliographical References (p. [182]-192) And Index.