A Primer of Probability Logic

1 Deduction and Probability: What Probability Logic Is About 1
1.1 Deduction and Certainty 1
1.2 Inference and Probability Change: Nonmonotonicity 3
1.3 Conditionals 4
1.4 Decision and Action: The Advantage of Being Right 5
1.5 Summary and Limitations of This Work 7
1.6 Glossary 9
2 Probability and Logic 11
2.1 Logical Symbolism and Basic Concepts 11
2.2 Fundamental Connections between Logic and Probability 14
2.3 Probability Functions and Algebra 20
2.4 Deductive Theory of Probability 21
2.5 Probabilistic Independence, Symmetry, and Randomness 28
2.6 Uncertainty and Its Laws 31
2.7 Glossary 34
3 Deduction and Probability Part I: Statics 37
3.1 Introduction 37
3.2 Two Hauptsatze of the Theory of Uncertainty Accumulation 38
3.3 Reinforcing Conclusions by Introducing Redundancy among Premises: Degrees of Essentialness 41
3.4 Remarks on Defaults and Other Assumptions That Limit Conclusion Uncertainties 48
3.5 Glossary 53
4 Conditional Probabilities and Conditionalization 55
4.1 Introduction: The Dependence of Probabilities on States of Knowledge 55
4.2 The Formalism of Conditional Probability 57
4.3 The Chain Rule: Probabilistic Dependence and Independence 59
4.4 Ersatz Formulas and Truth Values 63
4.5 Probability Change and Bayes' Principle 65
4.6 On Induction 70
4.7 Induction and Symmetry 76
4.8 Critique of Bayesian Epistemology 84
4.9 Glossary 85
5 Deduction and Probability Part II: Dynamics 89
5.1 Introduction 89
5.2 Deductive Dynamics, General 90
5.3 Nonmonotonicity and Scope: The Two-premise Case 92
5.4 What Follows from a Contradiction 96
5.5 More General Inferences 97
5.6 Defaults: The Logic of Conversation Again 101
5.7 Posterior Premise Uncertainty 107
5.8 Glossary 110
6 Probability Conditionals: Basics 113
6.1 Introduction 113
6.2 The Paradoxes of Material Implication 115
6.3 Intuitive Analysis of Other Inference Patterns 117
6.4 Probabilistic Validity 131
6.5 Order-of-Magnitude Orderings and Distributions 133
6.6 Enthymemes and Other Defaults 141
6.7 Problems of Inferential Dynamics 145
6.8 Glossary 148
7* Formal Theory of Probability Conditionals: Derivations and Related Matters 149
7.1 Aspects of Derivation: Basic Definitions and Equivalences 149
7.2 Rules of Conditional Inference 153
7.3 Derived Rules of Inference: Shortcut Derivations 159
7.4 Quasi-conjunction 164
7.5 An Ersatz Truth-table Test for Probabilistic Validity 166
7.6 Validity and Completeness, with Sketches of Their Proofs 171
7.7 Other Aspects of Metatheory 175
7.8 P-tautology, Equivalence, and Consistency 179
7.9 On Probabilistic Certainty Formulas 184
7.10 Glossary 186
8* Truth, Triviality, and Controversy 189
8.1 Problems, and Some History 189
8.2 Truth-functionality 190
8.3 Truth-conditionality: Stalnaker's Theory 192
8.4 The Problem of Probability 196
8.5 Triviality Results 197
8.6 The Controversy about Truth: Theories of Truth 200
8.7 Glossary 203
9 Practical Reason 205
9.1 Introduction 205
9.2 Practical Inference: A Qualitative Representation 207
9.3 A Pragmatic Principle 208
9.4 Decision Matrices 210
9.5 First Generalization: Degrees of Confidence and Utility 212
9.6 Critical Degrees of Confidence 216
9.7 Measuring Confidence and Utility 220
9.8 Decision Making in the Nonindependence Case 223
9.9 Unorthodox Reflections on Practical Probabilities 227
9.10 Glossary 236
A1 Coherent Degrees of Confidence: Axiomatics and Rational Betting Odds 239
1.1 Introduction 239
1.2 The Axiomatic Approach 240
1.3 Rational Betting Odds and Dutch Books 242
1.4 Dynamical Dutch Books 247
1.5 Critique 248
A2 Infinitesimal Probabilities and Popper Functions 251
2.1 Introduction 251
2.2 Two Objections to the Stipulation 251
2.3 Intuitive Infinitesimals and Non-standard Probabilities 253
2.4 Popper Functions 256
A3 David Lewis's Triviality Results 260
3.1 Introduction 260
3.2 The Results 260
3.3 A Way Out 262
3.4 Discussion 265
A4 The Problem of Embedded Conditionals 268
4.1 Introduction 268
4.2 Some Partial Theories 269
A5 A Geometrical Argument for the 'Statistical Reasonableness' of Contraposition 276
A6 Counterfactual Conditionals 281
A7 Probabilistic Predicate Logic 285
7.1 Introduction 285
7.2 Language, Worlds, and Probabilities 285
7.3 Probabilistic Validity 287
7.4 Dynamics and Scope of Universal Statements: Confirmation 289
A8 Probabilistic Identity Logic 294
8.1 Generalities 294
8.2 Complications: Connections with Frege's Puzzle 296
A9 Approximate Generalizations 300
9.1 Introduction 300
9.2 Approximate Monadic Generalizations 301
9.3 Multi-variate Approximate Generalizations 304
9.4 Generalizations and Conditionals: Degrees of Truth and Probability 305
Answers to Selected Exercises 311
References 361
Name Index 371
Subject Index 373