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Contents Preface ix 1 Introduction and Overview 1 Notes 5 2 The HP Deﬁnition of Causality 9 2.1 Causal Models 10 2.2 A Formal Deﬁnition of Actual Cause 20 2.2.1 A language for describing causality 20 2.2.2 The HP deﬁnition of actual causality 22 2.3 Examples 28 2.4 Transitivity 41 2.5 Probability and Causality 46 2.6 Sufﬁcient Causality 53 2.7 Causality in Nonrecursive Models 56 2.8 AC2(bo ) vs. AC2(bu ) 58 2.9 Causal Paths 62 2.10 Proofs 64 2.10.1 Proof of Theorem 2.2.3 64 2.10.2 Proof of Proposition 2.4.6 65 2.10.3 Proof of Proposition 2.9.2 67 Notes 70 3 Graded Causation and Normality 77 3.1 Defaults, Typicality, and Normality 77 3.2 Extended Causal Models 79 3.3 Graded Causation 85 3.4 More Examples 86 3.4.1 Knobe effects 86 3.4.2 Bogus prevention 88 3.4.3 Voting examples 91 3.4.4 Causal chains 92 3.4.5 Legal doctrines of intervening causes 94 3.5 An Alternative Approach to Incorporating Normality 97 Notes 102 vii viii Contents 4 The Art of Causal Modeling 107 4.1 Adding Variables to Structure a Causal Scenario 108 4.2 Conservative Extensions 114 4.3 Using the Original HP Deﬁnition Instead of the Updated Deﬁnition 116 4.4 The Stability of (Non-)Causality 117 4.5 The Range of Variables 123 4.6 Dependence and Independence 124 4.7 Dealing With Normality and Typicality 126 4.8 Proofs 127 4.8.1 Proof of Lemma 4.2.2 127 4.8.2 Proof of Theorem 4.3.1 128 4.8.3 Proofs and example for Section 4.4 131 Notes 135 5 Complexity and Axiomatization 139 5.1 Compact Representations of Structural Equations 140 5.2 Compact Representations of the Normality Ordering 142 5.2.1 Algebraic plausibility measures: the big picture 143 5.2.2 Piggy-backing on the causal model 148 5.3 The Complexity of Determining Causality 151 5.4 Axiomatizing Causal Reasoning 154 5.5 Technical Details and Proofs 158 5.5.1 Algebraic plausibility measures: the details 158 5.5.2 Proof of Theorems 5.3.1(c) and 5.3.2(b) 160 5.5.3 Proof of Theorems 5.4.1, 5.4.2, and 5.4.4 161 Notes 167 6 Responsibility and Blame 169 6.1 A Naive Deﬁnition of Responsibility 171 6.2 Blame 173 6.3 Responsibility, Normality, and Blame 179 Notes 183 7 Explanation 187 7.1 Explanation: The Basic Deﬁnition 188 7.2 Partial Explanations and Explanatory Power 192 7.3 The General Deﬁnition of Explanation 199 Notes 201 8 Applying the Deﬁnitions 203 8.1 Accountability 204 8.2 Causality in Databases 205 8.3 Program Veriﬁcation 207 8.4 Last Words 211 Notes 212 References 215 Index 225 ...