For the Sake of the Argument For the Sake of the Argument Ramsey Test Conditionals, Inductive Inference, and Nonmonotonic Reasoning ISAAC LEVI Columbia University I CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521497138 © Cambridge University Press 1996 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1996 This digitally printed version 2007 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Levi, Isaac, 1930- For the sake of the argument: Ramsey test conditionals, inductive inference, and nonmonotonic reasoning / Isaac Levi. p. cm. Includes bibliographical references (p. ). ISBN 0-521-49713-2 (hbk.) 1. Hypothesis. 2. Commonsense reasoning. 3. Conditionals (Logic) 4. Induction (Logic) 5. Inference. I. Title. BC183.L48 1996 160-dc20 95-2436 ISBN 978-0-521-49713-8 hardback ISBN 978-0-521-03901-7 paperback For Jonathan and David Contents Preface xi 1 Introduction 1 1.1. Supposing and Believing 1 1.2. Supposition and Conditionals 8 1.3. Belief-Contravening and Inductive Nonmonotonicity 15 2 Unextended Ramsey Tests 18 2.1. Contractions 18 2.2. Revisions for Ramsey Tests 27 2.3. Informational-Value-Based Ramsey Tests 34 2.4. Consistent Expandability and Preservation 40 2.5. Consensus-Based Ramsey Tests 42 2.6. Attitudes, Their Representations, and Manifestations 43 2.7. Conditionals and Hypothetical 46 3 Modality without Modal Ontology 51 3.1. Truth, Information, and Full Belief 51 3.2. Are Judgments of Serious Possibility Truth Valued? 53 3.3. Truth and Probability 55 3.4. Conditional Judgments of Possibility 57 3.5. Reduction of Acceptance of Conditionals to Full Belief 58 3.6. Trivializing the Ramsey Test 60 3.7. Iteration 69 3.8. Imaging 74 3.9. Conclusion 82 4 Aspects of Conditional Logic 84 4.1. Logical Principles 84 4.2. Validity in the Logic of Conditionals 88 4.3. Logic of Noniterated Ramsey Test Conditionals 101 4.4. The Demise of Modus Ponens 105 4.5. Undermining and Undercutting 112 4.6. Reasoning from Suppositions 113 vii viii Contents 5 Nonmonotonicity in Belief Change and Suppositional Reasoning 120 5.1. Inference and Transformation of Belief States 120 5.2. Expansive Transformations 127 5.3. Monotonic and Nonmonotonic Expansive Transformations 132 5.4. Nonmonotonic Inferences 136 5.5. Inductively Extended Ramsey Tests 141 5.6. Ampliativity 142 5.7. Ampliativity and Inclusion 146 5.8. Nonmonotonicity and Restricted Weakening 151 5.9. Monotonicity, Undermining, and Undercutting 154 5.10. Ampliativity and Nonmonotonicity 158 6 Inductive Expansion 160 6.1. Expansion and Context 160 6.2. Potential Answers 161 6.3. New Error-Free Information 165 6.4. Freedom from Error 166 6.5. Informational Value 167 6.6. Boldness and Inductive Expansion 171 6.7. Inductively Extended Belief Revision Systems 173 6.8. High Probability, Partition Sensitivity, Deductive Closure 176 6.9. Belief and Disbelief 180 6.10. Bookkeeping 188 6.11. What Is Left of Conditional Logic? 191 6.12. Induction as a Generalization of Deduction 193 Appendix 195 7 Defaults 200 7.1. Defaults and Nonmonotonic Implication 200 7.2. What Is a Default? 207 7.3. Simple Normal Defaults and Nonmonotonicity 214 7.4. The Default Assumption 216 7.5. Foundationalism 223 7.6. Foundationalism and Defaults 227 7.7. Defaults and Bookkeeping 229 7.8. Defaults and Partition Sensitivity 230 8 Matters of Degree 234 8.1. Default Conclusions as Degrees of Belief 234 8.2. Probability as Maximizing Support 239 Contents ix 8.3. Probability as Satisficing Support 243 8.4. Satisficing Support as a Shackle Measure 256 8.5. Degrees of Incorrigibility 261 9 Normality and Expectation 269 9.1. Defaults and Normal Worlds 269 9.2. Expectations and Inductive Expansion 278 9.3. Probability-Based Induction 285 10 Agents and Automata 289 Notes 292 Bibliography 326 Name Index 333 Subject Index 335