STUDIES IN INDUCTIVE PROBABILITY AND RATIONAL EXPECTATION
SYNTH ESE LIBRARY MONOGRAPHS ON EPISTEMOLOGY, LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE, SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE, AND ON THE MATHEMATICAL METHODS OF SOCIAL AND BEHAVIOURAL SCIENCES Managing Editor: J A A K K 0 HI N TI K K A, A cademy of Finland and Stanford University Editors: ROBERT S. COHEN, Boston University f)onald DAVIDSON, University of Chicago (iabriel NUCHELMANS, University of Leyden WESLEY (' SALVlON, University of Arizona VOLUME 123
STUDIES IN INDUCTIVE PROBABILITY AND RATIONAL EXPECT ATION by THEO A. F. KUIPERS Depilrrmell{ of Philosophy. Ulliversitl' of Groningen D, REIDEL PUBLISHING COMPANY DORDRECHl : HOLLAND BOSTON: U.S.A.
Library of Congress Cataloging in Publication Data Kuipers, Theodorus A. F. 1947- Studies in inductive probability and rational expectation. (Synthese library; v. 123) Bibliography: p. Includes indexes. I. Probabilities. 2. Induction (Mathematics). 3. Knowledge, Theory of. 4. Analysis (Philosophy). l. Title. QA273.K796 519.2 78-677 ISBN-13: 978-94-009-9832-2 e-isbn-13: 978-94-009-9830-8 001: 10.1007/978-94-009-9830-8 Published by D. Reidel Publishing Company, P.O. Box 17, Dordrecht, Holland Sold and distributed in the U.S.A., Canada, and Mexico by D. Reidel Publishing Company, Inc. Lincoln Building, 160 Old Derby Street, Hingham, Mass. 02043, U.S.A. All Rights Reserved Copyright 1978 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1978 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage and retrieval system, without written permission from the copyright owner
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ACK NOWLEDGEMENTS During the research for this book and its completion I have been greatly indebted to Prof. J. J. A. Mooij and Prof. A. J. Starn. The personal encouragement and the philosophical criticism by Prof. Mooij combined with the criticism by and great help on mathematical aspects from Prof. Starn constituted the best conditions one could wish for individual, interdisciplinary research in the common domain of philosophy and mathematics. Without them this book would never have been written. I would also like to thank Prof. E. M. Barth, Dr. 1. F. A. K. van Benthem, Dr. G. Berger, Dr. R. Cooke, Prof. W. K. Essler, Prof. R. C. Jeffrey, Prof. I. N iiniluoto and M. E. de Zoeten for their criticism and suggestions at particular stages of the research. The main point of departure for the research has been the relevant work by Prof. R. Carnap and Prof. J. Hintikka. Without their writings this book could not have been written. From a certain stage on, a new system, invented by Prof. J. Hintikka and Prof. r. Niiniluoto, played a decisive role in the research. The publications by Prof. W. Stegmiiller, Prof. R. Hilpinen and Dr. J. Pietarinen have also been a great help to me. Finally I am very grateful to Prof. Hintikka and Prof. Niiniluoto for their recommendation to have the book published as volume 123 in the Synthese Library. Department 4 Philosophy University of GroninRen The Netherlands August, 1977 vii
T ABLE OF CONTENTS ACKNOWLEDGEMENTS vii I. INTRODUCTION 1. Concept Explication 1 2. Objectives and Survey 4 2. COGNITIVE RATIONALITY 8 1. On the Explication of the Concept of Rationality 8 2. Cognitive Rationality and Patterns of Expectation 10 3. lnductive Reasoning and Inductive Probability Theory 12 3. LOGICO-MATHEMATICAL PRELIMINARIES 16 1. Logical Vocabulary 16 2. Set-theoretical Vocabulary 17 3. Some Elements of Probability Theory 19 4. FORMALLY RATIONAL EXPECTATION IN A PARADIGMATIC CONTEXT 24 I. Paradigmatic Contexts 24 2. Two Conditions for Rational Expectation 25 3. A Framework for a Paradigmatic Context 27 4. First Analysis of a Rational Expectation Pattern 29 5. A Framework for a Paradigmatic Context (continued) 33 6. Third Formal Condition for Rational Expectation 35 7. Decidable Contexts 36 IX
x T ABLE OF CONTENTS 5. GENERALIZED CARNAPIAN SYSTEMS 38 I. Introduction 2. Constitutive Principles and Definition of GC-systems 3. General Analysis of GC-systems I. Some Direct Consequences 2. Generalized Special Values 3. First I nterpretation of GC-systems: the Urn-model (\I < :1) 46 4. Mathematical Expectations According to GC-systems 49 5. Non-inductive (A = ± CD) and Extreme-inductive (A = 0) GC-systems 51 6. Carnapian Systems (C-systems) 52 4. Analysis of Positive Inductive GC-systems (0 < A < (0) 54 1. Possible Reformulations 54 2. Generalized Special Values as Weighted Means 56 3. Second Interpretation of GC-systems: Repeated Experiments Governed by a Density-function (w < (0) 57 4. Principle of Structural Indifference (w < (0): C*- systems (A C~ \1') 59 5. Analysis of Negative Inductive GC-systems (A < 0) 61 I. Possible Reformulations 61 2. Generalized Special Values as Weighted Means (continued) 62 3. Hypergeometric Systems 64 Appendix to Section 2 (Proof of T2) 66 38 38 44 44 45 6. HINTIKKA Al';D UNIVERSALIZED CARNAPIAN SYSTEMS 71 1. Introduction 71 2. NH-systems 71 3. Hintikka-systems (H-systems) 75 4. Some Fundamental Properties of H-systems 79
TABLE OF CONTENTS XI 5. An Urn-model for H-systems 82 6. The Equivalence of NH- and SH-systems: Universalized Carnapian systems (UC-systems) 83 7. Analysis of UC-,ystems 90 I. General 90 2. Structurally Indifferent UC-systems: UC*-systems (p = I) 93 3. Extreme UC-:,ystems: p = 00. p = 0 94 8. Fundamental Discussion Related to Applications 96 9. Finite Parameters for H-systems 99 10. Reformulation of H-systems; k. x 101 II. G H-systems and G UC-systems 104 12. Survey of Systell1~ 106 Appendix to Section 2 (Proof of TI) 109 7. RATIONAL EXPEClATlON I", MULTINOMIAL CONTEXTS 112 1. Carnap's Intended Application 112 2. The Multinomial Context 114 3. Formally Rational Patterns for Open Multinomial Contexts 116 4. Material Conditions of Adequacy; UC-systems as Expectation Pattern for Open Multinomial Contexts 117 5. Constitutional Distri butions for Open Multinomial Contexts 123 6. The Hypergeometric Context 127 8. SOME PROBLEMS AND RELATED TOPICS 129 I. PER-systems 129 2. On Weakening WPERR 131 3. *UC*-systems and k * ~~. 132 4. Confirmation Theory 133 5. Falsification 135 6. Rules of Acceptance in UC-systems 136
xii T ABLE OF CONTENTS 9. CONCLUDING REMARKS 139 REFERENCES INDEX OF NAMES INDEX OF SUBJECTS Recurring Symbols Conditions/Principles/Axioms Definition of Systems 141 143 144 145 145 145