REASONING IN QUANTUM THEORY TRENDS IN LOGIC Studia Logica Library VOLUME 22 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany SCOPE OF THE SERIES Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, com- parisons and sources of inspiration is open and evolves over time. Volume Editor Ryszard Wójcicki The titles published in this series are listed at the end of this volume. REASONING IN QUANTUM THEORY Sharp and Unsharp Quantum Logics by M. DALLA CHIARA University of Florence, Italy R. GIUNTINI University of Cagliari, Italy and R. GREECHIE Louisiana Tech University, Ruston, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6562-9 ISBN 978-94-017-0526-4 (eBook) DOI 10.1007/978-94-017-0526-4 Printed on acid-free paper All Rights Reserved ©2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Contents List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi PART I Mathematical and Physical Background . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 1. The mathematical scenario of quantum theory and von Neumann’s axiomatization . . . . . . . . . . . . . . . . . 9 1.1. Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . 9 1.2. The geometry of quantum theory . . . . . . . . . . . . . . . . 24 1.3. The axiomatization of orthodox QT. . . . . . . . . . . . . . . 31 1.4. The “logic” of the quantum events . . . . . . . . . . . . . . . 34 1.5. The logico-algebraic approach to QT . . . . . . . . . . . . . . 38 Chapter 2. Abstract axiomatic foundations of sharp QT . . . . . . . 41 2.1. Mackey’s minimal axiomatization of QT . . . . . . . . . . . . 42 2.2. Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3. Event-state systems . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4. Event-state systems and preclusivity spaces . . . . . . . . . . 55 Chapter 3. Back to Hilbert space . . . . . . . . . . . . . . . . . . . . 65 3.1. Events as closed subspaces . . . . . . . . . . . . . . . . . . . . 65 3.2. Events as projections . . . . . . . . . . . . . . . . . . . . . . . 67 3.3. Hilbert event-state systems . . . . . . . . . . . . . . . . . . . 68 3.4. From abstract orthoposets of events to Hilbert lattices . . . . 70 Chapter 4. The emergence of fuzzy events in Hilbert space quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.1. The notion of effect . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2. Effect-Brouwer Zadeh posets . . . . . . . . . . . . . . . . . . . 77 4.3. Mac Neille completions . . . . . . . . . . . . . . . . . . . . . . 81 4.4. Unsharp preclusivity spaces . . . . . . . . . . . . . . . . . . . 82 Chapter 5. Effect algebras and quantum MV algebras . . . . . . . . 87 v vi CONTENTS 5.1. Effect algebras and Brouwer Zadeh effect algebras . . . . . . . 87 5.2. The L(cid:3) ukasiewicz operations . . . . . . . . . . . . . . . . . . . 94 5.3. MV algebras and QMV algebras . . . . . . . . . . . . . . . . . 96 5.4. Quasi-linear QMV algebras and effect algebras. . . . . . . . . 107 Chapter6. Abstractaxiomaticfoundationsofunsharpquantumtheory115 6.1. A minimal axiomatization of unsharp QT . . . . . . . . . . . 115 6.2. The algebraic structure of abstract effects . . . . . . . . . . . 119 6.3. The sharply dominating principle . . . . . . . . . . . . . . . . 124 6.4. Abstract unsharp preclusivity spaces . . . . . . . . . . . . . . 127 6.5. Sharp and unsharp abstract quantum theory . . . . . . . . . . 131 Chapter 7. To what extent is quantum ambiguity ambiguous? . . . . 137 7.1. Algebraic notions of “sharp” . . . . . . . . . . . . . . . . . . . 137 7.2. Probabilistic definitions of “sharpness” . . . . . . . . . . . . . 142 PART II Quantum Logics as Logic . . . . . . . . . . . . . . . . 147 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Chapter 8. Sharp quantum logics . . . . . . . . . . . . . . . . . . . . 155 8.1. Algebraic and Kripkean semantics for sharp quantum logics . 155 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 8.3. The implication problem in quantum logic . . . . . . . . . . . 164 8.4. Five polynomial conditionals . . . . . . . . . . . . . . . . . . . 165 8.5. Thequantumlogicalconditionalasacounterfactualconditional167 8.6. Implication-connectives . . . . . . . . . . . . . . . . . . . . . . 168 Chapter 9. Metalogical properties and anomalies of quantum logic . 171 9.1. The failure of the Lindenbaum property . . . . . . . . . . . . 171 9.2. A modal interpretation of sharp quantum logics . . . . . . . . 174 Chapter 10. An axiomatization of OL and OQL . . . . . . . . . . . 179 10.1. The calculi for OL and OQL. . . . . . . . . . . . . . . . . . 179 10.2. The soundness and completeness theorems . . . . . . . . . . 181 Chapter 11. The metalogical intractability of orthomodularity . . . . 185 11.1. Orthomodularity is not elementary . . . . . . . . . . . . . . 186 11.2. The embeddability problem . . . . . . . . . . . . . . . . . . . 188 11.3. Hilbert quantum logic and the orthomodular law . . . . . . . 189 Chapter 12. First-order quantum logics and quantum set theories . . 193 12.1. First-order semantics . . . . . . . . . . . . . . . . . . . . . . 193 12.2. Quantum set theories . . . . . . . . . . . . . . . . . . . . . . 198 Chapter 13. Partial classical logic, the Lindenbaum property and the hidden variable problem . . . . . . . . . . . . . . . . . . 201 13.1. Partial classical logic . . . . . . . . . . . . . . . . . . . . . . 201 CONTENTS vii 13.2. Partial classical logic and the Lindenbaum property . . . . . 208 13.3. States on partial Boolean algebras . . . . . . . . . . . . . . . 210 13.4. The Lindenbaum property and the hidden variable problem. 214 Chapter 14. Unsharp quantum logics . . . . . . . . . . . . . . . . . . 217 14.1. Paraconsistent quantum logic . . . . . . . . . . . . . . . . . . 217 14.2. ε-Preclusivity spaces . . . . . . . . . . . . . . . . . . . . . . . 220 14.3. An aside: similarities of PQL and historiography . . . . . . 222 Chapter 15. The Brouwer Zadeh logics . . . . . . . . . . . . . . . . . 225 15.1. The weak Brouwer Zadeh logic . . . . . . . . . . . . . . . . . 225 15.2. The pair semantics and the strong Brouwer Zadeh logic . . . 228 15.3. BZL3-effect realizations . . . . . . . . . . . . . . . . . . . . 234 Chapter 16. Partial quantum logics and L(cid:3) ukasiewicz’ quantum logic 237 16.1. Partial quantum logics . . . . . . . . . . . . . . . . . . . . . 237 16.2. L(cid:3) ukasiewicz quantum logic . . . . . . . . . . . . . . . . . . . 241 16.3. The intuitive meaning of the L(cid:3) ukasiewicz’ quantum logical connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Chapter 17. Quantum computational logic . . . . . . . . . . . . . . . 249 17.1. Quantum logical gates. . . . . . . . . . . . . . . . . . . . . . 252 17.2. The probabilistic content of the quantum logical gates . . . . 259 17.3. Quantum computational semantics. . . . . . . . . . . . . . . 262 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Synoptic tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 Index of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Index of Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 List of Figures 1.1.1 The Benzene ring . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 MO2: the smallest OML that is not a BA . . . . . . . . 15 1.1.3 The Greechie diagram of G12 . . . . . . . . . . . . . . . 17 1.1.4 The Hasse diagram of G12 . . . . . . . . . . . . . . . . . 18 1.4.1 Failure of bivalence in QT . . . . . . . . . . . . . . . . . 37 2.2.1 J18: the smallest OMP that is not an OML . . . . . . . 51 2.4.1 The Greechie diagram of GGM410 . . . . . . . . . . . . 59 2.4.2 The state s0 on GGM410 . . . . . . . . . . . . . . . . . . 60 5.1.1 WT: the smallest OA that is not an OMP . . . . . . . 91 5.3.1 M4: the smallest QMV that is not an MV. . . . . . . . 106 5.4.1 The operation ⊕ of M . . . . . . . . . . . . . . . . . . 108 wl 5.4.2 The Hasse diagram of M . . . . . . . . . . . . . . . . 109 wl 6.5.1 The Greechie diagram of G52 . . . . . . . . . . . . . . . 134 6.5.2 The Greechie diagram of G58 . . . . . . . . . . . . . . . 135 9.1.1 Quasi-model for γ in R2 . . . . . . . . . . . . . . . . . . 173 11.3.1 The Greechie diagram of G30 . . . . . . . . . . . . . . . 190 17.1.1 A noncontinuous fuzzy square root of the negation . . . 260 ix