Reasoning in Quantum Theory

Reasoning in Quantum Theory : Sharp and Unsharp Quantum Logics

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Description

"Is quantum logic really logic?" This book argues for a positive answer to this question once and for all. There are many quantum logics and their structures are delightfully varied. The most radical aspect of quantum reasoning is reflected in unsharp quantum logics, a special heterodox branch of fuzzy thinking.
For the first time, the whole story of Quantum Logic is told; from its beginnings to the most recent logical investigations of various types of quantum phenomena, including quantum computation. Reasoning in Quantum Theory is designed for logicians, yet amenable to advanced graduate students and researchers of other disciplines.
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Product details

  • Paperback | 298 pages
  • 160 x 240 x 17.02mm | 498g
  • Dordrecht, Netherlands
  • English
  • Softcover reprint of hardcover 1st ed. 2004
  • XXII, 298 p.
  • 9048165628
  • 9789048165629

Table of contents

List of Figures. List of Tables. Preface. Acknowledgements. I: Mathematical and Physical Background. Introduction. 1. The mathematical scenario of quantum theory and von Neumann's axiomatization. 1.1. Algebraic structures. 1.2. The geometry of quantum theory. 1.3. The axiomatization of orthodox QT. 1.4. The 'logic' of the quantum events. 1.5. The logico-algebraic approach to QT. 2. Abstract axiomatic foundations of sharp QT. 2.1. Mackey's minimal axiomatization of QT. 2.2. Events. 2.3. Event-state systems. 2.4. Event-state systems and preclusivity spaces. 3. Back to Hilbert space. 3.1. Events as closed subspaces. 3.2. Events as projections. 3.3. Hilbert event-state systems. 3.4. From abstract orthoposets of events to Hilbert lattices. 4. The emergence of fuzzy events in Hilbert space quantum theory. 4.1. The notion of effect. 4.2. Effect-Brouwer Zadeh posets. 4.3. MacNeille completions. 4.4. Unsharp preclusivity spaces. 5. Effect algebras and quantum MV algebras. 5.1. Effect algebras and Brouwer Zadeh effect algebras. 5.2. The Lukasiewwicz operations. 5.3. MV algebras and QMV algebras. 5.4. Quasi-linear QMV algebras and effect algebras. 6. Abstract axiomatic foundations of unsharp quantum theory. 6.1. A minimal axiomatization of unsharp QT. 6.2. The algebraic structure of abstract effects. 6.3. The sharply dominating principle. 6.4. Abstract unsharp preclusivity spaces. 6.5. Sharp and unsharp abstract quantum theory. 7. To what extent is quantum ambiguity ambiguous? 7.1. Algebraic notions of 'sharp'. 7.2. Probabilistic definitions of 'sharpness'. II: Quantum Logics as Logic. Introduction. 8. Sharp quantum logics. 8.1. Algebraic and Kripkean semantics for sharp quantum logics. 8.2. Algebraic and Kripkean realizations of Hilbert event-state systems. 8.3. The implication problem in quantum logic. 8.4. Five polynomial conditions. 8.5. The quantum logical conditional as a counterfactual conditional. 8.6. Implication-connectives. 9. Metalogical properties and anomalies of quantum logic. 9.1. The failure of the Lindenbaum properety. 9.2. A modal interpretation of sharp quantum logics. 10. An axiomatization of OL and OQL. 10.1. The calculi of OL and OQL. 10.2. The soundness and completeness theorems. 11. The metalogical intractability of orthomodularity. 11.1. Orthomodularity is not elementary. 11.2. The embeddability problem. 11.3. Hilbert quantum logic and the orthomodular law. 12. First-order quantum logics and quantum set theories. 12.1. First-order semantics. 12.2. Quantum set theories. 13. Partial classical logic, the Lindenbaum property and the hidden variable problem. 13.1. Partial classical logic. 13.2. Partial classical logic and the Lindenbaum property. 13.3. States on partial Boolean algebras. 13.4. The Lindenbaum property and the hidden variable problem. 14. Unsharp quantum logics. 14.1.
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Review Text

From the reviews:

"The aim of the book is to present various logical investigations of quantum phenomena including ... quantum computation. ... The book is accomplished with a bibliography of 180 items. It is written in a very nice and fresh style ... . The authors are keeping the interest of the reader permanently on his trip through the book. ... The monograph is welcome in the quantum logic community, and it surely enriches the literature on quantum structures ... ." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1059 (10), 2005)

"The book tries to delineate 'what we know of' quantum logics ... . The richness of the material of this book is impressive ... . this book is really important and deserves a place in any philosophical or technical library." (Décio Krause, Mathematical Reviews, Issue 2005 j)

"This book is included as Volume 22 in the distinguished Book Series on 'Trends in Logic' published by Kluwer. ... The book is written for multidisciplinary audience. ... the presentation is human friendly, combining a simple introduction to essential mathematical concepts with a historical overview of their relevance to quantum theory. For anyone interested in quantum theory, this book is an essential reference." (George J. Klir, International Journal of General Systems, Vol. 34 (3), 2005)
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Review quote

From the reviews:





"The aim of the book is to present various logical investigations of quantum phenomena including ... quantum computation. ... The book is accomplished with a bibliography of 180 items. It is written in a very nice and fresh style ... . The authors are keeping the interest of the reader permanently on his trip through the book. ... The monograph is welcome in the quantum logic community, and it surely enriches the literature on quantum structures ... ." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1059 (10), 2005)


"The book tries to delineate 'what we know of' quantum logics ... . The richness of the material of this book is impressive ... . this book is really important and deserves a place in any philosophical or technical library." (Decio Krause, Mathematical Reviews, Issue 2005 j)


"This book is included as Volume 22 in the distinguished Book Series on 'Trends in Logic' published by Kluwer. ... The book is written for multidisciplinary audience. ... the presentation is human friendly, combining a simple introduction to essential mathematical concepts with a historical overview of their relevance to quantum theory. For anyone interested in quantum theory, this book is an essential reference." (George J. Klir, International Journal of General Systems, Vol. 34 (3), 2005)
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