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Reasoning in Quantum Theory : Sharp and Unsharp Quantum Logics - Maria Luisa Dalla Chiara

Reasoning in Quantum Theory

Sharp and Unsharp Quantum Logics

Hardcover Published: 31st March 2004
ISBN: 9781402019784
Number Of Pages: 298

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"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.

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)

List of Figures
List of Tables
Preface
Acknowledgements
Mathematical and Physical Background
Introduction
The mathematical scenario of quantum theory and von Neumann's axiomatization
Algebraic structures
The geometry of quantum theory
The axiomatization of orthodox QT
The "logic" of the quantum events
The logico-algebraic approach to QT
Abstract axiomatic foundations of sharp QT
Mackey's minimal axiomatization of QT
Events
Event-state systems
Event-state systems and preclusivity spaces
Back to Hilbert space
Events as closed subspaces
Events as projections
Hilbert event-state systems
From abstract orthoposets of events to Hilbert lattices
The emergence of fuzzy events in Hilbert space quantum theory
The notion of effect
Effect-Brouwer Zadeh posets
MacNeille completions
Unsharp preclusivity spaces preclusivity spaces
Effect algebras and quantum MV algebras
Effect algebras and Brouwer Zadeh effect algebras
The Lukasiewwicz operations
MV algebras and QMV algebras
Quasi-linear QMV algebras and effect algebras
Abstract axiomatic foundations of unsharp quantum theory
A minimal axiomatization of unsharp QT
The algebraic structure of abstract effects
The sharply dominating principle
Abstract unsharp preclusivity spaces
Sharp and unsharp abstract quantum theory
To what extent is quantum ambiguity ambiguous
Algebraic notions of "sharp"
Probabilistic definitions of "sharpness"
Quantum Logics as Logic
Introduction
Sharp quantum logics
Algebraic and Kripkean semantics for sharp quantum logics
Algebraic and Kripkean realizations of Hilbert event-state systems
The implication problem in quantum logic
Five polynomial conditions
The quantum logical conditional as a counterfactual conditional
Implication-connectives
Metalogical properties and anomalies of quantum logic
The failure of the Lindenbaum properety
A modal interpretation of sharp quantum logics
An axiomatization of OL and OQL
The calculi of OL and OQL
The soundness and completeness theorems
The metalogical intractability of orthomodularity
Orthomodularity is not elementary
The embeddability problem
Hilbert quantum logic and the orthomodular law
First-order quantum logics and quantum set theories
First-order semantics
Quantum set theories
Partial classical logic, the Lindenbaum property and the hidden variable problem
Partial classical logic
Partial classical logic and the Lindenbaum property
States on partial Boolean algebras
The Lindenbaum property and the hidden variable problem
Unsharp quantum logics
Paraconsistent quantum logic
&egr;-Preclusivity spaces
An aside: similarities of PQL and historiography
The Brouwer Zadeh logics
The weak Brouwer Zadeh logic
The pair semantics and the strong Brouwer Zadeh logic
ZL3-effect realizations
Partial quantum logics and Łukasiewicz's quantum logic
Partial quantum logics
Łukasiewicz' quantum logic
The intuitive meaning of the Łukasiewicz' quantum logical connectives
Quantum computational logic
Quantum logical gates
The probabilistic content of the quantum logical gates
Quantum computational semantics
Conclusions
Synoptic Tables
Bibliography
Index of Symbols
Subject Index
Index of Names
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9781402019784
ISBN-10: 1402019785
Series: Trends in Logic
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 298
Published: 31st March 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 27.9 x 21.0  x 2.54
Weight (kg): 1.39