Quantum mechanics is said to be the most successful physical theory ever. It is, in fact, unique in its success when applied to concrete physical problems. On the other hand, however, it raises profound conceptual problems that are equally unprecedented. Quantum logic, the topic of this volume, can be described as an attempt to cast light on the puzzle of quantum mechanics from the point of view of logic.
Since its inception in the famous 1936 paper by Birkhoff and von Neumann entitled, "The logic of quantum mechanics," quantum logic has undergone an enormous development. Various schools of thought and approaches have emerged, and there are a variety of technical results. The chapters of this volume constitute a comprehensive presentation of the main schools, approaches and results in the field of quantum logic.
Author(s): Kurt Engesser; Dov M. Gabbay; Daniel Lehmann
Publisher: Elsevier Science
Year: 2008
Language: English
Pages: 715
Front Cover
Handbook of Quantum Logic and Quantum Structures – Quantum Logic
Copyright Page
Table of Contents
Foreword
Editorial Preface
Chapter 1 The Birkhoff–von Neumann Concept of Quantum Logic
1 INTRODUCTION
2 QUANTUM LOGIC: LOGICIZATION OF NON-BOOLEAN ALGEBRAIC STRUCTURES. THE STANDARD VIEW
3 THE BIRKHOFF-VON NEUMANN CONCEPT OF QUANTUM LOGIC
4 VON NEUMANN'S POST 1936 STRUGGLE WITH QUANTUM LOGIC
5 SUMMARY AND CONCLUDING REMARKS
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 2 Is Quantum Logic a Logic?
1 INTRODUCTION
2 LATTICES
3 LOGICS
4 THE SOUNDNESS OF QL: ORTHOMODULARITY LOST
5 THE SOUNDNESS OF CL: DISTRIBUTIVITY LOST
6 THE COMPLETENESS OF QL FOR WOML MODELS: NON-ORTHOMODULARITY CONFIRMED
7 THE COMPLETENESS OF CL FOR WDOL MODELS: NON-DISTRIBUTIVITY CONFIRMED
8 THE COMPLETENESS OF QL FOR OML MODELS: ORTHOMODULARITY REGAINED
9 THE COMPLETENESS OF CL FOR BOOLEAN ALGEBRA MODELS: DISTRIBUTIVITY REGAINED
10 DISCUSSION
ACKNOWLEDGEMENT
BIBLIOGRAPHY
Chapter 3 Is Logic Empirical?
1 INTRODUCTION
2 QUANTUM MECHANICS IN A NUTSHELL
3 QUANTUM LOGIC IN A NUTSHELL
4 STANDARD INTERPRETATION AND MEASUREMENT PROBLEM OF QUANTUM MECHANICS
5 QUANTUM LOGIC AND THE STANDARD INTERPRETATION OF QUANTUM MECHANICS
6 QUANTUM LOGIC AND CLASSICAL PROPOSITIONS
7 QUANTUM LOGIC IN OTHER APPROACHES TO QUANTUM MECHANICS
8 CONCLUSION
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 4 Quantum Axiomatics
1 INTRODUCTION
2 STATE PROPERTY SPACES
3 A SET OF AXIOMS
4 MORPHISMS
5 DECOMPOSITION OF A STATE PROPERTY SPACE
6 ADDITIONAL AXIOMS
7 SUPERPOSITION
8 HILBERT SPACE REPRESENTATIONS
BIBLIOGRAPHY
Chapter 5 Quantum Logic and Nonclassical Logics
1 INTRODUCTION
2 HOW QUANTUM LOGIC EMERGES FROM QUANTUM EVENT-STATE SYSTEMS
3 QUANTUM LOGIC VS POSITIVE, MINIMAL AND INTUITIONISTIC LOGIC
4 THE "LINDENBAUM-ANOMALY"
5 QUANTUM LOGIC AND THE HIDDEN VARIABLE PROBLEM
6 THE UNSHARP APPROACHES TO QT
7 UNSHARP QUANTUM LOGICS
8 QUANTUM LOGIC AND PARACONSISTENT LOGIC
9 QUANTUM LOGIC AND LINEAR LOGIC
10 QUANTUM LOGIC AND QUANTUM COMPUTATION
11 MATHEMATICAL APPENDIX
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 6 Gentzen Methods in Quantum Logic
1 INTRODUCTION
2 MINIMAL QUANTUM LOGIC IN GENTZEN STYLE
3 THE CONTRAPOSITION THEOREM
4 THE DUALITY THEOREM
5 THE CUT-ELIMINATION THEOREM
6 THE COMPLETENESS THEOREM
BIBLIOGRAPHY
Chapter 7 Categorical Quantum Mechanics
1 INTRODUCTION
2 REVIEW OF QUANTUM MECHANICS AND TELEPORTATION
3 COMPACT CLOSED CATEGORIES AND THE LOGIC OF ENTANGLEMENT
4 STRONGLY COMPACT CLOSED CATEGORIES AND 2-DIMENSIONAL DIRAC NOTATION
5 BIPRODUCTS, BRANCHING AND MEASUREMENTS
6 ABSTRACT QUANTUM MECHANICS: AXIOMATICS AND QUANTUM PROTOCOLS
7 EXTENSIONS AND FURTHER DEVELOPMENTS
BIBLIOGRAPHY
Chapter 8 Extending Classical Logic for Reasoning about Quantum Systems
1 INTRODUCTION
2 EXOGENOUS APPROACH
3 DECIDABLE FRAGMENT OF EQPL
4 COMPLETENESS AND DECIDABILITY
5 APPLICATION EXAMPLES
6 CONCLUDING REMARKS
ACKNOWLEDGMENTS
BIBLIOGRAPHY
Chapter 9 Solèr's Theorem
1 PRELIMINARIES AND STRUCTURE OF THE PROOF
2 CONSTRUCTION OF THE SEMI-ORDERING ON F
3 ARCHIMEDEANITY AND COMPLETENESS OF (F,≤)
4 THE NON-COMMUTATIVE CASE
BIBLIOGRAPHY
Chapter 10 Operational Quantum Logic: A Survey and Analysis
1 INTRODUCTION
2 SOME HISTORICAL REMARKS
3 QUANTUM OBJECTS
4 MORPHISMS
5 DYNAMICAL EVOLUTION
6 FROM BEING TO BECOMING
7 COMPOSITE QUANTUM OBJECTS
8 QUANTUM PROCESSES
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 11 Test Spaces
0 INTRODUCTION AND OVERVIEW
1 BACKGROUND ON QUANTUM LOGICS
2 TEST SPACE BASICS
3 LOGICS OF TEST SPACE
4 SUPPORTS AND ENTITIES
5 TENSOR PRODUCTS OF QUANTUM LOGICS
6 SYMMETRIC TEST SPACES
7 TOPOLOGICAL CONSIDERATIONS
ACKNOWLEDGEMENTS
BIBLIOGRAPHY
Chapter 12 Contexts in Quantum, Classical and Partition Logic
1 MOTIVATION
2 CLASSICAL CONTEXTS
3 QUANTUM CONTEXTS
4 AUTOMATA AND GENERALIZED URN LOGIC
5 SUMMARY
BIBLIOGRAPHY
Chapter 13 Nonmonotonicity and Holicity in Quantum Logic
INTRODUCTION
1 STRUCTURE OF THIS CHAPTER
2 BASICS OF NONMONOTONIC LOGIC
3 CONSEQUENCE REVISION SYSTEMS
4 HOLISTIC LOGICS: EVERYTHING IS ENCODED IN EVERYTHING
5 SOME HILBERT SPACE THEORY
6 TOWARDS HILBERT SPACE
BIBLIOGRAPHY
Chapter 14 A Quantum Logic of Down Below
1 INTRODUCTION
2 AGENCY
3 SEMANTIC SPACE: COMPUTATIONAL APPROXIMATIONS OF CONCEPTUAL SPACE
4 BRIDGING SEMANTIC SPACE AND QUANTUM MECHANICS
5 QUANTUM LOGIC AND CONCEPTUAL GENERALIZATION
6 SUMMARY AND CONCLUSIONS
ACKNOWLEDGMENTS
BIBLIOGRAPHY
Chapter 15 A Completeness Theorem of Quantum Set Theory
1 INTRODUCTION
2 COMPLETE ORTHOMODULAR LATTICES
3 QUANTUM LOGIC QL
4 QUANTUM SET THEORY QZFZ
5 A COMPLETENESS THEOREM OF QZFZ
BIBLIOGRAPHY
Index
