Reasoning in Quantum Theory: Sharp and Unsharp Quantum Logics
Chiara Manfriani, Roberto Giuntini, Richard J. Greechie
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.
•Mar 14, 2014•112 citations