Book contents
- Frontmatter
- Contents
- Editor's Statement
- Foreword
- Preface
- The Logic of Quantum Mechanics
- Part I HILBERT-SPACE QUANTUM MECHANICS
- Part II BASIC STRUCTURES IN THE DESCRIPTION OF QUANTUM SYSTEMS
- Chapter 10 The Typical Mathematical Structure of Propositions: Orthomodular AC Lattices
- Chapter 11 Probability Measures on Orthomodular Posets and Lattices
- Chapter 12 Characterization of Commutativity
- Chapter 13 States and Propositions of a Physical System
- Chapter 14 Quantum-Mechanical Features in Terms of the Logic of the Physical System
- Chapter 15 On the Hidden-Variables Issue
- Chapter 16 Proposition-State Structure and Idealized Measurements
- Chapter 17 Superpositions of States and Closure Spaces
- Chapter 18 Transition-Probability Spaces and Quantum Systems
- Chapter 19 On the Convex-Set Approach
- Chapter 20 Introduction to a Quantum Logic
- Part III RECONSTRUCTION OF HILBERT-SPACE QUANTUM MECHANICS
- Appendix A Trace-Class Operators
- Appendix B The Spectral Theorem
- Appendix C Proofs for Chapter 11
- Subject Index
- Miscellaneous Endmatter
Chapter 20 - Introduction to a Quantum Logic
Published online by Cambridge University Press: 05 June 2013
- Frontmatter
- Contents
- Editor's Statement
- Foreword
- Preface
- The Logic of Quantum Mechanics
- Part I HILBERT-SPACE QUANTUM MECHANICS
- Part II BASIC STRUCTURES IN THE DESCRIPTION OF QUANTUM SYSTEMS
- Chapter 10 The Typical Mathematical Structure of Propositions: Orthomodular AC Lattices
- Chapter 11 Probability Measures on Orthomodular Posets and Lattices
- Chapter 12 Characterization of Commutativity
- Chapter 13 States and Propositions of a Physical System
- Chapter 14 Quantum-Mechanical Features in Terms of the Logic of the Physical System
- Chapter 15 On the Hidden-Variables Issue
- Chapter 16 Proposition-State Structure and Idealized Measurements
- Chapter 17 Superpositions of States and Closure Spaces
- Chapter 18 Transition-Probability Spaces and Quantum Systems
- Chapter 19 On the Convex-Set Approach
- Chapter 20 Introduction to a Quantum Logic
- Part III RECONSTRUCTION OF HILBERT-SPACE QUANTUM MECHANICS
- Appendix A Trace-Class Operators
- Appendix B The Spectral Theorem
- Appendix C Proofs for Chapter 11
- Subject Index
- Miscellaneous Endmatter
Summary
Sentences Associated with a Quantum System and Their Truth Values
In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject.
Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.
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- The Logic of Quantum Mechanics , pp. 217 - 226Publisher: Cambridge University PressPrint publication year: 1984