K. Svozil. Quantum Logic, Springer-Verlag, Singapore, 1998, xx + 214 pp. ISBN 981-4021-07-5. US$ 49 softcover.
"Quantum Logic" deals with the foundations of quantum mechanics and, related to it, the behaviour of finite, discrete deterministic systems. The quantum logical approach is particularly suitable for the investigation and exclusion of certain hidden parameter models of quantum mechanics. Conversely, it can be used to embed quantum universes into classical ones. It is also highly relevant for the characterization of finite automation. This book has been written with a broad readership in mind. Great care has been given to the motivation of the concepts and to the explicit and detailed discussions of examples.
In the first part of the book, quantum logic is introduced as pioneered by Garrett Birkhoff and John von Neumann in the thirties. They organized it top-down: The starting point is von Neumann's Hilbert space formalism of quantum mechanics. In a second step, certain entities of Hilbert spaces are identified with propositions, partial order relations and lattice operations---Birkhoff's field of expertise. These relations and operations can then be associated with the logical implication relation and operations such as 'and,' 'or,' and 'not.' Thereby, a "nonclassical,'' nonboolean logical structure is induced which originates in theoretical physics. If theoretical physics is taken as a faithful representation of our experience, such an "operational'' logic derives its justification by the phenomena themselves. In this sense, one of the main ideas behind quantum logic is the quasi-inductive construction of the logical and algebraic order of events from empirical facts.
This is very different from the "classical'' logical approach, which is also top-down: There, the system of symbols, the axioms, as well as the rules of inference are mentally constructed, abstract objects of our thought. Insofar as our thoughts can pretend to exist independent from the physical Universe, such "classical'' logical systems can be conceived as totally independent from the world of the phenomena.
The applicability of such mentally constructed objects of our thoughts to the natural sciences often appears unreasonable. Quantum logic is an example that indeed it is unreasonable to naively apply abstractly invented concepts to the phenomena. As it turns out, neither is "classical'' Boolean logic a faithful representation of the relations and operations among physical information, nor can it a priori be expected that the "classical'' logical tautologies correspond to any physical truth.
The second part of the book deals mainly with the Kochen-Specker theorem(s) and the embeddability of quantum logical structures into classical ones. The idea of embedding a quantum world into a classical one resembles Plato's cave metaphor insofar as the quantum phenomena are informally conceived as "shadows'' of the "true classical arena'' permanently hidden to us.
The final part deals with several quasiclassical analogues of quantum logic, in particular Wright's generalized urn model, the experimental logic of automata, and of complementarity games.
Contents: Hilbert Space Quantum Mechanics; Comeasurable Observables; Complementarity; Hilbert Lattices; Composite Systems; Probabilities; Contexuality; Quantum Tautologies; What Price Value-definiteness?; Quasi-classical Analogies; Appendix; References; Index.
Essay Review by M. Redei