Abstract
This paper has a beginning, a middle, and an end. If these parts are to follow the dramatic unities, they will lead from suffering through recognition to reversal; but of this ideal they may fall short.
The research for this paper was supported by the John Simon Guggenheim Memorial Foundation and Canada Council grant S71-0546. While I have many debts, some of which are indicated at various points below, my debts to Nancy Delaney Cartwright in Part III and in the Appendix are so pervasive that they cannot be adequately chronicled. I hereby acknowledge them gratefully.
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Notes
List of special symbols. E Section I.1. Note that the symbol λ plays different roles in Parts II and III.
For a general treatment see Chapter 2 of my Formal Semantics and Logic New York, Macmillan, 1971.
Cf. Formal Semantics and Logic (Note 3), Chapter 5, Section 2a.
This may be of some relevance to Professor Greechie’s remarks on implication in quantum logics in this volume.
See R. Stalnaker and R. H. Thomason, ‘A Semantic Analysis of Conditional Logic’ Theoria 36 (1970), 23–42; for alternatives see D. Lewis, ‘Completeness and Decidability oi Three Logics of Counterfactual Conditionals’, Theoria 37 (1971), 74-85.
For a general exposition of the actual developments in quantum logic, see the paper by Professors Gudder and Greechie in this volume. More general discussions of quantum logics of various stripe are also found in my ‘The Labyrinth of Quantum Logics’ and ‘The Formal Representation of Physical Quantities’ in Boston Studies in Philosophy of Science, D. Reidel Publ. Co., Dordrecht, Holland, forthcoming
Cf. J. M. Jauch, Foundations of Quantum Mechanics Addison-Wesley, Reading, Massachusetts, 1968, p. 100.
S. Kochen and E. P. Specker, The Problem of Hidden Variables in Quantum Mechanics’ Journal of Mathematics and Mechanics 17 (1968), 59–87. See also N. Zierler and M. Schlesinger, ‘Boolean Embeddings of Orthomodular Sets and Quantum Logic’ Duke Mathematical Journal 32 (1965), 251-162.
The deduction I have in mind is that of G. W. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, New York, (1963); see also M. J. Maczynski, ‘A Remark on Mackey’s Axiom System for Quantum Mechanics’ Bulletin de l’cadémie Polonaise des Sciences (Série des sciences mathématique astronomique et physique) 15 (1967), 583-7.
Specifically, see the definition of ‘projective logic’ and Theorem 7.44 in Chapter VII of V. S. Varadarajan, Geometry of Quantum Theory, Vol. I, Van Nostrand, Princeton, 1968.
Cf. S. Gudder, ‘Hidden Variables in Quantum Mechanics Reconsidered’, Review of Modern Physics 40 (1968), 229–31.
J. C. T. Pool, ‘Baer*-Semigroups and the Logic of Quantum Mechanics’, Communications of Mathematical Physics 9 (1968), 118–41, and’ semi-Modularity and the Logic of Quantum Mechanics’ ibid., 212-28.
This use of’ standard’ is from Varadarajan, op. cit. For a more detailed motivation for the problems discussed in this part see the papers by C. Hooker and myself in the volume referred to in Note 14.
In Part II of my ‘A Formal Approach to the Philosophy of Science’ (henceforth FAPS), pp. 303-366 in Paradigms and Paradoxes (ed. by R. Colodny), University of Pittsburgh Press, Pittsburgh, 1972.
In FAPS, these were called □-sentences and O-sentences, and written ‘□ (m, E )’ and ‘O(w, E )’
J. Bub, ‘What is a Hidden Variable Theory of Quantum Phenomena?’, International Journal of Theoretical Physics 2 (1969), 101–23.
J. Bub, ‘Hidden Variables and Quantum Logic’, Boston Studies in the Philosophy of Science, D. Reidel Publ. Co., Dordrecht, Holland, forthcoming.
Op. cit. (Note 8).
In different contexts, different Hilbert spaces are used to represent the states of the same system. The present assertion of identity can be similarly understood as having the tacit rider ‘relative to all purposes in a given context.’ It can be taken as a metaphysical assertion only if it is presupposed that for each system there is a state-space which is peculiarly appropriate to it.
See the definition in Kochen and Specker, op. cit., p. 63; note that they follow this with the assumption that if for all states α then m = m’, which we here deny.
After the completion of this paper I found ideas similar to those of what I call the modal interpretation in a paper by A. Fine presented at the Fourth International Congress of Logic, Methodology and Philosophy of Science, Bucharest, August 1971.
Except for the remarks on the statistical interpretation this appendix is a summary of some parts of FAPS; mathematical details may be found in the appendix of FAPS.
With the qualification that in FAPS a single (orthogonal) decomposition of a mixture β was always specified, with the remainder of Uβ ignored. Note added in proof. The question discussed in Section III-2 concerning Equation (11) is settled by a theorem in E. Schrödinger, ‘Probability Relations Between Separated Systems’, Proc. Cambridge Philosophical Society 32 (1936), 446–52.
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Van Fraassen, B.C. (1973). Semantic Analysis of Quantum Logic. In: Hooker, C.A. (eds) Contemporary Research in the Foundations and Philosophy of Quantum Theory. The University of Western Ontario Series in Philosophy of Science, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2534-8_3
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