Abstract
Taking as starting point two familiar interpretations of probability, we develop these in a perhaps unfamiliar way to arrive ultimately at an improbable claim concerning the proper axiomatization of probability theory: the domain of definition of a point-valued probability distribution is an orthomodular partially ordered set. Similar claims have been made in the light of quantum mechanics but here the motivation is intrinsically probabilistic. This being so the main task is to investigate what light, if any, this sheds on quantum mechanics. In particular it is important to know under what conditions these point-valued distributions can be thought of as derived from distribution-pairs of upper and lower probabilities on boolean algebras. Generalising known results this investigation unsurprisingly proves unrewarding. In the light of this failure the next topic investigated is how these generalized probability distributions are to be interpreted.
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References
Bell, J. L. and M. Machover:A Course in Mathematical Logic, North-Holland, 1977.
Bell, J. S.: ‘On the Problem of Hidden Variables in Quantum Mechanics’,Rev. Mod. Phys. 38 (1966). Reprinted inSpeakable and Unspeakable in Quantum Mechanics, CUP, 1987.
Beltrametti, E. and G. Cassinelli:The Logic of Quantum Mechanics, Addison-Wesley, 1981.
de Finetti, B.:Theory of Probability, Vol I, Wiley, 1974.
de Finetti, B.: ‘Foresight: Its Logical Laws, Its Subjective Sources’, in H. Kyburg and H. Smokler (eds.),Studies in Subjective Probability (second edition), Krieger, 1980.
Feynman, R. P.: ‘Negative Probability’, in B. J. Hiley and F. D. Peat (eds.),Quantum Implications, Routledge and Kegan Paul, 1987.
Fine, T.:Theories of Probability, Academic Press, 1973.
Forrest, P.:Quantum Metaphysics, Blackwell, 1988.
Gleason, A. M.: ‘Measures on the Closed Subspaces of a Hilbert Space’,J. Math. Mech. 6 (1957). Reprinted in Hooker (ed.).
Hooker, C. A. (ed.):The Logico-Algebraic Approach to Quantum Mechanics, Vol. I, Reidel, 1979.
Jammer, M.:The Philosophy of Quantum Mechanics, Wiley, 1974.
Kalmbach, G.:Orthomodular Lattices, Academic Press, 1983.
Kunen, K.:Set Theory, North-Holland, 1980.
Suppes, P.: ‘The Probabilistic Agrument for a Nonclassical Logic of Quantum Mechanics’,Phil. Sci. 33 (1966). Reprinted in Hooker (ed.).
van Fraassen, B. C. and C. A. Hooker: ‘A Semantic Analysis of Niels Bohr's Philosophy of Quantum Theory’, in W. Harper and C. A. Hooker (eds.),Foundations of Probability, Statistical Inference, and Statistical Theories of Science, Vol. III, Reidel, 1976.
von Mises, R.:Mathematical Theory of Probability and Statistics, Academic Press, 1964.
Zierler, N. and M. Schlessinger: ‘Boolean Embeddings of Orthomodular Sets and Quantum Logic’,Duke Math. J. 32 (1965). Reprinted in Hooker (ed.).
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Milne, P. The foundations of probability and quantum mechanics. J Philos Logic 22, 129–168 (1993). https://doi.org/10.1007/BF01049259
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DOI: https://doi.org/10.1007/BF01049259