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Bell’s Theorem, Quantum Theory and Conceptions of the Universe pp 37–49Cite as

From George Boole To John Bell — The Origins of Bell’s Inequality

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 37))

Abstract

We consider a combinatorial problem which was enunciated by George Boole (1854) and explore its identity with fundamental puzzles in a diverse range of fields such as statistical theory, propositional logic, the theory of computational complexity, the Einstein-Podolsky-Rosen paradox in quantum mechanics, the theory of neural networks, and the Ising spin model.

Acknowledgements: This research is supported by the Edelstein Center for the History and Philosophy of Science at the Hebrew University

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Notes and References

  1. Steiner, M., Philos. Studies 34, 133 (1978); Jour. of Philos. 80, 502 (1983).

    Google Scholar 

  2. Bell, J. S., Physics 1, 195 (1964). Wigner E.P. Amer. J. Phys. 38, 1005 (1970).

    Google Scholar 

  3. Clauser, J.F., and Home, M.A., Phys. Rev. D 10 526 (1974).

    Google Scholar 

  4. Boole, George, The Laws of Thought, Dover (original edition 1854 ).

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  5. Hailperin, T., Amer. Math. Monthly 72 343 (1965). For more details see Hailperin, T., Boole’s Logic and Probability North Holland, 2nd edition (1986). The relations between this type of problem and linear programming are also indicated in Renyi, A., Foundation of probability,Holden-Day (1970).

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  6. Poincaré, H., Calcul De Probabilités Gauthier-Villars (1912).

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  7. Bonferroni, C.E., Theorie statistica delle classi e calcolo dell probabilita, Pubblic. Ist. Sup. Sc. Ec. e. Comm. di Firenze 8, 1 (1936).

    Google Scholar 

  8. Chung, K. L., Annals. Math. Stat. 12, 328 (1941)

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  9. This statement must be qualified. In some cases applying complementation yields back the same inequality. For example, in inequality (3) for n = 3, complementing all three events yields (3) again.

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  10. Fréchét, M., Les Probabilités Associées a un Systeme D’événements Compatibles et Dépandants, Hermann (Vol I 1940, Vol II 1943 ).

    Google Scholar 

  11. Feller, W., An Introduction to Probability-Theory and Its Applications, Wiley, (1970).

    Google Scholar 

  12. Some recent references: Kounias, S., and Marin, J., SIAM J. Appl. Math. 30, 307 (1976); Hunter, D., J. Appl. Prob. 13, 597 (1976); Cerasoli, M., Discre te Math. 44, 221 (1983); Maurer, W., Discre-ate Appl. Math. 6, 157 (1983).

    Google Scholar 

  13. Correlation polytopes were introduced in Pitowsky, I., J. Math. Phys. 27 1556 (1986). The discussion below follows Pitowsky, I., Quantum Probability Quantum Logic,Lecture Notes In Physics, Springer (1989).

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  14. This is proved in Fine, A., Phys. Rev. Lett. 48,291 (1982), see also my book cited in note 12.

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  15. These operations generate a group of n!2’y elements which acts as a symmetry group of c(n, S n ). This is of course also the symmetry group of the n-dimensional hyercube but also of the n-dimensional hyper octahedron. see e.g. Todd J. A., Proc. Cambridge Phil. Soc. 27,212 (1931).

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  16. The proof in I. Pitowsky, Correlation Polytopes, Their Geometry and Complexity Math. programming (forthcoming)

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  17. For details on terminology and basic results see: Garey, M. R., and Johnson, D. S., Computers and Intractability, A guide to the Theory of NP-Completeness W. H. Freeman (1979).

    Google Scholar 

  18. also Gudder, S., Quantum Probability Academic Press N.Y. (1988).

    Google Scholar 

  19. The literature on neural networks is vast. The basic model relevant for our concern was introduced in Hopfield, J. J., Proc. Natl. Acad. Sci. U.S.A. 79, 2554 (1982).

    Google Scholar 

  20. Barahona, F., J. Phys. A. 15, 3241 (1982),and my article cited in note (15).

    Google Scholar 

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Authors and Affiliations

  1. Department of Philosophy, The Hebrew University, Jerusalem, Israel

    Itamar Pitowsky

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  1. Itamar Pitowsky
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Editors and Affiliations

  1. Department of Physics, George Mason University, Fairfax, Virginia, USA

    Menas Kafatos

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© 1989 Springer Science+Business Media Dordrecht

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Pitowsky, I. (1989). From George Boole To John Bell — The Origins of Bell’s Inequality. In: Kafatos, M. (eds) Bell’s Theorem, Quantum Theory and Conceptions of the Universe. Fundamental Theories of Physics, vol 37. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0849-4_6

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  • DOI: https://doi.org/10.1007/978-94-017-0849-4_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4058-9

  • Online ISBN: 978-94-017-0849-4

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