Quantum Probability Theory and Non-Boolean Logic

  • Sisir RoyEmail author


Since the very inception of quantum theory, the corresponding logic for quantum entities has attracted much attention. The logic underlying the quantum theory is shown to be non-Boolean in character. Boolean logic is a two/valued logic which is used for the description of everyday objects. Modern computers are based on this logic. The existence of an interference term for microscopic entities or quantum entities clearly indicates the existence of three-valued or non-Boolean logic. This is popularly known as quantum logic. It is mathematically shown that a set of propositions which satisfies the different axiomatic structures for the non-Boolean logic generates Hilbert space structures. The quantum probability associated with this type of quantum logic can be applied to decision-making problems in the cognitive domain. It is to be noted that, until now, no quantum mechanical framework is taken as a valid description of the anatomical structures and functions of the brain. This framework of quantum probability is very abstract and devoid of any material content. So it can be applied to any branch of knowledge like biology, social science, etc. Of course, it is necessary to understand the issue of contextualization, i.e., here, in the case of the brain.


Non-Boolean logic Quantum logic Quantum probability Hilbert space Contextualization 


  1. Aristotle (born 384 bce, Stagira, Chalcidice, Greece—died322, Chalcis, Euboea);
  2. Birkhoff Garret; Von Neumann, John (1936) Annals of Mathematics; 37 (4): 823–843.Google Scholar
  3. Bell, J.L. (1986) Brit. J. Phil.Sci.; 37, 83-99.Google Scholar
  4. Engesser, K, Gabbay, D.M. & Lehman, D. (2011); “Handbook of Quantum Logic and Quantum structures: Quantum Structures”; Eds. by Kurt Engesser, Dov M. Gabbay, Daniel Lehmann; Elsevier, Radarweg 29, PO Box 211,1000 AE Amsterdam, the Netherlands.Google Scholar
  5. Friedrich, Jacob Fries (1773-1843); Chisholm, Hugh, ed. (1911). Fries, Jacob Friedrich; Encyclopædia Britannica; (11th ed.); Cambridge University Press.Google Scholar
  6. Hacck, Susan(1978) Deviant Logic, Fuzzy Logic: Beyond the Formalism;The University of Chicago.Google Scholar
  7. Hilbert David & Corant, Richard (1953); “Methods of Mathematical Physics”;Vol I, Interscience.Google Scholar
  8. Husserel Edmond, Moran Dermut (2001) Logical Investigations, Vol.1, abridged; reprint; revised; Psychology Press.Google Scholar
  9. Jauch, J.M. & Piron, C. (1969) On the structure of quantal Proposition systems; Helvetica Physica Acta; 36, 827-837.Google Scholar
  10. Mill, John Stuart (1806-1873) Bulletin of Symbolic logic; 19 th century Logic Between Philosophy and Mathematics; 6(04); Dec 1999; 433-450;Google Scholar
  11. Mill, J. Stuart (1806-1873) On Liberty and other Writings; (Extends the 1974 Deviant Logic, with some additional essays published between 1973 and 1980, particularly on fuzzy logic, cf. The Philosophical Review, 107: 3, 468–471.Google Scholar
  12. Payne, J.W., Bettman, J.R. & Johnson, E.J. (1992) Behavioral decision research: A constructive processing perspective; Annual Review of Psychology, 43, 87-131.Google Scholar
  13. Putnam, H. (1974) How to think quantum-logically”; Synthese, 29, 55-61. Reprinted in P. Suppes (ed.); Logic and Probability in Quantum Mechanics (Dordrecht: Reidel, 1976); pp. 47-53.Google Scholar
  14. Simon Herbert (1957); 1957. “Models of Man”; John Wiley; ibid; (1972); (with Allen Newell): “Human Problem Solving”; Prentice Hall, Englewood Cliffs, NJ, (1972).Google Scholar
  15. Simon Herbert, A. (1972); “Decision and Organization”; by C.M. Mcguire & Ray Radner (eds.); North Holland Publishing companyGoogle Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  1. 1.National Institute of Advanced Studies, IISc CampusBengaluruIndia

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