From Classical Logic to Fuzzy Logic and Quantum Logic: A General View

  • Sorin Nadaban

Abstract

The aim of this article is to offer a concise and unitary vision upon the algebraic connections between classical logic and its generalizations, such as fuzzy logic and quantum logic. The mathematical concept which governs any kind of logic is that of lattice. Therefore, the lattices are the basic tools in this presentation. The Hilbert spaces theory is important in the study of quantum logic and it has also been used in the present paper.

References

[1] Atanassov, K.T. (1986). Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 87-96, 1986.
https://doi.org/10.1016/S0165-0114(86)80034-3

[2] Atanassov, K.T.; Gargov, G. (1989). Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31(3), 343-349, 1989.
https://doi.org/10.1016/0165-0114(89)90205-4

[3] Birkhoff, G. (1973) Lattice Theory, American Mathematical Society: Providence, RI, USA, 1973.

[4] Bustince, H.; Barrenechea, E.; Pagola, M.; Fernandez, J.; Xu, Z., Bedregal, B.; Montero, J.; Hagras, H., Herrera, F.; De Baets, B. (2016). A historical account of types of fuzzy sets and their relationships, IEEE Trans. Fuzzy Syst., 24, 179-194, 2016.
https://doi.org/10.1109/TFUZZ.2015.2451692

[5] Dann, J.M. (1996). Generalized Ortho-Negation. In book: Negation: A Concept in Focus, 3-26, 1996.

[6] Dvurecenskij, A. (1996). Fuzzy set representations of some quantum structures, Preprint Series of the Mathematical Institute of Slovak Academy of Sciences, 2, 1996.

[7] Ejegwa, P.A. (2019). Pythagorean fuzzy set and its application in career placements based on academic performance using max-min-max composition, Complex & Intelligent Systems, 5, 165-175, 2019.
https://doi.org/10.1007/s40747-019-0091-6

[8] Foulis, D.J.; Bennet, M.K. (1994). Effect algebras and unsharp quantum logics, Foundations on Physics, 24, 1331-1352, 1994.
https://doi.org/10.1007/BF02283036

[9] Goguen, J.A. (1967). L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174, 1967.
https://doi.org/10.1016/0022-247X(67)90189-8

[10] Greenhoe, D.J. (2015). Boolean and ortho fuzzy subset logic, arXiv: 1409.4222, 2015.

[11] Husimi, K. (1937). Studies on the foundations of quantum mechanics I, Proc. Physico- Mathematical Soc. Japan, 9, 766-789, 1937.

[12] Klement, E.P.; Mesiar, R. (2018). L-Fuzzy Sets and Isomorphic Lattices: Are All the "New" Results Really New?, Mathematics, 6, 146, 2018.
https://doi.org/10.3390/math6090146

[13] Klement, E.P.; Mesiar, R.; Pap, E. (2000). Triangular norms, Kluwer Academic Publisher, 2000.
https://doi.org/10.1007/978-94-015-9540-7

[14] Mac Lane, S.; Birkhoff, G. (1999). Algebra, 3rd Edition, Chelsea Publishing, Providence, 1999.

[15] M¬łaczynscki, M.J. (1973). On some numerical characterization of Boolean algebras, Colloquium Mathematicum, 27, 207-210, 1973.
https://doi.org/10.4064/cm-27-2-207-210

[16] M¬łaczynscki, M.J. (1974). Functional properties of quantum logics, International Journal of Theoretical Physics, 11, 149-156, 1974.
https://doi.org/10.1007/BF01809565

[17] Melnichenko, G. (2010). Energy discriminant analysis, quantum logic and fuzzy sets, Journal of Multivariate Analysis, 101, 68-76, 2010.
https://doi.org/10.1016/j.jmva.2009.04.011

[18] Menger, K. (1942). Statistical metrics, Proc. Nat. Acad. of Sci., U.S.A. 28, 535-537, 1942.
https://doi.org/10.1073/pnas.28.12.535

[19] Pavicic, M.; Megill, N.D. (2009). Is quantum logic a logic?, Handbook of quantum logic and quantum structures: quantum logic, Eds. K. Engesser, D.M. Gabbay, D. Lehmann, Elsevier, 2009.
https://doi.org/10.1016/B978-0-444-52869-8.50005-0

[20] Pykacz, J. (1987). Quantum logics as families of fuzzy subsets of the set of physical states, Preprints of the Second International Fuzzy Systems Association Congress, Tokyo, July 20-25, Vol. 2, 437-440, 1987.

[21] Pykacz, J. (1994). Fuzzy quantum logics and infinite-valued Lukasiewicz logic, International Journal of Theoretical Physics, 33, 1403-1416, 1994.
https://doi.org/10.1007/BF00670685

[22] Pykacz, J. (2007). Quantum structures and fuzzy set theory, Handbook of Quantum Logic and Quantum Structures: Quantum Structures, Edited by K. Engesser, D.M. Gabbay and D. Lehmann, Elsevier, 55-74, 2007.
https://doi.org/10.1016/B978-044452870-4/50024-8

[23] Schweizer, B., Sklar, A. (1960). Statistical metric spaces, Pacific Journal of Mathematics, 10, 314-334, 1960.
https://doi.org/10.2140/pjm.1960.10.313

[24] Smarandache, F. (1999). A unifying field in logics. Neutrosophy: neutrosophic probability, set and logic, American Research Press Rehoboth, 1999.

[25] Yager, R.R. (2013). Pythagorean fuzzy subsets, Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), Edmonton, AB, 57-61, 2013.
https://doi.org/10.1109/IFSA-NAFIPS.2013.6608375

[26] Zadeh, L.A. (1965). Fuzzy sets, Information and Control, 8(3), 338-353, 1965.
https://doi.org/10.1016/S0019-9958(65)90241-X

[27] Zadeh, L.A. (1975). The concept of a linguistic variable and its applications to approximate reasoning, Part I, Inform. Sci., 8, 199-251, 1975.
https://doi.org/10.1016/0020-0255(75)90036-5

[28] Zeman, J.J. (1978). Generalized normal logic, Journal of Philosophical Logic, 7(1), 225-243, 1978.
https://doi.org/10.1007/BF00245929
Published
2021-01-14
How to Cite
NADABAN, Sorin. From Classical Logic to Fuzzy Logic and Quantum Logic: A General View. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL, [S.l.], v. 16, n. 1, jan. 2021. ISSN 1841-9844. Available at: <http://univagora.ro/jour/index.php/ijccc/article/view/4125>. Date accessed: 12 apr. 2021. doi: https://doi.org/10.15837/ijccc.2021.1.4125.