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Information Technologies and Control

The Journal of Institute of Information and Communication Technologies of Bulgarian Academy of Sciences

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1312-2622
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An Essay on Complex Valued Propositional Logic

V. Sgurev
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  • Department Inteligent Systems Institute of Information and Communication Technologies Bulgarian Academy of Sciences Acad G. Bonchev St., bl. 25A, 1113 Sofia
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Published Online: 2014-06-17 | DOI: https://doi.org/10.2478/itc-2013-0001

Abstract

In decision making logic it is often necessary to solve logical equations for which, due to the features of disjunction and conjunction, no admissible solutions exist. In this paper an approach is suggested, in which by the introduction of Imaginary Logical Variables (ILV), the classical propositional logic is extended to a complex one. This provides a possibility to solve a large class of logical equations.The real and imaginary variables each satisfy the axioms of Boolean algebra and of the lattice. It is shown that the Complex Logical Variables (CLV) observe the requirements of Boolean algebra and the lattice axioms. Suitable definitions are found for these variables for the operations of disjunction, conjunction, and negation. A series of results are obtained, including also the truth tables of the operations disjunction, conjunction, negation, implication, and equivalence for complex variables. Inference rules are deduced for them analogous to Modus Ponens and Modus Tollens in the classical propositional logic. Values of the complex variables are obtained, corresponding to TRUE (T) and FALSE (F) in the classic propositional logic. A conclusion may be made from the initial assumptions and the results achieved, that the imaginary logical variable i introduced hereby is “truer” than condition “T” of the classic propositional logic and i - “falser” than condition “F”, respectively. Possibilities for further investigations of this class of complex logical structures are pointed out

Keywords: Propositional logic; logical equations; complex propositional logic; Boolean algebra; imaginary logical variable; lattice

References

  • 1. Kleene, S. C. Mathematical Logic. N.Y., J. Wiley & Sons, 1967.Google Scholar

  • 2. Mendelson, E. Introduction to Mathematical Logic. D.von Nostrand Com. Inc., Princeton, 1975.Google Scholar

  • 3. Kowalski, R. Logic for Problem Solving. Elsevier North Holland Inc., 1979.Google Scholar

  • 4. Johnson, B. Topics in Universal Algebra. N.Y., Springer-Verlag, 1972.Google Scholar

  • 5. Ionov, A. S., G. A. Petrov. Quaternion Logic as a Base of New Computational Logic. http://zhurnal.ape.relarn.ru/articles/2007/047.pdf (in Russian).Google Scholar

  • 6. Hung, T. Nguyen, Vladik Kreinovich, Valery Shekhter. On the Possibility of Using Complex Values in Fuzzy Logic for Representing Inconsistencies. http://www.cs.utep.edu/vladik/1996/tr96-7b.pdf.Google Scholar

  • 7. Tamir, D. E., A. Kandel. Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes. - Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844, VI, 2011, No. 3 (September), 562-576.Google Scholar

  • 8. Shoup, Richard G. A Complex Logic for Computation with Simple Interpretations for Physics. CA, Interval Research Palo Alto, 94304, http://www.boundarymath.org/papers/CompLogic.pdf.Google Scholar

  • 9. Aizenberg, Igor. Complex-Valued Neural Networks with Multi- Valued Neurons. Springer, 2011.Google Scholar

  • 10. Lucas, Chris. A Logic of Complex Values. Proceedings of the First International Conference on Neutrosophy, Neutrosophic Logic, Set, Probability and Statistics, 1-3 December 2001, University of New Mexico, 121-138. ISBN 1-931233-55.Google Scholar

  • 11. Kauffman, Louis H. Virtual Logic - The Flagg Resolution, Cybernetics & Human Knowing, 6, 1999, No.1, 87-96.Google Scholar

About the article

Received: 2013-01-03

Published Online: 2014-06-17

Published in Print: 2013-03-01


Citation Information: Information Technologies and Control, Volume 11, Issue 1, Pages 2–7, ISSN (Online) 1312-2622, DOI: https://doi.org/10.2478/itc-2013-0001.

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© 2014. This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. BY-NC-ND 3.0

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