# An Essay on Complex Valued Propositional Logic

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## 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

## References

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

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

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

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

• 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).

• 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.

• 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.

• 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.

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

• 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.

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

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