Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes

  • Dan E. Tamir Texas State University, CS Department 601 University Drive San Marcos, Texas 78666 E-mail:
  • Abraham Kandel University of South Florida, CSE Department 4202 E. Fowler Ave Tampa, Florida 33620


Complex fuzzy sets, classes, and logic have an important role in applications, such as prediction of periodic events and advanced control systems, where several fuzzy variables interact with each other in a multifaceted way that cannot be represented effectively via simple fuzzy operations such as union, intersection, complement, negation, conjunction and disjunction. The initial formulation of these terms stems from the definition of complex fuzzy grade of membership. The problem, however, with these definitions are twofold: 1) the complex fuzzy membership is limited to polar representation with only one fuzzy component. 2) The definition is based on grade of membership and is lacking the rigor of axiomatic formulation. A new interpretation of complex fuzzy membership enables polar and Cartesian representation of the membership function where the two function components carry uncertain information. Moreover, the new interpretation is used to define complex fuzzy classes and develop an axiomatic based theory of complex propositional fuzzy logic. Additionally, the generalization of the theory to multidimensional fuzzy grades of membership has been demonstrated. In this paper we propose an axiomatic framework for first order predicate complex fuzzy logic and use this framework for axiomatic definition of complex fuzzy classes. We use these rigorous definitions to exemplify inference in complex economic systems. The new framework overcomes the main limitations of current theory and provides several advantages. First, the derivation of the new theory is based on axiomatic approach and does not assume the existence of complex fuzzy sets or complex fuzzy classes. Second, the new form significantly improves the expressive power and inference capability of complex fuzzy logic and class theory. The paper surveys the current state of complex fuzzy sets, complex fuzzy classes, and complex fuzzy logic; and provides an axiomatic basis for first order predicate complex fuzzy logic and complex class theory.


[1] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning - Part I. Information Sciences, 1975, 7: p. 199-249.

[2] Klir, G.J., Tina, A., Fuzzy sets, uncertainty, and information. 1988, Upper Saddle River, NJ: Prentice Hall.

[3] Kandel, A., Fuzzy Mathematical Techniques with Applications. 1987, New York: Addison Wesley.

[4] Hossu D., Fagarasan I., Hossu A., Iliescu S.S., Evolved Fuzzy Control System for a Steam Generator, International Journal of Computers Communications & Control, 2010, 5(2):P. 179-192.

[5] Mhalla A., Jerbi N., Dutilleul S.C., Craye E., Benrejeb M., Fuzzy Filtering of Sensors Signals in Manufacturing Systems with Time Constraints, International Journal of Computers Communications & Control, 2010, 5(3):p. 362-374.

[6] Cordova F.M., Leyton G., A Fuzzy Control Heuristic Applied to Non-linear Dynamic System Using a Fuzzy Knowledge Representation, International Journal of Computers Communications & Control, 2010, 5(5): P. 664-674

[7] Tamir, D.E. and A. Kandel, An axiomatic approach to fuzzy set theory. Information Sciences, 1990, 52: p. 75-83.

[8] Tamir, D.E., Kandel, A., Fuzzy semantic analysis and formal specification of conceptual knowledge. Information Sciences, Intelligent systems, 1995, 82(3-4): p. 181-196.

[9] Ronen, M., Shabtai, R., Guterman, H., Hybrid model building methodology using unsupervised fuzzy clustering and supervised neural networks. Biotechnology and Bioengineering, 2002, 77(4): p. 420-429.

[10] Bhattacharya M., Das A., Genetic Algorithm Based Feature Selection In a Recognition Scheme Using Adaptive Neuro Fuzzy Techniques, International Journal of Computers Communications & Control, 2010, 5(4): p. 458-468.

[11] Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning - Part I. Information Sciences, 1975, 7: p. 199-249.

[12] Mendel, J.M., Uncertainty, fuzzy logic, and signal processing. Signal Processing, 2000, 80: 913-933

[13] Tamir, D.E., Kandel, A., "The Pyramid Fuzzy C-means Algorithm," The International Journal of Computational Intelligence in Control, December 2010.

[14] Agarwal, D., Tamir, D. E., Last, M., Kandel, A., A comparative study of software testing using artificial neural networks and Info-Fuzzy networks. 2011, (Submitted).

[15] Ramot, D., Milo, R., Friedman, M., Kandel, A., Complex fuzzy sets. IEEE Transactions on Fuzzy Systems 2002, 10(2): p. 171-186.

[16] Ramot, D., Friedman, M., Langholz, G., Kandel, A., Complex fuzzy logic. IEEE Transactions on Fuzzy Systems, 2003, 11(4): p. 450-461.

[17] Dick, S., Towards complex fuzzy logic. IEEE Transaction on Fuzzy Systems, 2005, 13: p. 405-414.

[18] Tamir, D.E., Lin, J., Kandel, A., A New Interpretation of Complex Membership Grade, Accepted for publication in the International Journal of Intelligent Systems, 2011.

[19] Tamir, D.E., Last, M., Kandel, A., Generalized Complex Fuzzy Propositional Logic. Accepted for publication in the World Conference on Soft computing, San Francisco, 2011.

[20] Baaz, M., Hajek, P., Montagna, F., Veith, H., Complexity of t-tautologies. Annals of Pure and Applied Logic, 2002, 113(1-3): p. 3-11.

[21] Cintula, P., Weakly implicative fuzzy logics. Archive for Mathematical Logic, 2006, 45(6): p. 673-704.

[22] Cintula, P., Advances in Ł and Ł1/2 logics. Archives of Mathematical Logic, 2003, 42: p. 449-468.

[23] Montagna, F., On the predicate logics of continuous t-norm BL-algebras. Archives of Mathematical Logic, 2005, 44: p. 97-114.

[24] Hájek, P., Fuzzy logic and arithmetical hierarchy. Fuzzy Sets and Systems, 1995, 3(8): p. 359–363.

[25] Moses, D., Degani, O., Teodorescu, H., Friedman, M., Kandel, A. Linguistic coordinate transformations for complex fuzzy sets. in 1999 IEEE International conference on Fuzzy Systems. 1999, Seoul, Korea.

[26] Zhang, G., Dillon, T. S., Cai, K., Ma, J., Lu, J., Operation properties and delta equalities of complex fuzzy sets. International Journal on Approximate Reasoning, 2009, 50(8): p. 1227- 1249.

[27] Nguyen, H.T., Kandel, A., Kreinovich, V. Complex fuzzy sets: towards new foundations. in Proceedings, 2000 IEEE International Conference on Fuzzy Systems. 2000, San Antonio, Texas.

[28] Deshmukh, A.Y., Bavaskar, A. B., Bajaj, P. R., Keskar A. G., Implementation of complex fuzzy logic modules with VLSI approach. International Journal on Computer Science and Network Security, 2008, 8: p. 172-178.

[29] Chen, Z., Aghakhani, S., Man, J., Dick, S., ANCFIS: A Neuro-Fuzzy Architecture Employing Complex Fuzzy Sets. 2009, (submitted).

[30] Man, J., Chen, Z., Dick, S. Towards inductive learning of complex fuzzy inference systems. in Proceedings of the International Conference of the North American Fuzzy Information Processing. 2007, San Diego, CA.

[31] Hirose, A., Complex-valued neural networks. 2006: Springer-Verlag.

[32] Michel, H.E., Awwal, A. A. S., Rancour, D. Artificial neural networks using complex numbers and phase encoded weights-electronic and optical implementations. in Proceedings of the International Joint Conference on Neural Networks. 2006, Vancouver, BC.

[33] Noest, A.J., Discrete-state phasor neural nets. Physics Review A, 1988, 38: p. 2196-2199.

[34] Leung, S.H., Hyakin, S., The complex back propagation algorithm. IEEE Transactions on Signal Processing, 1991, 39(9): p. 2101-2104.

[35] Iritani, N.T., Sakakibara, K., Improvements of the traffic signal control by complex-valued Hopfield networks. in Proceedings of the International Joint Conference on Neural Networks. 2006, Vancouver, BC.

[36] Buckley, J.J., Fuzzy complex numbers. Fuzzy Sets and Systems, 1989, 33: p. 333-345.

[37] Buckley, J.J., Qu, Y., Fuzzy complex analysis I: differentiation. Fuzzy Sets and Systems, 1991, 41: p. 269-284.

[38] Zhang, G., Fuzzy limit theory of fuzzy complex numbers. Fuzzy Sets and Systems, 1992, 46(2): p. 227-2352.

[39] Wu, C., Qiu, J., Some remarks for fuzzy complex analysis. Fuzzy Sets and Systems, 1999, 106: p. 231-238.

[40] Behounek, L., Cintula, P., Fuzzy class theory. Fuzzy Sets and Systems, 2005, 154(1): p. 34-55.
How to Cite
TAMIR, Dan E.; KANDEL, Abraham. Axiomatic Theory of Complex Fuzzy Logic and Complex Fuzzy Classes. INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROL, [S.l.], v. 6, n. 3, p. 562-576, sep. 2011. ISSN 1841-9844. Available at: <>. Date accessed: 30 nov. 2020. doi:


Fuzzy Logic, Fuzzy Class Theory, Complex Fuzzy Logic, Complex fuzzy Class theory