The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems

Author(s)
Jiancheng Zhang

ABSTRACT

Based on the analysis of the properties of Γ-conclusion by means of deduction theorems, completeness theorems and the theory of truth degree of formulas, the present papers introduces the concept of the membership degree of formulas A is a consequence of Γ (or Γ-conclusion) in Lukasiewicz n-valued propositional logic systems, Godel n-valued propositional logic system and the R_{0} n-valued propositional logic systems. The condition and related calculations of formulas A being Γ-conclusion were discussed by extent method. At the same time, some properties of membership degree of formulas A is a Γ-conclusion were given. We provide its algorithm of the membership degree of formulas A is a Γ-conclusion by the constructions of theory root.

Based on the analysis of the properties of Γ-conclusion by means of deduction theorems, completeness theorems and the theory of truth degree of formulas, the present papers introduces the concept of the membership degree of formulas A is a consequence of Γ (or Γ-conclusion) in Lukasiewicz n-valued propositional logic systems, Godel n-valued propositional logic system and the R

Cite this paper

J. Zhang, "The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems,"*American Journal of Operations Research*, Vol. 2 No. 2, 2012, pp. 147-152. doi: 10.4236/ajor.2012.22017.

J. Zhang, "The Theory of Membership Degree of Γ-Conclusion in Several n-Valued Logic Systems,"

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[1] H. W. Liu and G. J. Wang, “Unified Forms of Fully Implicational Restriction Methods for Fuzzy Reasoning,” Information Sciences, Vol. 177, No. 3, 2007, pp. 956-966. doi:10.1016/j.ins.2006.08.012

[2] J. Pavelka, “On Fuzzy Logic II-Enriched Residuated Lattices and Semantics of Propositional Calculi,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Vol. 25, 2011, pp. 119-134.

[3] G. J. Wang and H. Wang, “Non-Fuzzy Versions of Fuzzy Reasoning in Classical Logic,” Information Sciences, Vol. 138, No. 1-4, 2011, pp. 211-236. doi:10.1016/S0020-0255(01)00131-1

[4] G. J. Wang, “On the Logic Foundation of Fuzzy Reasoning,” Information Sciences, Vol. 117, No. 1-2, 1999, pp. 47-88. doi:10.1016/S0020-0255(98)10103-2

[5] M. S. Ying, “Compactness, the L?wenheim-Skolem Property and the Direct Product of Lattices of Truth Values,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Vol. 38, 1992, pp. 521-524.

[6] F. Esteva and L. Godo, “Monoidal t-Norm Based Logic: Towards a Logic for Left-Continuous t-Norms,” Fuzzy Set and Systems, Vol. 124, No. 3, 2001, pp. 271-288. doi:10.1016/S0165-0114(01)00098-7

[7] P. Hájek, “Metamathematics of Fuzzy Logic,” Kluwer Academic Publishers, Dordrecht, 1998.

[8] G. J. Wang and H. J. Zhou, “Introduction to Mathematical Logic and Resolution Principle,” 2nd Edition, Science in China Press, Beijing, 2006 (in Chinese).

[9] G. J. Wang, “A Formal Deductive System for Fuzzy Propositional Calculus,” Chinese Science Bulletin, Vol. 42, No. 14, 1997, pp. 1521-1525.

[10] J. C. Zhang, “Some Properties of the Roots of Theories in Propositional Logic Systems,” Computers and Mathematics with Applications, Vol. 55, No. 9, 2008, pp. 2086- 2093. doi:10.1016/j.camwa.2007.08.035

[11] J. C. Zhang and X. Y. Yang, “Some Properties of Fuzzy Reasoning in Propositional Fuzzy Logic Systems,” Information Sciences, Vol. 180, No. 23, 2010, pp. 4661- 4671. doi:10.1016/j.ins.2010.07.035

[12] S. Gottwald, “A Treatise on Many-Valued Logics, Studies in Logic and Computation,” Research Studies Press, Baldock, 2001.

[13] G. J. Wang, “Theory of Non-Classical Mathematical Logic and Approximate Reasoning,” Science in China Press, Beijing, 2000.

[14] D. Dubois, J. Lang and H. Prade, “Fuzzy Set in Approximate Reasoning,” Fuzzy Sets and Systems, Vol. 40, No. 1, 1991, pp. 143-244. doi:10.1016/0165-0114(91)90050-Z