JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2017, Vol. 52 ›› Issue (11): 71-81.doi: 10.6040/j.issn.1671-9352.0.2017.164

Previous Articles     Next Articles

(T,N)-implication and its basic properties

YU Jun-hong, ZHOU Hong-jun*   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xian 710119, Shaanxi, China
  • Received:2017-04-17 Online:2017-11-20 Published:2017-11-17

PDF (PC)

374

Abstract: Fuzzy implications play a fundamental role in fuzzy logic and approximate reasoning. According to construction methods, there are mainly four classes of fuzzy implications, namely,(S,N)-implications, R-implications, QL-implications and Yagers generated implications. We introduce a new class of implications, called(T,N)-implications, generated from t-norms T and fuzzy negations N inspired by the classical tautology p→q≡(p∧q). We discuss the properties of(T,N)-implications and study some classical logic tautologies(e.g., law of importation and distributivity over t-norms or t-conorms)for (T,N)-implications. And the relationships of (T,N)-implications to f-implications, g-implications,(S,N)-implications and R-implications are investigated.

Key words: triangular norm, triangular conorm, (T,N)-implication, R-implication, g-implication, fuzzy implication, f-implication, (S,N)-implication, distributivity equation, fuzzy negation

CLC Number: 

  • O142
[1] 王国俊. 非经典数理逻辑与近似推理 [M]. 2 版. 北京: 科学出版社, 2008. WANG Guojun. Non-classical mathematical logic and approximate reasoning[M]. 2rd ed. Beijing: Science Press, 2008.
[2] HÁJEK P. Metamathematics of fuzzy logic[M]. Dordrecht: Kluwer Academic Publishers, 1998.
[3] GOTTWALD S. A treatise on many-valued logic[M]. Baldock: Research Studies Press, 2001.
[4] PEI Daowu. Formalization of implication based fuzzy reasoning method[J]. International Journal of Approximate Reasoning, 2012, 53:837-846.
[5] 李玉, 王贵君. 一类正则蕴涵算子诱导的DISO模糊系统的构造[J]. 山东大学学报(理学版), 2014, 49(6):57-63. LI Yu, WANG Guijun. Structures of DISO fuzzy system induced by a class of regular implication operators[J]. Journal of Shandong University(Natural Science), 2014, 49(6):57-63.
[6] BACZYNSKI M, JAYARAM B. On the characterizations of(S,N)-implication[J]. Fuzzy Sets and Systems, 2007, 158:1713-1727.
[7] MAS M, MONSERRAT M, TORRENS J, et al. A survey on fuzzy implication functions[J]. IEEE Transactions on Fuzzy Systems, 2007, 15(6):1107-1121.
[8] BACZYNSKI M, JAYARAM B. QL-implications: some properties and intersections[J]. Fuzzy Sets and Systems, 2010, 161:158-188.
[9] YAGER R R. Some new classes of implication operators and their role in approximate reasoning[J]. Information Science, 2004, 167(23):193-216.
[10] KLEMENT E P, MESIA R. Triangular norms[M]. Dordrecht: Kluwer Academic Publisher, 2000.
[11] BACZYNSKI M, JAYARAM B. Fuzzy implication[M]. Berlin: Springer, 2008.
[12] FODOR J, ROUBENS M. Fuzzy preference modeling and multicriteria decision support[M]. Dordrecht: Kluwer Academic Publisher, 1994.
[13] SCHWEIZER B, SKLAR A. Probabilistic metric spaces[M]. New York: Dover Publications, 2005.
[14] JAYARM B. On the law of importation (x∧y)→z≡x→(y→z) in fuzzy logic[J]. IEEE Transactions on Fuzzy Systems, 2008, 16:130-144.
[15] BERTOLUZZA C. On the distributivity between t-norms and t-conorms[C] // Proc znd IEEE Int Conf. San Francisco: IEEE, 1993: 140-147.
[16] BERTOLUZZA C, DOLDI V. On the distributivity between t-norms and t-conorms[J]. Fuzzy Sets and Systems, 2004, 142(1):85-104.
[17] ZHANG Fengxia, LIU Huawen. On a new class of implication:(g,u)-implications and distributive equations[J]. International Journal of Approximate Reasoning, 2013, 54:1049-1065.
[1] HAN Liang, LIU Hua-wen. On the solutions of the distributive equations of #br# logic operators on a finite chain [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(2): 29-35.
[2] LIU Chun-hui1,2. Theory of filters in Fuzzy implication algebras [J]. J4, 2013, 48(09): 73-77.
[3] SONG Jia, LUO Minnan, LI Yongming. [J]. J4, 2009, 44(3): 67-70 .
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd