JACIII Vol.19 No.6 pp. 861-866
doi: 10.20965/jaciii.2015.p0861


An Algebraic Aspect of Correspondences Between Implicational Fragment Logics and Fuzzy Logics

Mayuka F. Kawaguchi* and Michiro Kondo**

*Division of Computer Science and Information Technology, Hokkaido University
Kita 14, Nishi 9, Sapporo 060-0814, Japan
**Tokyo Denki University
2-1200 Muzai Gakuendai, Inzai, Chiba 270-1382, Japan

May 13, 2015
August 28, 2015
Online released:
November 20, 2015
November 20, 2015
implicative algebras, implicational fragment logics, monotone BI-algebras, weakly-associative conjunctive algebras

This research report treats a correspondence between implicational fragment logics and fuzzy logics from the viewpoint of their algebraic semantics. The authors introduce monotone BI-algebras by loosening the axiomatic system of BCK-algebras. We also extend the algebras of fuzzy logics with weakly-associative conjunction from the case of the real unit interval to the case of a partially ordered set. As the main result of this report, it is proved that the class of monotone BI-algebras with condition (S) coincides with the class of weakly-associative conjunctive algebras.

  1. [1] T. Hosoi, “On the implicational fragment logics,” J. Tsuda College, No.7, pp. 1-5, 1975.
  2. [2] K. Iseki and S. Tanaka, “An introduction to the theory of BCK-algebras,” Mathematica Japonica, Vol.23, No.1, pp. 1-26, 1978.
  3. [3] Y. Komori, “BCK-daisuu to BCI-daisuu no go no mondai,” RIMS Kokyuroku, Kyoto Univ., Vol.786, pp. 27-31, 1992 (in Japanese).
  4. [4] H. Rasiowa, “An Algebraic Approach to Non-Classical Logics,” North-Holland, 1974.
  5. [5] P. Háajek, “Metamathematics of Fuzzy Logic,” Trends in Logic, Vol.4, Kluwer Academic Publishers, 1998.
  6. [6] F. Esteva and L. Godo, “Monoidal t-norm based logic: towards a logic for left-continuous t-norms,” Fuzzy Sets and Systems, Vol.124, No.3, pp. 271-288, 2001.
  7. [7] A. Iorgulescu, “Iséeki algebras. Connection with BL algebras,” Soft Computing, Vol.8, pp. 449-463, 2003.
  8. [8] M. Kondo and M. F. Kawaguchi, “Partially ordered set with residuated t-norm,” Proc. of 35th Int. Symp. on Multiple-Valued Logic, pp. 26-29, 2005.
  9. [9] M. F. Kawaguchi and M. Miyakoshi, “L-fuzzy logic with non-associative conjunctions,” Int. J. Multiple-Valued Logic, Vol.4, No.4, pp. 281-306, 1999.
  10. [10] M. F. Kawaguchi and M. Miyakoshi, “Weakly associative functions on [0, 1] as logical connectives,” Proc. of 34th Int. Symp. on Multiple-Valued Logic, pp. 44-48, 2004.
  11. [11] M. F. Kawaguchi, Y. Koike, and M. Miyakoshi, “An algebraic structure of fuzzy logics with weakly associative conjunctors,” Integrated Uncertainty Management and Applications, Advances in Intelligent and Soft-Computing, Vol.68, Springer-Verlag, pp. 349-359, 2010.
  12. [12] K. Iseki, “BCK-algebras with condition (S),” Mathematica Japonica, Vol.24, No.1, pp. 107-119, 1979.

*This site is desgined based on HTML5 and CSS3 for modern browsers, e.g. Chrome, Firefox, Safari, Edge, IE9,10,11, Opera.

Last updated on May. 26, 2017