누승적 미아놈 논리 IMIAL의 몇몇 공리적 확장

Some axiomatic extensions of the involutive mianorm Logic IMIAL

  • 양은석 (전북대학교 철학과, 비판적사고와논술연구소)
  • Yang, Eunsuk (Department of Philosophy & Institute of Critical Thinking and Writing, Chonbuk National University)
  • 투고 : 2017.07.24
  • 심사 : 2017.09.17
  • 발행 : 2017.10.31

초록

이 글에서 우리는 누승적 미아놈 논리 IMIAL의 몇몇 공리적 확장 체계의 표준 완전성을 다룬다. 이를 위하여, 먼저 누승적 미아놈에 바탕을 둔 일곱 개의 논리 체계를 소개한다. 각 체계에 상응하는 대수적 구조를 정의한 후, 이들 체계가 대수적으로 완전하다는 것을 보인다. 다음으로, 이 논리 체계들 중 네 체계가 표준적으로 완전하다는 것 즉 단위 실수 [0, 1]에서 완전하다는 것을 제네이-몬테그나 방식의 구성을 사용하여 보인다.

In this paper, we deal with standard completeness of some axiomatic extensions of the involutive mianorm logic IMIAL. More precisely, first, seven involutive mianorm-based logics are introduced. Their algebraic structures are then defined, and their corresponding algebraic completeness is established. Next, standard completeness is established for four of them using construction in the style of Jenei-Montagna.

키워드

참고문헌

  1. Cintula, P. (2006), "Weakly Implicative (Fuzzy) Logics I: Basic properties", Archive for Mathematical Logic, 45, pp. 673-704. https://doi.org/10.1007/s00153-006-0011-5
  2. Cintula, P., Horcik, R., and Noguera, C. (2013), "Non-associative substructural logics and their semilinear extensions: axiomatization and completeness properties", Review of Symbol. Logic, 12, pp. 394-423.
  3. Cintula, P., Horcik, R., and Noguera, C. (2015), "The quest for the basic fuzzy logic", Mathematical Fuzzy Logic, P. Hajek (Ed.), Springer.
  4. Cintula, P. and Noguera, C. (2011), A general framework for mathematical fuzzy logic, Handbook of Mathematical Fuzzy Logic, vol 1, P. Cintula, P. Hajek, and C. Noguera (Eds.), London, College publications, pp. 103-207.
  5. Esteva, F., Gispert, L., Godo, L., and Montagna, F. (2002), "On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic", Studia Logica, 71, pp. 393-420.
  6. Galatos, N., Jipsen, P., Kowalski, T., and Ono, H. (2007), Residuated lattices: an algebraic glimpse at substructural logics, Amsterdam, Elsevier.
  7. Hajek, P. (1998), Metamathematics of Fuzzy Logic, Amsterdam, Kluwer.
  8. Horcik, R. (2011), Algebraic semantics: semilinear FL-algebras, Handbook of Mathematical Fuzzy Logic, vol 1, P. Cintula, P. Hajek, and C. Noguera (Eds.), London, College publications, pp. 283-353.
  9. Jenei, S. and Montagna, F. (2002), "A Proof of Standard completeness for Esteva and Godo's Logic MTL", Studia Logica, 70, pp. 183-192. https://doi.org/10.1023/A:1015122331293
  10. Metcalfe, G., and Montagna, F. (2007), "Substructural Fuzzy Logics", Journal of Symbolic Logic, 72, pp. 834-864. https://doi.org/10.2178/jsl/1191333844
  11. Yang, E. (2009), "On the standard completeness of an axiomatic extension of the uninorm logic", Korean Journal of Logic, 12/2, pp. 115-139.
  12. Yang, E. (2013), "Standard completeness for MTL", Korean Journal of Logic, 16/3, pp. 437-452.
  13. Yang, E. (2014), "An Axiomatic Extension of the Uninorm Logic Revisited", Korean Journal of Logic, 17/2, pp. 323-348.
  14. Yang, E. (2015), "Weakening-free, non-associative fuzzy logics: micanorm-based logics", Fuzzy Sets and Systems, 276, pp. 43-58. https://doi.org/10.1016/j.fss.2014.11.020
  15. Yang, E. (2016), "Basic substructural core fuzzy logics and their extensions: Mianorm-based logics", Fuzzy Sets and Systems, 301, pp. 1-18. https://doi.org/10.1016/j.fss.2015.09.007
  16. Yang, E. (2017a), "Involutive basic substructural core fuzzy logics: Involutive mianorm-based logics", Fuzzy Sets and Systems, 320, pp. 1-16. https://doi.org/10.1016/j.fss.2017.03.013
  17. Yang, E. (2017b), "Involutive Micanorm Logics with the n-potency axiom", Korean Journal of Logic, 20/2, pp. 273-292.