약한 결합 원리를 갖는 준구조 퍼지 논리를 위한 집합 이론적 크립키형 의미론

Set-theoretic Kripke-style Semantics for Weakly Associative Substructural Fuzzy Logics

  • 양은석 (전북대학교 철학과, 비판적사고와논술연구소)
  • Yang, Eunsuk (Department of Philosophy & Institute of Critical Thinking and Writing, Chonbuk National University)
  • 투고 : 2018.10.30
  • 심사 : 2018.12.05
  • 발행 : 2019.02.28

초록

이 글에서 우리는 (곱 연언 &의) 약한 형식의 결합 원리를 갖는 준구조 퍼지 논리를 위한 집합 이론적 크립키형 의미론을 연구한다. 이를 위하여 먼저 약한 결합 원리를 갖는 세 준구조 퍼지 논리체계들을 상기한 후 이 체계들에 상응하는 크립키형 의미론을 소개한다. 다음으로 집합 이론적 방식을 이용하여 이 체계들이 완전하다는 것을 보인다.

This paper deals with Kripke-style semantics, which will be called set-theoretic Kripke-style semantics, for weakly associative substructural fuzzy logics. We first recall three weakly associative substructural fuzzy logic systems and then introduce their corresponding Kripke-style semantics. Next, we provide set-theoretic completeness results for them.

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. https://doi.org/10.1017/S1755020313000099
  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. Diaconescu, D. and Georgescu, G. (2007), "On the forcing semantics for monoidal t-norm based logic", Journal of Universal Computer Science 13: pp. 1550-1572.
  6. Esteva, F. and Godo, L. (2001), "Monoidal t-norm based logic: towards a logic for left-continuous t-norms", Fuzzy Sets and Systems 124: pp. 271-288. https://doi.org/10.1016/S0165-0114(01)00098-7
  7. Galatos, N., Jipsen, P., Kowalski, T., and Ono, H., (2007). Residuated lattices: an algebraic glimpse at substructural logics, Amsterdam, Elsevier.
  8. Kripke, S. (1963). "Semantic analysis of modal logic I: normal modal propositional calculi", Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 9, pp. 67-96. https://doi.org/10.1002/malq.19630090502
  9. Kripke, S. (1965a). "Semantic analysis of intuitionistic logic I", Formal systems and Recursive Functions, J. Crossley and M. Dummett (Eds.), Amsterdam, North-Holland Publ. Co., pp. 92-129.
  10. Kripke, S. (1965b). "Semantic analysis of modal logic II", The theory of models, J. Addison, L. Henkin and A. Tarski (Eds.), Amsterdam, North-Holland Publ. Co., pp. 206-220
  11. Metcalfe, G., and Montagna, F. (2007), "Substructural Fuzzy Logics", Journal of Symbolic Logic, 72, pp. 834-864. https://doi.org/10.2178/jsl/1191333844
  12. Montagna, F. and Ono, H. (2002) "Kripke semantics, undecidability and standard completeness for Esteva and Godo's Logic $MTL{\forall}$", Studia Logica, 71, pp. 227-245. https://doi.org/10.1023/A:1016500922708
  13. Montagna, F. and Sacchetti, L. (2003) "Kripke-style semantics for many-valued logics", Mathematical Logic Quaterly, 49, pp. 629-641. https://doi.org/10.1002/malq.200310068
  14. Montagna, F. and Sacchetti, L. (2004) "Corrigendum to "Kripke-style semantics for many-valued logics", Mathematical Logic Quaterly, 50, pp. 104-107. https://doi.org/10.1002/malq.200310081
  15. Yang, E. (2009) "Non-associative fuzzy-relevance logics: strong t-associative monoidal uninorm logics", Korean Journal of Logic, 12(1), pp. 89-110.
  16. Yang, E. (2012) "Kripke-style semantics for UL", Korean Journal of Logic, 15(1), pp. 1-15. https://doi.org/10.22860/KAFL.2012.15.1.1
  17. Yang, E. (2014a) "Algebraic Kripke-style semantics for weakening-free fuzzy logics", Korean Journal of Logic, 17(1), pp. 181-195. https://doi.org/10.22860/KAFL.2014.17.1.181
  18. Yang, E. (2014b) "Algebraic Kripke-style semantics for relevance logics", Journal of Philosophical Logic, 43, pp. 803-826. https://doi.org/10.1007/s10992-013-9290-6
  19. Yang, E. (2015a) "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
  20. Yang, E. (2015b) "Some an axiomatic extension of the involutive micanorm logic IMICAL", Korean Journal of Logic, 18(2), pp. 197-215. https://doi.org/10.22860/KAFL.2015.18.2.197
  21. Yang, E. (2016a) "Algebraic Kripke-style semantics for substructural fuzzy logics", Korean Journal of Logic 19(2), pp. 295-322. https://doi.org/10.22860/KAFL.2016.19.2.295
  22. Yang, E. (2016b) "Weakly associative fuzzy logics", Korean Journal of Logic 19(3), pp. 437-461. https://doi.org/10.22860/KAFL.2016.19.3.437
  23. Yang, E. (2016c) "Basic substructural core fuzzy logics and their extensions: Mianorm-based logics", Fuzzy Sets and Systems 276, pp. 43-58. https://doi.org/10.1016/j.fss.2014.11.020
  24. 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
  25. Yang, E. (2017b) "A non-associative generalization of continuous t-norm-based logics", Journal of Intelligent & Fuzzy Systems 33, pp. 3743-3752. https://doi.org/10.3233/JIFS-17623
  26. Yang, E. (2018a) "Set-theoretical Kripke-style semantics for an extension of HpsUL, CnHpsUL*", Korean Journal of Logic 21(1), pp. 39-57. https://doi.org/10.22860/KAFL.2018.21.1.39
  27. Yang, E. (2018b) "Algebraic Kripke-style semantics for weakly associative fuzzy logics", Korean Journal of Logic 21(2), pp. 155-173. https://doi.org/10.22860/KAFL.2018.21.2.155