Algebraic Kripke-style semantics for substructural fuzzy logics

준구조 퍼지 논리를 위한 대수적 크립키형 의미론

  • Yang, Eunsuk (Department of Philosophy & Institute of Critical Thinking and Writing, Chonbuk National University)
  • 양은석 (전북대학교 철학과, 비판적사고와논술연구소)
  • Received : 2016.03.05
  • Accepted : 2016.06.13
  • Published : 2016.06.30

Abstract

This paper deals with Kripke-style semantics, which will be called algebraic Kripke-style semantics, for fuzzy logics based on uninorms (so called uninorm-based logics). First, we recall algebraic semantics for uninorm-based logics. In the general framework of uninorm-based logics, we next introduce various types of general algebraic Kripke-style semantics, and connect them with algebraic semantics. Finally, we analogously consider particular algebraic Kripke-style semantics, and also connect them with algebraic semantics.

이 글에서 우리는 유니놈에 기반한 퍼지 논리를 위한 대수적 크립키형 의미론을 다룬다. 이를 위하여 먼저 유니놈에 기반한 논리체계들을 위한 대수적 의미론을 재고한다. 다음으로 유니놈에 기반한 체계들의 일반적 구조에서 다양한 종류의 일반적 대수적 크립키형 의미론을 소개하고 그것들을 대수적 의미론과 연관 짓는다. 마지막으로 우리는 유사하게 특수한 대수적 의미론을 소개하고 이를 또한 대수적 의미론과 연관 짓는다.

Keywords

References

  1. Bimbo, K. and Dunn, J. M. (2002), "Four-valued logic", Notre Dame Journal of Formal Logic, 42, pp. 171-192.
  2. Blok, W. J. and Pigozzi, D. (1989), Algebraizable Logic, Memoirs of the American Methematical Society, vol. 386. American Methematical Society, Providence.
  3. Ciabattoni, A., Esteva, F., and Godo, L. (2002), "T-norm-based logics with n-contraction", Special Issue on SOFSEM2002 of Neural Network World, 12, pp. 453-460.
  4. Cintula, P. (2006), "Weakly Implicative (Fuzzy) Logics I: Basic properties", Archive for Mathematical Logic, 45, pp. 673-704.
  5. Cintula, P., Esteva, F., Gispert, J., Godo, L., Montagna, F., and Noguera, C. (2009), "Distinguished algebraic semantics for t-norm-based Fuzzy Logics", Annals of Pure and Applied Logic, 160, pp. 53-81. https://doi.org/10.1016/j.apal.2009.01.012
  6. Cintula, P. and Noguera, C. (2011), A general framework for mathematical fuzzy logic, in P. Cintula, P. Hajek, and C. Noguera (eds.) Handbook of Mathematical Fuzzy Logic, vol 1, London: College publications, pp. 103-207.
  7. 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.
  8. Dunn, J. M. (1976) "A Kripke-style semantics for R-Mingle using a binary accessibility relation", Studia Logica, 35, pp. 163-172. https://doi.org/10.1007/BF02120878
  9. Dunn, J. M. (2000), "Partiality and its Dual", Studia Logica, 66, pp. 5-40. https://doi.org/10.1023/A:1026740726955
  10. Gabbay, D. and Metcalfe, G. (2007), "Fuzzy Logics based on [0, 1)-continuous uninorms", Archive for Mathematical Logic, 46, pp. 425-449. https://doi.org/10.1007/s00153-007-0047-1
  11. Kripke, S. (1963), "Semantic analysis of modal logic I: normal modal propositional calculi", Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 9, pp. 67-96.
  12. Kripke, S. (1965a), "Semantic analysis of intuitionistic logic I", in J. Crossley and M. Dummett (eds.) Formal systems and Recursive Functions, Amsterdam: North-Holland Publ Co, pp. 92-129.
  13. Kripke, S. (1965b), "Semantic analysis of modal logic II", in J. Addison, L. Henkin, and A. Tarski (eds.) The theory of models, Amsterdam: North-Holland Publ Co, pp. 206-220.
  14. Metcalfe, G. and Montagna, F. (2007), "Substructural Fuzzy Logics", Journal of Symbolic Logic, 72, pp. 834-864. https://doi.org/10.2178/jsl/1191333844
  15. Metcalfe, G., Olivetti, N., and Gabbay, D. (2009) Proof Theory for Fuzzy Logics, Springer.
  16. Montagna, F., Noguera, C., and Horcik, R. (2006), "On weakly cancellative fuzzy logics", Journal of Logic and Computation, 16, pp. 423-450. https://doi.org/10.1093/logcom/exl002
  17. 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
  18. 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
  19. 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
  20. Novak, V. (1990), "On the syntactico-semantical completeness of first-order fuzzy logic I, II", Kybernetika, 26, pp. 47-66.
  21. Thomason, R. H. (1969), "A semantic study of constructive falsity", Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 15, pp. 247-257. https://doi.org/10.1002/malq.19690151602
  22. Tsinakis, C. and Blount, K. (2003), "The structure of residuated lattices", International Journal of Algebra and Computation, 13, pp. 437-461. https://doi.org/10.1142/S0218196703001511
  23. Urquhart, A. (1986), "Many-valued logic". in D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic, vol. 3, Dordrecht: Reidel Publ Co, pp. 71-116.
  24. Wang, S., Wang, B., and Pei, D. (2005), "A fuzzy logic for an ordinal sum t-norm", Fuzzy Sets and Systems, 149, pp. 297-307. https://doi.org/10.1016/j.fss.2004.01.005
  25. Wang, S., Wang, B., and Ren. F. (2005), "NML, a schematic extension of F. Esteva and L. Godo's logic MTL", Fuzzy Sets and Systems, 149, pp. 285-295. https://doi.org/10.1016/j.fss.2003.12.005
  26. Yager, R. R., and Rybalov, A., (1996), "Uninorm aggregation operators", Fuzzy Sets and Systems, 80, 111-120. https://doi.org/10.1016/0165-0114(95)00133-6
  27. Yang, E. (2009), "(Star-based) four-valued Kripke-style Semantics for some neighbors of E, R, T", Logique et Analyse, 207, pp. 255-280.
  28. Yang, E. (2012a), "(Star-based) three-valued Kripke-style semantics for pseudo- and weak-Boolean logics", Logic Journal of the IGPL, 20, pp. 187-206. https://doi.org/10.1093/jigpal/jzr030
  29. Yang, E. (2012b), "Kripke-style semantics for UL", Korean Journal of Logic, 15 (1), pp. 1-15.
  30. Yang, E. (2014a), "Algebraic Kripke-Style Semantics for Relevance Logics", Journal of Philosophical Logic, 43, pp. 803-826. https://doi.org/10.1007/s10992-013-9290-6
  31. Yang, E. (2014b), "Algebraic Kripke-style semantics for weakening-free fuzzy logics", Korean Journal of Logic, 17 (1), pp. 181-195.
  32. Yang, E. (2015), "Basic substructural core fuzzy logics and their extensions: Mianorm-based logics", Fuzzy Sets and Systems, http://dx.doi.org/10.1016/j.fss.2015.09.007. Published online: 25 September 2015.