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The distributivity on bi-approximation semantics. (English) Zbl 1436.03317
Summary: In this paper, we give a possible characterization of the distributivity on bi-approximation semantics. To this end, we introduce new notions of special elements on polarities and show that the distributivity is first-order definable on bi-approximation semantics. In addition, we investigate the dual representation of those structures and compare them with bi-approximation semantics for intuitionistic logic. We also discuss that two different methods to validate the distributivity – by the splitters and by the adjointness – can be explicated with the help of the axiom of choice as well.
MSC:
03G10 Logical aspects of lattices and related structures
03G25 Other algebras related to logic
03G27 Abstract algebraic logic
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[1] Ciabattoni, A., N. Galatos, and K. Terui, “Algebraic proof theory for substructural logics: Cut-elimination and completions,” Annals of Pure and Applied Logic , vol. 163 (2012), pp. 266-90. · Zbl 1245.03026
[2] Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order , 2nd ed., Cambridge University Press, New York, 2002. · Zbl 1002.06001
[3] Galatos, N., and P. Jipsen, “Residuated frames with applications to decidability,” Transactions of the American Mathematical Society , vol. 365 (2013), pp. 1219-49. · Zbl 1285.03077
[4] Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated lattices: An algebraic glimpse at substructural logics , vol. 151 of Studies in Logic and the Foundations of Mathematics , Elsevier, Amsterdam, 2007. · Zbl 1171.03001
[5] Gehrke, M., “Generalized Kripke frames,” Studia Logica , vol. 84 (2006), pp. 241-75. · Zbl 1115.03013
[6] Ghilardi, S., and G. Meloni, “Constructive canonicity in non-classical logics,” Annals of Pure and Applied Logic , vol. 86 (1997), pp. 1-32. · Zbl 0949.03019
[7] Goldblatt, R. I., “Semantic analysis of orthologic,” Journal of Philosophical Logic , vol. 3 (1974), pp. 19-35. · Zbl 0278.02023
[8] Hartonas, C., “Duality for lattice-ordered algebras and for normal algebraizable logics,” Studia Logica , vol. 58 (1997), pp. 403-50. · Zbl 0886.06002
[9] Hartonas, C., and J. M. Dunn, “Stone duality for lattices,” Algebra Universalis , vol. 37 (1997), pp. 391-401. · Zbl 0902.06008
[10] Restall, G., An Introduction to Substructural Logics , Routledge, London, 2000. · Zbl 1028.03018
[11] Suzuki, T., “Bi-approximation semantics for substructural logic at work,” pp. 411-33 in Advances in Modal Logic (Nancy, France, 2008) , edited by L. Beklemishev, V. Goranko, and V. Shehtman, vol. 8 of Advances in Modal Logic , College Publications, London, 2010.
[12] Suzuki, T., “Canonicity results of substructural and lattice-based logics,” Review of Symbolic Logic , vol. 4 (2011), pp. 1-42. · Zbl 1229.03023
[13] Suzuki, T., “Morphisms on bi-approximation semantics,” pp. 494-515 in Advances in Modal Logic , edited by T. Bolander, T. Bra√ľner, S. Ghilardi, and L. Moss, vol. 9 of Advances in Modal Logic , College Publications, London, 2012.
[14] Suzuki, T., “A Sahlqvist theorem for substructural logic,” Review of Symbolic Logic , vol. 6 (2013), pp. 229-53. · Zbl 1282.03014
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