Notre Dame Journal of Formal Logic

The Distributivity on Bi-Approximation Semantics

Tomoyuki Suzuki

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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.

Article information

Notre Dame J. Formal Logic, Volume 57, Number 3 (2016), 411-430.

Received: 23 April 2013
Accepted: 12 May 2014
First available in Project Euclid: 20 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03G10: Lattices and related structures [See also 06Bxx] 03G25: Other algebras related to logic [See also 03F45, 06D20, 06E25, 06F35]
Secondary: 03G27: Abstract algebraic logic

lattice-based logics relational semantics canonicity


Suzuki, Tomoyuki. The Distributivity on Bi-Approximation Semantics. Notre Dame J. Formal Logic 57 (2016), no. 3, 411--430. doi:10.1215/00294527-3542442.

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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.
  • [2] Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, 2nd ed., Cambridge University Press, New York, 2002.
  • [3] Galatos, N., and P. Jipsen, “Residuated frames with applications to decidability,” Transactions of the American Mathematical Society, vol. 365 (2013), pp. 1219–49.
  • [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.
  • [5] Gehrke, M., “Generalized Kripke frames,” Studia Logica, vol. 84 (2006), pp. 241–75.
  • [6] Ghilardi, S., and G. Meloni, “Constructive canonicity in non-classical logics,” Annals of Pure and Applied Logic, vol. 86 (1997), pp. 1–32.
  • [7] Goldblatt, R. I., “Semantic analysis of orthologic,” Journal of Philosophical Logic, vol. 3 (1974), pp. 19–35.
  • [8] Hartonas, C., “Duality for lattice-ordered algebras and for normal algebraizable logics,” Studia Logica, vol. 58 (1997), pp. 403–50.
  • [9] Hartonas, C., and J. M. Dunn, “Stone duality for lattices,” Algebra Universalis, vol. 37 (1997), pp. 391–401.
  • [10] Restall, G., An Introduction to Substructural Logics, Routledge, London, 2000.
  • [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.
  • [13] Suzuki, T., “Morphisms on bi-approximation semantics,” pp. 494–515 in Advances in Modal Logic, edited by T. Bolander, T. Braüner, 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.