## Notre Dame Journal of Formal Logic

### Stable Formulas in Intuitionistic Logic

#### Abstract

In 1995 Visser, van Benthem, de Jongh, and Renardel de Lavalette introduced NNIL-formulas, showing that these are (up to provable equivalence) exactly the formulas preserved under taking submodels of Kripke models. In this article we show that NNIL-formulas are up to frame equivalence the formulas preserved under taking subframes of (descriptive and Kripke) frames, that NNIL-formulas are subframe formulas, and that subframe logics can be axiomatized by NNIL-formulas. We also define a new syntactic class of ONNILLI-formulas. We show that these are (up to frame equivalence) the formulas preserved in monotonic images of (descriptive and Kripke) frames and that ONNILLI-formulas are stable formulas as introduced by Bezhanishvili and Bezhanishvili in 2013. Thus, ONNILLI is a syntactically defined set of formulas axiomatizing all stable logics. This resolves a problem left open in 2013.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 59, Number 3 (2018), 307-324.

Dates
Accepted: 2 December 2015
First available in Project Euclid: 4 May 2018

https://projecteuclid.org/euclid.ndjfl/1525420860

Digital Object Identifier
doi:10.1215/00294527-2017-0030

Mathematical Reviews number (MathSciNet)
MR3832083

Zentralblatt MATH identifier
06939322

#### Citation

Bezhanishvili, Nick; de Jongh, Dick. Stable Formulas in Intuitionistic Logic. Notre Dame J. Formal Logic 59 (2018), no. 3, 307--324. doi:10.1215/00294527-2017-0030. https://projecteuclid.org/euclid.ndjfl/1525420860

#### References

• [1] Bezhanishvili, G., and N. Bezhanishvili, “An algebraic approach to canonical formulas: Intuitionistic case,” Review of Symbolic Logic, vol. 2 (2009), pp. 517–49.
• [2] Bezhanishvili, G., and N. Bezhanishvili, “An algebraic approach to canonical formulas: Modal case,” Studia Logica, vol. 99 (2011), pp. 93–125.
• [3] Bezhanishvili, G., and N. Bezhanishvili, “Canonical formulas for wK4,” Review of Symbolic Logic, vol. 5 (2012), pp. 731–62.
• [4] Bezhanishvili, G., and N. Bezhanishvili, “Locally finite reducts of Heyting algebras and canonical formulas,” Notre Dame Journal of Formal Logic, vol. 58 (2017), pp. 21–45.
• [5] Bezhanishvili, G., N. Bezhanishvili, and R. Iemhoff, “Stable canonical rules,” Journal of Symbolic Logic, vol. 81 (2016), pp. 284–315.
• [6] Bezhanishvili, G., and S. Ghilardi, “An algebraic approach to subframe logics: Intuitionistic case,” Annals of Pure and Applied Logic, vol. 147 (2007), pp. 84–100.
• [7] Bezhanishvili, N., “Lattices of intermediate and cylindric modal logics,” Ph.D. dissertation, University of Amsterdam, Amsterdam, 2006. Available at http://www.illc.uva.nl/Research/Publications/Dissertations/DS-2006-02.text.pdf.
• [8] Bezhanishvili, N., “Frame based formulas for intermediate logics,” Studia Logica, vol. 90 (2008), pp. 139–59.
• [9] Bezhanishvili, N., and S. Ghilardi, “Multiple-conclusion rules, hypersequents syntax and step frames,” pp. 54–61 in Advances in Modal Logic (AiML, 2014), edited by R. Goré, B. Kooi, and A. Kurucz, College Publications, London, 2014.
• [10] Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2001.
• [11] Chagrov, A., and M. Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, New York, 1997.
• [12] de Jongh, D. H. J., “Investigations on the intuitionistic propositional calculus,” PhD dissertation, University of Wisconsin-Madison, Madison, Wis., 1968.
• [13] Fine, K., “Logics containing K4, II,” Journal of Symbolic Logic, vol. 50 (1985), pp. 619–51.
• [14] Jankov, V. A., “On the relation between deducibility in intuitionistic propositional calculus and finite implicative structures,” Doklady Akademii Nauk., vol. 151 (1963), pp. 1293–94.
• [15] van Dalen, D., Intuitionistic Logic, pp. 225–339 in Handbook of Philosophical Logic, Vol. III: Alternatives to Classical Logic, edited by D. Gabbay and F. Guenthner, vol. 3 of Synthese Library, Reidel, Dordrecht, 1986.
• [16] Visser, A., J. van Benthem, D. de Jongh, and G. R. Renardel de Lavalette, “$\mathrm{NNIL}$, a study in intuitionistic propositional logic,” pp. 289–326 in Modal logics and Process Algebra: A Bisimulation Perspective (Amsterdam, 1994), edited by A. Ponse, M. de Rijke, and Y. Venema, vol. 53 of CSLI Lecture Notes, CSLI, Stanford, CA, 1995.
• [17] Yang, F., “Intuitionistic subframe formulas, NNIL-formulas and n-universal models,” MSc thesis, University of Amsterdam, Amsterdam, The Netherlands, 2008, http://www.illc.uva.nl/Research/Publications/Reports/MoL-2008-12.text.pdf.
• [18] Zakharyaschev, M., “Syntax and semantics of superintuitionistic logics,” Algebra and Logic, vol. 28 (1989), pp. 262–82.
• [19] Zakharyaschev, M., “Canonical formulas for K4, II: Cofinal subframe logics,” Journal of Symbolic Logic, vol. 61 (1996), pp. 421–49.