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STABLE CANONICAL RULES

Published online by Cambridge University Press:  09 March 2016

GURAM BEZHANISHVILI ,
NICK BEZHANISHVILI  and
ROSALIE IEMHOFF
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES NM 88003, USAE-mail: guram@math.nmsu.edu
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM P.O. BOX 94242, 1090 GE AMSTERDAM THE NETHERLANDSE-mail: N.Bezhanishvili@uva.nl
ROSALIE IEMHOFF
Affiliation:
DEPARTMENT OF PHILOSOPHY UTRECHT UNIVERSITY JANSKERKHOFF 13A, 3512 BL UTRECHT THE NETHERLANDSE-mail: r.iemhoff@uu.nl
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Abstract

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We introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.

Keywords

03B45Modal logicmulti-conclusion consequence relationaxiomatizationfiltrationmodal algebravarietyuniversal class
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 1 , March 2016 , pp. 284 - 315
Copyright
Copyright © The Association for Symbolic Logic 2016 

References

REFERENCES

Bezhanishvili, G. and Bezhanishvili, N., An algebraic approach to canonical formulas: Modal case. Studia Logica, vol. 99 (2011), no. 1–3, pp. 93–125.CrossRefGoogle Scholar
Bezhanishvili, G. and Bezhanishvili, N., Canonical formulas for wK4. Review of Symbolic Logic, vol. 5 (2012), no. 4, pp. 731–762.Google Scholar
Bezhanishvili, G. and Bezhanishvili, N., Locally finite reducts of Heyting algebras and canonical formulas. Notre Dame Journal of Formal Logic, available as Utrecht University Logic Group Preprint Series Report 2013-3052015, to appear.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., and Ilin, J., Cofinal stable logics, 2015, available as ILLC Prepublication Series Report, PP-2015-08, submitted.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., and Ilin, J., Stable modal logics, 2015, in preparation.Google Scholar
Bezhanishvili, G., Ghilardi, S., and Jibladze, M., An algebraic approach to subframe logics. Modal case. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 2, pp. 187–202.Google Scholar
Bezhanishvili, G., Mines, R., and Morandi, P. J., Topo-canonical completions of closure algebras and Heyting algebras. Algebra Universalis, vol. 58 (2008), no. 1, pp. 1–34.Google Scholar
Bezhanishvili, N., Lattices of Intermediate and Cylindric Modal Logics, PhD thesis, University of Amsterdam, Amsterdam, 2006.Google Scholar
Bezhanishvili, N., Frame based formulas for intermediate logics. Studia Logica, vol. 90 (2008), no. 2, pp. 139–159.Google Scholar
Bezhanishvili, N., Gabelaia, D., Ghilardi, S., and Jibladze, M, Admissible bases via stable canonical rules, 2015, available as ILLC Prepublication Series Report, PP-2015-14, submitted.Google Scholar
Bezhanishvili, N. and Ghilardi, S., Multiple-conclusion rules, hypersequents syntax and step frames, Advances in Modal Logic (AiML 2014) (Gore, R., Kooi, B., and Kurucz, A., editors), College Publications, London, 2014, pp. 54–61. An extended version available as ILLC Prepublication Series Report PP-2014-05.Google Scholar
Blackburn, P., de Rijke, M., and Venema, Y., Modal Logic, Cambridge University Press, Cambridge, 2001.Google Scholar
Blok, W., On the degree of incompleteness of modal logics. Bulletin of Symbolic Logic, vol. 7 (1978), no. 4, pp. 167–175.Google Scholar
Burris, R. and Sankappanavar, H., A Course in Universal Algebra, Springer, Berlin, 1981.Google Scholar
Chagrov, A. and Zakharyaschev, M., Modal Logic, The Clarendon Press, New York, 1997.Google Scholar
Conradie, W., Morton, W., and van Alten, C., An algebraic look at filtrations in modal logic. Logic Journal of IGPL, vol. 21 (2013), no. 5, pp. 788–811.Google Scholar
Esakia, L., On the theory of modal and superintuitionistic systems, Logical Inference (Moscow, 1974), Nauka, Moscow, 1979, pp. 147–172 (in Russian).Google Scholar
Ghilardi, S., Continuity, freeness, and filtrations. Journal of Applied Non-Classical Logics, vol. 20 (2010), no. 3, pp. 193–217.CrossRefGoogle Scholar
Goranko, V. and Passy, S., Using the universal modality: gains and questions. Journal of Logic and Computation, vol. 2 (1992), no. 1, pp. 5–30.CrossRefGoogle Scholar
Iemhoff, R., Consequence relations and admissible rules, to appear in the Journal of Philosophical Logic, available at http://dspace.library.uu.nl/handle/1874/282623, DOI: 10.1007/s10992-015-9380-8.Google Scholar
Jeřábek, E., Canonical rules, this JOURNAL, vol. 74 (2009), no. 4, pp. 1171–1205.Google Scholar
Kracht, M., Tools and Techniques in Modal Logic, North-Holland, Amsterdam, 1999.Google Scholar
Kracht, M., Modal consequence relations, Handbook of Modal Logic (Blackburn, P., van Benthem, J., and Wolter, F., editors), pp. 491–545, Elsevier, Amsterdam, 2007.Google Scholar
Lemmon, E. J., Algebraic semantics for modal logics. I, this Journal, vol. 31 (1966), pp. 46–65.Google Scholar
Lemmon, E. J., Algebraic semantics for modal logics. II, this Journal, vol. 31 (1966), pp. 191–218.Google Scholar
Lemmon, E. J., An Introduction to Modal Logic. The “Lemmon Notes” (Segerberg, Krister, editor), American Philosophical Quarterly Monograph Series, Vol. 11, Basil Blackwell, Oxford, 1977. In collaboration with Dana Scott.Google Scholar
McKenzie, R., Equational bases and nonmodular lattice varieties. Transactions of the American Mathematical Society, vol. 174 (1972), pp. 1–43.CrossRefGoogle Scholar
McKinsey, J. C. C., A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.Google Scholar
McKinsey, J. C. C. and Tarski, A., The algebra of topology. Annals of Mathematics, vol. 45 (1944), pp. 141–191.Google Scholar
Rautenberg, W., Splitting lattices of logics. Archive of Mathematical Logik, Grundlagen, vol. 20 (1980), no. 3–4, pp. 155–159.Google Scholar
Rybakov, V. V., Admissibility of Logical Inference Rules, North-Holland, Amsterdam, 1997.Google Scholar
Sambin, G., Subdirectly irreducible modal algebras and initial frames. Studia Logica, vol. 62 (1999), no. 2, pp. 269–282.Google Scholar
Segerberg, K., An Essay in Classical Modal Logic. Vols. 1, 2, 3, Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet, Uppsala, 1971.Google Scholar
Shehtman, V., “Everywhere” and “here”. Journal of Applied Non-Classical Logics, vol. 9 (1999), no. 2–3, pp. 369–379.Google Scholar
Venema, Y., A dual characterization of subdirectly irreducible BAOs. Studia Logica, vol. 77 (2004), no. 1, pp. 105–115.Google Scholar
Venema, Y., Algebras and coalgebras, Handbook of Modal Logic (Blackburn, P., van Benthem, J., and Wolter, F., editors), Elsevier, Amsterdam, 2007, pp.331–426.Google Scholar
Wronski, A., Intermediate logics and the disjunction property. Reports on Mathematical Logic, vol. 1 (1973), pp. 39–51.Google Scholar
Zakharyaschev, M., Canonical formulas for K4. I. Basic results, this Journal, vol. 57 (1992), no. 4, pp. 1377–1402.Google Scholar