Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-16T01:24:41.027Z Has data issue: false hasContentIssue false
  • English
  • Français

CANONICAL FORMULAS FOR wK4

Published online by Cambridge University Press:  19 September 2012

GURAM BEZHANISHVILI  and
NICK BEZHANISHVILI
GURAM BEZHANISHVILI*
Affiliation:
Department of Mathematical Sciences, New Mexico State University
NICK BEZHANISHVILI*
Affiliation:
Department of Computing, Imperial College London
*
*DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITY LAS CRUCES, NM 88003-8001 E-mail: gbezhani@math.nmsu.edu
†DEPARTMENT OF COMPUTING IMPERIAL COLLEGE LONDON SOUTH KENSINGTON CAMPUS LONDON, SW7 2AZ, UK E-mail: nbezhani@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize the theory of canonical formulas for K4, the logic of transitive frames, to wK4, the logic of weakly transitive frames. Our main result establishes that each logic over wK4 is axiomatizable by canonical formulas, thus generalizing Zakharyaschev’s theorem for logics over K4. The key new ingredients include the concepts of transitive and strongly cofinal subframes of weakly transitive spaces. This yields, along with the standard notions of subframe and cofinal subframe logics, the new notions of transitive subframe and strongly cofinal subframe logics over wK4. We obtain axiomatizations of all four kinds of subframe logics over wK4. We conclude by giving a number of examples of different kinds of subframe logics over wK4.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 5 , Issue 4 , December 2012 , pp. 731 - 762
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Bezhanishvili, N. (2006). Lattices of intermediate and cylindric modal logics. PhD Thesis, University of Amsterdam.Google Scholar
Bezhanishvili, N. (2008). Frame based formulas for intermediate logics. Studia Logica, 90(2), 139–159.CrossRefGoogle Scholar
Bezhanishvili, G., & Bezhanishvili, N. (2009). An algebraic approach to canonical formulas: Intuitionistic case. Review of Symbolic Logic, 2(3), 517–549.Google Scholar
Bezhanishvili, G., & Bezhanishvili, N. (2011). An algebraic approach to canonical formulas: Modal case. Studia Logica, 99(1–3), 93–125.Google Scholar
Bezhanishvili, G., Esakia, L., & Gabelaia, D. (2011). Spectral and T 0-spaces in d-semantics. In Bezhanishvili, N., Löbner, S., Schwabe, K., and Spada, L., editors. 8th International Tbilisi Symposium on Logic, Language, and Computation. Selected Papers, Lecture Notes in Artificial Intelligence. New York: Springer, pp. 16–29.Google Scholar
Bezhanishvili, G., & Ghilardi, S. (2007). An algebraic approach to subframe logics. Intuitionistic case. Annals of Pure and Applied Logic, 147(1–2), 84–100.CrossRefGoogle Scholar
Bezhanishvili, G., Ghilardi, S., & Jibladze, M. (2011). An algebraic approach to subframe logics. Modal case. Notre Dame Journal of Formal Logic, 52(2), 187–202.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2008). Modal Logic. Cambridge, UK: Cambridge University Press.Google Scholar
Blok, W. (1978). On the degree of incompleteness of modal logics. Bulletin of the Section of Logic, 7(4), 167–175.Google Scholar
Blok, W. (1980). The lattice of modal logics: An algebraic investigation. Journal of Symbolic Logic, 45(2), 221–236.CrossRefGoogle Scholar
Bull, R. A. (1966). That all normal extensions of S4.3 have the finite model property. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 12, 341–344.Google Scholar
Chagrov, A., & Zakharyaschev, M. (1997). Modal Logic, Vol. 35 of Oxford Logic Guides. New York: The Clarendon Press Oxford University Press.Google Scholar
Esakia, L. (1985). Heyting Algebras I. Duality Theory (Russian). Tbilisi, Georgia: Metsniereba.Google Scholar
Esakia, L. (2001). Weak transitivity—a restitution. In Logical Investigations, No. 8 (Moscow, 2001). Moscow, Russia: Nauka (in Russian), pp. 244–255.Google Scholar
Fine, K. (1971). The logics containing S4.3. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 17, 371–376.Google Scholar
Fine, K. (1974). An ascending chain of S4 logics. Theoria, 40(2), 110–116.Google Scholar
Fine, K. (1985). Logics containing K4. II. Journal of Symbolic Logic, 50(3), 619–651.Google Scholar
Jankov, V. (1963). On the relation between deducibility in intuitionistic propositional calculus and finite implicative structures. Doklady Akademii Nauk SSSR, 151, 1293–1294(in Russian).Google Scholar
Kracht, M. (1993a). Prefinitely axiomatizable modal and intermediate logics. Mathematical Logic Quarterly, 39(3), 301–322.CrossRefGoogle Scholar
Kracht, M. (1993b). Splittings and the finite model property. Journal of Symbolic Logic, 58(1), 139–157.Google Scholar
Kracht, M. (1999). Tools and Techniques in Modal Logic, Vol. 142 of Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Co.Google Scholar
McKenzie, R. (1972). Equational bases and nonmodular lattice varieties. Transactions of the American Mathematical Society, 174, 1–43.Google Scholar
Rautenberg, W. (1980). Splitting lattices of logics. Archiv für mathematische Logik und Grundlagenforschung, 20(3–4), 155–159.CrossRefGoogle Scholar
Scroggs, S. J. (1951). Extensions of the Lewis system S5. Journal of Symbolic Logic, 16, 112–120.Google Scholar
Wolter, F. (1993). Lattices of modal logics. PhD thesis, Free University of Berlin.Google Scholar
Wolter, F. (1997). The structure of lattices of subframe logics. Annals of Pure and Applied Logic, 86(1), 47–100.Google Scholar
Zakharyaschev, M. (1989). Syntax and semantics of superintuitionistic logics. Algebra and Logic, 28(4), 262–282.Google Scholar
Zakharyaschev, M. (1992). Canonical formulas for K4. I. Basic results. Journal of Symbolic Logic, 57(4), 1377–1402.CrossRefGoogle Scholar
Zakharyaschev, M. (1996). Canonical formulas for K4. II. Cofinal subframe logics. Journal of Symbolic Logic, 61(2), 421–449.Google Scholar
Zakharyaschev, M., & Alekseev, A. (1995). All finitely axiomatizable normal extensions of K4.3 are decidable. Mathematical Logic Quarterly, 41(1), 15–23.Google Scholar