Skip to main content
Log in

A Splitting Logic in NExt(KTB)

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

It is shown that the normal modal logic of two reflexive points jointed with a symmetric binary relation splits the lattice of normal extensions of the logic KTB. By this fact, it is easily seen that there exists the third largest logic in the class of all normal extensions of KTB.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Blackburn, P., de Rijke M., and Venema Y., Modal Logic, Cambridge University Press, 2001.

  • Blok, W., ’On the degree of incompleteness in modal logics and the covering relation in the lattice of modal logics’, Tech. Rep. 78-07, Department of Mathematics, University of Amsterdam, 1978.

  • Blok W. (1980) ’The lattice of modal logics: an algebraic investigation’. Journal of Symbolic Logic 45: 221–236

    Article  Google Scholar 

  • Blyth, S.N., and Sankappanavar, H.P., a course in universal algebra, Splinger Verlag, 1981.

  • Fine K. (1974) ’An ascending chain of S4 logics’. Theoria 40: 110–116

    Article  Google Scholar 

  • Jankov V.A. (1968) ’The The construction of a sequence of strongly independent superintuitionistic propositional calculi’. Soviet Mathematics Doklady 9: 806–807

    Google Scholar 

  • Kracht M. (1990) ’An almost general splitting theorem for modal logic’. Studia Logica 49: 455–470

    Article  Google Scholar 

  • Kracht M. (1993) ’Splittings and the finite model property’. Journal of Symbolic Logic 58: 139–157

    Article  Google Scholar 

  • Kripke S.A. (1963) ’Semantical analysis of modal logic I’. Zeitschrift fúr mathematische Logik und Grundlagen der Mathematik 9: 61–96

    Article  Google Scholar 

  • Makinson D.C. (1971) ’Some embedding theorems for modal logic’. Notre Dame Journal of Formal Logic 12: 252–254

    Article  Google Scholar 

  • McKenzie R. (1972) ’Equational bases and nonmodular lattice varieties’. Transactions of the American Mathematical Society 174: 1–43

    Article  Google Scholar 

  • Miyazaki, Y., ’Normal modal logics containing KTB with some finiteness conditions’,in Advances in Modal Logic, vol. 5, King’s College Publications, 2005, pp. 171–190.

  • Rautenberg W. (1977) ’Der Verband der normalen verzweigten Modallogiken’. Mathematische Zeitschrift 156: 123–140

    Article  Google Scholar 

  • Rautenberg W. (1980) ’Splitting lattices of logics’. Archiv für Mathematische Logik 20: 155–159

    Article  Google Scholar 

  • Sambin G. (1999) ’Subdirectly irreducible modal algebras and initial frames’. Studia Logica 62: 269–282

    Article  Google Scholar 

  • Whitman P. (1943) ’Splittings of a lattice’. American Journal of Mathematics 65:179–196

    Article  Google Scholar 

  • Wolter, F., Lattices of Modal Logics, Ph.D. thesis, Freie Universität, Berlin, 1993.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yutaka Miyazaki.

Additional information

Received February 17, 2006

Rights and permissions

Reprints and permissions

About this article

Cite this article

Miyazaki, Y. A Splitting Logic in NExt(KTB). Stud Logica 85, 381–394 (2007). https://doi.org/10.1007/s11225-007-9039-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-007-9039-8

Keywords

Navigation