Advertisement

Studia Logica

, Volume 83, Issue 1–3, pp 133–155 | Cite as

Modal Frame Correspondences and Fixed-Points

  • Johan Van Benthem
Article

Abstract

Taking Löb's Axiom in modal provability logic as a running thread, we discuss some general methods for extending modal frame correspondences, mainly by adding fixed-point operators to modal languages as well as their correspondence languages. Our suggestions are backed up by some new results – while we also refer to relevant work by earlier authors. But our main aim is advertizing the perspective, showing how modal languages with fixed-point operators are a natural medium to work with.

Keywords

Löb's Axiom fixed-point frame correspondence modal μ-calculus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    ACZEL, P., ‘An Introduction to Inductive Definitions’, in J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 739–782.Google Scholar
  2. [2]
    D'AGOSTINO, G., J. VAN BENTHEM, A. MONTANARI, and A. POLICRITI, ‘Modal Deduction in Second-Order Logic and Set Theory. Part I’, Logic and Computation 7 (1997), 251–265.CrossRefGoogle Scholar
  3. [3]
    D'AGOSTINO, G., and M. HOLLENBERG, ‘Logical Questions concerning the /i-Calculus: Interpolation, Lyndon&Los-Tarski’, Journal of Symbolic Logic 65 (2000), 310–332.Google Scholar
  4. [4]
    VAN BENTHEM, J., ‘Some Correspondence Results in Modal Logic’, Report 74–05, Mathematisch Instituut, Universiteit van Amsterdam, 1974.Google Scholar
  5. [5]
    VAN BENTHEM, J., Modal Logic and Classical Logic, Bibliopolis, Napoli, 1983.Google Scholar
  6. [6]
    VAN BENTHEM, J., ‘Toward a Computational Semantics’, in P. GARDENFORS (ed.), Generalized Quantifiers: Linguistic and Logical Approaches, Reidel, Dordrecht, 1987, pp. 31–71.Google Scholar
  7. [7]
    VAN BENTHEM, J., ‘The R,ange of Modal Logic’, Journal of Applied Non-Classical Logics 9, 2/3 (1999), 407–442.Google Scholar
  8. [8]
    VAN BENTHEM, J., ‘Minimal Predicates, Fixed-Points, and Definability’, R.eport PP-2004–01, ILLC Amsterdam. Appeared in Journal of Symbolic Logic 70, 3 (2005), 696–712.CrossRefGoogle Scholar
  9. [9]
    VAN BENTHEM, J., S. VAN OTTERLOO, O. ROY, ‘Preference Logic, Conditionals, and Solution Concepts in Games’, R.eport PP-2005–28, ILLC Amsterdam, to appear in Festschrift for Krister Segerberg, University of Uppsala, 2006.Google Scholar
  10. [10]
    VAN BENTHEM, J., J. VAN BUCK, B. Kooi ‘Logics for Communication and Change’, ILLC Preprint DARE 14 8524, University of Amsterdam, in R., van der Meijden, (ed.), Proceedings of TARK 2005, National University of Singapore, 2005.Google Scholar
  11. [11]
    BEZEM, J-W., and R. DE VRUER (eds.), (‘Terese’), Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science, Vol. 55, Cambridge University Press, 2003.Google Scholar
  12. [12]
    BLACKBURN, P., M. DE RIJKE and Y. VENEMA, Modal Logic, Cambridge University Press, 2001.Google Scholar
  13. [13]
    TEN GATE, B., Model Theory for Extended Modal Languages, ILLC Dissertation Series DS-2005–01, University of Amsterdam, 2005.Google Scholar
  14. [14]
    BLOK, W., and J. VAN BENTHEM, ‘Transitivity Follows from Dummett's Axiom’, Theoria 44, 2 (1978), 117–118.Google Scholar
  15. [15]
    EBBINGHAUS, H-D., and J. FLUM, Finite Model Theory, Springer, Berlin, 1995.Google Scholar
  16. [16]
    GABBAY, D., and H-J. OHLBACH, ‘Quantifier Elimination in Second-Order Predicate Logic’, South African Computer Journal 7 (1992), 35–43.Google Scholar
  17. [17]
    GORANKO, V., and D. VAKARELOV, ‘Elementary Canonical Formulas I. Extending Sahlqvist's Theorem’, Department of Mathematics, Rand Afrikaans University, Johannesburg&Faculty of Mathematics and Computer Science, Kliment Ohridski University, Sofia, 2003.Google Scholar
  18. [18]
    HAREL, D., D. KOZEN, and J. TIURYN, Dynamic Logic, The MIT Press, Cambridge (Mass.), 2000.Google Scholar
  19. [19]
    HOLLENBERG, M., Logic and Bisimulation, Dissertation Series Vol. XXIV, Zeno Institute of Philosophy, University of Utrecht, 1998.Google Scholar
  20. [20]
    NONNENGART, A., and A. SZALAS, ‘Fixed-Point Approach to Second-Order Quantifier Elimination with Applications to Modal Correspondence Theory’, in E. Orlowska (ed.), Logic at Work, Physica-Verlag, Heidelberg, 1999, pp. 89–108.Google Scholar
  21. [21]
    SAHLQVIST, H., ‘Completeness and Correspondence in First and Second Order Semantics for Modal Logic’, in S. Kanger (ed.), Proceedings of the Third Scandinavian Logic Symposium, Amsterdam, North Holland, 1975, pp. 110–143.Google Scholar
  22. [22]
    SMORYNSKI, C., ‘Modal Logic and Self-Reference’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. II, Reidel, Dordrecht, 1984, pp. 441–495.Google Scholar
  23. [23]
    VENEMA, Y., Many-Dimensional Modal Logics, Dissertation, Institute for Logic, Language and Computation, University of Amsterdam, 1991.Google Scholar
  24. [24]
    VISSER, A.,‘Semantics and the Liar Paradox’, in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Vol. IV, Reidel, Dordrecht, 1984, pp. 617–706.Google Scholar
  25. [25]
    VISSER, A., ‘Lob's Logic Meets the /i-Calculus’, Philosophical Institute, University of Utrecht. To appear in Liber Amicorum for Jan-Willem Klop, 2005.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.ILLCUniversity of AmsterdamAmsterdamNetherlands
  2. 2.Department of PhilosophyStanford UniversityStanfordUSA

Personalised recommendations