Journal of Logic, Language and Information

, Volume 14, Issue 3, pp 263–279 | Cite as

Guards, Bounds, and Generalized Semantics

  • Johan van Benthem


Some initial motivations for the Guarded Fragment still seem of interest in carrying its program further. First, we stress the equivalence between two perspectives: (a) satisfiability on standard models for guarded first-order formulas, and (b) satisfiability on general assignment models for arbitrary first-order formulas. In particular, we give a new straightforward reduction from the former notion to the latter. We also show how a perspective shift to general assignment models provides a new look at the fixed-point extension LFP(FO) of first-order logic, making it decidable. Next, we relate guarded syntax to earlier quantifier restriction strategies for achieving effective axiomatizability in second-order logic – pointing at analogies with ‘persistent’ formulas, which are essentially in the Bounded Fragment of many-sorted first-order logic. Finally, we look at some further unexplored directions, including the systematic use of ‘quasi-models’ as a semantics by itself.


Artificial Intelligence Quantifier Restriction Assignment Model General Assignment Restriction Strategy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Andréka, H., van Benthem, J., and Németi, I., 1998, “Modal languages and bounded fragments of predicate logic,” Journal of Philosophical Logic 27, 217–274.Google Scholar
  2. Andréka, H., Hodkinson, I., and Németi, I., 1999, “Finite algebras of relations are representable on finite sets,” Journal of Symbolic Logic 64, 243–267.Google Scholar
  3. Andréka, H., Madarasz, J., and Németi, I., to appear, “Logics of relativistic space-time,” in M. Aiello, J. van Benthem and I. Pratt (eds.), Handbook of Spatial Reasoning, Kluwer-Springer Academic Publishers, Dordrecht.Google Scholar
  4. Andréka, H. and Németi, I., 2005, private communication.Google Scholar
  5. Feferman, S., 1969, “Persistent and invariant formulas for outer extensions,” Compositio Mathematica 20, 29–52.Google Scholar
  6. Grädel, E., 1999A, “Decision procedures for guarded logics,” in Automated Deduction – Proceedings CADE 16, Lecture Notes in Computer Science 1632, Springer Verlag, Berlin, 31–51.Google Scholar
  7. Grädel, E., 1999B, “On the restraining power of guards,” Journal of Symbolic Logic 64, 1719-1742.Google Scholar
  8. Grädel, E., 1999C, “The decidability of guarded fixed point logic,” in J. Gerbrandy, M. Marx, M. de Rijke, and Y. Venema (eds.), JFAK Essays Dedicated to Johan van Benthem on the Occasion of his 50th Birthday, CD-ROM, Amsterdam University Press.
  9. Hoogland, E. and Marx, M., 2002, “Interpolation and definability in guarded fragments,” Studia Logica 70, 373–409.CrossRefGoogle Scholar
  10. Hoogland, E., Marx, M., and Otto, M., 1999, “Beth definability for the guarded fragment,” in H. Ganzinger, D. McAllester, and A. Voronkov (eds.), Logic for Programming and Automated Reasoning, LPAR 6, Springer Lecture Notes in AI 1705, 273–285, Springer, Berlin.Google Scholar
  11. Kerdiles, G., 2001, Saying it with Pictures: A Logical Landscape of Conceptual Graphs, Dissertation DS-2001-09, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  12. Marx, M., 2001, “Tolerance logic,” Journal of Logic, Language and Information 10, 353–373.Google Scholar
  13. Marx, M. and Venema, Y., 1997, Multi-Dimensional Modal Logic, Kluwer, Dordrecht.Google Scholar
  14. Montague, R., 1970, “Pragmatics and intensional logic,” Synthese 22, 68–94.CrossRefGoogle Scholar
  15. Németi, I., 1985, “Cylindric-relativized set algebras have strong amalgamation,” Journal of Symbolic Logic 50(3), 689–700.Google Scholar
  16. Németi, I., 1995, “Decidability of weakened versions of first-order logic,” in Logic Colloquium 92, CSLI Publications, Stanford, pp. 177–241.Google Scholar
  17. Németi, I., 1996, “Fine-structure analysis of first-order logic,” in M. Marx, M. Masuch, and L. Pólos (eds.), Arrow Logic and Multi-Modal Logic, CSLI Publications, Stanford, 221–247.Google Scholar
  18. ten Cate, B., 2005, Model Theory for Extended Modal Languages, Dissertation, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  19. van Benthem, J., 1983, Modal Logic and Classical Logic, Bibliopolis, Napoli.Google Scholar
  20. van Benthem, J., 1996A, “Complexity of contents versus complexity of wrappings,” in M. Marx, M. Masuch, and L. Pólos (eds.), Arrow Logic and Multimodal Logic, Studies in Logic, Language and Information, CSLI Publications, Stanford and Cambridge University Press, 203–219.Google Scholar
  21. van Benthem, J., 1996B, Exploring Logical Dynamics, CSLI Publications, Stanford.Google Scholar
  22. van Benthem, J., 1997, “Dynamic bits and pieces,” Report LP-97-01, Institute for Logic, Language and Computation, University of Amsterdam.Google Scholar
  23. van Benthem, J., 1999, “The range of modal logic,” Journal of Applied Non-Classical Logics 9(2/3), 407–442.Google Scholar
  24. van Benthem, J., 2001, “Modal logic in two gestalts,” in M. de Rijke, H. Wansing and M. Zakharyashev (eds.), Advances in Modal Logic, Vol. II, Uppsala 1998, CSLI Publications, Stanford, 73–100.Google Scholar
  25. van der Hoek, W. and de Rijke, M., 1993, “Generalized quantifiers and modal logic,” Journal of Logic, Language, and Information 2, 19–50.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Johan van Benthem
    • 1
    • 2
  1. 1.Amsterdam Institute for Logic Language and Computation (ILLC)University of AmsterdamAmsterdam
  2. 2.Stanford UniversityStanford

Personalised recommendations