Journal of Logic, Language and Information

, Volume 7, Issue 3, pp 265–296 | Cite as

Modal Pure Type Systems

  • Tijn Borghuis


We present a framework for intensional reasoning in typed λ-calculus. In this family of calculi, called Modal Pure Type Systems (MPTSs), a “propositions-as-types”-interpretation can be given for normal modal logics. MPTSs are an extension of the Pure Type Systems (PTSs) of Barendregt (1992). We show that they retain the desirable meta-theoretical properties of PTSs, and briefly discuss applications in the area of knowledge representation.

Knowledge representation natural deduction normal modal logics typed λ-calculus 


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  1. Ahn, R., 1994, “Communicating contexts: A pragmatic approach to information change,” pp. 1–13 in Types for Proofs and Programs: Selected Papers International Workshop TYPES '94, P. Dybjer, B. Nordström, and J. Smith, eds., Berlin: Springer-Verlag.Google Scholar
  2. Ahn, R. and Kolb, H.P., 1990, “Discourse representation meets constructive mathematics,” in Papers from the Second Symposium on Logic and Language, L. Kálmán and L. Pólós, eds., Budapest: Akadémia Kiadó.Google Scholar
  3. Ahn, R.M.C., Beun, R.J., Borghuis, T., Bunt, H.C., and van Overveld, C.W.A.M., 1994, “The DenKarchitecture: A fundamental approach to user interfaces,” Artificial Intelligence Review 8, 431–455.Google Scholar
  4. Barendregt, H., 1992, “Lambda calculi with types,” pp. 117–309 in Handbook of Logic in Computer Science, A. Abramsky, D.M. Gabbay, and T.S.E. Maibaum, eds., Oxford: Oxford University Press.Google Scholar
  5. Barendregt, H. and Hemerik, K., 1990, “Types in lambda calculus and programming languages,” pp. 1–36 in European Symposium on Programming, N. Jones, ed., Lecture Notes in Computer Science, Vol. 432, Berlin: Springer-Verlag.Google Scholar
  6. Van Benthem, J., 1991, “Reflections on epistemic logic,” Logique & Analyse 133–134, 5–14.Google Scholar
  7. Van Benthem Jutting, L.S., 1977, “Checking Landau's ‘Grundlagen’ in the Automath system,” Ph.D. Thesis, Eindhoven University of Technology.Google Scholar
  8. Berardi, S., 1988, “Towards a mathematical analysis of the Coquand—Huet calculus of constructions and the other systems in Barendregt's cube,” Department of Computer Science, Carnegie-Mellon University and Dipartimento Matematica, Unversità di Torino, Italy.Google Scholar
  9. Borghuis, T., 1993, “Interpreting modal natural deduction in type theory,” pp. 67–102 in Diamonds and Defaults, M. De Rijke, ed., Dordrecht: Kluwer Academic Publishers.Google Scholar
  10. Borghuis, T., 1994, “Coming to terms with modal logic: On the interpretation of modalities in typed lambda calculus,” Ph.D. Thesis, Eindhoven University of Technology.Google Scholar
  11. Borghuis, T., 1995, “Contexts in dialogue,” pp. 37–48 in Proceedings of the International Conference on Cooperative Multimodal Communication CMC/95, H. Bunt, R.—J. Beun, and T. Borghuis, eds., Tilburg: Samenwerkings Orgaan Brabantse Universiteiten.Google Scholar
  12. Chellas, B.F., 1980, Modal Logic: An Introduction, Cambridge: Cambridge University Press.Google Scholar
  13. Fitch, F.B., 1952, Symbolic Logic, An Introduction, New York: The Ronald Press Company.Google Scholar
  14. Fitting, M., 1983, Proof Methods for Modal and Intuitionistic Logics, Dordrecht: Reidel Publishing Company.Google Scholar
  15. Gabbay, D.M., 1996, Labeled Deductive Systems, Oxford: Clarendon Press.Google Scholar
  16. Geuvers, H., 1993, “Logics and type systems,” Ph.D. Thesis, University of Nijmegen.Google Scholar
  17. Geuvers, H. and Nederhof, M.—J., 1991, “Modular proof of strong normalization for the calculus of constructions,” Journal of Functional Programming 1(2), 155–189.Google Scholar
  18. Hintikka, J., 1962, Knowledge and Belief. An Introduction to the Logic of the Two Notions, Ithaca, NY: Cornell University Press.Google Scholar
  19. Kraus, S. and Lehmann, D., 1986, “Knowledge, belief and time,” pp. 186–195 in Proceedings ICALP 1986, L. Kott, ed., Lecture Notes in Computer Science, Vol. 226, Berlin: Springer-Verlag.Google Scholar
  20. Martini, S. and Masini, A., 1996, “A computational interpretation of modal proofs,” pp. 213–241 in Proof Theory of Modal Logic, H. Wansing, ed., Dordrecht: Kluwer Academic Publishers.Google Scholar
  21. Nederpelt, R.P., 1977, “Presentation of natural deduction,” pp. 115–125 in Symposium: Set Theory, Foundations of Mathematics, Recueil des Travaux de l'Institut Mathématique, Nouv. série, tome 2 (10), Beograd: Faculty of Mathematics, University of Beograd.Google Scholar
  22. Nederpelt, R.P., 1990, “Type systems — Basic ideas and applications,” pp. 367–384 in Proceedings CSN 1990, A.J. van de Goor, ed., Amsterdam: CWI.Google Scholar
  23. De Queiroz, R.J.G.B. and Gabbay, D.M., 1992, “Extending the Curry—Howard interpretation to linear, relevant and other resource logics,” Journal of Symbolic Logic 57(4), 1319–1365.Google Scholar
  24. De Queiroz, R.J.G.B. and Gabbay, D.M., 1995, “The functional interpretation of modal necessity,” in Advances in Intensional Logic, M. De Rijke, ed., Dordrecht: Kluwer Academic Publishers.Google Scholar
  25. Thijsse, E.C.G., 1992, “Partial logic and knowledge representation,” Ph.D. Thesis, Tilburg, Eburon Delft.Google Scholar
  26. Van Westrhenen, S.C., Sommerhalder, R., and Tonino, J.F.M., 1993, LOGICA: Een Inleiding met Toepassingen in de Informatica, Schoonhoven: Academic Service.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Tijn Borghuis
    • 1
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

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