Combining Logics in Simple Type Theory

  • Christoph Benzmüller
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6245)


Simple type theory is suited as framework for combining classical and non-classical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics. Combinations of modal logics and other logics are particularly relevant for multi-agent systems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, P.B.: General models and extensionality. Journal of Symbolic Logic 37, 395–397 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andrews, P.B.: General models, descriptions, and choice in type theory. Journal of Symbolic Logic 37, 385–394 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  4. 4.
    Andrews, P.B., Brown, C.E.: TPS: A hybrid automatic-interactive system for developing proofs. Journal of Applied Logic 4(4), 367–395 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Giesl, J., Haehnle, R. (eds.) IJCAR 2010 - 5th International Joint Conference on Automated Reasoning, Edinburgh, UK. LNCS (LNAI). Springer, Heidelberg (2010)Google Scholar
  6. 6.
    Baldoni, M.: Normal Multimodal Logics: Automatic Deduction and Logic Programming Extension. PhD thesis, Universita degli studi di Torino (1998)Google Scholar
  7. 7.
    Benzmüller, C.: Automating access control logic in simple type theory with LEO-II. In: Gritzalis, D., López, J. (eds.) Proceedings of 24th IFIP TC 11 International Information Security Conference, SEC 2009, Emerging Challenges for Security, Privacy and Trust, Pafos, Cyprus, May 18-20. IFIP, vol. 297, pp. 387–398. Springer, Heidelberg (2009)Google Scholar
  8. 8.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. Journal of Symbolic Logic 69, 1027–1088 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Benzmüller, C., Paulson, L.: Exploring Properties of Normal Multimodal Logics in Simple Type Theory with LEO-II. In: Festschrift in Honor of Peter B. Andrews on His 70th Birthday. Studies in Logic, Mathematical Logic and Foundations. College Publications (2008)Google Scholar
  10. 10.
    Benzmüller, C., Paulson, L.C.: Quantified Multimodal Logics in Simple Type Theory. SEKI Report SR–2009–02, SEKI Publications, DFKI Bremen GmbH, Safe and Secure Cognitive Systems, Cartesium, Enrique Schmidt Str. 5, D–28359 Bremen, Germany (2009), ISSN 1437-4447,
  11. 11.
    Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL (2010)Google Scholar
  12. 12.
    Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II — A Cooperative Automatic Theorem Prover for Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Benzmüller, C., Pease, A.: Progress in automating higher-order ontology reasoning. In: Konev, B., Schmidt, R., Schulz, S. (eds.) Workshop on Practical Aspects of Automated Reasoning (PAAR 2010), Edinburgh, UK, July 14. CEUR Workshop Proceedings (2010)Google Scholar
  14. 14.
    Benzmüller, C., Pease, A.: Reasoning with embedded formulas and modalities in SUMO. In: Bundy, A., Lehmann, J., Qi, G., Varzinczak, I.J. (eds.) The ECAI 2010 Workshop on Automated Reasoning about Context and Ontology Evolution (ARCOE 2010), Lisbon, Portugal, August 16-17 (2010)Google Scholar
  15. 15.
    Benzmüller, C., Rabe, F., Sutcliffe, G.: THF0 — The Core TPTP Language for Classical Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 491–506. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Blackburn, P., Marx, M.: Tableaux for quantified hybrid logic. In: Egly, U., Fermüller, C.G. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 38–52. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  17. 17.
    Braüner, T.: Natural deduction for first-order hybrid logic. Journal of Logic, Language and Information 14(2), 173–198 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Fitting, M.: Interpolation for first order S5. Journal of Symbolic Logic 67(2), 621–634 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. Studies in Logic, vol. 148. Elsevier Science, Amsterdam (2003)zbMATHGoogle Scholar
  21. 21.
    Garg, D., Abadi, M.: A Modal Deconstruction of Access Control Logics. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 216–230. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2009)Google Scholar
  23. 23.
    Gödel, K.: Eine interpretation des intuitionistischen aussagenkalküls. Ergebnisse eines Mathematischen Kolloquiums 8, 39–40 (1933); Also published in Gödel [24], pp. 296–302Google Scholar
  24. 24.
    Gödel, K.: Collected Works, vol. I. Oxford University Press, Oxford (1986)zbMATHGoogle Scholar
  25. 25.
    Goldblatt, R.: Logics of Time and Computation. Center for the Study of Language and Information - Lecture Notes, vol. 7. Leland Stanford Junior University (1992)Google Scholar
  26. 26.
    Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15, 81–91 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. Journal of Logic, Language and Information 18(4), 437–464 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    McKinsey, J.C.C., Tarski, A.: Some theorems about the sentential calculi of lewis and heyting. Journal of Symbolic Logic 13, 1–15 (1948)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  30. 30.
    Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: Proceedings 3rd International Conference on Knowledge Representation and Reasoning, pp. 165–176 (1992)Google Scholar
  31. 31.
    Segerberg, K.: Two-dimensional modal logic. Journal of Philosophical Logic 2(1), 77–96 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sutcliffe, G.: TPTP, TSTP, CASC, etc. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 6–22. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  33. 33.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reasoning 43(4), 337–362 (2009)zbMATHCrossRefGoogle Scholar
  34. 34.
    Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1–27 (2010)Google Scholar
  35. 35.
    Sutcliffe, G., Benzmüller, C., Brown, C., Theiss, F.: Progress in the development of automated theorem proving for higher-order logic. In: Schmidt, R.A. (ed.) Automated Deduction – CADE-22. LNCS, vol. 5663, pp. 116–130. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christoph Benzmüller
    • 1
  1. 1.Articulate SoftwareAngwinU.S.

Personalised recommendations