Skip to main content
SpringerLink
Log in

Combining Logics in Simple Type Theory

  • Conference paper
Computational Logic in Multi-Agent Systems (CLIMA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6245))

Included in the following conference series:

  • 308 Accesses

Abstract

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.

A previous version of this paper has been presented at the World Congress and School on Universal Logic III (UNILOG’2010), Lisbon, Portugal, April 18-25, 2010.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews, P.B.: General models and extensionality. Journal of Symbolic Logic 37, 395–397 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andrews, P.B.: General models, descriptions, and choice in type theory. Journal of Symbolic Logic 37, 385–394 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  3. Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Baldoni, M.: Normal Multimodal Logics: Automatic Deduction and Logic Programming Extension. PhD thesis, Universita degli studi di Torino (1998)

    Google Scholar 

  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. Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. Journal of Symbolic Logic 69, 1027–1088 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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, http://arxiv.org/abs/0905.2435

  11. Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL (2010)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  17. Braüner, T.: Natural deduction for first-order hybrid logic. Journal of Logic, Language and Information 14(2), 173–198 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56–68 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fitting, M.: Interpolation for first order S5. Journal of Symbolic Logic 67(2), 621–634 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  22. Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2009)

    Google Scholar 

  23. Gödel, K.: Eine interpretation des intuitionistischen aussagenkalküls. Ergebnisse eines Mathematischen Kolloquiums 8, 39–40 (1933); Also published in Gödel [24], pp. 296–302

    Google Scholar 

  24. Gödel, K.: Collected Works, vol. I. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  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. Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15, 81–91 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  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. Segerberg, K.: Two-dimensional modal logic. Journal of Philosophical Logic 2(1), 77–96 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  33. Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reasoning 43(4), 337–362 (2009)

    Article  MATH  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Articulate Software, Angwin, CA, U.S.

    Christoph Benzmüller

Authors
  1. Christoph Benzmüller
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Clausthal University of Technology, Julius-Albert-Str. 4, 38678, Clausthal-Zellerfeld, Germany

    Jürgen Dix

  2. Departamento de Informática, Universidade Nova de Lisboa, Faculdade de Ciências de Tecnologia, 2829-516, Caparica, Portugal

    João Leite

  3. NICTA,, P.O. Box 6030, 4067, St Lucia, QLD, Australia

    Guido Governatori

  4. University of Luxembourg, Campus Kirchberg, 6, rue, Richard Coudenhove-Kalergi, 1359, Luxembourg, Luxembourg

    Wojtek Jamroga

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Benzmüller, C. (2010). Combining Logics in Simple Type Theory. In: Dix, J., Leite, J., Governatori, G., Jamroga, W. (eds) Computational Logic in Multi-Agent Systems. CLIMA 2010. Lecture Notes in Computer Science(), vol 6245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14977-1_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-14977-1_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14976-4

  • Online ISBN: 978-3-642-14977-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation