Skip to main content
SpringerLink
Log in

Combining and automating classical and non-classical logics in classical higher-order logics

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Numerous classical and non-classical logics can be elegantly embedded in Church’s simple type theory, also known as classical higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. 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 embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, P.B.: General models and extensionality. J. Symb. Log. 37, 395–397 (1972)

    Article  MATH  Google Scholar 

  2. Andrews, P.B.: General models, descriptions, and choice in type theory. J. Symb. Log. 37, 385–394 (1972)

    Article  MATH  Google Scholar 

  3. Andrews, P.B.: Theorem proving via general matings. J. ACM 28(2), 193–214 (1981)

    Article  MATH  Google Scholar 

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

  5. Andrews, P.B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., Xi, H.: TPS: a theorem-proving system for classical type theory. J. Autom. Reason. 16(3), 321–353 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Andrews, P.B., Brown, C.: TPS: a hybrid automatic-interactive system for developing proofs. J. Appl. Log. 4(4), 367–395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Giesl, J., Hähnle, R. (eds.) Proceedings of Automated Reasoning: 5th International Joint Conference, IJCAR 2010, Edinburgh, UK, 16–19 Jul 2010. Lecture Notes in Artificial Intelligence, vol. 6173, pp. 76–90. Springer (2010)

  8. Baldoni, M.: Normal multimodal logics: automatic deduction and logic programming extension. PhD thesis, Universita degli studi di Torino (1998)

  9. Benzmüller, C.: Extensional higher-order paramodulation and RUE-resolution. In: In Proc. of CADE-16. LNAI, number 1632, pp. 399–413. Springer (1999)

  10. Benzmüller, C.: Automating access control logic in simple type theory with LEO-II. In: Gritzalis, D., López, J. (eds.) Emerging Challenges for Security, Privacy and Trust, Proceedings of the 24th IFIP TC 11 International Information Security Conference, SEC 2009, Pafos, Cyprus, 18–20 May 2009. IFIP, vol. 297, pp. 387–398. Springer (2009)

  11. Benzmüller, C.: Combining logics in simple type theory. In: Dix, J., Leite, J., Governatori, G., Jamroga, W. (eds.) Computational Logic in Multi-Agent Systems, Proceedings of the 11th International Workshop, CLIMA XI, Lisbon, Portugal, 16–17 Aug 2010. Lecture Notes in Artificial Intelligence, vol. 6245, pp. 33–48. Springer, Lisbon, Portugal (2010)

    Google Scholar 

  12. Benzmüller, C.: Verifying the modal logic cube is an easy task (for higher-order automated reasoners). In: Siegler, S., Wasser, N. (eds.) Verification, Induction, Termination Analysis—Festschrift for Christoph Walther on the Occasion of His 60th Birthday. LNCS, vol. 6463, pp. 117–128. Springer (2010)

  13. Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. J. Symb. Log. 69, 1027–1088 (2004)

    Article  MATH  Google Scholar 

  14. Benzmüller, C., Kohlhase, M.: LEO—a higher-order theorem prover. In: Kirchner, C., Kirchner, H. (eds.) Automated Deduction—CADE-15, Proceedings of the 15th International Conference on Automated Deduction, Lindau, Germany, 5–10 Jul 1998. Lecture Notes in Artificial Intelligence, number 1421, pp. 139–143. Springer, Lindau, Germany (1998)

    Google Scholar 

  15. Benzmüller, C., Paulson, L.C.: Festschrift in Honor of Peter B. Andrews on His 70th Birthday, chapter Exploring Properties of Normal Multimodal Logics in Simple Type Theory with LEO-II. Studies in Logic, Mathematical Logic and Foundations. College Publications (2008)

  16. Benzmüller, C., Paulson, L.C.: Quantified Multimodal Logics in Simple Type Theory. SEKI Report SR–2009–02 (ISSN 1437–4447). SEKI Publications, DFKI Bremen GmbH, Safe and Secure Cognitive Systems, Cartesium, Enrique Schmidt Str. 5, D–28359 Bremen, Germany http://arxiv.org/abs/0905.2435 (2009)

  17. Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. Log J IGPL 18, 881–892 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Benzmüller, C., Pease, A.: Progress in automating higher-order ontology reasoning. In: Konev, B., Schmidt, R., Schulz, S. (eds.) Proceedings of the Workshop on Practical Aspects of Automated Reasoning (PAAR-2010), Edinburgh, UK. CEUR Workshop Proceedings (2010)

  19. Benzmüller, C., Pease, A.: Reasoning with embedded formulas and modalities in SUMO. In: Bundy, A., Lehmann, J., Qi, G., Varzinczak, I.J. (eds.) Proceedings of the ECAI-10 Workshop on Automated Reasoning about Context and Ontology Evolution (ARCOE-10), Lisbon, Portugal, 16–17 Aug 2010

  20. Benzmüller, C., Rabe, F., Sutcliffe, G.: THF0—the core TPTP language for classical higher-order logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) Proceedings of the 4th International Joint Conference on Automated Reasoning. Lecture Notes in Artificial Intelligence, number 5195, pp. 491–506. Springer (2008)

  21. Benzmüller, C., Theiss, F., Paulson, L., Fietzke, A.: LEO-II—a cooperative automatic theorem prover for higher-order logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds.), Automated Reasoning, Proceedings of the 4th International Joint Conference, IJCAR 2008, Sydney, Australia, 12–15 Aug 2008. Lecture Notes in Artificial Intelligence, vol. 5195, pp. 162–170. Springer (2008)

  22. Blackburn, P., van Benthem, J.F.A.K., Wolter, F.: Handbook of Modal Logic, vol. 3 (Studies in Logic and Practical Reasoning). Elsevier Science Inc., New York, NY, USA (2006)

    Google Scholar 

  23. Blackburn, P., de Rijke, M.: Why combine logics? Stud. Log. 59, 5–27 (1995)

    Article  Google Scholar 

  24. Blackburn, P., Marx, M.: Tableaux for quantified hybrid logic. In: Egly, U., Fermüller, C.G. (eds.) Proceedings of the Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2002, Copenhagen, Denmark, 30 Jul–1 Aug 2002. Lecture Notes in Computer Science, vol. 2381, pp. 38–52. Springer (2002)

  25. Braüner, T.: Natural deduction for first-order hybrid logic. J. Logic Lang. Inf. 14(2), 173–198, (2005)

    Article  MATH  Google Scholar 

  26. Brown, C.: Automated Reasoning in Higher-Order Logic: Set Comprehension and Extensionality in Church’s Type Theory. Studies in Logic: Logic and Cognitive Systems, number 10. College Publications (2007)

  27. Carnielli, W., Coniglio, M.E.: Combining logics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2008 edition (2008)

  28. Carnielli, W.A., Coniglio, M., Gabbay, D.M., Gouveia, P., Sernadas, C.: Analysis and synthesis of logics: how to cut and paste reasoning systems; electronic version. Applied Logic Series. Springer, Dordrecht (2008)

    Google Scholar 

  29. Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  30. Fitting, M.: Interpolation for first order S5. J. Symb. Log. 67(2), 621–634 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gabbay, D.M.: Fibred semantics and the weaving of logics, part 1: Modal and intuitionistic logics. J. Symb. Log. 61(4), 1057–1120 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Gabbay, D.M., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. Studies in Logic, vol. 148. Elsevier Science (2003)

  33. Garg, D., Abadi, M.: A modal deconstruction of access control logics. In: Amadio, R. (ed.) Proceedings of the 11th International Conference on the Foundations of Software Science and Computational Structures. Lecture Notes in Computer Science, number 4962, pp. 216–230 (2008)

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

  35. Gödel, K.: Eine interpretation des intuitionistischen aussagenkalküls. Ergebnisse eines Mathematischen Kolloquiums, 8, 39–40 (1933) Also published in Gödel, K.: Collected Works, Volume I, pp. 296–302. Oxford University Press (1986)

  36. Goldblatt, R.: Logics of time and computation. Center for the Study of Language and Information - Lecture Notes, number 7. Leland Stanford Junior University (1992)

  37. Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15, 81–91 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Munoz, C. (eds.) Proceedings of the 1st International Workshop on Design and Application of Strategies/Tactics in Higher Order Logics. NASA Technical Reports, number NASA/CP-2003-212448, pp. 56–68 (2003)

  39. Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. J. Logic Lang. Inf. 18(4), 437–464 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. McKinsey, J.C.C., Tarski, A.: Some theorems about the sentential calculi of lewis and heyting. J. Symb. Log. 13, 1–15 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  41. Miller, D.: A Compact Representation of Proofs. Stud. Log. 46(4), 347–370 (1987)

    Article  MATH  Google Scholar 

  42. Nguyen, L.A.: MProlog: an extension of prolog for modal logic programming. In: Demoen, B., Lifschitz, V. (eds.) Logic Programming, Proceedings of the 20th International Conference, ICLP 2004, Saint-Malo, France, 6–10 Sept 2004. Lecture Notes in Computer Science, vol. 3132, pp. 469–470. Springer (2004)

  43. Nguyen, L.A.: Multimodal logic programming. Theor. Comp. Sci. 360, 247–288 (2006)

    Article  MATH  Google Scholar 

  44. Nguyen, L.A.: Modal logic programming revisited. J. Appl. Non-Class. Log. 19(2), 167–181 (2009)

    Article  MATH  Google Scholar 

  45. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Lecture Notes in Computer Science, number 2283. Springer (2002)

  46. Prior, A.N.: Past, Present and Future. Clarendon Press, Oxford (1967)

    Book  MATH  Google Scholar 

  47. 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)

  48. Schulz, S.: E—a Brainiac theorem prover. Journal of AI Commun. 15(2/3), 111–126 (2002)

    MATH  Google Scholar 

  49. Segerberg, K.: Two-dimensional modal logic. J. Philos. Logic 2(1), 77–96 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  50. Sutcliffe, G.: TPTP, TSTP, CASC, etc. In: Diekert, V., Volkov, M., Voronkov, A. (eds.) Proceedings of the 2nd International Computer Science Symposium in Russia. Lecture Notes in Computer Science, number 4649, pp. 7–23. Springer (2007)

  51. Sutcliffe, G.: The TPTP World—infrastructure for automated reasoning. In: Clarke, E.M., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning—16th International Conference, LPAR-16, Dakar, Senegal, 25 Apr–1 May 2010, Revised Selected Papers. Lecture Notes in Computer Science, vol. 6355, pp. 1–12. Springer (2010)

  52. Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1–27 (2010)

    MathSciNet  MATH  Google Scholar 

  53. Thomason, R.H.: Combinations of tense and modality. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic. Extensions of Classical Logic, vol. 2, pp. 135–165. D. Reidel (1984)

  54. Venema, Y.: In: Goble, L. (ed.) The Blackwell Guide to Philosophical Logic, chapter Venema, Yde, Temporal Logic. Blackwell Publishing http://www.blackwellreference.com/public/book?id=g9780631206934_9780631206934 (2001).

Download references

Author information

Authors and Affiliations

  1. Articulate Software, Angwin, CA, USA

    Christoph Benzmüller

Authors
  1. Christoph Benzmüller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christoph Benzmüller.

Additional information

This work has been funded by the German Research Foundation (DFG) under grant BE 2501/6-1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benzmüller, C. Combining and automating classical and non-classical logics in classical higher-order logics. Ann Math Artif Intell 62, 103–128 (2011). https://doi.org/10.1007/s10472-011-9249-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-011-9249-7

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation