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

  • Christoph BenzmüllerEmail author


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.


Classical and non-classical logics Quantified multimodal logics Logic combinations Classical higher-order logic Semantic embeddings Knowledge representation Higher-order automated theorem proving 

Mathematics Subject Classifications (2010)

03B62 03B42 03B44 03B45 03B35 03B20 03B15 68T27 68T30 68T15 


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  1. 1.
    Andrews, P.B.: General models and extensionality. J. Symb. Log. 37, 395–397 (1972)zbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, P.B.: General models, descriptions, and choice in type theory. J. Symb. Log. 37, 385–394 (1972)zbMATHCrossRefGoogle Scholar
  3. 3.
    Andrews, P.B.: Theorem proving via general matings. J. ACM 28(2), 193–214 (1981)zbMATHCrossRefGoogle Scholar
  4. 4.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers (2002)Google Scholar
  5. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Andrews, P.B., Brown, C.: TPS: a hybrid automatic-interactive system for developing proofs. J. Appl. Log. 4(4), 367–395 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 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)Google Scholar
  8. 8.
    Baldoni, M.: Normal multimodal logics: automatic deduction and logic programming extension. PhD thesis, Universita degli studi di Torino (1998)Google Scholar
  9. 9.
    Benzmüller, C.: Extensional higher-order paramodulation and RUE-resolution. In: In Proc. of CADE-16. LNAI, number 1632, pp. 399–413. Springer (1999)Google Scholar
  10. 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)Google Scholar
  11. 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. 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)Google Scholar
  13. 13.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. J. Symb. Log. 69, 1027–1088 (2004)zbMATHCrossRefGoogle Scholar
  14. 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. 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)Google Scholar
  16. 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 (2009)
  17. 17.
    Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. Log J IGPL 18, 881–892 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 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)Google Scholar
  19. 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 2010Google Scholar
  20. 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)Google Scholar
  21. 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)Google Scholar
  22. 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. 23.
    Blackburn, P., de Rijke, M.: Why combine logics? Stud. Log. 59, 5–27 (1995)CrossRefGoogle Scholar
  24. 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)Google Scholar
  25. 25.
    Braüner, T.: Natural deduction for first-order hybrid logic. J. Logic Lang. Inf. 14(2), 173–198, (2005)zbMATHCrossRefGoogle Scholar
  26. 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)Google Scholar
  27. 27.
    Carnielli, W., Coniglio, M.E.: Combining logics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy, Winter 2008 edition (2008)Google Scholar
  28. 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. 29.
    Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Fitting, M.: Interpolation for first order S5. J. Symb. Log. 67(2), 621–634 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 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)Google Scholar
  33. 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)Google Scholar
  34. 34.
    Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy. Winter 2009 edition (2009)Google Scholar
  35. 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)Google Scholar
  36. 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)Google Scholar
  37. 37.
    Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15, 81–91 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 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)Google Scholar
  39. 39.
    Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. J. Logic Lang. Inf. 18(4), 437–464 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    McKinsey, J.C.C., Tarski, A.: Some theorems about the sentential calculi of lewis and heyting. J. Symb. Log. 13, 1–15 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Miller, D.: A Compact Representation of Proofs. Stud. Log. 46(4), 347–370 (1987)zbMATHCrossRefGoogle Scholar
  42. 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)Google Scholar
  43. 43.
    Nguyen, L.A.: Multimodal logic programming. Theor. Comp. Sci. 360, 247–288 (2006)zbMATHCrossRefGoogle Scholar
  44. 44.
    Nguyen, L.A.: Modal logic programming revisited. J. Appl. Non-Class. Log. 19(2), 167–181 (2009)zbMATHCrossRefGoogle Scholar
  45. 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)Google Scholar
  46. 46.
    Prior, A.N.: Past, Present and Future. Clarendon Press, Oxford (1967)zbMATHCrossRefGoogle Scholar
  47. 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)Google Scholar
  48. 48.
    Schulz, S.: E—a Brainiac theorem prover. Journal of AI Commun. 15(2/3), 111–126 (2002)zbMATHGoogle Scholar
  49. 49.
    Segerberg, K.: Two-dimensional modal logic. J. Philos. Logic 2(1), 77–96 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 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)Google Scholar
  51. 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)Google Scholar
  52. 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)MathSciNetzbMATHGoogle Scholar
  53. 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)Google Scholar
  54. 54.
    Venema, Y.: In: Goble, L. (ed.) The Blackwell Guide to Philosophical Logic, chapter Venema, Yde, Temporal Logic. Blackwell Publishing (2001).

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Articulate SoftwareAngwinUSA

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