International Conference on Automated Reasoning with Analytic Tableaux and Related Methods

Automated Reasoning with Analytic Tableaux and Related Methods pp 213-220 | Cite as

Invited Talk: On a (Quite) Universal Theorem Proving Approach and Its Application in Metaphysics

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9323)


Classical higher-order logic is suited as a meta-logic in which a range of other logics can be elegantly embedded. Interactive and automated theorem provers for higher-order logic are therefore readily applicable. By employing this approach, the automation of a variety of ambitious logics has recently been pioneered, including variants of first-order and higher-order quantified multimodal logics and conditional logics. Moreover, the approach supports the automation of meta-level reasoning, and it sheds some new light on meta-theoretical results such as cut-elimination. Most importantly, however, the approach is relevant for practice: it has recently been successfully applied in a series of experiments in metaphysics in which higher-order theorem provers have actually contributed some new knowledge.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, C.: Some emendations of Gödel’s ontological proof. Faith and Philosophy 7(3) (1990)Google Scholar
  2. 2.
    Benzmüller, C.: Automating access control logics in simple type theory with LEO-II. In: Gritzalis, D., Lopez, J. (eds.) SEC 2009. IFIP AICT, vol. 297, pp. 387–398. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Benzmüller, C.: Combining and automating classical and non-classical logics in classical higher-order logic. Annals of Mathematics and Artificial Intelligence 62(1-2), 103–128 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Benzmüller, C.: Automating quantified conditional logics in HOL. In: Rossi, F. (ed.) IJCAI 2013, Beijing, China, pp. 746–753 (2013)Google Scholar
  5. 5.
    Benzmüller, C.: Cut-free calculi for challenge logics in a lazy way. In: Clint van Alten, C.N., Cintula, P. (eds.) Proceedings of the International Workshop on Algebraic Logic in Computer Science (2013)Google Scholar
  6. 6.
    Benzmüller, C.: A top-down approach to combining logics. In: ICAART 2013, Barcelona, Spain, pp. 346–351. SciTePress Digital Library (2013)Google Scholar
  7. 7.
    Benzmüller, C.: Higher-order automated theorem provers. In: Delahaye, D., Paleo, B.W. (eds.) All about Proofs, Proof for All, Mathematical Logic and Foundations, pp. 171–214. College Publications, London (2015)Google Scholar
  8. 8.
    Benzmüller, C.: HOL provers for first-order modal logics — experiments. In: Benzmuüller, C., Otten, J. (eds.) ARQNL@IJCAR 2014, EPiC Series. EasyChair (2015, to appear)Google Scholar
  9. 9.
    Benzmüller, C., Claus, M., Sultana, N.: Systematic verification of the modal logic cube in Isabelle/HOL. In: Kaliszyk, C., Paskevich, A. (eds.) PxTP 2015, Berlin, Germany. EPTCS (2015, to appear)Google Scholar
  10. 10.
    Benzmüller, C., Gabbay, D., Genovese, V., Rispoli, D.: Embedding and automating conditional logics in classical higher-order logic. Annals of Mathematics and Artificial Intelligence 66(1-4), 257–271 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Benzmüller, C., Miller, D.: Automation of higher-order logic. In: Gabbay, D.M., Siekmann, J.H., Woods, J. (eds.) Handbook of the History of Logic. Computational Logic, vol. 9, pp. 215–254. North Holland, Elsevier (2014)Google Scholar
  12. 12.
    Benzmüller, C., Otten, J., Raths, T.: Implementing and evaluating provers for first-order modal logics. In: Raedt, L.D., Bessiere, C., Dubois, D., Doherty, P., Frasconi, P., Heintz, F., Lucas, P. (eds.) ECAI 2012. Frontiers in Artificial Intelligence and Applications, Montpellier, France, vol. 242, pp. 163–168. IOS Press (2012)Google Scholar
  13. 13.
    Benzmüller, C., Paleo, B.W.: Gödel’s God in Isabelle/HOL. Archive of Formal Proofs (2013)Google Scholar
  14. 14.
    Benzmüller, C., Paleo, B.W.: Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers. In: Schaub, T., Friedrich, G., O’Sullivan, B. (eds.) ECAI 2014. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 93–98. IOS Press (2014)Google Scholar
  15. 15.
    Benzmüller, C., Paulson, L.: Exploring properties of normal multimodal logics in simple type theory with LEO-II. In: Benzmüller, C., Brown, C., Siekmann, J., Statman, R. (eds.) Reasoning in Simple Type Theory — Festschrift in Honor of Peter B. Andrews on His 70th Birthday, Studies in Logic, Mathematical Logic and Foundations, pp. 386–406. College Publications (2008)Google Scholar
  16. 16.
    Benzmüller, C., Paulson, L.: Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL 18(6), 881–892 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Benzmüller, C., Paulson, L.: Quantified multimodal logics in simple type theory. Logica Universalis (Special Issue on Multimodal Logics) 7(1), 7–20 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Benzmüller, C., Paulson, L.C., Sultana, N., Theiß, F.: The higher-order prover LEO-II. Journal of Automated Reasoning, (2015, to appear)Google Scholar
  19. 19.
    Benzmüller, C., Pease, A.: Higher-order aspects and context in SUMO. Journal of Web Semantics (Special Issue on Reasoning with context in the Semantic Web) 12-13, 104–117 (2012)Google Scholar
  20. 20.
    Benzmüller, C., Raths, T.: HOL based first-order modal logic provers. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 127–136. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  21. 21.
    Benzmüller, C., Weber, L., Paleo, B.W.: Computer-assisted analysis of the Anderson-Hájek ontological controversy. In: Silvestre, R.S., Béziau, J.-Y. (eds.) Handbook of the 1st World Congress on Logic and Religion, Joao Pessoa, Brasil, pp. 53–54 (2015)Google Scholar
  22. 22.
    Benzmüller, C., Woltzenlogel Paleo, B.: Higher-order modal logics: Automation and applications. In: Faber, W., Paschke, A. (eds.) Reasoning Web 2015. LNCS, vol. 9203, pp. 32–74. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  23. 23.
    Benzmüller, C., Woltzenlogel Paleo, B.: Interacting with modal logics in the coq proof assistant. In: Beklemishev, L.D. (ed.) CSR 2015. LNCS, vol. 9139, pp. 398–411. Springer, Heidelberg (2015)Google Scholar
  24. 24.
    Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development. Springer (2004)Google Scholar
  25. 25.
    Blanchette, J., Böhme, S., Paulson, L.: Extending Sledgehammer with SMT solvers. Journal of Automated Reasoning 51(1), 109–128 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Blanchette, J.C., Nipkow, T.: Nitpick: A counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  27. 27.
    Brown, C.E.: Satallax: An automatic higher-order prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 111–117. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  28. 28.
    Bundy, A.: The use of explicit plans to guide inductive proofs. In: Lusk, E., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310, pp. 111–120. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  29. 29.
    Gabbay, D.M.: Labelled Deductive Systems. Clarendon Press (1996)Google Scholar
  30. 30.
    Gödel, K.: Appx.A: Notes in Kurt Gödel’s Hand. In: [37], pp. 144–145 (2004)Google Scholar
  31. 31.
    Melis, E., Meier, A., Siekmann, J.H.: Proof planning with multiple strategies. Artif. Intell. 172(6-7), 656–684 (2008)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHGoogle Scholar
  33. 33.
    Ohlbach, H.J.: Semantics-based translation methods for modal logics. Journal of Logic and Computation 1(5), 691–746 (1991)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Ohlbach, H.J., Schmidt, R.A.: Functional translation and second-order frame properties of modal logics. Journal of Logic and Computation 7(5), 581–603 (1997)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Scott, D.: Appx.B: Notes in Dana Scott’s Hand. In: [37], pp. 145–146 (2004)Google Scholar
  36. 36.
    Siekmann, J.H., Benzmüller, C., Autexier, S.: Computer supported mathematics with omega. J. Applied Logic 4(4), 533–559 (2006)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Sobel, J.: Logic and Theism: Arguments for and Against Beliefs in God. Cambridge U. Press (2004)Google Scholar
  38. 38.
    Sutcliffe, G.: The TPTP problem library and associated infrastructure. Journal of Automated Reasoning 43(4), 337–362 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1–27 (2010)MathSciNetMATHGoogle Scholar
  40. 40.
    Wisnieski, M., Steen, A.: Embedding of quantified higher-order nominal modal logic into classical higher-order logic. In: Benzmuüller, C., Otten, J. (eds.) Proceedings on the 1st International Workshop on Automated Reasoning in Quantified Non-Classical Logics, ARQNL (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Freie UniversitätBerlinGermany

Personalised recommendations