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

  • Christoph BenzmüllerEmail author
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.


Modal Logic Automate Reasoning Proof Assistant Ontological Argument Conditional Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  33. 33.
    Ohlbach, H.J.: Semantics-based translation methods for modal logics. Journal of Logic and Computation 1(5), 691–746 (1991)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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

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