Interacting with Modal Logics in the Coq Proof Assistant

  • Christoph Benzmüller
  • Bruno Woltzenlogel PaleoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)


This paper describes an embedding of higher-order modal logics in the Coq proof assistant. Coq’s capabilities are used to implement modal logics in a minimalistic manner, which is nevertheless sufficient for the formalization of significant, non-trivial modal logic proofs. The elegance, flexibility and convenience of this approach, from a user perspective, are illustrated here with the successful formalization of Gödel’s ontological argument.


Modal Logic Natural Deduction Proof Assistant Elimination Rule Automate Theorem Prover 
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.



We thank Cedric Auger and Laurent Théry, for their answers to our questions about Ltac in the Coq-Club mailing-list.


  1. 1.
    Benzmüller, C., Paleo, B.W.: Formalization, mechanization and automation of Gödel’s proof of god’s existence. CoRR, abs/1308.4526 (2013)Google Scholar
  2. 2.
    Benzmüller, C., Paleo, B.W.: Gödel’s God inIsabelle/HOL. Archive of Formal Proofs, 2013 (2013)Google Scholar
  3. 3.
    Benzmüller, C., Paulson, L.C.: 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, pp. 386–406. College Publications (2008)Google Scholar
  4. 4.
    Benzmüller, C., Paulson, L.C.: Quantified multimodal logics in simple type theory. Logica Universalis (Special Issue on Multimodal Logics) 7(1), 7–20 (2013)zbMATHCrossRefGoogle Scholar
  5. 5.
    Benzmüller, C.E., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II - a cooperative automatic theorem prover for classical higher-order logic (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  6. 6.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher-order semantics and extensionality. J. Symb. Log. 69(4), 1027–1088 (2004)zbMATHCrossRefGoogle Scholar
  7. 7.
    Benzmüller, C., Otten, J., Raths, T.: Implementing and evaluating provers for first-order modal logics. In: ECAI 2012–20th European Conference on Artificial Intelligence, pp. 163–168 (2012)Google Scholar
  8. 8.
    Blackburn, P., Benthem, J.v., Wolter, F.: Handbook of Modal Logic. Elsevier, Amsterdam (2006)Google Scholar
  9. 9.
    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
  10. 10.
    Corazzon, R.: Contemporary bibliography on the ontological proof. (
  11. 11.
    Fitting, M.: Types, Tableaux and Gödel’s God. Kluver Academic Press, Dordrecht (2002)Google Scholar
  12. 12.
    Gabbay, D.M.: Labelled Deductive Systems. Clarendon Press, Oxford (1996) zbMATHGoogle Scholar
  13. 13.
    Gödel, K.: Ontological proof. In: Gödel, K.: Collected Works, Unpublished Essays and Letters. vol. 3, pp. 403–404. Oxford University Press, Oxford (1970)Google Scholar
  14. 14.
    Harrison, J.: HOL light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  15. 15.
    Nipkow, T., Paulson, L.C., Wenzel, M. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002) zbMATHGoogle Scholar
  16. 16.
    Oppenheimer, P.E., Zalta, E.N.: A computationally-discovered simplification of the ontological argument. Australas. J. Philos. 2(89), 333–349 (2011)CrossRefGoogle Scholar
  17. 17.
    Paleo, B.W., Benzmüller, C.: Formal theology repository. (
  18. 18.
    Paulin-Mohring, C.: Introduction to the calculus of inductive constructions. In: Delahaye, D., Paleo, B.W. (eds.) All about Proofs, Proofs for All. Mathematical Logic and Foundations. College Publications, London (2015) Google Scholar
  19. 19.
    Rushby, J.: The ontological argument in PVS. In: Proceedings of CAV Workshop Fun With Formal Methods, St. Petersburg, Russia (2013)Google Scholar
  20. 20.
    Schmidt, R.: List of modal provers. ( schmidt/tools/)
  21. 21.
    Scott, D.: Appendix B. Notes in Dana Scott’s hand. In: [24] (2004)Google Scholar
  22. 22.
    Siders, A., Paleo, B.W.: A variant of Gödel’s ontological proof in a natural deduction calculus. ( ralDeduction/GodProof-ND.pdf?raw=true)
  23. 23.
    Sobel, J.H.: Gödel’s ontological proof. In: On Being and Saying. Essays for Richard Cartwright, pp. 241–261. MIT Press, Cambridge (1987)Google Scholar
  24. 24.
    Sobel, J.H.: Logic and Theism: Arguments for and Against Beliefs in God. Cambridge University Press, Cambridge (2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christoph Benzmüller
    • 1
  • Bruno Woltzenlogel Paleo
    • 2
    Email author
  1. 1.Freie Universität BerlinBerlinGermany
  2. 2.Vienna University of TechnologyViennaAustria

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