The Ontological Modal Collapse as a Collapse of the Square of Opposition

  • Christoph BenzmüllerEmail author
  • Bruno Woltzenlogel Paleo
Part of the Studies in Universal Logic book series (SUL)


The modal collapse that afflicts Gödel’s modal ontological argument for God’s existence is discussed from the perspective of the modal square of opposition.


Higher-order logics Interactive and automated theorem proving Modal logics Ontological argument 

Mathematics Subject Classification (2000)

Prim. 03B15; Sec. 68T15 



We would like to thank Paul Weingartner, André Fuhrmann and Melvin Fitting for several discussions about Gödel’s ontological argument.

Note About Authorship Alphabetic order has been used for the authors’ names. The extent and kind of contribution of each author cannot be inferred from the order.


  1. 1.
    C.A. Anderson, Some emendations of Gödel’s ontological proof. Faith Philos. 7 (3), 291–303 (1990)CrossRefGoogle Scholar
  2. 2.
    A.C. Anderson, M. Gettings, Gödel ontological proof revisited, in Gödel’96: Logical Foundations of Mathematics, Computer Science, and Physics. Lecture Notes in Logic, vol. 6 (Springer, Berlin, 1996), pp. 167–172Google Scholar
  3. 3.
    C. Benzmüller, HOL based universal reasoning, in Handbook of the 4th World Congress and School on Universal Logic, ed. by J.Y. Beziau, A. Buchsbaum, A. Costa-Leite, A. Altair (Rio de Janeiro, 2013), pp. 232–233.
  4. 4.
    C. Benzmüller, B. Woltzenlogel Paleo, Gödel’s God in Isabelle/HOL. Archive of Formal Proofs (2013).
  5. 5.
    C. Benzmüller, B. Woltzenlogel Paleo, Gödel’s God on the computer, in Proceedings of the 10th International Workshop on the Implementation of Logics, ed. by S. Schulz, G. Sutcliffe, B. Konev. EPiC Series. EasyChair (2013). Invited abstractGoogle Scholar
  6. 6.
    C. Benzmüller, B. Woltzenlogel Paleo, Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers, in ECAI 2014, ed. by T. Schaub, G. Friedrich, B. O’Sullivan. Frontiers in Artificial Intelligence and Applications, vol. 263 (IOS Press, Amsterdam, 2014), pp. 163–168Google Scholar
  7. 7.
    C. Benzmüller, L.C. Paulson, 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 (College Publications, London, 2008), pp. 386–406Google Scholar
  8. 8.
    C. Benzmüller, L.C. Paulson, Quantified multimodal logics in simple type theory. Log. Univers. (Special Issue on Multimodal Logics) 7 (1), 7–20 (2013)Google Scholar
  9. 9.
    C. Benzmüller, F. Theiss, L. Paulson, A. Fietzke, LEO-II - a cooperative automatic theorem prover for higher-order logic, in Proceedings of IJCAR 2008. LNAI, vol. 5195 (Springer, Berlin, 2008), pp. 162–170Google Scholar
  10. 10.
    Y. Bertot, P. Casteran, Interactive Theorem Proving and Program Development (Springer, Berlin, 2004)CrossRefGoogle Scholar
  11. 11.
    F. Bjørdal, Understanding Gödel’s Ontological Argument, in The Logica Yearbook 1998, ed. by T. Childers (Filosofia, Prague, 1999)Google Scholar
  12. 12.
    J.C. Blanchette, T. Nipkow, Nitpick: a counterexample generator for higher-order logic based on a relational model finder, in Proceeding of ITP 2010. LNCS, vol. 6172 (Springer, Berlin, 2010), pp. 131–146Google Scholar
  13. 13.
    C.E. Brown, Satallax: an automated higher-order prover, in Proceedings of IJCAR 2012. LNAI, vol. 7364 (Springer, Berlin, 2012), pp. 111–117Google Scholar
  14. 14.
    S. Chatti, F. Schang, The cube, the square and the problem of existential import. Hist. Philos. Log. 34 (2), 101–132 (2013)CrossRefGoogle Scholar
  15. 15.
    R. Corazzon, Contemporary bibliography on the ontological proof.
  16. 16.
    M. Fitting, Types, Tableaux and Gödel’s God (Kluver Academic Press, Dordrecht, 2002)CrossRefGoogle Scholar
  17. 17.
    A. Fuhrmann, Existenz und notwendigkeit — Kurt Gödels axiomatische theologie, in Logik in der Philosophie, ed. by W. Spohn et al. (Synchron, Heidelberg, 2005)Google Scholar
  18. 18.
    K. Gödel, Appendix A. Notes in Kurt Gödel’s hand, in Logic and Theism: Arguments for and Against Beliefs in God (Cambridge University Press, Cambridge, 2004), pp. 144–145Google Scholar
  19. 19.
    P. Hajek, A new small emendation of Gödel’s ontological proof. Stud. Log. 71 (2), 149–164 (2002)CrossRefGoogle Scholar
  20. 20.
    P. Hajek, Ontological proofs of existence and non-existence. Stud. Log. 90 (2), 257–262 (2008)CrossRefGoogle Scholar
  21. 21.
    R. Koons, Sobel on Gödel’s ontological proof. Philos. Christi 2, 235–248 (2006)Google Scholar
  22. 22.
    T. Nipkow, L.C. Paulson, M. Wenzel, Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283 (Springer, Berlin, 2002)Google Scholar
  23. 23.
    P.E. Oppenheimera, E.N. Zalta, A computationally-discovered simplification of the ontological argument. Australas. J. Philos. 89 (2), 333–349 (2011)CrossRefGoogle Scholar
  24. 24.
    B. Woltzenlogel Paleo, Automated verification and reconstruction of Gödel’s proof of God’s existence. OCG J. 04, 4–7 (2013)Google Scholar
  25. 25.
    J. Rushby, The ontological argument in PVS, in Proceedings of CAV Workshop “Fun With Formal Methods”, St. Petersburg, Russia (2013)Google Scholar
  26. 26.
    D. Scott, Appendix B. Notes in Dana Scott’s hand, in Logic and Theism: Arguments for and Against Beliefs in God (Cambridge University Press, Cambridge, 2004), pp. 145–146Google Scholar
  27. 27.
    J.H. Sobel, Gödel’s ontological proof, in On Being and Saying. Essays for Richard Cartwright (MIT, Cambridge, 1987), pp. 241–261Google Scholar
  28. 28.
    J.H. Sobel, Logic and Theism: Arguments for and Against Beliefs in God (Cambridge University Press, Cambridge, 2004)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Christoph Benzmüller
    • 1
    Email author
  • Bruno Woltzenlogel Paleo
    • 2
  1. 1.Department of Mathematics and Computer ScienceFreie Universität BerlinBerlinGermany
  2. 2.Institut für ComputersprachenVienna University of TechnologyWienAustria

Personalised recommendations