Skip to main content
SpringerLink
Log in

Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic

  • Conference paper
  • First Online:
KI 2017: Advances in Artificial Intelligence (KI 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10505))

Abstract

A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel’s God in order to discuss his emendation of Gödel’s ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting’s textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument which avoid the modal collapse, which is a strongly criticized side-effect in Gödel’s resp. Scott’s original work.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In this paper we work with the Isabelle/HOL proof assistant [22], which explains the chosen abbreviation. Generally, however, the work presented here can be mapped to any other system implementing Church’s simple type theory [13].

  2. 2.

    This term was originally coined by Fitelson and Zalta in [14] and describes an emerging, interdisciplinary field aiming at the rigorous formalization and deep logical assessment of philosophical arguments in an automated reasoning environment.

  3. 3.

    More loosely related work studied Anselm’s older, non-modal version of the ontological argument directly in Prover9 [23] and PVS [24].

  4. 4.

    In contrast to deep semantic embeddings, where the embedded logic is presented as an abstract datatype, our shallow semantic embeddings avoid inductive definitions and maximize the reuse of logical operations from the meta-level. In particular, tedious new binding mechanisms are avoided in our approach.

  5. 5.

    Possibilist and actualist quantification can be seen as the semantic counterparts of the concepts of possibilism and actualism in the metaphysics of modality. They relate to natural-language expressions such as ‘there is’, ‘exists’, ‘is actual’, etc.

  6. 6.

    The notion of rigid designation was introduced by Kripke in [21], where he discusses its many interesting ramifications in logic and the philosophy of language.

  7. 7.

    We prove theorems in Isabelle by using the keyword ‘by’ followed by the name of a proof method. Some methods used here are: simp (term rewriting), blast (tableaus), meson (model elimination), metis (ordered resolution and paramodulation), auto (classical reasoning and term rewriting) and force (exhaustive search trying different tools). In our computer-formalization and assessment of Fitting’s textbook [17], we provide further evidence that our embedded logic works as intended by verifying the book’s theorems and examples.

  8. 8.

    We utilize here (counter-)model finder Nitpick [12] for the first time. For the conjectured lemma, Nitpick finds a countermodel (not shown here), i.e. a model satisfying all the axioms which falsifies the given formula.

  9. 9.

    The de dicto/de re distinction is used regularly in the philosophy of language for disambiguation of sentences involving intensional contexts.

  10. 10.

    Implication can also be proven in the reverse direction (which is not needed for our purposes). Using these definitions, we can derive axioms for the most common modal logics (see also [5]). Thereby we are free to use either the semantic constraints or the related Sahlqvist axioms. Here we provide both versions. In what follows we use the semantic constraints for improved performance.

  11. 11.

    To prove T1, the fact is used that positive properties cannot entail negative ones (A2), from which the possible instantiation of positive properties follows. A computer-formalization of Leibniz’s theory of concepts can be found in [1], where the notion of concept containment in contrast to ordinary property entailment is discussed.

  12. 12.

    We provide a proof in Isabelle/Isar, a language specifically tailored for writing proofs that are both computer- and human-readable. We refer the reader to [17] for other proofs not shown in this article.

  13. 13.

    Essence is defined here (and in Fitting’s variant) in the version of Scott; Gödel’s original version leads to the inconsistency reported in [9, 10].

  14. 14.

    In what follows, the ‘’ parentheses are used to convert an extensional object into its ‘rigid’ intensional counterpart (e.g. ).

  15. 15.

    Fitting’s original treatment in [15] left several details unspecified and we had to fill in the gaps by choosing appropriate formalization variants (see [17] for details).

  16. 16.

    Gödel’s original axioms (without Scott’s addition) are proven inconsistent in [9].

  17. 17.

    This theorem’s proof could be completely automatized for Gödel’s and Fitting’s variants. For Anderson’s version however, we had to reproduce in Isabelle/HOL the original natural-language proof given by Anderson (see [3], Theorem 2*, p. 296).

References

  1. Alama, J., Oppenheimer, P.E., Zalta, E.N.: Automating Leibniz’s theory of concepts. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 73–97. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_4

    Chapter  Google Scholar 

  2. Anderson, A., Gettings, M.: Gödel ontological proof revisited. In: Hajek, P. (ed.) Gödel 1996: Logical Foundations of Mathematics, Computer Science, and Physics. Lecture Notes in Logic, vol. 6, pp. 167–172. Springer (2001)

    Google Scholar 

  3. Anderson, C.: Some emendations of Gödel’s ontological proof. Faith Philos. 7(3), 291–303 (1990)

    Article  Google Scholar 

  4. Benzmüller, C.: Universal reasoning, rational argumentation and human-machine interaction. arXiv (2017). http://arxiv.org/abs/1703.09620

  5. Benzmüller, C., Claus, M., Sultana, N.: Systematic verification of the modal logic cube in Isabelle/HOL. In: Kaliszyk, C., Paskevich, A. (eds.) PxTP 2015, EPTCS, Berlin, Germany, vol. 186, pp. 27–41 (2015)

    Google Scholar 

  6. Benzmüller, C., Paulson, L.: Quantified multimodal logics in simple type theory. Logica Univers. (Special Issue on Multimodal Logics) 7(1), 7–20 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Benzmüller, C., Weber, L., Woltzenlogel-Paleo, B.: Computer-assisted analysis of the Anderson-Hájek controversy. Logica Univers. 11(1), 139–151 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Benzmüller, C., Woltzenlogel Paleo, B.: 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 

  9. Benzmüller, C., Woltzenlogel Paleo, B.: The inconsistency in Gödel’s ontological argument: a success story for AI in metaphysics. In: IJCAI 2016 (2016)

    Google Scholar 

  10. Benzmüller, C., Woltzenlogel Paleo, B.: An object-logic explanation for the inconsistency in Gödel’s ontological theory (extended abstract). In: Helmert, M., Wotawa, F. (eds.) KI 2016: Advances in Artificial Intelligence. LNAI, vol. 9904, pp. 205–244. Springer, Berlin (2016)

    Google Scholar 

  11. Bjørdal, F.: Understanding Gödel’s ontological argument. In: Childers, T. (ed.) The Logica Yearbook 1998. Filosofia (1999)

    Google Scholar 

  12. 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). doi:10.1007/978-3-642-14052-5_11

    Chapter  Google Scholar 

  13. Church, A.: A formulation of the simple theory of types. J. Symbol. Logic 5, 56–68 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fitelson, B., Zalta, E.N.: Steps toward a computational metaphysics. J. Philos. Logic 36(2), 227–247 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fitting, M.: Types, Tableaus and Gödel’s God. Kluwer, Dordrecht (2002)

    Book  MATH  Google Scholar 

  16. Fitting, M., Mendelsohn, R.: First-Order Modal Logic. Synthese Library, vol. 277. Kluwer, Dordrecht (1998)

    Google Scholar 

  17. Fuenmayor, D., Benzmüller, C.: Types, Tableaus and Gödel’s God in Isabelle/HOL. Archive of Formal Proofs (2017). Formally verified with Isabelle/HOL

    Google Scholar 

  18. Gallin, D.: Intensional and Higher-Order Modal Logic. N.-Holland, Amsterdam (1975)

    MATH  Google Scholar 

  19. Gödel, K.: Appx. A: notes in Kurt Gödel’s hand. In: [27], pp. 144–145 (2004)

    Google Scholar 

  20. Hájek, P.: A new small emendation of Gödel’s ontological proof. Studia Logica 71(2), 149–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kripke, S.: Naming and Necessity. Harvard University Press, Cambridge (1980)

    Google Scholar 

  22. Nipkow, T., Wenzel, M., Paulson, L.C. (eds.): Isabelle/HOL — A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  23. Oppenheimera, P., Zalta, E.: A computationally-discovered simplification of the ontological argument. Australas. J. Philos. 89(2), 333–349 (2011)

    Article  Google Scholar 

  24. Rushby, J.: The ontological argument in PVS. In: Proceedings of CAV Workshop “Fun With Formal Methods”, St. Petersburg, Russia (2013)

    Google Scholar 

  25. Scott, D.: Appx.B: notes in Dana Scott’s hand. In: [27], pp. 145–146 (2004)

    Google Scholar 

  26. Sobel, J.: Gödel’s ontological proof. In: On Being and Saying. Essays for Richard Cartwright, pp. 241–261. MIT Press (1987)

    Google Scholar 

  27. Sobel, J.: Logic and Theism: Arguments for and Against Beliefs in God. Cambridge U. Press, Cambridge (2004)

    Google Scholar 

  28. Wisniewski, M., Steen, A., Benzmüller, C.: Einsatz von Theorembeweisern in der Lehre. In: Schwill, A., Lucke, U. (eds.) Hochschuldidaktik der Informatik: 7. Fachtagung des GI-Fachbereichs Informatik und Ausbildung/Didaktik der Informatik, 13–14 September 2016 an der Universität Potsdam, Commentarii informaticae didacticae (CID), Potsdam, Germany (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Freie Universität Berlin, Berlin, Germany

    David Fuenmayor & Christoph Benzmüller

  2. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Christoph Benzmüller

Authors
  1. David Fuenmayor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Benzmüller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to David Fuenmayor or Christoph Benzmüller .

Editor information

Editors and Affiliations

  1. Fakultät für Informatik, Technische Universität Dortmund, Dortmund, Germany

    Gabriele Kern-Isberner

  2. FB Informatik, TU Darmstadt, Darmstadt, Hessen, Germany

    Johannes Fürnkranz

  3. FB Informatik, Universität Koblenz, Koblenz, Rheinland-Pfalz, Germany

    Matthias Thimm

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Fuenmayor, D., Benzmüller, C. (2017). Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic. In: Kern-Isberner, G., Fürnkranz, J., Thimm, M. (eds) KI 2017: Advances in Artificial Intelligence. KI 2017. Lecture Notes in Computer Science(), vol 10505. Springer, Cham. https://doi.org/10.1007/978-3-319-67190-1_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-67190-1_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67189-5

  • Online ISBN: 978-3-319-67190-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation