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Computer Science and Metaphysics: A Cross-Fertilization

Daniel Kirchner / Christoph Benzmüller / Edward N. Zalta
Published Online: 2019-08-23 | DOI: https://doi.org/10.1515/opphil-2019-0015


Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.

Keywords: Computational Metaphysics; Abstract Object Theory; Shallow Semantic Embedding; Gödel’s Ontological Argument; Theorem Proving; Automated Reasoning


  • Alama, Jesse, Oppenheimer, Paul E., and Zalta, Edward N. “Automating Leibniz’s Theory of Concepts.” In Automated DeductionCADE-2525th International Conference on Automated Deduction, Berlin, Germany, August 17, 2015, Proceedings, edited by Amy P. Felty and Aart Middeldorp, 9195:73–97. Springer: Lecture Notes in Computer Science, 2015.Google Scholar

  • Anderson, C. Anthony, “Some Emendations of Gödel’s Ontological Proof.” Faith and Philosophy 7:3 (1990), 291–303.Google Scholar

  • Anderson, C. Anthony, and Gettings, Michael. “Gödel’s Ontological Proof Revisited.” In Gödel’96: Logical Foundations of Mathematics, Computer Science, and Physics, 6:167–172. Springer: Lecture Notes in Logic, 1996.Google Scholar

  • Andrews, Peter B. “On Connections and Higher-Order Logic.” Journal of Automated Reasoning 5:3 (1989), 257–291.CrossrefGoogle Scholar

  • Bentert, Matthias, Benzmüller, Christoph, Streit, David and Woltzenlogel Paleo, Bruno. “Analysis of an Ontological Proof Proposed by Leibniz.” In Death and Anti-Death, Volume 14: Four Decades After Michael Polanyi, Three Centuries After G.w. Leibniz, edited by Charles Tandy. Ria University Press, 2016.Google Scholar

  • Benzmüller, Christoph. “Universal (Meta-)Logical Reasoning: Recent Successes.” Science of Computer Programming 172 (2019), 48–62.Google Scholar

  • Benzmüller, Christoph, Claus, Maximilian, and Sultana, Nik. “Systematic Verification of the Modal Logic Cube in Isabelle/HOL.” In PxTP 2015, edited by Cezary Kaliszyk and Andrei Paskevich, 186:27–41. EPTCS, 2015.Google Scholar

  • Benzmüller, Christoph, Farjami, Ali, and Parent, Xavier. “A Dyadic Deontic Logic in HOL.” In Deontic Logic and Normative Systems — 14th International Conference, Deon 2018, Utrecht, the Netherlands, 3-6 July, 2018, edited by Jan Broersen, Cleo Condoravdi, Shyam Nair, and Gabriella Pigozzi, 33–50. College Publications, 2018.Google Scholar

  • Benzmüller, Christoph, Farjami, Ali, and Parent, Xavier. “Åqvist’s Dyadic Deontic Logic E in HOL.” Journal of Applied Logics, 2019. To appear, preprint: URL = http://orbilu.uni.lu/handle/10993/37014

  • Benzmüller, Christoph, and Fuenmayor, David. “Can Computers Help to Sharpen Our Understanding of Ontological Arguments?” In Mathematics and Reality, Proceedings of the 11th All India Students’ Conference on Science & Spiritual Quest (Aissq), 6-7 October, 2018, Iit Bhubaneswar, Bhubaneswar, India, edited by Sudipto Gosh, Ramgopal Uppalari, K. Vasudeva Rao, Varun Agarwal, and Sushant Sharma, 195–226. The Bhaktivedanta Institute, Kolkata, 2018.Google Scholar

  • Benzmüller, Christoph, Parent, Xavier and van der Torre, Leendert. “A Deontic Logic Reasoning Infrastructure.” In 14th Conference on Computability in Europe, Cie 2018, Kiel, Germany, July 30-August, 2018, Proceedings, edited by Florin Manea, Russell G. Miller, and Dirk Nowotka, 10936:60–69. Springer: Lecture Notes in Computer Science, 2018.Google Scholar

  • Benzmüller, Christoph, and Paulson, Lawrence. “Quantified Multimodal Logics in Simple Type Theory.” Logica Universalis 7:1 (2013), 7–20.Google Scholar

  • Benzmüller, Christoph, and Scott, Dana. “Axiom Systems for Category Theory in Free Logic.” Archive of Formal Proofs. 2018. URL = https://www.isa-afp.org/entries/AxiomaticCategoryTheory.html

  • Benzmüller, Christoph, and Scott, Dana. “Automating Free Logic in HOL, with an Experimental Application in Category Theory.” Journal of Automated Reasoning. 2019. 10.1007/s10817-018-09507-7Google Scholar

  • Benzmüller, Christoph, Sultana, Nik, Paulson, Lawrence C. and Theiß, Frank. “The Higher-Order Prover LEO-II.” Journal of Automated Reasoning 55:4 (2015), 389–404.Google Scholar

  • Benzmüller, Christoph, Weber, Leon and Woltzenlogel Paleo, Bruno. “Computer-Assisted Analysis of the Anderson-Hájek Controversy.” Logica Universalis 11:1 (2017), 139–151.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “Automating Gödel’s Ontological Proof of God’s Existence with Higher-Order Automated Theorem Provers.” In ECAI 2014, edited by Torsten Schaub, Gerhard Friedrich, and Barry O’Sullivan, 263:93–98. IOS Press: Frontiers in Artificial Intelligence and Applications, 2014.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “Higher-Order Modal Logics: Automation and Applications.” In Reasoning Web 2015, edited by Adrian Paschke and Wolfgang Faber, 32–74. Springer: Lecture Notes in Computer Science, 2015.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “Interacting with Modal Logics in the Coq Proof Assistant.” In Computer Science — Theory and Applications — 10th International Computer Science Symposium in Russia, CSR 2015, Listvyanka, Russia, July 13-17, 2015, Proceedings, edited by Lev D. Beklemishev and Daniil V. Musatov, 9139:398–411. Springer: Lecture Notes in Computer Science, 2015.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “An Object-Logic Explanation for the Inconsistency in Gödel’s Ontological Theory (Extended Abstract, Sister Conferences).” In KI 2016: Advances in Artificial Intelligence, Proceedings, edited by Malte Helmert and Franz Wotawa, 244–250. Springer: Lecture Notes in Computer Science, 2016.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “The Inconsistency in Gödel’s Ontological Argument: A Success Story for AI in Metaphysics.” In IJCAI 2016, edited by Subbarao Kambhampati, 936–942. AAAI Press, 2016.Google Scholar

  • Benzmüller, Christoph, and Woltzenlogel Paleo, Bruno. “The Modal Collapse as a Collapse of the Modal Square of Opposition.” In The Square of Opposition: A Cornerstone of Thought (Collection of Papers Related to the World Congress on the Square of Opposition Iv, Vatican, 2014), edited by Jean-Yves Béziau and Gianfranco Basti, 307–313. Springer: Studies in Universal Logic, 2016.Google Scholar

  • Bertot, Yyes, and Casteran, Pierre. Interactive Theorem Proving and Program Development - Coq’Art: The Calculus of Inductive Constructions. Springer: Texts in Theoretical Computer Science, 2004.Google Scholar

  • Bjørdal, Frode. “Understanding Gödel’s Ontological Argument.” In The Logica Yearbook 1998, edited by T. Childers, 214–217. Filosofia, 1999.Google Scholar

  • Blanchette, Jasmin Christian, Böhme, Sascha and Paulson, Lawrence Charles. “Extending Sledgehammer with SMT Solvers.” Journal of Automated Reasoning 51:1 (2013), 109–128.CrossrefGoogle Scholar

  • Blanchette, Jasmin Christian, and Nipkow, Tobias. “Nitpick: A Counterexample Generator for Higher-Order Logic Based on a Relational Model Finder.” In Interactive Theorem Proving, First International Conference, ITP 2010, Edinburgh, Uk, July 11-14, 2010. Proceedings, edited by Matt Kaufmann and Lawrence C. Paulson, 6172:131–146. Springer: Lecture Notes in Computer Science, 2010.Google Scholar

  • Blanchette, Jasmin Christian, Popescu, Andrei, Wand, Daniel, and Weidenbach, Christoph. “More SPASS with Isabelle – Superposition with Hard Sorts and Configurable Simplification.” In Interactive Theorem Proving — Third International Conference, ITP F2012, Princeton, Nj, Usa, August 13-15, 2012. Proceedings, edited by Lennart Beringer and Amy P. Felty, 7406:345–360. Springer: Lecture Notes in Computer Science, 2012.Google Scholar

  • Brown, Chad E. “Satallax: An Automatic Higher-Order Prover.” In Automated Reasoning — 6th International Joint Conference, IJCAR 2012, Manchester, Uk, June 26-29, 2012. Proceedings, edited by Bernhard Gramlich, Dale Miller, and Uli Sattler, 7364:111–117. Springer: Lecture Notes in Computer Science, 2012.Google Scholar

  • Cruanes, Simon, and Blanchette, Jasmin Christian. “Extending Nunchaku to Dependent Type Theory.” In Proceedings First International Workshop on Hammers for Type Theories, Hatt@IJCAR 2016, 3–12. EPTCS, 2016.Google Scholar

  • de Moura, Leonardo, and Bjørner, Nikolaj. “Z3: An Efficient Smt Solver.” In Tools and Algorithms for the Construction and Analysis of Systems, edited by C. R. Ramakrishnan and Jakob Rehof, 337–340. Springer, 2008.Google Scholar

  • Deters, Morgan, Reynolds, Andrew, King, Tim, Barrett, Clark W., and Tinelli, Cesare. “A Tour of CVC4: How It Works, and How to Use It.” In Formal Methods in Computer-Aided Design, FMCAD 2014, edited by Koen Claessen and Viktor Kuncak, 7. IEEE, 2014.Google Scholar

  • Farjami, Ali, Meder, Paul, Parent, Xavier, and Benzmüller, Christoph. 2018. “I/O Logic in HOL.” Journal of Applied Logics, 2019. To appear, preprint: URL = https://orbilu.uni.lu/handle/10993/37013

  • Fitelson, Branden, and Zalta, Edward N. “Steps Toward a Computational Metaphysics.” Journal Philosophical Logic 36:2 (2007), 227–47.Google Scholar

  • Fitting, Melvin. Types, Tableaus, and Gödel’s God. Kluwer, 2002.Google Scholar

  • Freyd, Peter, and Scedrov, Andre. Categories, Allegories. North Holland, 1990.Google Scholar

  • Fuenmayor, David, and Benzmüller, Christoph. “Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic.” In KI 2017: Advances in Artificial Intelligence 40th Annual German Conference on AI, Dortmund, Germany, September 25-29, 2017, Proceedings, 10505:114–127. Springer: Lecture Notes in Artificial Intelligence, 2017.Google Scholar

  • Fuenmayor, David, and Benzmüller, Christoph. “Types, Tableaus and Gödel’s God in Isabelle/HOL.” Archive of Formal Proofs, 2017. URL = https://www.isa-afp.org/entries/Types_Tableaus_and_Goedels_God.html

  • Fuenmayor, David, and Benzmüller, Christoph. “A Case Study on Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument.” Journal of Applied Logics (Special Issue on Formal Approaches to the Ontological Argument) 5:7 (2018), 1567–1603.Google Scholar

  • Fuenmayor, David, and Benzmüller, Christoph. “Formalisation and Evaluation of Alan Gewirth’s Proof for the Principle of Generic Consistency in Isabelle/HOL.” Archive of Formal Proofs, 2018. URL = http://isa-afp.org/entries/GewirthPGCProof.html

  • Gödel, Kurt. “Appendix A. Notes in Kurt Gödel’s Hand.” In Logic and Theism: Arguments for and Against Beliefs in God, edited by J. H. Sobel, 144–145. Cambridge University Press, 1970.Google Scholar

  • Hájek, Petr. “Magari and Others on Gödel’s Ontological Proof.” In Logic and Algebra, edited by A. Ursini and P. Agliano, 125–135. Dekker, 1996.Google Scholar

  • Hájek, Petr. “Der Mathematiker und die Frage der Existenz Gottes.” In Kurt Gödel. Wahrheit und Beweisbarkeit, edited by B. Buldt et al., 325–336. öbv & hpt Verlagsgesellschaft mbH, 2001.Google Scholar

  • Hájek, Petr. “A New Small Emendation of Gödel’s Ontological Proof.” Studia Logica 71:2 (2002), 149–164.Google Scholar

  • Huffman, Brian, and Kuncar, Ondrej. “Lifting and Transfer: A Modular Design for Quotients in Isabelle/HOL.” In Certified Programs and Proofs — Third International Conference, CPP 2013, Melbourne, Vic, Australia, December 11-13, 2013, Proceedings, edited by Georges Gonthier and Michael Norrish, 8307:131–146. Springer: Lecture Notes in Computer Science, 2013.Google Scholar

  • Kirchner, Daniel. “Representation and Partial Automation of the Principia Logico-Metaphysica in Isabelle/HOL.” Archive of Formal Proofs, 2017. URL = https://www.isa-afp.org/entries/PLM.html

  • Kovács, Laura, and Voronkov, Andrei. “First-Order Theorem Proving and Vampire.” In Computer Aided Verification - 25th International Conference, CAV 2013, Saint Petersburg, Russia, July 13-19, 2013. Proceedings, edited by Natasha Sharygina and Helmut Veith, 8044:1–35. Springer: Lecture Notes in Computer Science, 2013.Google Scholar

  • Lowe, Edward Jonathan. “A Modal Version of the Ontological Argument.” In Debating Christian Theism, edited by J. P. Moreland, K. A. Sweis, and C. V. Meister, 61–71. Oxford University Press, 2013.Google Scholar

  • MacLane, Saunders. “Groups, Categories and Duality.” Proceedings of the National Academy of Sciences 34:6 (1948), 263–267.Google Scholar

  • Nipkow, Tobias, Paulson, Lawrence C., and Wenzel, Markus. Isabelle/HOL — a Proof Assistant for Higher-Order Logic. Springer: Lecture Notes in Computer Science, 2002.CrossrefGoogle Scholar

  • Oppenheimer, Paul E., and Zalta, Edward N. “On the Logic of the Ontological Argument.” Philosophical Perspectives 5 (1991), 509–529.Google Scholar

  • Oppenheimer, Paul E., and Zalta, Edward N. “A Computationally-Discovered Simplification of the Ontological Argument.” Australasian Journal of Philosophy 89:2 (2011), 333–49.Google Scholar

  • Oppenheimer, Paul E., and Zalta, Edward N. “Relations Versus Functions at the Foundations of Logic: Type-Theoretic Considerations.” Journal of Logic and Computation 21:2 (2011), 351–74.Google Scholar

  • Pelletier, Francis J., and Zalta, Edward N. “How to Say Goodbye to the Third Man.” Noûs 34:2 (2000), 165–202.Google Scholar

  • Schulz, Stephan. “System Description: E 1.8.” In Logic for Programming, Artificial Intelligence, and Reasoning – 19th International Conference, Lpar-19, Stellenbosch, South Africa, December 14-19, 2013. Proceedings, edited by Kenneth L. McMillan, Aart Middeldorp, and Andrei Voronkov, 8312:735–743. Springer: Lecture Notes in Computer Science, 2013.Google Scholar

  • Scott, Dana S. “Appendix B: Notes in Dana Scott’s Hand.” In Logic and Theism: Arguments for and Against Beliefs in God, edited by J. H. Sobel, 145–146. Cambridge University Press, 1972.Google Scholar

  • Scott, Dana S. “Identity and Existence in Intuitionistic Logic.” In Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9–21, 1977, edited by Michael Fourman, Christopher Mulvey, and Dana Scott, 752:660–696. Springer: Lecture Notes in Mathematics, 1979.Google Scholar

  • Sobel, Jordan H. “Gödel’s Ontological Proof.” In On Being and Saying. Essays for Richard Cartwright, edited by Judith Jarvis Tomson, 241–61. MIT Press, 1987.Google Scholar

  • Sobel, Jordan H. Logic and Theism: Arguments for and Against Beliefs in God. Cambridge University Press, 2004.Google Scholar

  • Steen, Alexander, and Benzmüller, Christoph. “The Higher-Order Prover Leo-III.” In Automated Reasoning. IJCAR 2018, edited by Didier Galmiche, Stephan Schulz, and Roberto Sebastiani, 10900:108–116. Springer: Lecture Notes in Computer Science, 2018.Google Scholar

  • Zalta, Edward N. Abstract Objects: An Introduction to Axiomatic Metaphysics. D. Reidel, 1983.Google Scholar

  • Zalta, Edward N. “Logical and Analytic Truths That Are Not Necessary.” The Journal of Philosophy 85:2 (1988), 57–74.Google Scholar

  • Zalta, Edward N. “Twenty-Five Basic Theorems in Situation and World Theory.” Journal of Philosophical Logic 22:4 (1993), 385–428.CrossrefGoogle Scholar

  • Zalta, Edward N. “Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege’s Grundgesetze in Object Theory.” Journal of Philosophical Logic 28:6 (1999), 619–660.CrossrefGoogle Scholar

About the article

Received: 2019-04-28

Accepted: 2019-06-03

Published Online: 2019-08-23

Published in Print: 2019-01-01

Citation Information: Open Philosophy, Volume 2, Issue 1, Pages 230–251, ISSN (Online) 2543-8875, DOI: https://doi.org/10.1515/opphil-2019-0015.

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© 2019 Daniel Kirchner et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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