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A Simplified Variant of Gödel's Ontological Argument [article]

Christoph Benzmüller
2022 arXiv   pre-print
The presented work thus exemplifies a fruitful human-computer interaction in computational metaphysics.  ...  The variant presented has been obtained as a side result of a series of theory simplification experiments conducted in interaction with a modern proof assistant system.  ...  The Isabelle/HOL sources of the conducted verification studies are presented in Figs. 1-4 in the Appendix.  ... 
arXiv:2202.06264v2 fatcat:cbum3ml4hfbqvgz27hougaosga

Computer Science and Metaphysics: A Cross-Fertilization

Daniel Kirchner, Christoph Benzmüller, Edward N. Zalta
2019 Open Philosophy  
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  ...  Fitting, "Types, Tableaus, and Gödel's God".  ...  In order to construct an SSE, one has to represent the general model structure of AOT in Isabelle/ HOL.  ... 
doi:10.1515/opphil-2019-0015 fatcat:wljwgnhj2zffxakqpghk4xdxzm

Computer Science and Metaphysics: A Cross-Fertilization [article]

Daniel Kirchner, Christoph Benzmüller, Edward N. Zalta
2019 arXiv   pre-print
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  ...  Anderson & Gettings, "Gödel's ontological proof revisited". 47 Fitting, "Types, Tableaus, and Gödel's God". 48 Fuenmayor & Benzmüller, "Types, Tableaus and Gödel's God in Isabelle/HOL". cost of introducing  ...  Fuenmayor & Benzmüller, "Types, Tableaus and Gödel's God in Isabelle/HOL".51 Benzmüller & Fuenmayor, "Can computers help to sharpen our understanding of ontological arguments?".  ... 
arXiv:1905.00787v3 fatcat:slgk2xqljva7jdgapn6nzkaile

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

David Fuenmayor, Christoph Benzmüller
2017 Lecture Notes in Computer Science  
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  ...  By employing our embedding of IHOML in Isabelle/HOL, we could not only verify Fitting's results, but also guarantee consistency of axioms.  ... 
doi:10.1007/978-3-319-67190-1_9 fatcat:tiflccsgsndsrgn5cpvweo4tdq

Computer-supported Analysis of Positive Properties, Ultrafilters and Modal Collapse in Variants of Gödel's Ontological Argument

Christoph Benzmüller, David Fuenmayor
2020 Bulletin of the Section of Logic  
In contrast to Scott's version of Gödel's argument the two variants contributed by Anderson and Fitting avoid modal collapse.  ...  Although they appear quite different on a cursory reading they are in fact closely related. This has been revealed in the computer-supported formal analysis presented in this article.  ...  As already mentioned, the technical results presented in this article have been summarized at abstract level in a joint article with Daniel Kirchner and Ed Zalta.  ... 
doi:10.18778/0138-0680.2020.08 fatcat:atxpcxhcqfd3fhllaj6xfwkcey

Computer-supported Analysis of Positive Properties, Ultrafilters and Modal Collapse in Variants of Gödel's Ontological Argument [article]

Christoph Benzmüller, David Fuenmayor
2020 arXiv   pre-print
Although they appear quite different on a cursory reading they are in fact closely related. This has been revealed in the computer-supported formal analysis presented in this article.  ...  In contrast to Scott's version of G\"odel's argument the two variants contributed by Anderson and Fitting avoid modal collapse.  ...  As already mentioned, the technical results presented in this article have been summarized at abstract level in a joint article with Daniel Kirchner and Ed Zalta.  ... 
arXiv:1910.08955v2 fatcat:gbpqs3hvgrfbpmx3ybuvkq2oky

Universal (meta-)logical reasoning: Recent successes

Christoph Benzmüller
2019 Science of Computer Programming  
A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.  ...  Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal  ...  In the running example discussed below the Isabelle/HOL proof assistant is used.  ... 
doi:10.1016/j.scico.2018.10.008 fatcat:xim3wn4jfbcsvl2hfdn26p6f4y

A (Simplified) Supreme Being Necessarily Exists, says the Computer: Computationally Explored Variants of Gödel's Ontological Argument

Christoph Benzmüller
2020 Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning  
Key to the gained simplifications of Gödel's original theory is the exploitation of a link to the notions of filter and ultrafilter in topology.  ...  An approach to universal (meta-)logical reasoning in classical higher-order logic is employed to explore and study simplifications of Kurt Gödel's modal ontological argument.  ...  Acknowledgements I am grateful to friends and colleagues who have supported this line of research in the past and/or directly contributed with comments and suggestions, including (in alphabetical order  ... 
doi:10.24963/kr.2020/80 dblp:conf/kr/Benzmuller20 fatcat:5f6a5rlgtzdd5jjcjnfleqyn2e

A (Simplified) Supreme Being Necessarily Exists, says the Computer: Computationally Explored Variants of Gödel's Ontological Argument [article]

Christoph Benzmüller
2020 arXiv   pre-print
The contributed material is also well suited to support teaching of non-trivial logic formalisms in classroom.  ...  Some argument premises are modified, others are dropped, modal collapse is avoided and validity is shown already in weak modal logics K and T.  ...  Acknowledgements I am grateful to friends and colleagues who have supported this line of research in the past and/or directly contributed with comments and suggestions, including (in alphabetical order  ... 
arXiv:2001.04701v10 fatcat:biplwnjbd5fetmfdvt3pul2664

Recent Successes with a Meta-Logical Approach to Universal Logical Reasoning (Extended Abstract) [chapter]

Christoph Benzmüller
2017 Lecture Notes in Computer Science  
argument, while the slightly modified variant by Dana Scott [32] was verified in the interactive proof assistants Isabelle/HOL [30] and Coq [20] .  ...  In the course of these experiments (cf. [17,15,16,18,19,14] for details), my prover LEO-II [10] detected an previously unnoticed inconsistency in Kurt Gödel's [26] prominent variant of the ontological  ...  The PLM has meanwhile been successfully encoded in Isabelle/HOL by my student Daniel Kirchner [27] .  ... 
doi:10.1007/978-3-319-70848-5_2 fatcat:5dhzxs57mzbi7ju2h2nrpkejoa

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

C. Benzmüller, L. Weber, B. Woltzenlogel Paleo
2017 Logica Universalis  
The particular focus here is on the ontological argument for the existence of God.  ...  For example, the theorem prover Leo-II [2] detected an inconsistency in Gödel's original script [12] of the argument, which was undetected by philosophers before.  ...  In our experiments we have utilized the proof assistant Isabelle/HOL [18] together with the external automated higher-order provers Leo-II [2] and Satallax [8] , called through Sledgehammer.  ... 
doi:10.1007/s11787-017-0160-9 fatcat:flqsjus6v5axtkj2neqwnr57si

Universal Reasoning, Rational Argumentation and Human-Machine Interaction [article]

Christoph Benzmüller
2017 arXiv   pre-print
In the future, universal logic reasoning in combination with appropriate, controlled forms of rational argumentation may serve as a communication layer between humans and intelligent machines.  ...  Universal logic reasoning in turn, as envisioned already by Leibniz, may support the rigorous formalisation and deep logical analysis of rational arguments within machines.  ...  This includes (in alphabetical order): Larry Paulson, Dana Scott, Geoff Sutcliffe, Alexander Steen, Max Wisniewski, Bruno Woltzenlogel-Paleo and Edward Zalta.  ... 
arXiv:1703.09620v1 fatcat:4kybqxjisvdnvfrrsvpxabcomi

The Higher-Order Prover Leo-II

Christoph Benzmüller, Nik Sultana, Lawrence C. Paulson, Frank Theiß
2015 Journal of automated reasoning  
Recent progress in this direction is reported for the Isabelle/HOL system.  ...  Leo-II's proof certificates are exploited in the prover's recent integration with Isabelle/HOL, through which Leo-II proofs can now be transformed and verified (Sect. 8).  ...  Arnaud Fietzke contributed to code in early phases of the project, and Chad Brown, who shared some Satallax code with Leo-II, has always been a good collaborator of the project.  ... 
doi:10.1007/s10817-015-9348-y pmid:30174358 pmcid:PMC6109767 fatcat:35dv765knbdsjnvfi6ihgbjhha

MECHANIZING PRINCIPIA LOGICO-METAPHYSICA IN FUNCTIONAL TYPE THEORY

DANIEL KIRCHNER, CHRISTOPH BENZMÜLLER, EDWARD N. ZALTA
2019 The Review of Symbolic Logic  
This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL.  ...  This approach enables the fruitful reuse of state-of-the-art higherorder proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT  ...  In previous applications of the SSE approach, this issue didn't arise. For example, in the context of the analysis of Gödel's modal ontological argument for the existence of God (cf.  ... 
doi:10.1017/s1755020319000297 fatcat:ka5aenuwabgsbnwzrbwve5hsw4

A Deontic Logic Reasoning Infrastructure [chapter]

Christoph Benzmüller, Xavier Parent, Leendert van der Torre
2018 Lecture Notes in Computer Science  
, assessed and compared in experimental ethics application studies.  ...  Our motivation is the development, study and provision of legal and moral reasoning competencies in future intelligent machines.  ...  In fact, our infrastructure already now Fig. 1 . 1 GDPR example scenario in Isabelle/HOL.  ... 
doi:10.1007/978-3-319-94418-0_6 fatcat:w2fcjeegu5eezclyciv6talrii
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