15. Modal Collapse in Gödel’s Ontological Proof

Kovač, Srećko

doi:10.1515/9783110325881.323

Cited by 9 publications

(8 citation statements)

References 19 publications

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“…As intended by Fitting, modal collapse is not provable anymore, which can be seen in line 25, where Nitpick reports a countermodel with two worlds and one entity.…”

Section: Fitting's Variant Of Gödel's Argumentmentioning

confidence: 92%

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

Benzmüller,

Fuenmayor

2019

Preprint

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Three variants of Kurt Gödel's ontological argument, as proposed by Dana Scott, C. Anthony Anderson and Melvin Fitting, are encoded and rigorously assessed on the computer.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, as our computer-supported formal analysis (conducted in the proof assistant system Isabelle/HOL) reveals. Key to our formal analysis is the utilization of suitably adapted notions of (modal) ultrafilters, and a careful distinction between extensions and intensions of positive properties.

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“…As intended by Fitting, modal collapse is not provable anymore, which can be seen in line 25, where Nitpick reports a countermodel with two worlds and one entity.…”

Section: Fitting's Variant Of Gödel's Argumentmentioning

confidence: 92%

“…One might conclude from it, that the premises of Gödel's argument imply that everything is determined, or alternatively, that there is no free will. Srécko Kovacš[25] argues that modal collapse was eventually intended byGödel.…”

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confidence: 99%

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

Benzmüller,

Fuenmayor

2019

Preprint

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“…Modal collapse (MC), formally notated as ∀s (s → 2s), expresses that "what holds that holds necessarily", which can also be interpreted as "there are no contingent truths" and that "everything is determined". The observation that Gödel's argument implies modal collapse has already been made by Sobel (1987), and Kovač (2012) argues that modal collapse may even have been intended by Gödel. Indeed, the study of modal collapse has been the catalyst for much recent research on the ontological argument.…”

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confidence: 94%

A Simplified Variant of Gödel's Ontological Argument

Benzmüller¹

2022

Preprint

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A simplified variant of Gödel's ontological argument is presented. The simplified argument is valid already in basic modal logics K or KT, it does not suffer from modal collapse, and it avoids the rather complex predicates of essence (Ess.) and necessary existence (NE) as used by Gödel. 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 starting point for these experiments was the computer encoding of Gödel's argument, and then automated reasoning techniques were systematically applied to arrive at the simplified variant presented. The presented work thus exemplifies a fruitful human-computer interaction in computational metaphysics. Whether the presented result increases or decreases the attractiveness and persuasiveness of the ontological argument is a question I would like to pass on to philosophy and theology.Gödel's (1970) ontological argument has attracted significant, albeit controversial, interest among philosophers, logicians and theologians (Sobel, 2004). In this article I present a simplified variant of Gödel's argument that was developed in interaction with the proof assistant system Isabelle/HOL (Nipkow et al., 2002), which is based on classical higherorder logic (Benzmüller & Andrews, 2019). My personal interest in Gödel's argument has been primarily of logical nature. In particular, this interest encompasses the challenge of automating and applying reasoning in quantified modal logics using an universal meta-logical reasoning approach (Benzmüller, 2019) in which (quantified) non-classical logics are semantically embedded in classical higher-order logic. The simplified ontological argument presented below is a side result of this research, which began with a computer encoding of Gödel's argument so that it became amenable to formal analysis and computer-assisted theory simplification experiments; cf. Benzmüller (2020) for more technical details on the most recent series of experiments. The simplified argument selected for presentation in this article has, I believe, the potential to further stimulate the philosophical and theological debate on Gödel's argument, since the simplifications achieved are indeed quite far-reaching:-Only minimal assumptions about the modal logic used are required. The simplified variant presented is indeed valid in the comparatively weak modal logics K or KT, which only use uncontroversial reasoning principles. 1

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“…Symmetry of the accessibility relation r associated with the modal 2-operator can be postulated alternatively in our metalogical framework.5 For more information on modal collapse (in logic S5) consultSobel (1987;,Fitting (2002) andKovač (2012); see also the references therein.…”

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confidence: 99%

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

Benzmüller

2020

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning

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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. Some argument premises are modified, others are dropped, modal collapse is avoided and validity is shown already in weak modal logics K and T. 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. The paper illustrates how modern knowledge representation and reasoning technology for quantified non-classical logics can contribute new knowledge to other disciplines. The contributed material is also well suited to support teaching of non-trivial logic formalisms in classroom.

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15. Modal Collapse in Gödel’s Ontological Proof

Cited by 9 publications

References 19 publications

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

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

A Simplified Variant of Gödel's Ontological Argument

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

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