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

Abstract

<jats:p>Three variants of Kurt Gödel's ontological argument, 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. This has been revealed in the computer-supported formal analysis presented in this article. 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.</jats:p>

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

“…However, modal collapse is still implied even in the weak logic KB. 5 Own prior work recently showed that different modal ultrafilter properties can be deduced from Gödel's premises (Benzmüller and Fuenmayor 2020). These insights are key to the argument simplifications developed and studied in this paper: If Gödel's premises entail that positive properties form a modal ultrafilter, then why not turning things around, and start out with an axiom U1 postulating ultrafilter properties for P?…”

“…A modal filter φ, see lines 14-17, is required to 1. be large: U ∈ φ, where U denotes the full set of γ-type objects we start with, Own prior work (Benzmüller and Fuenmayor 2020) studied two different notions of modal ultrafilter (termed γand δ-ultrafilter), which are defined on intensions and extensions of properties, respectively. This distinction is not needed in this paper; what we call modal ultrafilter here corresponds to our prior notion of γ-ultrafilter.…”

“…Variants of Kurt Gödel's (1970), resp. Dana Scott's (1972), modal ontological argument have previously been studied and verified on the computer by Benzmüller and Woltzenlogel Paleo (2014; and Benzmüller and Fuenmayor (2020), and some previously unknown issues were revealed in these works, 1 and it was shown that logic KB, instead of S5, is already sufficient to derive from Gödel's axioms that a supreme being necessarily exists.…”

A Simplified Variant of Gödel's Ontological Argument

“…For example, variants of Gödel's argument that avoid modal collapse have been presented by Anderson (1990Anderson ( , 1996 and Fitting (2002), among others, cf. also the formal verification and comparison of these works by Benzmüller and Fuenmayor (2020). In the following, however, it is shown that modal collapse can in fact be avoided by much simpler means.…”

Symbolic Ai and Gödel's Ontological Argument

“…Principle ultrafilters have a smallest element, which satisfies all properties of the ultrafilter; such a smallest element in our context is associated with the concept/property of a God‐like being. It is relevant to remark that the notion of filter/ultrafilter needed to be adapted for the modal context of the ontological argument; for more details, see the discussion in Benzmüller and Fuenmayor (2020); in the latter work is has been shown how seemingly rather different variants of the modal ontological argument are nevertheless closely related from the perspective of the ultrafilter structures they axiomatize.…”