Computer-supported Analysis of Positive Properties, Ultrafilters and Modal Collapse in Variants of G\"odel's Ontological Argument

TL;DR

This paper uses computer-assisted formal analysis to evaluate three variants of G"odel's ontological argument, revealing their structural similarities and differences, especially regarding modal collapse and the role of ultrafilters.

Contribution

It introduces a formal, computer-supported method to analyze variants of G"odel's ontological argument, highlighting the differences in modal collapse and the use of ultrafilters.

Findings

Anderson and Fitting variants avoid modal collapse unlike Scott's version

The variants are closely related despite superficial differences

Ultrafilters and distinctions between extensions and intensions are key to analysis

Abstract

Three variants of Kurt G\"odel'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 G\"odel'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.