A Simplified Variant of Gödel's Ontological Argument

Abstract

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