Applying Second-Order Quantifier Elimination in Inspecting G\"odel's Ontological Proof

TL;DR

This paper uses automated first-order logic tools and second-order quantifier elimination to analyze G"odel's ontological proof, revealing new insights and practical application potentials.

Contribution

It introduces a modeling approach with macros and automated reasoning to analyze G"odel's proof, highlighting previously unnoticed aspects and demonstrating practical elimination applications.

Findings

Uncovered new details of G"odel's proof

Demonstrated practical second-order quantifier elimination

Provided potential benchmarks for elimination tasks

Abstract

In recent years, G\"odel's ontological proof and variations of it were formalized and analyzed with automated tools in various ways. We supplement these analyses with a modeling in an automated environment based on first-order logic extended by predicate quantification. Formula macros are used to structure complex formulas and tasks. The analysis is presented as a generated type-set document where informal explanations are interspersed with pretty-printed formulas and outputs of reasoners for first-order theorem proving and second-order quantifier elimination. Previously unnoticed or obscured aspects and details of G\"odel's proof become apparent. Practical application possibilities of second-order quantifier elimination are shown and the encountered elimination tasks may serve as benchmarks.