Logic of the Ontological Argument

TL;DR

This paper examines the logical foundations of Gödel's ontological argument, highlighting issues with the application of necessitation in second-order modal logic and proposing conditions to preserve the S5 logic.

Contribution

It analyzes the impact of necessitation on the logic of the ontological argument and suggests that certain axioms must be assumed in necessitated form to maintain S5 consistency.

Findings

Necessitation application can compromise the consequence relation.

Assuming axioms in necessitated form preserves S5 logic.

The argument's logical structure depends on how necessitation is applied.

Abstract

In his ontological argument G\"{o}del says nothing about its underlying logic. The argument is modal and at least of second-order and since S5 axiom is used so it is widely accepted that the logic of the argument is the S5 second-order modal logic. However, there is a step in the proof in which G\"{o}del applies the necessitation rule on the assumptions of the argument. This is repeated by all of his followers. This application of the necessitation rule can seriously harm the consequence relation of the logic of the ontological argument. It seems that the only way to preserve the modal logic S5 for the ontological argument is to assume some of its axioms in the necessitated form.