Gödel's Ontological Proof Revisited

Summary

Gödel's version of the modal ontological argument for the existence of God has been criticized by J. Howard Sobel [5] and modified by C. Anthony Anderson [1]. In the present paper we consider the extent to which Anderson's emendation is defeated by the type of objection first offered by the Monk Gaunilo to St. Anselm's original Ontological Argument. And we try to push the analysis of this Gödelian argument a bit further to bring it into closer agreement with the details of Gödel's own formulation. Finally, we indicate what seems to be the main weakness of this emendation of Gödel's attempted proof.

Gaunilo observed against St. Anselm that his form of argument, if cogent, could be used to “prove” all sorts of unwelcome conclusions - for example, that there is somewhere a perfect island. It would seem to even follow that there are near-perfect, but defective, demi-gods and all matter of other theologically repugnant entities. Gaunilo concluded, reasonably enough, that something must be wrong with the argument.

Kurt Gödel's modern version of the Ontological Argument [12] involves an attempt to complete the details of Leibniz's proof that it is possible that there is a perfect being or a being with all and only “positive” attributes. Given this conclusion, other assumptions about positive properties, and, well, a second-order extension of the modal logic S5, Gödel successfully deduced the actual existence, indeed the necessary existence, of the being having all and only positive attributes. Alas, or “Oh, joy!”, depending on ones’ theological prejudices, J. Howard Sobel showed that Gödel's assumptions lead also to the conclusion that whatever is true is necessarily true. Followers of Spinoza aside, this casts quite considerable doubt on the premisses of the argument. We shall consider here Anderson's emendation which does not suffer from the mentioned defect and which is still recognizably closely related to Gödel's argument.

Here are the assumptions and definitions -the notion of a positive attribute is taken as a primitive by Gödel and in the present version. We hasten to add that the idea is not crystal clear; Gödel's own explanations are extremely terse and somewhat cryptic. A property's being positive is supposed to be a good thing, such properties being characteristic of a completely and necessarily non-defective being.