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Pruss and Rasmussen on the Gödelian Ontological Argument

Pruss and Rasmussen's eighth chapter focuses on the Gödelian ontological argument, which is so-called because it is based on some unpublished notes by the mathematician Kurt Gödel. Pruss has already written extensively on this argument, and it is safe to say that he is recognized as its foremost proponent today. The argument here (in contrast to Pruss's previous treatments, and also to the original Gödel notes, and the influential treatment by Sobel) is presented with the bare minimum of technical apparatus and should be accessible to anyone familiar with the basics of sentential logic. Along the way, the chapter includes interesting discussions of purported conflicts between excellences or perfections. Finally, the chapter defends the interesting and original claim that objections to theism, even if successful, do not rebut the Gödelian ontological argument for a necessary being. Here I'll first give an overview of the argument, then discuss the argument's relation to theism.

Pruss and Rasmussen present the argument as follows (italic capital letters are names of properties; slightly rephrased):

1. If A is positive, then ~A is not positive.


2. If A is positive and A entails B, then B is positive.


3. Existing necessarily (N) is positive.


4. Possibly causing something (C) is positive.


5. Therefore, if A and B are positive, then A&B is positive. (From (1) and (2))


6. Therefore, there is a necessarily existing entity that possibly causes something. (From (3), (4), (5), and S5) (p. 151)

The notion of property entailment employed in (2) is defined as follows: "a property A entails a property B provided that necessarily anything that has A has B" (p. 151). Given this definition, (1) and (2) do indeed entail (5): if A&B were impossible, so that nothing could have both A and B, then A would entail ~B. But then, according to (1) and (2), B would not be positive. This implies, more generally, that possibly there is a being with all positive properties. But according to (3) and (4) existing necessarily and being concrete (causally capable) are both positive properties. According to S5, however, these properties are possessed necessarily if at all. Therefore, there is a necessary concrete thing.

The key difficulty with this argument is to find an interpretation of 'positive' that makes the premises true. Pruss and Rasmussen discuss several possibilities, and argue that there is more than one conception that will do the trick. They seem to prefer the analysis on which a positive property is one whose negation is negative, where a negative properties are those "that limit their possessors, that detract from their excellence" (p. 154).

The key weakness of the argument (as the authors recognize) is that it needs to rely on some kind of logic of 'positivity', but the needed notion of positivity can't just be a directly intuitive one, because attempts to plug in our intuitive notion generates counterexamples. The hope is, by using intuitions about limiting or detracting from excellence we can get a enough grip on the concept of positivity to have intuitions in favor of the premises. As the authors also recognize, the most obvious way in which (1) might turn out to be false is that everything might be necessarily limited.

Here's an alternative form of that worry, which the authors don't mention. Some Neoplatonists and classical theists have thought that the possession of any property at all would necessarily be limiting—to attribute a property is to say that something is this and therefore not that—and used this notion to motivate the strong doctrine of the simplicity of God/the One. The authors give a nominalist variant of the argument (sect. 8.6), formulated in terms of predicates instead of properties, which might survive some versions of this move, since all but the most extreme versions of the view hold that many things can be truly predicated of God/the One.

The nominalist version has an additional worry, though: the idea that some predicate could be truly applied to something that does not exist (i.e., that a subject-predicate sentence with an empty name in subject position could be true) is considerably more plausible than the idea that non-existent things could have properties. This means that the nominalist version needs to put extra heavy weight on the peculiarities of the predicate 'exists necessarily', which would in turn strengthen the position of an objector who wanted to claim that, in light of those peculiarities, the general principles about positive predicates shouldn't apply to that predicate.

In any event, I think the argument shows pretty convincingly that intuitive principles about properties and positivity lead right to the existence of a necessary being with all positive properties. The real question is how far we should trust these intuitive principles in this unusual context. To answer that question, we'd need to work out a general account of the metaphysics and logic of properties, and also to examine the principles carefully for untoward consequences or conflicts with other intuitive principles. In the meantime, the argument certainly makes some contribution to the plausibility of the claim that there is a necessary being.

I turn now to the argument's relation to theism. This book purports to be an examination of the thesis that there is a necessary being, without regard to whether that being might be God. In obedience to this limitation of scope, the previous chapters did not really address the question of whether the necessary being is God. Here, though, the necessary being is said to have all positive properties, and the positivity of properties is connected with intuitions about 'excellence'. The inference to God thus looks too obvious to be ignored.

Pruss and Rasmussen concede that there is a short argument from the conclusion of the Gödelian argument to God, and a reader (especially one who knows the authors' other work) is apt to suspect that the authors endorse this argument. Nevertheless, the authors say, if someone thinks that considerations about evil (or whatever) render the existence of God (as traditionally conceived) absurd, and therefore goes looking for flaws in the argument, it turns out that the weakest point is in establishing divine attributes like uniqueness, omnipotence, omniscience, and moral perfection (sect. 8.7). This, the authors say, is because the assumptions in this stage of the argument, like the assumptions in the earlier stage, rely on intuition. However, the later stage intuitions are more complex and therefore more error-prone than the first-stage intuitions (p. 169). Without these additional conclusions, standard objections to theism like the problem of evil gain no purchase.

I'm unsure about the relative security of the intuitions. The first stage intuitions are highly abstract, and one might be suspicious about highly abstract intuitions, just as one is suspicious of intuitions about complex matters. (Gödel and Pruss are mathematicians, and the stage one intuitions have a simplicity that looks similar to set-theoretic axioms or something, so perhaps these are the kinds of intuitions that seem most secure to mathematicians.) Nevertheless, I do think the point is extremely well-taken that if anyone thinks they have compelling arguments against God as traditionally conceived, it would be a mistake to jump immediately to a rejection of this argument for a necessary being, because the later steps here are open to question just as the earlier ones are. In this way, the Gödelian ontological argument is on the same footing as most cosmological arguments, and the existence of some sort of necessary being is on firmer footing than the existence of the traditional God.