A Gödelian ontological argument

I’ve never really had a nice relationship with the ontological argument from Anselm. When I first heard of it, it seemed strange that existence would be greater than non-existence, so I pushed it aside. About 2 years later, I realised that existence could maybe be bootstrapped from other properties, like power. But by then I had come to realise the distinction between epistemology and ontology, and struggled to believe that this argument wasn’t confusing the two at some point. That’s where I’m at at the moment: existence-in-mind just doesn’t seem comparable to existence-in-reality in the way that’s needed for the argument to work. Maybe I’m wrong, but that’s where I’m at[1].

Many people talk about Anselm’s ontological argument as the ontological argument. But, like many theistic arguments (and arguments in general, I suppose), to call it the ontological argument is a bit misleading. There are a number of ontological arguments out there, and Anselm’s one is but one of them. Descartes had another ontological argument which Leibniz worked on a bit, and in the 20th century we’ve had modal ontological arguments coming from Norman Malcolm, Charles Hartshorne, and Alvin Plantinga. Another “class” of ontological arguments are the so-called “Gödelian” ontological arguments. Kurt Gödel, the famous mathematician of Gödel’s Incompleteness Theorems, developed his argument using the primitive idea of a “positive property”. The arguments that follow this approach, like Gödel’s before them, are developed as formal axiomatic system with a theorem at the end that says that there is a God-like being who exists. Jordan Sobel showed, in 1987, that Gödel’s axioms also imply that every true proposition is necessarily true. This argument from Sobel is called the “modal collapse argument”, and it shows that Gödel’s argument is unsound. However, since then, there have been a number of Gödelian ontological arguments which have been formulated so as not to fall prey to the modal collapse argument. These have come from Curtis Anderson, Allen Hazen, Robert Koons, and Petr Hajek, to name four. And, then there’s the recent “Modal Perfection Argument” from Robert Maydole.

Of prime importance to this blog post is yet another Gödelian ontological argument formulated by Alexander Pruss[2]. While I’m not convinced by Anselm’s, Descartes, and many of the other ontological arguments, this one does certainly seem plausible to me. I’ll sketch it briefly in this post.

Pruss’ Gödelian ontological argument

As with all Gödelian ontological arguments, we need a concept of a “positive” property. We’ll also define the analogous concept of a “negative” property and throughout we’ll note equivalences between these two (wherever you see the phrase “if and only if”)[3]. There are actually a number of candidates for how we might understand these ideas, but I think it’s sufficient to go with the usual great-making theme (even though I don’t agree with Anselm’s argument, I still like the ideas in it). So we’ll stipulate that a property is positive if it is better to have than not to have, and a property is negative if it is worse to have than not. So positive properties make a being greater and negative properties make a being lesser, limiting it somehow. Further, we note that a property is positive if and only if its negation is negative and that there are properties that are neither positive nor negative.

On these stipulations, it’s plausible that, omniscience, omnipotence, omnibenevolence, and omnipresence are positive properties. In fact, mere omnibenevolence might actually be a negative property, since it is greater to be the paradigm of moral goodness than to merely exemplify moral goodness. For the sake of simplicity of presentation, though, I’ll use omnibenevolence to describe the greatest conceivable property involving moral attributes, be it moral perfection or being the paradigm of moral goodness. Nevertheless, I think we get the point.

Given this idea of positivity of properties, we can state the argument. We start with the following two formal axioms (as they’re called):

F1. If A is a positive property, then ~A is not a positive property.

F2. If A and B are properties, A is positive, and A entails B, then B is positive.

These pretty much follow by definition, as far as I can tell. From these two axioms, we already have the following Lemma:

Lemma 1 Any two positive properties are compossible. That is, for any two positive properties, there is a possible world in which a being exemplifies both.

Proof Assume, to the contrary, that there are two positive properties, A and B, that are not compossible. Then A entails ~B. By F2, this means that ~B is a positive property, and by F1 this means that B is not positive, which is a contradiction. Thus A and B are compossible.

So far so good. Now, we stipulate the following non-formal axiom:

N1. Necessary existence is positive.

This is an interesting axiom. First, it’s important to note that it doesn’t assume that existence is a property. Once something exists, it has a property of the kind of existence it has (ie. contingent or necessary), but the existence itself is not what we’re concerned with here. Second, to some N1 is intuitive in and of itself, but not to me. I need additional reasons for thinking N1 to be true. Off the top of my head I can think of two: aseity and being the paradigm of moral goodness. Aseity, on Christian theology, involves God’s self-existence and independence from anything else. He doesn’t depend on anything but himself for his existence and everything else in creation depends on him. Clearly aseity is positive, and aseity entails necessity, so by F2 necessity must be positive too. Another way we might get there is an insight from the Euthryphro objection to Divine Command Theory. In the objection, it is assumed (quite plausibly) that, if something were good, then it would be necessarily so. Now if something is the paradigm of moral value/goodness (which is a positive property), then is seems that it is the only thing that could be the paradigm and so, since good things are necessarily good, the paradigm would need to be necessary. So something being the paradigm of moral value entails that it exists necessarily. We’ll come back to N1 later, when we’re talking about the gap problem.

Right, so we’ve got the positive property that’s going to enable us to move our God-like being from being merely possible, to it being actual (if you haven’t seen something like this before, we’ll do this in a bit and we require N1 to do it). But before we get there, Pruss introduces another class of properties: strongly positive and strongly negative properties. A property is strongly positive if having it essentially is positive, and strongly negative if possibly having it is negative (note: a property P is an essential property of B if B cannot fail to exemplify it. That is, every possible world in which B exists, B exemplifies P. This captures the idea of the property belonging to the essence of B). Again, note that a property is strongly positive if and only if its negation is strongly negative. The examples I gave earlier are plausibly strongly positive properties: it is better to be essentially omniscient than merely accidentally omniscient, for instance. Moreover, as I noted above, being the paradigm of moral goodness seems like an essential property anyway. So, putting this into a non-formal axiom we have:

N2. Essential omniscience, omnipotence and omnibenevolence are positive properties.

There’s one more classification of properties we need: “uniqualizing” properties. A property is uniqualizing if there can be at most one being that has that property. Pruss gives the following example: being the tallest woman is uniqualizing. So if A is uniqualizing and both x and y have A, then x = y. Analogously, we can talk about “nearly universal” properties, which are properties that must be had by all except at most one being. Again, a property is uniqualizing if and only if its negation is nearly universal. Our next non-formal axiom links uniqualizing properties with the previous types of properties:

N3. There is at least one uniqualizing strongly positive property.

In fact, we can give examples: being the paradigm of moral goodness, being greater than every other being, being the creator of every other being, aseity, and omnipotence. That last one might come as a surprise. Pruss explains as follows:

Omnipotence requires perfect freedom and an efficacious will. But there cannot be two beings with perfect freedom and an efficacious will. For if they are perfectly free, they will be able to will incompatible propositions to be true, and then one of their wills shall have to fail to be efficacious.

[If you’re not too interested in the axioms of modal logic and their applicability to metaphysical possibility, you can skip this bit. I’ve made the font lighter so you know where to carry on reading from below]

At this point we can state the theorem that we want to prove, but before we can fully appreciate the proof of it we need some background regarding the modal axiom (5) in S5. S5 is a system of axioms about modality and, as far as I can tell, many philosophers think that it captures the modality of metaphysical possibility (ie. the relationships between possible worlds). We’ll talk about this more next time; for now I just want to speak about the axiom (5). The axiom, in everyday language, states that what could be the case is not dependent on what actually is the case. Put another way, if both A and B are possible, then were A to be actual, B would still be possible. Put another way, no matter how things could have gone, if things could have gone a specific way, they always could have gone that way. If we’re thinking about possible worlds, (5) says that there is only one collection of possible worlds. Of these the first one seems most intuitively obvious to me: surely what could happen is logically prior to what does happen. Anyway, (5) can be written in symbolic logic as,

5. ◊p→□◊p,

where p is any proposition. Now, since (5) is true it follows that if some proposition is possibly necessarily true, then it is necessarily true:

◊□p →◊□~~p (law of double-negation)

         →◊~◊~p (since □p↔~◊~p)

         →~□◊~p (since □p↔~◊~p)

         →~◊~p (by (5))

         →□p (since □p↔~◊~p).

Ok, given these axioms, we have the following theorem:

Theorem 2 There exists a necessary, essentially omniscient, omnipotent, omnibenevolent being.

Proof By N1, N3 and Lemma 1 it follows that there is possibly a necessarily existing being G which has some property A essentially, such that A is uniqualizing. By S5, G necessarily exists. By N2, N3 and Lemma 1 there is a being who possibly has A and is essentially omniscient. But since A is uniqualizing and G is necessary, it follows that it is impossible for any other being to have A and not be G (in other words, in the possible world in which this omniscient being exists, the only being who has A is G). Thus G is essentially omniscient. Similarly, G is essentially omnipotent and essentially omnibenevolent. Thus there exists a necessary, essentially omniscient, omnipotent, omnibenevolent being.

That’s a pretty cool result. Also, this theorem can easily be extended to include all strongly positive properties that we’ve mentioned.


  1. A number of people will note that the problem with Anselm’s argument is that it makes existence a property. I don’t know what to think about this, but I remember reading a critique of this objection and finding it compelling. Perhaps I should reread that.
  2. He’s formulated and defended this argument in two papers: “A Gödelian Ontological Argument Improved” in Religious Studies (2009) and “A Gödelian Ontological Improved Even More” in M. Szatkowski (ed.), Ontological Proofs Today (2012).
  3. There’s a good reason for this: one of the criticisms of Pruss’ original paper (which he attended to in his second) was that the idea of positive property led to some counter-intuitive facts. For example, since knowing 2+2=4 is positive, it follows that knowing 2+2=4 or being foolish is positive. This seems somewhat strange. However, because we require positive and negative properties to be such that they’re related via equivalence, it follows that knowing 2+2=4 or being foolish is positive if and only if not knowing that 2+2=4 and not being foolish is negative. The fact that this is a negative property isn’t counter-intuitive at all and so mitigates any counter-intuitiveness in the positive case (since they’re equivalent statements). At the end of the day, complementing the idea of positive properties with the idea of negative properties makes the whole endeavour less gerrymandered. Of particular importance, though we’ll do everything in this post using positive, strongly positive or uniqualizing properties, the entire process can be done with negative, strongly negative or nearly universal properties instead. That’s why each time we introduce these ideas, we are keen to note their complementary version and the relevant equivalences. It’s left as an exercise for the reader to state the axioms and results using only negative properties.

2 thoughts on “A Gödelian ontological argument

  1. Hey Roland, these sorts of things are way out of my league unfortunately, but I do like to try and understand them and I find them interesting. I think the most damning critique of Anselm as I understand it is that he makes ‘existence’ a positive attribute within his vague category of perfection (which means that he is able to move from ‘possible world’ to ‘necessary existence’). It seems to me that existence has no business in this category (for example, it is not to Hitler’s credit that he at least existed, rather than merely being an ‘evil concept’; his existence is not a positive attribute; I don’t think it’s even arguable that existence is an intensifier of good an evil such that it can be a beneficial attribute of perfect things but not evil things).

    This category mistake (if that is not misusing Kant’s idea) means that we should be able to posit the necessary existence of any concept of existence, such as the perfect island.

    I haven’t investigated any of the newer ones that you have, but in spite of the sophistication of philosophical presentations, I still suspect that there is too much smuggling of incompatible ideas such as perfection and existence through back doors. Even our concepts of omnipotence, omnibeneficience etc. are easy tags for difficult grey areas.

    I suspect that this argument is always going to seem like sleight of hand to skeptics. It always does to me. I must give it a better read through though.

    • Hey Jordan, I didn’t realise that you read my blog 😛

      If I understand you correctly you’re talking about the objection from Kant that I mentioned in the first footnote in the post, namely that existence is not a property/predicate. Like I said in the footnote, I’m not actually too convinced that existence isn’t a property (I mean, people have responded to Kant), but even granting Kant his point, I don’t think it’s a problem for Anselm (and thus, not the main reason I dislike the Anselmian argument). Assuming we can compare beings who exist in the mind with beings who exist in reality (which *is* the assumption that I have a problem with) it seems quite natural to say that the being in reality can be greater than the being who is merely in the mind. Why? Well, for one, there’s potency: the being who actually exists can actually affect things in reality, whereas the being who merely exists in the mind cannot do anything, it can only be understood to do something. This is what I was talking about when I mentioned “bootstrapping” originally. This approach to the argument doesn’t actually treat existence as a property, it merely notes that something needs to exist to have other great-making properties.

      Of course it’s not always clear that a given being would be greater if it existed in reality. The Hitler case is a good example: the Hitler-in-reality is surely more potent than the Hitler-in-mind, but he’s also more evil (the Hitler-in-mind can only be understood to be evil, instead of actually being evil). So while, for a given property, the Hitler-in-reality is greater, it might be that on the whole, he isn’t. In this case, however, I’m using Hitler to describe a being that is by nature evil. It seems unclear that a being who has merely the possibility of doing evil things is less in reality than not. So if we were actually talking about Adolf Hitler, then I’d say the Adolf-in-reality is greater than the Adolf-in-mind. Again, it could be that existence enables some incommeasurable good, which would always make existence-in-reality greater than existence-in-mind. But these are mere details, the point is Anselm’s argument needn’t require that we say *existence* is great-making, but that it’s required for some other great-making properties.

      Also, as far as I’m aware, Anselm didn’t talk about *necessary existence* as a property, and that’s not what Kant had a problem with (for “necessary” describes the kind of existence an already existent thing has, and is thus a property, but, on Kant’s view anyway, existence isn’t). In Anselm’s argument, necessity came as a corollary, for the argument could be run in any possible world and therefore the being exists in every possible world. He never really had modality in mind when he formulated it originally.

      The modern ontological arguments (that is, the modal, GÖdelian, and modal perfection) do not treat existence as a property, and they don’t deal with existence-in-mind. They start and finish with metaphysical possibilities. I presume some are susceptible to parodies, but some aren’t. I don’t think the GÖdelian argument I’ve gone through here is susceptible to parodies (this is actually the reason we defined both negative and positive properties and kept commenting on their relationship, I should probably make a footnote about that).

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