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.

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The possibility premise of the simple existential cosmological argument

In my previous post I showed that even without the plausible S5 or Brouwer axioms, we can move from the possible explanation of the maximal simple existential fact (we called it K there), to the existence of a necessary being. Now I see no reason for thinking such an explanation is impossible, and it seems to me that since we’re merely asking for possibility, that premise should get the benefit of the doubt. Regardless, consider this argument for it

  1. If each fact in a collection of simple existential facts possibly has an explanation, then possibly the conjunction of those facts has an explanation.
  2. Any simple existential fact possibly has an explanation.
  3. Therefore, possibly, K has an explanation.

I can’t think of any counter examples to (1) and (2) seems plausible, at least until someone can give us an example of something for which it is impossible to explain it’s existence or non-existence (that is, not merely something that doesn’t happen to have an explanation, something that couldn’t have an explanation). Good luck with that.

A cosmological argument from simple existential facts

There are loads of different cosmological arguments out there and hopefully someday I’ll be able to write blog posts about some of them. Right now, however, I want to share an interesting version I came up with, thanks to an argument from Alexander Pruss: define a “simple existential fact” to be a true proposition reporting simply the existence or non-existence of a specific being[1,2]. So U doesn’t exist, where U is any specific unicorn and Roland exists, where Roland is me, are simple existential facts of the actual world. Now, let K be the conjunction of all simple existential facts of the actual world that don’t involve necessary beings. So, K only reports the existence or non-existence of beings which possibly fail to exist.

Now assume that possibly, K has an explanation. From this it follows that a necessary being exists. We now show this: Let α denote the actual world[3]. Since possibly, K has an explanation, there is some possible world w, in which K is true and there is some fact q that explains K. Since K involves existential facts, q must involve the causal activity of some being that exists in w[4], call it N. Now either N exists in α or it doesn’t. Assume it doesn’t. Then N doesn’t exist is a conjunct of K. Since K is true at w, it follows that N doesn’t exist in w. But this is a contradiction. Therefore, N exists in α. Is N necessary? Well, assume N isn’t necessary. Then N exists is a conjunct of K, and in w, N explains its own existence. But surely, nothing contingent can explain its own existence. So N is a necessarily existent being.

What’s really nice about this argument is that it doesn’t assume S5 or even the Brouwer axiom. Furthermore, even though I think the Principle of Sufficient Reason is true, to say that K has an explanation doesn’t commit one to the full-blown PSR. In fact, to say that possibly, K has an explanation doesn’t even entail that K actually has an explanation: while N could explain K, N needn’t actually explain K.

Notes

  1. “Specific” has a specific meaning here: rigidly designated. A rigid designator always refers to the same entity and is different from a definite description which changes depending on the current state of affairs. For example, “the president of the United States” is a definite description, whereas “Richard Nixon” is a rigid designator (of course, if there were more than one Richard Nixon we could be talking about, then some more specificity would be needed).
  2. I wonder if I couldn’t simplify this even further by saying “a specific being or class of beings” where class could be something like “unicorns”?
  3. That is, α is the rigid designator for this possible world.
  4. This was discussed very briefly here. It doesn’t seem possible to explain the existence of contingent beings “conceptually”.

Explanations

Recently[1] I’ve been doing some reading on (amoung other things) Leibnizian Cosmological Arguments and the Principle of Sufficient Reason. One thing that’s involved in these arguments is the idea of an “explanation”. We generally have a firm grasp or intuition of whether something is an explanation for some fact or not. Consider the following statements involving explanations:

  1. John sent his children to school A rather than school B because school A has good sports facilities
  2. The kettle is boiling because John turned it on and it was working properly
  3. The kettle is boiling because the heat of the flame is being conducted via the copper bottom of the kettle to the water, increasing the kinetic energy of the water molecules, such that they vibrate so violently that they break the surface tension of the water and are thrown off in the form of steam[2]
  4. We die because the process of natural selection has selected organisms that die over the course of the earth’s evolutionary history

In each of the above cases, the explanation is italicised and the explanandum (the fact to be explained) comes before the “because”. Cases (1) and (2) involve personal or libertarian explanations, case (3) involves a scientific explanation and case (4) an evolutionary explanation (we’ll see later why such evolutionary explanations, including case (4), often fail to explain their explanandum).

Now, as far as I can tell, no-one has been able to give a complete reductive analysis[3] of explanations. Nonetheless, we know a number of things about explanations; two of which I wish to share here.

1. Explanations can be non-entailing

An explanation entails its explanadum if it is impossible for the facts in the explanation to be true and not the facts in the explanadum. Consider example (2) above: it is not possible for the facts John has turned the kettle on and the kettle is working properly to be true without the kettle is boiling following. Simply put, the facts in the explanation entail the facts in the explanadum. We can see another example of this in (3).

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A failed analysis of would-counterfactuals

I was thinking about “would-counterfactuals” the other day and wondering how they’re meant to be understood on a libertarian account of free will that holds to contrary choice as a necessary condition for a free choice. I thought I had come up with some way of giving meaning to statements of the form “Agent S would do action A if put in circumstance C”. However, I realised that I had failed. Nevertheless, it was interesting thinking about metaphysics and mathematics so I thought I’d share it.

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