Goldbach’s Conjecture and the Uselessness of the Modal Ontological Argument

The ontological argument for God is most commonly associated with the 11th-century Saint Anselm of Canterbury, who defined God as “something than which nothing greater can be thought.” Anselm then argued that if God existed only in thought and not in reality, then a greater being actually could be thought, namely one that existed in reality. And since that would be in contradiction to the original definition, it cannot be true, and so God must necessarily exist in reality. Not many people find this argument convincing, of course, even though it can be difficult to explain exactly why it is so wrong. It seems to rely on some kind of confusion between the idea of real existence and the idea of that idea of real existence, which is hard to keep track of. But one way or another, even if it might be true that God necessarily exists, the mere idea that he necessarily exists simply doesn’t constitute a proof of that idea.

More recently, some philosophers, most notably Alvin Plantinga, have come up with various modal forms of what is basically this same old ontological argument. Modality in this context refers to the ideas of possibility and necessity within the constraints of logic, regardless of what is actually the case. For example, even though it is almost certainly not the case that there is a giant teapot in orbit around the sun beyond the range of our telescopes (Bertrand Russell’s famous example), it is logically possible that such a thing could exist. In the jargon of philosophers, there are possible worlds that contain such a teapot, even though the actual world does not. But it is not logically possible for there to be a spherical cube in such an orbit. There is no possible world where such a logical contradiction exists. In other words, it is possibly true that the teapot exists and possibly true that the teapot does not exist, but it is not possibly true that the spherical cube exists, and it is necessarily true that the spherical cube does not exist. This is the language we need to understand in order to see through the modal ontological argument.

What is the modal ontological argument? I can only present a simplified form and not the specific variations of it, but this will be sufficient to understand the idea. First, it defines God as a being who has the property of necessary existence. That is, unlike the giant teapot, this being does not exist in some possible worlds but not others. His existence or non-existence applies equally to all possible worlds. Unlike the classic ontological argument, the modal form does not insist that this definition itself makes it certain that this being really exists. Instead, it relies on the observation that if some truth is logically necessary in any possible world, then it is true in every possible world. Therefore, if a God matching this definition exists in any possible world, he exists in all possible worlds. The crucial implication we are apparently supposed to draw is that if we consider it even remotely possible that God exists, then we must conclude that God does in fact exist.

This implication is false, of course, but relies on a subtle confusion that some philosophers apparently have trouble putting their fingers on. The confusion is between two entirely distinct notions of possibility that I will call modal possibility and epistemic possibility. The kind of possibility we’ve been discussing so far is modal possibility. The teapot exists in some possible worlds but not others, so it is modally possible that the teapot exists. For example, if it exists in 0.01% of all possible worlds, then in the absence of any further information there would be a 0.01% chance that it exists in the actual world. But this is a completely different concept from epistemic possibility, which is the kind of possibility we are more directly concerned with in real life. To say a thing is epistemically possible is just to say we cannot know for certain whether it is true or false, regardless of any implications that might have on whether it is true in any possible world or every possible world or none.

To illustrate this distinction in terms so simple not even a philosopher could reasonably argue with, consider the case of Goldbach’s conjecture. This is a famous idea proposed by the 18th-century mathematician Christian Goldbach that every even integer greater than two can be expressed as the sum of two primes. Although there are some good reasons for thinking this is probably true, it has never been proved. We simply do not know for certain whether it is true or not. We have to admit there is an epistemic possibility that it may be false. But when we say there is an epistemic possibility that Goldbach’s conjecture may be false, one thing we are absolutely not saying is that there is a modal possibility that it is false. We are not saying that it is true in some possible worlds and false in others. In fact, if it is true in any possible world it is true in all possible worlds, and if it is false in any possible world it is false in all possible worlds. This is because mathematical truths are necessary truths: it is not logically possible that they could be false.

Once we understand this distinction between modal and epistemic possibility, the uselessness of the modal ontological argument becomes clear. Yes, it is true that if a necessary being we call God exists in any possible world, then he exists in all possible worlds. But that is not at all the same thing as saying that if we think it is possible that God might exist, we must therefore conclude that he does in fact exist. Just as we may have a great deal of uncertainty about an untold number of mathematical propositions such as Goldbach’s conjecture, so we may have a great deal of uncertainty about the proposition that God exists, and the fact that such propositions are necessarily true if true in any possible world has absolutely no relevance to our assessment of them.

To be clear, my objection here is not to the idea that God necessarily exists. There may well be good, rational reasons for believing that the causal chain for all existence probably terminates in a necessary being that we could perhaps call God, just as there are good, rational reasons for believing Goldbach’s conjecture probably expresses a necessary truth. But silly little “proofs” of the idea that God necessarily exists, like the ontological argument in both its classic and modal forms, simply do not help make that case.

This page copyright © 2011 Edward A. Morris.  Created June 20, 2011.  Last updated October 21, 2016.

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