From the archives: Plantinga’s ontological argument, take three

Plantinga’s ontological argument is an annoying argument in that it relies heavily on jargon that’s unintelligible to the average person, including using “possible” in a somewhat unintuitive way. As a result, even though I think it’s a terrible argument, someone without training in the relevant philosophy is likely to make wrong guesses about what’s wrong with it. Thus, I present an attempted explanation of the problems with the argument that I wrote in December of last year:

Rather than respond directly to comments on my previous post,I’m rewriting it, taking the issue “from the top” so to speak. The last four paragraphs are what I’d most like people to read and comment on, but the earlier parts are changed quite a bit too by adding a discussion of William Lane Craig.

I’ll begin with what William Lane Craig says about Plantinga’s argument in the third edition of his book Reasonable Faith (just because I’m writing this with a view towards including it in the book, which will have a whole chapter on Craig):

Now in his version of the argument, Plantinga conceives of God as a being which is “maximally excellent” in every possible world. Plantinga takes maximal excellence to entail such excellent-making properties as omniscience, omnipotence, and moral perfection. A being which has maximal excellence in every possible world would have what Plantinga calls “maximal greatness”…1) It is possible that a maximally great being exists.
2) If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3) If a maximally great being exists in some possible world, then it exists in every possible world.
4) If a maximally great being exists in every possible world, then it exists in the actual world.
5) If a maximally great being exists in the actual world, then a maximally great being exists.
6) Therefore, a maximally great being exists.

In my view, the jargon Plantinga uses in this argument is needlessly confusing. Instead of talking about “maximal excellence” and “maximal greatness,” he could just define God as “an omnipotent, omniscient, morally perfect, necessarily existent being.” Similarly, there’s no need to invoke possible world talk, we can state the argument talking just about possibility and necessity (where “necessary” just means “couldn’t possibly be otherwise”). That condenses the argument to:

1) It is possible that God exists. (Craig’s premise 1)
2) If it is possible that God exists, then it is necessary that God exists. (Craig’s premise 3)
3) If it is necessary that God exists, God exists. (Craig’s premise 4)
4) Therefore, God exists.

Working backwards: (3) is obvious once you know what “necessary” means. If God couldn’t possibly have not existed, then of course God exists. Premise (2) is supposed to come from a combination of two things: the definition of God as a being who exists necessarily, and the S5 axioms of modal logic.

Some people are going to object about simply defining God as a necessary being. However, among atheist philosophers, the attitude generally seems to be, “Oh, theists can define ‘God’ however they want. For example, if they want to define ‘God’ as ‘the greatest possible being,’ they can do that.”

Now, after granting theists their definition of God as a starting point, some atheist philosophers, Michael Martin for example, might argue that the concept of a greatest possible being is incoherent, or incompatible with other things commonly believed of God, but the point of that sort of attack isn’t to show that it’s wrong to define God that way, it’s to show that if we define God that way, then God can’t possibly exist.

In fact, in the philosophy world it seems to be generally regarded as OK to just announce that you’re going to use some word to mean such-and-such. As long as it isn’t needlessly confusing, and you don’t equivocate between two different meanings of a word, you can define words however you want. So I think most philosophers are going to give Plantinga the OK on the first piece of support for premise two.

Now the other piece: in a system of formal logic, axioms are things you’re allowed to just assume when working within the system. So for example, when I was in graduate school, the homework problems for my formal logic class generally involved proving some theorem or other. If we were working within a particular system of logic, we were allowed to have a step in our proofs be writing down an axiom with a note indicating “this is an axiom.”

The S5 axioms for modal logic–the logic of possibility and necessity–have the important consequence that if something is possibly necessarily true is necessarily true. This, when combined with the definition of God as necessarily existing, is where the otherwise bizarre-looking premise (2) of my restatement of Plantinga’s argument comes from. Philosophers have disagreed on which axioms are the right axioms to use when doing modal logic. And I don’t know of any decisive argument to show that the S5 axioms are the right axioms.

However, my understanding is that most philosophers nowadays accept the S5 axioms, and Plantinga’s key claim seems plausible enough to me. To say that what’s possibly necessarily true is necessarily true is to say that it makes no sense to think the following: “well, this could be false, but it could also be such that it couldn’t possibly be false.” And I don’t see how that makes any sense, if we’re talking about genuine possibility (what Plantinga calls “broadly logical possibility”) rather than possibility-for-all-I-know (often called “epistemic possibility,” from the Greek word for knowledge).

At this point, Plantinga’s argument may look pretty good. He’s got the first two key claims, and of course it’s at least possible that God exists, right? Not so fast. Once you accept the S5 axioms, it becomes completely crazy to think you can just assume things are possible. Or at least, it becomes completely crazy to assume things are possibly necessary. This is because S5 allows for Plantinga-style arguments for any purported necessary truth. The fact that the argument involves God isn’t actually an important feature of the argument.

So for example, philosophers generally claim that mathematical truths are, if true, necessarily true. Two plus two not only equals four, it could not possibly equal anything other than four. Because of this, if you accept S5 and also are willing to just assume a given mathematical claim is possibly true, you can “prove” that mathematical claim through a Plantinga-style argument.

For example: the Goldbach conjecture is an oft-cited example of a mathematical claim that nobody has been able to prove or disprove. If you accept the S5 axioms, and also assume that the Goldbach conjecture is possibly true, you can reason like this: “Possibly the Goldbach conjecture is true. But it is if true, necessarily true. So possibly the Goldbach conjecture is necessarily true. Therefore, by S5, the Goldbach conjecture is true!” Obviously, it is absurd to think you can prove the Goldbach conjecture that way.

It’s important to be clear on where the absurdity comes from. It does not come from the S5 axioms alone, nor does it come solely from assuming that the Goldbach conjecture is possibly true. Rather, it comes from the combination of those two things. S5 and taking possible necessities for granted are two things that do not go well together. My inclination is to accept S5, but reject assuming such possibilities. (Note that you could claim that while it’s not okay to assume mathematical claims are possible, but is okay to assume God is possible. But why would you think that?)

It’s also important to emphasize the distinction between genuine (“broadly logical”) possibility and (“epistemic”) possibility-for-all-we-know. I suspect that’s where part of the appeal of just assuming possibilities comes from. The Goldbach conjecture might be true for all we know, but it might be false for all we know. In that sense, both are possibilities. But, according to the conventional wisdom about mathematics, if the Goldbach conjecture turns out to be true, there was never a genuine possibility of it being false. There was only “possibility for all we knew.”

Craig does not deal with the Goldbach Conjecture objection, but he does deal with the objection that you might use a Plantinga-style argument to prove the existence of “a necessarily existent lion.” In response, Craig argues that “does not seem even remotely incoherent,” which means we should think it is possible that God exists. In contrast:

The idea of something like a necessarily existent lion also seems incoherent. For as a necessary being, such a beast would have to exist in every possible world we can conceive. But any animal which could exist in a possible world in which the universe is composed wholly of a singularity of infinite spacetime curvature, density, and temperature just is not a lion.

But it makes just as much sense to argue that we can conceive of a world containing only physical objects is not a world with a god in it, therefore it is possible that God does not exist. This, incidentally, entails that if God is defined as existing necessarily, we have just proved that God does not exist. Incidentally, this makes me think that theists ought not define God as existing necessarily, because it makes the existence of God too easy to attack.

To see that this is a problem with the ontological argument, though, you do not have to agree with the argument that God is possibly nonexistent, and therefore nonexistent. You need only think we have no more business assuming a possibly necessarily existent God than we do assuming a possibly necessarily existent lion.

Now unlike Craig, Plantinga is not so crazy as to claim that his argument actually proves the existence of God, or to insist people must grant his assumption that God is possible. Instead, he says, of ontological arguments:

They cannot, perhaps, be said to prove or establish their conclusion. But since it is rational to accept their central premiss, they do show that it is rational to accept that conclusion.

But again, by analogy with mathematics, we can see that this is a silly way to argue.

Imagine two mathematicians, Alice and Bob, arguing over whether it’s reasonable to believe the Goldbach conjecture. Alice argues that the Goldbach conjecture is unproven, and we should not believe unproven mathematical claims. Bob concedes that it is unproven, but says the Goldbach conjecture seems true to him, and it’s reasonable for him to believe it on that basis.

Now, you may agree with Alice here, or you may agree with Bob, but imagine Bob tried to strengthen his position by saying, “Well, surely you agree that it’s at least reasonable for me to believe that the Goldbach conjecture is possibly true. But if I believe the Goldbach conjecture is possibly true, S5 allows me to infer that it is true. So it’s reasonable for me to believe the Goldbach conjecture.” This is a silly argument. Even if you think Bob is reasonable to believe the Goldbach conjecture, this can’t be the reason why.

Once again, we need to be very clear on what the problem is. The problem is not necessarily that it is unreasonable to think that the Goldbach conjecture is possibly true. Maybe Bob is right about that. The problem, instead, is that Bob cannot expect Alice to agree. Given that Alice thinks it is unreasonable to accept the Goldbach conjecture, she probably will not think it is reasonable to believe the Goldbach conjecture is possibly true, especially if she accepts the S5 modal axioms. Bob’s argument is, if not quite circular, an example of an argument that would be bad even if it were deductively sound.

So, not only does Plantinga’s argument fail to prove the existence of God, it fails even in Plantinga’s stated goal of showing that belief in God is reasonable. Nor, I think is it especially insightful in any other way. Plantinga did not invent the S5 axioms, he was not the first person to suggest they are the right modal axioms, and I do not think he provided any decisive argument for them (I don’t think such a decisive argument exists.)

The argument could work as a clever illustration of the S5 axioms–the sort of thing a professor might mention to his student while explaining modal logic, or that might end up on a whiteboard of a grad student lounge as a joke. But it does nothing whatsoever to establish the intellectual respectability of theism.

Bill O'Reilly's argument for the existence of God
Dissolving the problem of induction
There are no good arguments for the existence of God
My looong review of William Lane Craig's book Reasonable Faith

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