(1) Humans have some property P; but (2) If AN is true, then humans could not have property P (because, e.g., such a property could not exist given AN); therefore, (3) AN is false.
Their argument is that these arguments cannot ever succeed. This would actually be quite significant, since, as can be seen on reading through their argument, the argument that theistic incompleteness arguments cannot succeed would be quite easily generalizable to any kind of incompleteness argument. Incompleteness arguments in general, however, are extraordinarily common rational instruments. I'll talk a bit more about the implications of this at the end of the post, but let's focus on the particular version they give. This is how they start out:
Consider two putative epistemically possible worlds, w and w*. These worlds are near duplicates of each other except in one of these worlds, w, God exists, but in the other world, w*, God does not exist. Moreover, one of these worlds is the actual world. In other words, if God exists, then we exist in w, but if God does not exist, then we exist in w*.
It is worrisome already to be talking about possible worlds being the actual world; there are good reasons for refusing to identify actual worlds with any given possible world. The obvious riposte is "But surely the actual world is possible, and thus a possible world?" But unless we are modal realists, holding that every possible world is actual, we actually hold that the actual world is able to be in different ways; which means that our account of a possible world is actually an incomplete account of the actual world insofar as it is capable of being this way rather than some other way. In this sense, every description of any possible world is a description of some particular possible way the actual world itself can be. It's illegitimate to lay out possible worlds and simple say, "This possible world is this actual world," because this possible world by definition can't be any other possible world. The actual world would be necessary. To be sure, you can get around this with some highly controversial assumptions; but they are highly controversial.
Megill and Linford then go on to talk a bit about epistemic possibility; I find their characterization also worrying since they put it in terms of certainty, which we know can be fragmentary and even in real life inconsistent. But I don't know that this affects the structure of the argument, beyond what I note below, so I will continue.
The atheist thinks that we live in w*, so the theist needs to convince the atheist that w* cannot be the actual world; and the way that incompleteness arguments try to do this is by arguing that some property P could not be in w* in the absence of God. There are two possibilities: either the atheist thinks P is in fact in w* or not. That is, either P is a property in AN’s ontology or it is not; this is an instance of excluded middle. Suppose that P is not in AN’s ontology; the atheistic naturalist does not even think that P exists. But if not, then the relevant incompleteness argument cannot succeed; if an atheist does not believe that property P exists in the first place, they will not be troubled by AN’s alleged failure to explain the existence of P. On this scenario, the theist might as well fault AN for failing to explain the existence of phlogiston.
So the idea is that we have these two epistemically possible worlds, possible ways the world could be given what we know about the world, w in which God exists, and w* in which there is no God and these are "near duplicates". The incompleteness argument is that there is a property that actually requires God to exist, despite the fact that there is no God in w*. So the first thing Megill and Linford do is consider the case in which the atheist holds that the property is not found in w* (i.e., the atheistic world). That is, the atheist holds that such a property does not exist. Then, say Megill and Linford, the incompleteness argument is not a problem for the atheist; he doesn't think P, which requires God, exists, so he's not committed by it to holding that God exists.
This is a slightly problematic characterization. An obvious minor worry is that the atheist's not admitting P just pushes the argument back one step; it does not stop the incompleteness argument, because in arguing that P requires God's existence, the theist is ipso facto arguing that everything that requires P requires God's existence, and it may, for all we know, be the case that the atheist admits the existence of one of these. If w* includes Q, and Q requires P which requires God's existence, not admitting the existence of P and God doesn't actually get the atheist out of anything. This worry is only minor, and doesn't affect the argument as a whole; it's actually a reflection of the fact that Megill and Linford are making a completely general argument against theistic incompleteness arguments, not objecting to one particular theistic incompleteness argument.
Far more worrisome is the implication that nobody need to be worried about anything that's not in their ontology, which, however, gets into more serious generalizability issues that I will talk about below. Let's move on and consider the next leg of the argument.
But suppose that the atheist does think that P is in w*. Now, the theist who advances an incompleteness argument has to show that w* is not epistemically possible because without God, P could not exist. The theist needs to convince the atheist to move from w* to w, to recognize that God must be added to w* because some property P in w* could not exist in a Godless world. Suppose that the theist produces an argument that P could not exist without God. This argument is designed to convince the atheist to reject w* for w; but in this scenario, w* and w will almost be perfect duplicates; they are qualitatively identical aside from the existence of God and the existence of the property P that God’s existence makes possible.... But this is problematic
Here is where I think the argument really starts to go off the rails, or at least to become puzzling in terms of how it is supposed to work. By the supposition previously noted, w and w* are near duplicates, differing as such only in whether God exists, and we are considering cases in which w* is thought to include P, having already considered the case in which it is thought not to include it. We run into the same worry I noted then, about the fact that in arguing that P requires God's existence, the theist is ipso facto arguing that anything that requires P requires God's existence. If w and w* are really possible worlds, they have to be consistent -- inconsistent worlds are impossible worlds, not possible ones. In the case of epistemic possibility, the consistency has to be with what we know, in some sense of the word. But precisely the theist's argument, in the case where the theist and the atheist both recognize the existence of P, is that any atheistic account of the world is not consistent, and thus not actually epistemically possible, even if the atheist mistakenly thinks it is.
Megill and Linford had previously set up apparatus to block this point by arguing that the atheist believes w* is the actual world and thus is epistemically possible; therefore, "for the atheist, w* is the actual world and so must be epistemically possible." They then claim that the only way it could turn out not to be epistemically possible is if we could be "completely certain of the truth of theism". But all of this is obviously not true, and appears to involve confusing "what someone believes follows from what they know" and "what follows from what someone knows". If atheists were logically omniscient, we could rest satisfied that what they think follows from what they know really does follow from what they know. But they aren't; none of us are. We can be mistaken about what is actually consistent with what we know. Not every view of the world someone may have describes an epistemically possible world; some only seem that way if you overlook some important inconsistency or contradiction. Worldviews are not hermetically sealed; they have to answer to the real world, and come from what we know of the real world, and they can be assessed for consistency in doing so.
So if we aren't assuming that atheists know all of the implications of all that they know, we aren't really dealing with what is epistemically possible, but with what they atheist thinks is epistemically possible. Not at all the same thing. And because of that, the atheist may well be inconsistent -- which is yet another way in which the atheistic claim could turn out not to be epistemically possible, besides being certain of the truth of theism. (It could turn out, for instance, that we are left uncertain whether theism is true, but that on the basis of the incompleteness argument, we can see that this particular form of atheism is untenable.)
Megill and Linford give three reasons why they think the incompleteness argument is problematic in the case where the atheist admits the existence of P.
First, from the atheist’s perspective, it looks like the theist is simply tacking God on to a world, and God is then somehow able to magically imbue the lives in that world with some important property without changing anything else about the world. Second, it appears that God is largely an epiphenomena in such a world; after all, nothing else about the world changes with the addition of God, who then somehow brings P into the world without changing anything else. Third, if two lives can be qualitatively identical aside from the existence of a property P, then it seems that P does not supervene on or depend upon any of the other properties of the lives. But how can we completely sever moral significance, rationality, meaning and so on from the content of particular lives?
There seems to be some strange slippage with regard to the notion of 'near duplicate possible worlds'. As noted above, despite the tendency of Megill and Linford to say that w and w* only differ in the existence of God and (depending on the situation) the existence of P, in reality they also differ in everything that genuinely requires the things in which they differ. Thus the most nearly duplicate epistemically possible worlds may not actually be very similar -- to claim that they are very similar is just to claim that the way in which they differ doesn't matter much at all. But the atheist is not entitled to assume that w and w* really are only slightly different; that is equivalent to saying that God's existence has almost no implications, a claim that is not conceded by the incompleteness argument itself nor follows from the mere difference of w and w*. That the theist is only arguing about P does not change the fact, noted above, that it would also apply to anything that requires P, if anything does -- and that chain might, for all we know at this general level, extend back to lots and lots and lots and lots of things. Thus the atheist is not in a position to determine whether God is just 'tacked on' as an arbitrary difference or not; the implications of God's existence or nonexistence might (as far as we know) be quite extensive, even in terms of what only follows from this particular argument itself. Only by following through all the implications would it be possible to determine how big a difference it might be; the atheist has no warrant, however, for assuming that nothing but P, alone, is on the table. It just happens to be P that the theist is trying to link to God's existence; if he succeeds, P brings along anything whatsoever that requires P, despite the fact that none of them have been explicitly discussed. It might also bring along other things that don't themselves require P but are similar enough to P to require parity of argument. P is just the central element of what may end up being a very large package of things (we can't tell without investigating P more closely), and thus the differences between these 'near duplicate worlds' might, as far as we know, be quite considerable. In fact, the three arguments trade on the oddity of thinking that that it would make very little difference. But that it would make very little difference is not at all an assumption theists are committed to in incompleteness arguments; the argument is not that P and nothing but P requires the existence of God, but that at least P requires it, and it is a direct implication of it that anything that requires P would also require it, just by transitivity.
All these are mere initial worries and might seem evadable. But the interesting thing about the Megill & Linford argument, and the reason for talking about it at all, is that it is highly generalizable. It doesn't depend on particular features of the arguments, but only on their general shape; it doesn't depend on precisely what kind of theism is in view or what God is or what the property in question is. This is because it is supposed to be a general argument against all theistic incompleteness arguments. But it is in fact easily generalizable to any situation with the following characteristics:
(1) Position A involves holding that something, X, does not exist.
(2) Position B involves holding that X does exist.
(3) The person who holds position B argues against position A that there is some property P in the world that requires the existence of X.
Let's take an example. Person A and person B disagree about whether there are gravitons, A holding there are no such things, and B holding that there are. B argues against A that there is some material property, P, that requires the existence of gravitons. In other words, B is arguing that A's position is incomplete. Suppose that A does not accept the existence of P. 'Then the relevant incompleteness argument cannot succeed; if A does not believe that property P exists in the first place, they will not be troubled by their position's alleged failure to explain the existence of P. On this scenario, B might as well fault A for failing to explain the existence of phlogiston.' Suppose that A does think that P exists. Then 'B needs to convince A to move from w* (the epistemically possible world without gravitons) to w (the one with), to recognize that gravitons must be added to w* because some property P in w* could not exist in a world without gravitons'. But then 'from A's perspective, it looks like gravitons are just tacked on to the world, and that gravitons are somehow able to magically imbue things in the world with some important property without changing anything else about the world.' Further, 'it appears that gravitons are largely an epiphenomena in such a world; after all, nothing else about the world changes with the addition of them.' Moreover, 'if two things can be qualitatively identical aside from the existence of a property P, then it seems that P does not supervene on or depend upon any of the other properties of the things.' But how is this supposed to work for gravitons as explanations of material phenomena? 'Either a given property used in an incompleteness argument is in A's ontology or it isn’t. If it isn’t, A has nothing to explain. If it is, then B is committed to divorcing these properties from anything else, along with other bizarre claims. A has no reason to accept any incompleteness argument.'
You can have infinite fun with this. Let's take the material world. A holds that there is no material world; B holds that there is, and argues that A's account of the world is incomplete without the existence of the material world.Suppose that A does not accept the existence of P. 'Then the relevant incompleteness argument cannot succeed; if A does not believe that property P exists in the first place, they will not be troubled by their position's alleged failure to explain the existence of P. On this scenario, B might as well fault A for failing to explain the existence of phlogiston.' Suppose that A does think that P exists. Then 'B needs to convince A to move from w* (the epistemically possible world without material things) to w (the one with), to recognize that matter must be added to w* because some property P in w* could not exist in an immaterial world'. But then 'from A's perspective, it looks like matter is just tacked on to the world, and that matter is somehow able to magically imbue things in the world with some important property without changing anything else about the world.' Further, 'it appears that matter is largely an epiphenomena in such a world; after all, nothing else about the world changes with the addition of it.' Moreover, 'if two things can be qualitatively identical aside from the existence of a property P, then it seems that P does not supervene on or depend upon any of the other properties of the things.' But how is this supposed to work for the material world itself? 'Either a given property used in an incompleteness argument is in A's ontology or it isn’t. If it isn’t, A has nothing to explain. If it is, then B is committed to divorcing these properties from anything else, along with other bizarre claims. A has no reason to accept any incompleteness argument.'
That's actually quite close to some of Berkeley's arguments, in fact. All standard arguments for the material world are incompleteness arguments: they claim that there is something in our experiences of the world such that any account of them is incomplete without bringing in matter. And it doesn't take much generalization at all from the Megill-Linford argument; given the structure of the argument, there is nothing to prevent the generalization. By parity, if we reject incompleteness-without-God arguments for Megill & Linford's specific reason, we should also reject incompleteness-without-matter arguments.
And we can generalize even further, without any difficulty at all. While not all arguments to the existence of a cause are incompleteness arguments, most are: they take some phenomenon E and argue that our account of E cannot be complete without the existence of such-and-such cause C.
When we reach this point it becomes easy to see an important feature of the Megill-Linford argument: it depends on a separability principle, and thus is analogous to Hume's separability argument about causation, although, of course, Hume doesn't use the elaborate apparatus of possible worlds. Hume's version of the separability principle is, "Wherever the imagination perceives a difference among ideas, it can easily produce a separation". The basic idea of this in the context of Hume's theory of ideas is that if two ideas are distinguishable, they are capable of existing separately. Hume then argues that because cause and effect are distinct, we can conceive of something coming into existence without any cause. There is thus no necessary connection in the usual sense of the word between effect and cause; if we're just considering the idea of the effect and the idea of the cause, nothing about the one actually requires the other. You can't demonstrate the existence of one from the other, or the existence of anything from the existence of any other thing. What this is doing is blocking incompleteness arguments -- that you can't completely account for the effect without the cause, or for the cause without the effect. Megill and Linford's argument works an analogous way, and assumes an analogous separability principle. The question is whether P requires the existence of God; in the argument, if we assume the existence of P, the world-with-P can be conceived on its own without God, so the world-with-P can't require that God exist. Despite the rationalist apparatus of possible worlds, it is an entirely Humean argument.
And it has an entirely Humean skepticism as its end result; since there is nothing to stop generalizing, it generalizes to the furthest extent, to any case where the existence of one thing is claimed to require the existence of another. And the reasons for thinking it can't be right are exactly all the reasons for denying separability principles and holding that given something's existence you can prove another thing's existence.