Where does the burden of proof lie in a philosophical debate? In a debate of the type we are imagining, the answer is clear--in fact, trivial. The burden of proof lies on the person who's trying to prove something to someone. If Norma is trying to turn agnostics into nominalists, she is the one who is trying to prove something to someone: she's trying to prove to the agnostics that there are no universals, or at least that it's more reasonable than not to believe that there are no universals....So, trivially, in the case we are imagining, the burden of proof lies on the nominalist.
I've come across this before, and it has an initial plausibility, but in the end it just won't do. For one thing, while verbally it may sound very reasonable that that person has the burden of proof who is trying to prove, it is not so trivial as van Inwagen is trying to make it out to be. It doesn't seem that if you have two examples of a rational debate, both involving Tom, who in each case is giving the same arguments, that in the one debate Tom can have the burden of proof because he is trying to prove a claim to someone and that in the other debate he doesn't because he is doing something different, e.g., putting forward the arguments as suggestions, even if he also thinks that they form a proof. If we put emphasis on trying to prove, we leave room for the principle of double effect: and whenever the principle of double effect can be applied that means that something essential depends purely on intention. But in a debate we aren't always trying to prove anything to someone; often we are simply voicing objections, or insisting on the importance of this or that argument, or, indeed, playing the devil's advocate (and when we really play the devil's advocate, proving to someone what we are arguing for is the very last thing we want to do). But people can and do speak about burden of proof in these contexts. In a court of law, whence the concept of 'burden of proof' originates, it's simply irrelevant whether anyone's trying to prove something or not; the burden of proof is fixed universally by an external set of rules, and it doesn't matter what you are trying to do.
OK, so perhaps the point is not, "Trivially, whenever you have the burden of proof you are trying to prove something to someone," but rather "Trivially, whenever you are trying to prove something to someone, you have the burden of proof." But this doesn't seem to work, either. For one thing, in the courtroom, again, it doesn't matter what you are trying to do: the burden of proof is fixed independently of anything you are trying to do. A lawyer could try to prove something for which he has no burden of proof: for instance, he might be incompetent, and not properly understand what he must prove, and so is trying to prove something entirely different. He might not even have any burden of proof at all, and is just laboriously trying to prove something that he simply doesn't have to prove. Burden of proof in the law, again, is fixed by an external standard and is assigned by role; what you are trying to do is just not to the point. But even outside the court, in ordinary debate, we can arbitrarily agree to reassign the burden of proof, and this serves much the same role.
Moreover, this would often make the burden symmetrical, because both people can be trying to prove their mutually exclusive view. Van Inwagen recognizes this and dismisses it for no good reason that I can see:
You will see that I have imagined our ideal debate as based on a certain division of labor or, better, a certain principle of dialectical organization. I have not imagined a nominalist and a realist as simultaneously attempting to convert the audience to their respective positions. That way dialectical anarchy. I am really imagining two debates, or imagining that each side in the gets what might be called its innings.
No doubt some debates are structured in this way, but it seems implausible that all of them must be, on pain of 'dialectical anarchy'. When Socrates 'wins' a debate in Plato or Xenophon he often does so not by scoring more points during his innings but, so to speak, by moving his opponents' points from their name to his on the scoreboard: their own arguments become arguments for Socrates's position. In the Gorgias, for instance, there are things that could be seen as 'innings', but the intent to convert (if possible) the audience can be attributed to all parties simultaneously. There is no real way to split it up in this idealized manner without losing something important. And this is not dialectical anarchy; it's just a debate.
So the trivial truth seems not-so-trivial, and, indeed, is at least arguably false.
Now, van Inwagen does notice what often completely slips by most people, namely, that there are crucial ways in which 'burden of proof' cannot work in exactly the same way in courts of law and in other debates; the way he puts it (pp. 158-159 n. 2), in court the rule for burden of proof rests on moral considerations and in other cases it does not. I'm not wholly sure that the rule for burden of proof rests on moral considerations unless we are using 'moral' in a very broad sense, but we can pass that by. What sets the burden of proof in a debate in which moral considerations are not the source of the rule? Van Inwagen says that A will have the burden of proof rather than T when it is A and not T "whose job it is to change someone's beliefs."
But if we are setting aside moral considerations, what makes it anyone's 'job' to change someone's beliefs? 'Burden of proof' is a normative notion; if it is not normative for moral reasons, however, it seems it would have to be normative as a matter of politeness (assuming we haven't included that in 'moral') or as a matter of practical strategy or as a matter of positive fiat (e.g., because we simply agreed that it is a given person's job). The first seems to be ruled out because we don't ever seem to insist that someone has the burden of proof because it's rude if they don't try to prove something. We will sometimes think a person is being rude if they have the burden of proof and don't bother, of course; but the direction in those cases goes the wrong way for explaining why someone has the burden of proof at all. Moreover, it's difficult to think of ways in which one would be rude simply by failing to try to prove something. So let's set that aside. That leaves practical strategy, in which you have the burden of proof because of some particular thing that you are trying to do. Van Inwagen's suggestion is a particular example of this. But the same problems arise that we noted before; for instance, one may be trying to do any number of things, and it will be hard to say what they are without knowing your intention well, and that means that in many cases where we would obviously say that there is a burden of proof, there isn't, because someone is really trying to do something else entirely, despite appearances. And even if we both agree that I should have the burden of proof for now, this wouldn't guarantee that you have the burden of proof as a matter of practical strategy: it could be your strategy is not to prove but something that leads you to make promises to try to prove without actually ever intending to do so. We could generalize and say that what really matters is what a reasonable and impartial spectator would judge, but that just raises the question again: what makes it so that the impartial spectator would say that it's A's job to change someone's belief (and can the impartial spectator say that it's so even if A doesn't have any interest in changing anyone's belief?). And that leaves the third option, which is that whether or not it is someone's job is simply posited -- e.g., by agreement. This is actually my own view, that burden of proof is assigned either by an external positive authority (as in the case of the law) or by a sort of contract among the parties involved. But it puts us pretty far from any situation in which van Inwagen wants to use the concept.
It's possible, of course, that there is some fourth option, but, whatever it may be, it would be different from van Inwagen's claim, which fell under option two, practical strategy. Practical strategy is a better ground for an account of burden of proof than the more common candidate, which is just the analogy with the court of law. But it still is not so obvious or straightforward as van Inwagen thinks.