The only serious argument for incompatibilism that I know is the Consequence argument due, most famously, to Peter van Inwagen. (An Essay on Free Will, OUP, 1983.)
The version I will discuss is due to David Lewis. ("Are We Free to Break the Laws?”, Theoria 47 (1981), 113-121)
He tells us to think of the argument as a reductio. A compatibilist is someone who claims that the truth of determinism is compatible with the existence of the kinds of abilities that we assume we have in typical situations in which we deliberate and make a choice. Let’s call these ‘ordinary abilities’. The Consequence argument claims that if we suppose that a deterministic agent has ordinary abilities, we are forced to credit her with incredible abilities as well.
Here is Lewis's argument.
Pretend that determinism is true, and that I did not raise my hand (at that department meeting, to vote on a proposal) but had the ordinary ability to do so. If I had exercised my ordinary ability – if I had raised my hand -- then either the remote past or the laws of physics would have been different (would have to have been different). But if that’s so, then I have at least one of two incredible abilities – the ability to change the remote past or the ability to change the laws. But to suppose that I have either of these incredible abilities is absurd. So we must reject the claim that I had the ordinary ability to raise my hand.
Van Inwagen doesn't object to Lewis's way of stating his argument. On the contrary, he has said that Lewis's paper is “the finest essay that has ever been written in defense of compatibilism – possibly the finest essay that has ever been written about any aspect of the free will problem”. ("How to Think about the Problem of Free Will”, Journal of Ethics (2008) 12, 337-341).
Van Inwagen now agrees that the Consequence argument fails as a reductio.
he claims that it has nevertheless succeeded in
"raising the price" of compatibilism. (Freedom to Break the Laws",
I disagree. I say that the argument neither succeeds as a reductio nor succeeds in "raising the price" of compatibilism - that is, the price of commonsense at a deterministic world. What the argument does achieve -- at least on Lewis's articulation of it -- is a clear statement of the counterfactuals to which the compatibilist is committed. The argument is valuable for this reason. It makes it clear that we need to understand counterfactuals in order to understand what's at stake in the free will/determinism debate. But as an argument for incompatibilism, it fails.
Lewis's criticism of the Consequence Argument was published in 1981. His criticism was impeccable but his timing was bad. Lewis had published Counterfactuals (his possible worlds semantics and logic for counterfactuals) only 8 years earlier, in 1973, and counterfactuals were still poorly understood, and apparently not understood at all by some of the critics of Lewis's reply who seemed to think that Lewis had invented "local miracles counterfactuals" for the express purpose of defending a new and bizarre kind of compatibilism - "Local Miracles Compatibilism". There was further confusion due to the fact that Lewis developed his theory of counterfactuals in two stages: the formal logic came first (in 1973); and it was not until 1979 ("Counterfactual Dependence and Time's Arrrow") that Lewis proposed a detailed similarity ranking for possible worlds, and showed how to apply this similarity ranking in a way that gets the right truth-conditions for counterfactuals. It was also well-known, by then, that Lewis hoped to use counterfactuals to provide a counterfactual analysis of causation. All this was wildly ambitious, and many people were skeptical that Lewis could pull it all off. But -- and this is my main point -- it was natural, and understandable, back then, to think that Lewis's theory of counterfactuals is a "package deal" which you can accept only if you accept other parts of Lewisian metaphysics. So many people hesitated, and this may explain why Lewis's critique did not have the effect it should have had.
But now that time has passed and the dust has settled, it's clear that this is not the case. Lewis's theory of counterfactuals is independent of most of his other views. You can accept his theory of counterfactuals (including everything he says in response to the Consequence Argument) without accepting any of the following: Lewis's controversial brand of realism about possible worlds, his counterfactual analysis of causation, his "Best System" version of a Humean account of laws, his thesis of Humean supervenience.
Furthermore, Lewis's criticism of the Consequence Argument doesn't depend on the truth of his theory of counterfactuals. (His theory is, I believe, correct, but even if it weren't his criticism would still stand.)
Lewis's formulation of the Consequence argument nicely highlights a point that the better known modal version of the argument glosses over. The argument relies on a claim about counterfactuals. The argument says that if determinism is true, then at least one of these counterfactuals is true:
Different Past: If I had raised my hand, the remote past would have been different (would have to have been different).
Different Laws: If I had raised my hand, the laws would have been different (would have to have been different).
Now I agree that both these counterfactuals strike many people as incredible. But there is a reason for that -- we are not used to thinking in terms of determinism and we are not accustomed to counterfactual speculation about what would have to have been the case if anything at a deterministic world had happened in any way other than the way it actually happened.
On the other hand, we are good at evaluating counterfactuals, or at least some counterfactuals, and we are especially good at evaluating those counterfactuals that we entertain in contexts of choice, when we ask questions about the causal upshots of our contemplated actions. (What would happen if... I struck this match, put my finger in the fire, threw this rock at that window, raised my hand?) And when we contemplate our options, we take for granted the existence of many facts - including facts about the laws and the past.
In other words, when we evaluate counterfactuals in real life, we do so by considering imaginary situations which are very like the situation we are actually in, and we do not suppose that there are any gratuitous departures from actuality. And to suppose a difference in the past or the laws is a gratuitous difference -- if determinism is false.
So it is no surprise that when our attention is directed to Different Past and Different Laws, these counterfactuals strike us as incredible, or at least odd. But that doesn’t mean that they are false, and if determinism is true, then either Different Past or Different Laws is true.
So the first point is that we all need a theory of counterfactuals, and if determinism is true, the true counterfactuals will include either Different Past or Different Laws.
The second point is that the details of the correct compatibilist solution to the free will/determinism problem will turn on the details of the correct theory of counterfactuals.
If David Lewis's theory of counterfactuals is correct, or even more or less correct, then the relevant counterfactuals about the past and laws, at a deterministic world, are:
1. Same Past: If I had raised my hand, the past would still have been exactly the same until shortly before the time of my decision.
2. Slightly Different Laws: If I had raised my hand, the laws would have been ever so slightly different in a way that permitted the occurrence of a lawful divergence from actual history shortly before the time of my decision.
On the other hand, if Lewis's theory is wrong, and counterfactuals are always evaluated by holding the laws constant, then the relevant counterfactuals, at a deterministic world, are:
1. Same Laws: If I had raised my hand, the laws would still have been exactly the same.
2. Completely Different Past: If I had raised my hand, the past would have been different all the way back to the Big Bang.
We've got to choose. We need a theory of counterfactuals that applies at deterministic worlds, and our choice is limited to a theory that accepts Slightly Different Laws or Completely Different Past. Which theory we choose has nothing to do with the free will/determinism problem and everything with how we evaluate counterfactuals (in standard contexts).
Having sorted this out, I will now explain Lewis's critique of the Consequence Argument in a way that doesn't require you to accept the truth of Lewis's theory of counterfactuals:
Lewis's response to the Consequence Argument goes as follows: The argument trades on an equivocation between two counterfactuals.
(C1) If I had raised my hand, the laws (or the past) would have been different.
(C2) If I had raised my hand, my decision or action would have caused the laws (or the past) to be different
There is a corresponding equivocation between two ability claims:
(A1) I have the ability to do something such that if I did it, the laws (or the past) would have been different.
(A2) I have the ability to do something such that if I did it, my decision or action would have caused the laws (or the past) to be different.
The problem with the argument, says Lewis, is that it equivocates between these two ability claims. To count as a reductio against the compatibilist, the argument must establish that the compatibilist is committed to A2. But the compatibilist is committed only to C1 and thus only to A1. The compatibilist is committed only to saying that if determinism is true, we have abilities which we would exercise only if the past (and/or the laws) had been different in the appropriate ways. And while this may sound odd, it is no more incredible than the claim that the successful exercise of our abilities depends, not only on us, but also on the co-operation of factors outside our control. Since we are neither superheroes nor gods, we are always in this position, regardless of the truth or falsity of determinism.
To sum up: The Consequence Argument was supposed to show that if we attribute ordinary abilities to deterministic agents, we are forced to credit them with incredible past or law-changing abilities as well. But no such incredible conclusion follows. All that follows is something that we must accept anyway, as the price of our non-godlike nature: that the exercise of our abilities depends partly on circumstances outside our control.