The basic incompatibalist argument can be expressed as follows:
1. X is free only if X could have done other than what he did
2. If determinism is true then X could not have done other than what he did
3. Therefore, If determinism is true then X is not free.
Our lecturer then analysed Nielsen's compatibalist argument in terms of a counter-argument to the above, suggesting that Nielsen denies premise 2. I disagree - and what follows will explain why...
The idea of "could have done otherwise" is a crucial to our concept of 'freedom', but the 'could' there can be interpreted in two different ways:
Which version of 'could' does the incompatibalist intend?
Suppose we go with the 'categorical' version: then premise 2 of the incompatibalist's argument is trivially (tautologically) true. Determinism simply means that the state of the universe at time T precisely determines its state at T+1. There are no alternative possibilities to choose from, no categorical 'coulds'. This premise, a logical truth, cannot be attacked.
How about the 'hypothetical' option? Well, then premise 2 is obviously false. If the initial circumstances are different, then of course a different situation can deterministically result! This premise cannot be defended.
So either way, premise 2 is not worth arguing about. The real disagreement is about the meaning of 'free', about whether it requires a categorical 'could', or merely a hypothetical one. Clearly the incompatibalist intends his use of 'could' to be interpreted categorically (his assertion of premise 2 would be idiotic otherwise!). Given that premise 2 is thus a logical truth, any counterargument must instead attack the first premise. This is precisely what compatibalists (including Nielsen) do when they argue that freedom requires F2 rather than F1. Put another way, it is a redefinition of freedom to mean "not coerced", rather than "not (deterministically) caused".
Note that F1 and F2 are both genuine concepts. The argument seems to be about little more than which concept should be called by the word "freedom". That's all. Described like this, it's really no big deal. We could always just make up a second word to describe the other (say, "Shfreedom"). Words are arbitrary, it's the concept that matters.
Is this whole debate really that trivial? Well, not exactly. After all, we do use the word 'freedom' a lot, so it's fairly important to be clear about which concept we are referring to. The really important question, then, is "which concept (F1 or F2) is most useful for our purposes (when using the word 'freedom')?"
That, I think, is what the free will debate is really all about.
Update: Kip Werking at The Garden of Forking Paths makes a similar point.