When people talk about free will, in the context of philosophy, without knowing the terminology of the field, they often seem to mean something like libertarian free will—a position not related (except etymologically) to political libertarianism: the belief that you “could have done otherwise”: your will is properly free if, and only if, having to re-make a prior decision under perfectly identical circumstances, you can choose to do otherwise the second time.
Unfortunately, I don’t think that libertarian free will is logically coherent. Any event—decision or otherwise—is either deterministic or non-deterministic. If it is deterministic, this means that it is causally determined: the state of the universe around me, along with my disposition in the form of knowledge or beliefs, opinions, desires, goals, and so forth, fully determine what I will do. If you were omniscient, you could in principle predict my every action. On the other hand, if the decision is non-deterministic, this means that there is an element of randomness to it: to some degree, my decision is not determined by reality around me, nor by what I think or want. Intuitively, this does not seem to me like “free will”: in fact, as much as the deterministic versions limits freedom, the non-deterministic version limits will.
As far as I can tell, libertarian free will is supposed to occupy some magical middle ground that’s neither deterministic nor non-deterministic. This violates the law of the excluded middle—that is, it requires propositional logic to be wrong! This seems absurd and prima facie wrong, and even if it were true we could ipso facto not reason about it.
Note that, although the terms often arise in free will discussions, I have not hitherto said anything about materialism and dualism. This is because I honestly don’t see that it particularly matters. As it happens, I am a materialist: I think that our minds are what our brains do. But my argument about free will does not depend on this. If you want to suppose that your mind is really some sort of non-material spirit stuff, this does not affect the dilemma between (non-free) determinism and (non-willed) non-determinism. Dualism does not solve the problem of free will, because the problem is not about physical versus non-physical causation, but rather about the logic of causality itself. Christian apologists often argue quite vehemently about this, because metaphysical free will is essential to their theology; but their arguments seem largely to amount to an assault on physical causation without ever addressing the true problem—and as a rule they are quite fond of the laws of logic, so the excluded middle remains a major problem. Put bluntly, they want to absolve their God of responsibility for the things that we do out of “free will” in spite of his supposed omnipotence and omniscience. It does not work.
If a logical exposition exists to get out of this quandary, I've failed to find it and would be fascinated to hear it, but I'm not holding my breath. As far as I can tell, attempts to salvage libertarian free will are less clear-headed philosophy than desperate attempts to justify what we all intuitively feel in the face of what is logically true.
Furthermore, although I can readily see the objections to free will raised by the spectre of determinism, it's not that clear that they have all their apparent force when you look more closely. Normally, I think of a free choice as one where no one is constraining or coercing me. It can be deterministic. It can even be predictable, which is much stronger than merely deterministic: if I strongly prefer chocolate to vanilla ice cream, and you know I do, I can still freely choose to have chocolate every time. The fact that you know doesn't constrain me. I could choose vanilla if I wanted to—the fact that, given that I don't want to, I never do, is precisely what makes my decision free, even though it is an explicitly determined choice!
In fact, every good decision is deterministic. If I choose according to my best knowledge and current beliefs, and make the choice that best aligns with my dispositions and desires, in the sense of (so far as my knowledge can tell) being optimal toward achieving my goals, that is a deterministic choice: but if I had some greater metaphysical freedom, it's still the one I’d hope to make. A non-deterministic component can only serve to randomly push me away from this optimal choice. Is that more free? And is it truly willed if it is random?
I’m not terribly excited about the term compatibilism, but I suppose that in effect, I largely am a compatibilist, and my ἀπολογία can be summarised as: The alternative to deterministic free will entails a freedom to randomly act against my own interest, which perverts the word freedom into incoherence. As Dennett might say, that kind of free will worth having is deterministic.
Perhaps the free will problem is best addressed by Ordinary Language Philosophy:
Non-ordinary uses of language are thought to be behind much philosophical theorizing, according to Ordinary Language philosophy: particularly where a theory results in a view that conflicts with what might be ordinarily said of some situation. Such ‘philosophical’ uses of language, on this view, create the very philosophical problems they are employed to solve. This is often because, on the Ordinary Language view, they are not acknowledged as non-ordinary uses, and attempt to be passed-off as simply more precise (or ‘truer’) versions of the ordinary use of some expression – thus suggesting that the ordinary use of some expression is deficient in some way. But according to the Ordinary Language position, non-ordinary uses of expressions simply introduce new uses of expressions.[Internet Encyclopedia of Philosophy]
Maybe the fundamental problem of the free will debate is that it has developed a problematic concept of free will that wouldn’t exist if we didn’t have the discussion. In practice, determinism is perfectly compatible with every kind of freedom we care about or can measure; but in philosophy, philosophers and theologians have defined a problematic concept into being. In that case, we can explain it as being a matter of two different things: We do have free will, in the OLP sense, and this is compatible with determinism; but we do not have [libertarian] free-will—which, however, does not actually matter in reality.