I am just having fun in this post. I will make claims that seem to have established something interest, but I don't think I'm doing anything interesting. Compatibilism seems to be the doctrine that there is a possible world where determinism holds and some being is free. But God exists necessarily, and God is, plainly, free in every possible world. Therefore, compatibilism is true if and only if there is a world where determinism holds. But there is a world where determinism holds, if by "determinism" we understand the following doctrine: The total state of the world at any given time, together with all the laws, entails the state of the world at all given times. For, it is possible that an angel exists and that every time t, God instantaneously announces all of the history (past, present and future) of the world to the angel (maybe God creates a book that contains that history). Since God cannot lie, it follows that, necessarily, the total state of the world at any given time entails the state of the world at all other times. So, determinism is possibly true, and compatibilism is, hence, true.
Yes, but I've cheated here. While one might incautiously say that compatibilism is the compossibility of determinism with freedom, one really means something like the compossibility of physical determinism with an embodied person's being free. But if that's what compatibilism is, it's still probably true. For either a given form of physical determinism rules out the possibility of extra-physical interventions that deviate from the laws or it does not. If it does rule out that possibility, it is an impossible doctrine, being, apparently, incompatible with God's omnipotence. If it does allow for the possibility of extra-physical intervention, then it is possible for the extra-physical parts or aspects of an embodied being to intervene in the physical order. Since a determinism that allows for such a possibility is possible, and since a determinism that does not allow for such a possibility is impossible, it seems reasonable to take this interenable determinism to be what the doctrine of compatibilism is talking about. And then compatibilism is true.
But maybe I've cheated again. Maybe the question is whether physical determinism is compatible with the free will of purely physical persons. But there the answer is, I think, pretty simple: such a compatibilism is false, because there are no possible worlds containing purely physical persons.
Notice that on all of the above proposed accounts of compatibilism, I've settled the problem (assuming necessary theism or assuming a form of dualism, both of which are doctrines that I can argue for) without saying anything much about responsibility. That suggests that none of the above definitions of compatibilism got at what we really mean by the word.
It's not all that easy to get a definition of compatibilism that escapes the above. Here is a somewhat messy one that might do the trick: It is possible that there is an embodied and free person in a universe with deterministic laws which that person lacks the power to go against. However, this definition might be skewed towards incompatibilism. For the compatibilist may wish to say that one is not free if one lacks the power to act otherwise, but then account for "lacks the power" in a compatibilist way. But if "lacks the power" is understood in a compatibilist way, then the definition of compatibilism is probably not very good. I think the definition is still defensible, but it needs some work.
Basically, what I'd like to have is a definition of compatibilism where neither its truth nor its falsity follows from considerations that don't get at the heart of the real issues with responsibility. And it's not quite as easy as it at first seems to do that.