I've noted before that I think two different things have been conflated in most discussions of free will: free choice, which I have already mentioned, and free decision. So this brings me to the fifth point, on free decision (sorry that it's a bit messy):
Point #5: The intellect is a free power.
We find in looking at reasoning, that not every rational inference we make is necessitated by what went before; some inferences are merely probabilistic. In such cases we nonetheless draw our conclusions. This occurs in speculation (e.g., in determining whether a given set of evidences best fits with position A or position B) and in deliberation (in determining what we shall do in contingent matters). Deliberation, in other words, is a nondeterministic inquiry. If we take as the antecedents the rational elements of the inference, this follows necessarily.
One can, however, argue that in such cases the mind is determined by things independent of the inference itself; and it is certainly the case that such things can have an influence. But if we argue that they determine the inference, we seem to be committed to one of the following:
(a) thought about contingent matters is determined by causes not rationally organized;
(b) thought about contingent matters is determined by causes rationally organized.
If (a), then we seem to have a view in which reasoning proceeds the way it does without regard for rational considerations. It is a form of irrationalism.
(B) is much more complicated. If thought about contingent matters is determined by causes that are rationally organized, then either:
(1) These rationally ordered determining causes are not determined by anything other than themselves.
(2) These rationally ordered determining causes are rational causes.
If (1) we have something like theological determinism. If (2), then either they are not determined (in which case (2) reduces to (1)) or they are determined by nonrational factors (in which case (b) reduces to (a)) or they are determined by rationally organized causes, in which case we just keep tracing the issue back. (The same issues arise with deliberation and speculative inference alike.) Can one have an infinite regress? That's an interesting question; but most determinists do not hold that our thought is determined by an infinite regress of rational agents. So if we are consistent determinists, it seems that either we are theological determinists of some sort, holding that things, including our own thoughts, are determined by a rational agent, or irrationalists.
This suggests a thought that I have often had about naturalistic determinism; namely, that it willy-nilly drives its proponents toward pantheism. If the state of the world prior to my rational thought determines my thought, then it is merely arbitrary for us to deny that the world is a rational agent.
But, whatever one's views on that, determinism gums up our notion of rationality in significant ways. For instance, it appears that alternative possibilities are necessary for intellectual responsibility; but determinism denies alternative possibilities. Frankfurt examples, even if they worked for the will (which they do not) don't tell us anything about intellectual responsibility. And, indeed, it seems that one reason Frankfurt examples seem plausible to some people is that they actually splice alternative possibilities at the intellectual level with denial of alternative possibilities at the volitional level. (One of the reasons I disagree with the claims of scholars like Stump that Aquinas would not accept PAP is that he clearly requires it for thought about contingent matters. Since actiones sunt suppositorum, it follows that even in the case where the will faces no alternative possibilities, there are still alternative possibilities available to practical reason. So all such cases will still uphold PAP.)