Assuming determinism, is truth-aiming reasoning possible? If our actions and beliefs have prior causes in every case, then a sound argument for a position will not persuade just because it is right. A person may be convinced by a sound argument, but there is no certainty of that based on their reason operating correctly, as how it operates will be itself determined.
Looks like there are two separate questions here:
1. Given determinism, does reason "hold" (with "hold" left undefined, but I'll take a stab at that).
By "hold", I think you mean things like: is reason reliable, can it do what it claims to be able to do, should we trust it as a means to knowledge, etc. Is that about right?
My answer is yes. If the universe is deterministic, then when we reason we are, ultimately, made to reason, but it is still reasoning nonetheless. When we find ourselves in the fortunate position of being made to reason (as opposed to being made to think or behave unreasonably), we tend to succeed much more in our attempts at effecting desired changes in our world. Reason--in the sense of logic, critical thinking--is the foundation of science, philosophy, math, jurisprudence, engineering...even literature. It also has much to do with successful personal relationships, educational performance, career success, and other key features of being a thriving person. I mention the glowing track record of reason to give credence to the idea that it "holds" in the way I think you mean it. That we may be forced by circumstance to engage in this highly successful mental activity is irrelevant to whether it holds.
2. Given determinism, might people sometimes make mistakes of reasoning?
Yes. But, given imperfect people, mistakes of reasoning will always occur, whether the universe is deterministic, indeterministic, or even if we somehow have contra-causal/libertarian free will.
I am inclined to say that reason, seen as a process of inference that uses the standard rules of logic, is deterministic. Take a simple case :
All A are B
All C are A
All C are B
As I work through the argument, once I have accepted the two premises there is nothing else to think but that All C are B. I don't accept the premises, then pause to make any kind of decision or choice : there is no alternative thought to the conclusion.
But how do I know that I am rationally determined as distinct from being causally determined (say, by genetic factors) that have nothing to do with rationality? Then I can turn that question back on itself and ask : How do I know that I am causally rather than rationally determined to ask how I know that I am rationally determined as distinct from being causally determined (say, by genetic factors) that have nothing to do with rationality? I can't see how this circle can be broken because the counter-reply on behalf of causal determinism so readily suggests itself - as does the counter-counter reply on behalf of rationality ad infinitum.
In a deterministic world the mind is deterministic, but goes through the motions of reasoning to reach its conclusions. In such a universe our actions and beliefs are predetermined, and the state of our mind at the time of reasoning is predetermined, and so the path our reasoning will take is predetermined. The argument being presented is predetermined, and whether or not the mind will accept the rationale of that argument is predetermined.
Taking this into account, an argument will only be persuasive if the deterministic mind is in a state where it is receptive to the argument. It relies on the listener having similar enough data patterns within its head for the sound arguments to be seen as sound - they may sound completely invalid if their worldviews are too far apart for the same conclusions to be drawn by each participant. What one mind sees as completely reasonable could be seen as completely insane by another mind.
I would say when pondering reason, you should really be pondering perspectives, as rationale is built from perspectives. People will always have their own rationale for what they believe, built off their subjective perspective, and the word unreasonable merely means 'I don't understand' as opposed to 'there is a flaw in your thinking'.
All this holds true in a non-deterministic universe as well, however, so deterministic or not is redundant when discussing reason.