There are three main sorts of positions on the relation between free will and determinism. There doesn't seem to be any really perspicuous nomenclature for talking about them, but here's what I'll use. (That's my story and I'm sticking to it.):
Hard determinism: Determinism and free will are incompatible, and determinism is true; therefore, there is no free will.
Libertarianism: Determinism and free will are incompatible, and there is free will; therefore, determinism is not true. Of course, as used here, libertarianism refers only to a position with respect to free will, not to a political position. (Sometimes, since compatibilists generally believe in free will, they call this metaphysical libertarianism to distinguish it from their own view which they think also deserves to be called libertarianism. )
Compatibilism: Determinism and free will are compatible.
Here are some other definitions to get on the table:
Determinism: the doctrine that every event has a sufficient cause. To wit, for every event, x, there is some prior event or state of affairs, y, sufficient to cause precisely that event. In other words, given the earlier y, and any relevant laws of nature, nothing other than x, in all its detail and particularity, could have occurred. (By the way, determinism is not the same as predictability. The behavior of a system might be unpredictable, even in principle, but still deterministic. For example, as Popper pointed out, we cannot -- no matter how much we know and no matter how deterministic the process involved -- predict the content of future scientific discoveries because to predict them would be already to have made them. More generally, precise predicion depends upon precise modelling of the process to be predicted, but when the predicting system is part of the process to be predicted, it cannot, so long as it is a finite system, contain a precise model of itself -- including having a precise model of itself having a precise model, and so on -- for that way lies infinite regress. In the absence of a precise model, prediction is bound to be, at best, of limited precision.)
Indeterminism: the denial of determinism. It is not true that every event has a sufficient cause. There are cases (at least one) in which two or more different outcome events may follow upon exactly the same antecedent state of affairs. Note that this is not the same as free will. Indeterminism would be true if there were a single event anywhere in the universe without a sufficient cause, even if there were no beings having or allegedly having free will about.
Free will: A being has free will if and only if some of its actions are freely chosen, and an action is freely chosen when it is such that it can be proper or correct to ascribe responsibility to the agent for the act.
Note that compatibilism doesn't necessarily involve believing in determinism. It says that determinism and free will are compatible that is, that they can both be true together. One could believe that while thinking that determinism is not true. I mention this because that happens to be my own view. (More is available on my flavor of compatibilism and on why I think determinism is not true.) Though I think determinism is (probably) not true, I also think that, if it were true, it could also be true that we have free will (and I believe we do -- though the truth of some doctrine of free will is also not strictly required by the logic of the compatibilist position). I suspect most contemporary compatibilists, at least in academia, agree: they accept compatibilism while rejecting determinism. (This is one reason that it's not an altogether happy usage to refer to their position as soft determinism: One can be a soft determinist without being a determinist.)
Some have tried to argue that causal determinism flows directly from the Law of Identity. Others, who believe in libertarian free will, have held that the Law of Identity rules out the occurrence of any non-determined events outside the realm of free choice.
As far as I can see, this argument, in either the more general or the more restricted form, has nothing going for it. Must every entity act according to, and never against, its nature? Of course. How is that supposed to tell us that causal determinism is part of an entity's nature? Quite simply, it doesn't. If acting in accordance with some deterministic law of nature is part of an entity's nature, then of course that's how the entity will act. If, on the other hand, the most fundamental relevant laws of nature admit the occurrence of chance or non-determined events, then acting in that way will be acting in accordance with the entity's nature.
Theres no substitute for examining the evidence of how things actually behave. So far as I understand what the quantum physicists say, our world is one in which there are genuine (and irreducible) chance events. So far as this argument goes, it may also be a world in which there are irreducibly non-determined events of other sorts -- free choices, perhaps.
There have been a number of attempts to argue that knowledge (or conceptual knowledge) implies libertarianism. More precisely, the argument is typically that knowledge is incompatible with determinism. The argument is adumbrated as early as Aquinas and has been endorsed or considered by a number of more recent thinkers. I first came across it in C.S. Lewis's book, Miracles. I have to confess to thinking that appropriate. Believing in libertarian free will seems to me closely akin to believing in miracles. (The accounts of agent-causation Ive seen seem to me filled with vague mush: How does the agent non-deterministically cause choices? Somehow.)
Plainly, if any knowledge argument is to work, it will have to depend on some conception or other of knowledge. (And, of course, that conception will have to be correct.) Surprisingly, those who make the argument are rarely very explicit about this. Branden, for example, quotes the Objectivist definition -- correct identification of the facts of reality -- but doesn't actually put it to much use in his argument. Perhaps this is because there isn't much use to which it could be put. It sounds a great deal like 'knowledge is true belief,' known to be inadequate at least since the time of Plato.
Anyhow, two things need to be done if the knowledge argument is to be sound (or powerful). There needs to be a conception of knowledge which is in fact incompatible with determinism. (Mere assertions of incompatibility are not enough.) And it also needs to be the case that that conception is correct (or at least better than any deterministic alternative). Since proponents of the knowledge argument have not been careful to make sure both of these conditions are met, a sufficient answer to them in the present context can be provided by offering a plausible conception of knowledge that doesn't have anti-deterministic implications.
What I propose is (roughly) that knowledge is true belief that is caused in the right way. What is the right way? A way that is sensitive to the fact of which the belief is true. Slightly more formally, the way the belief is caused has to be such that the belief would vary counter-factually with the fact. There are, in other words, two ways that a true belief could be insensitive to the relevant fact or facts:
It could be a belief that would (continue to) be held to be true even if the fact or evidence regarding it were relevantly different. I believe (truly) that McCrud is guilty, and there is adequate evidence available to me to establish that, but I would still believe in his guilt, in the absence of that evidence (or even if my evidence pointed to his innocence), because I hate McCrud. If that is how my belief is caused, then I do not know that he is guilty.
It could be a belief that, though true and adequately supported, would be rejected in relevantly different situations in which it is still true and adequately supported. I believe that there is no largest prime number and have seen and understood the proof, but that's not why I keep on believing it. I keep on believing it because I was told it was so by Lovable (whom I love). If Lovable had told me that there was a largest prime number, I would believe that instead. Again, if that is how my belief is caused, then I do not know that there is no largest prime number.
Plainly, an account of knowledge along these lines is compatible with determinism, since beliefs that count as knowledge are just a special case of caused beliefs. Moreover, it seems to gets things right about the kinds of claims that, pre-theoretically, we do or do not want to count as instances of knowledge. For example, it allows us to say that beings we don't suppose to have free will do have knowledge, as in the case of a dog knowing that his master is at the door. Still, it is not dispositive. The proponent of the knowledge argument still might make a case that some other conception of knowledge is both superior and incompatible with determinism. However, the burden of proof is on him. I'll leave this in the form of a question: If this isn't good enough to count as knowledge why isn't it? What more do you want?
One response to my suggestion does not directly contest the claim that knowledge can be understood as true belief caused in the right way but asks, if determinism is true, how we can know that our beliefs are caused in the right way. After all, if they were not, we'd still hold the beliefs.
Matters tend to get rather vague at this point, but apparently, the thought is that if we don't know that our beliefs are caused in the right way, then we also don't have knowledge in holding those beliefs. (So, the anti-determinist argument from the reality of knowledge stands.)
But this won't do. Suppose I'm typing now. (I do suppose that.) Now, consider my knowing that I'm typing now. That's a different fact than my typing now, though of course I could not know that I am typing now if I were not typing now. According to my analysis, if I know it, then (a) it's true, (b) I believe it, and (c) I was caused to believe it in the right way. Is there more involved in knowing that I know that I'm typing now? Yes, for to know that I know it, I (d) have to believe that I know it, I have to actually know it (which requires satisfying conditions (a) through (c) above), and (e) my belief that I know it has to be caused in the right way. And the same thing, iterated, goes for knowing that I know that I know it, knowing that I know that I know that I know it, etc.
At each level, knowing something focuses on one fact, while knowing that one knows it focuses on a different fact. However common or natural the steps may be, the move from knowing to knowing that one knows is never just trivial, and, it is, therefore, never guaranteed that if one knows, one also knows that one knows. And, of course, we all do give out somewhere. I may be able to keep track of what it means to say I know it, that I know that I know it, and so forth, for several steps beyond anything I would normally be tempted to say, but, at some point, I no longer even understand what all the iterated "know thats" mean, and therefore, at that point, no longer have any belief at all that the whole string terminating in, say, the claim that I am now typing is true. Since I don't believe that the whole string expresses a truth, I also dont know it.
Perhaps this point could be made more clearly if you think of it in the first person. Pick something that you'd claim to know, say, that 2 + 2 = 4. I'm willing to assume you do know that. Do you also know that you know it? I'd guess the answer is probably so, but knowing that you know it involves more than just knowing it: it also involves your having some access to and evidence about why you believe it. Take it up another level or two: Do you know that you know that you know it? How about knowing that you know that you know that you know it? My guess is that, if you've followed me this far, things are getting pretty fuzzy. With each additional level of "knowing that," there are more things to keep track of in order even to understand the question, much less to be sure what the answer is. At some point, your capacities to keep track of it will give out (if they haven't already). When you get there, you will not be able to truthfully say that you know that ... you know that 2 + 2 = 4. That doesnt mean you dont know that twice two is four.
Accordingly, the fact -- if it is a fact about a particular belief -- that I don't know it to be caused in the right way provides no argument at all that I do not know it and, therefore, no fuel at all for an anti-determinist argument from the reality of knowledge.
Another response holds that for anyone to know something, she has to be able to check it, to reconsider, to assess the grounds for believing it and so on. But, if determinism is true, she may not be able to do this. She may be caused to believe something (that is in fact true), and caused in such a way that the belief varies counterfactually with the fact believed (as I claimed was needed for knowledge), but nonetheless, there may be, in the actual world, no causal path that leads from her state in believing it to anything that might be called checking or testing it. Since, in a deterministic world, there is no physical possibility of her testing it, she does not really know it.
In effect, this argument proposes an additional condition, that checking must be possible in order for something to count as knowledge. Now, I do not think this a necessary condition for knowledge, but I will not try to show that it is not. Instead, I will claim that, even if it is necessary, it may be satisfied within a deterministic world, or, at least, that no argument has been given why it cannot be.
Note that anyone who makes this objection is already conceding a lot. Above, I spoke of the circumstances in a deterministic world in which there is no causal path that leads from the state in which someone holds a true belief to her checking the truth of that belief. Plainly, that admits the possibility of other circumstances in which there is such a causal path. For the latter cases, there is no argument at all, at least on this basis, that, in those other circumstances, there is no knowledge. So, in raising this objection, it is already being admitted that knowledge is compatible with determinism. The sweeping claim that determinism makes knowledge impossible has tacitly been given up.
But consider further those cases in which there is no causal path leading to checking. Is it true for those cases that checking is impossible -- and therefore that knowledge is not possible? It appears the argument would go like this (remember, we're talking about a restricted class of cases):
1. If a person knows p, she must be able to check whether p.
2. If determinism is true, she is not able to check whether p.
3. Therefore, if determinism is true, she does not know p.
On the face of it, that appears valid. For the sake of argument, I'm willing to grant that both premises are true, so it would appear I'm committed to regarding it as sound also. And if it is sound, then the conclusion must be true.
Is that conclusion really inescapable? It depends on whether "able to check whether p" is used in the same sense in both premises. Compare this rather silly argument: "Rivers have banks. Banks are financial institutions. Therefore, rivers have financial institutions." Plainly, that argument is worthless because "banks" has to be interpreted differently in the two premises in order for both to be true.
So, is "able to check whether p" univocal in the two premises? Let's see.
Consider the second premise. Under what interpretation would it be the case that if determinism is true, she is not able to check whether p? Since a determinist holds that every event has a sufficient cause, then any event that actually occurs is necessary (given what has happened in its past together with the laws of nature). Only that event could occur, and any alternative is impossible. If an event, like checking whether p, does not actually occur, then it could not occur. The possible and the actual turn out to be identical (and so do the impossible and the non-actual). Call this deterministic possibility.
Is deterministic possibility what is asserted in the first premise? Does it mean if a person knows p, it must be deterministically possible for her to check whether p i.e., that she must actually check whether p? We might have doubts because there are other notions of possibility. Consider this one: It is possible (right now) for the chair in my kitchen to support a 300-pound man. As far as I know, thats true. But there's no 300-pound man within range. So, it's not deterministically possible for the chair to support a 300-pound man (right now). Does that undermine the earlier claim that it is possible? No, because that wasn't a claim about deterministic possibility. What it meant, roughly, was that, because of the chair's materials and construction, if a 300-pound man were sitting on it right now (even though that's not deterministically possible), it would not collapse. The claim refers to a complicated property that the chair does have right now, not to whether it is deterministically possible for it to support 300 pounds right now.
So, sometimes, possibility claims can be true even when deterministic possibility claims about the same subject are false. Moreover, such an analysis is at least initially plausible with regard to checking. To say a person is able to check seems to refer to a capacity which can be present whether exercised or not. We would not say, I don't think, that if Helen has rigorously proved a mathematical theorem (starting with premises known to be true and reasoning correctly from them) that the fact that she died shortly afterward -- and thus, never actually checked her result -- means that she didn't have the ability to check her result or that she didn't (briefly) know it to be correct. At the very least, the availability of other conceptions of possibility shifts the burden of proof. An argument is needed from the other side as to why deterministically possible checking is necessary for knowledge, not just for why ability to check is necessary.
(By the way, the above also contains the germ, if not the details, of the answer I'd give to the libertarian about how the necessity of genuine options if there is to be free choice can be consistent with the fact that, if determinism is true, only a single choice is (causally) possible. If the senses of possibility involved are not the same, then it is consistent to hold both that there is free choice from among a plurality of possibilities and that there is only a single possible outcome of a choice.)