Mackay makes a fairly conplicated argument, but I will attempt to simplify it as much as possible. This may be a slightly confused rendition of his argument, for it has been some time since I read his book. I simply set forth this argument because it is highly relevant to my discussion. If there are any questions, or if you think that I have straw manned Mackay, please direct your concerns to the complaints department.
Mackay's conclusion is that, even if we accept that the human brain works completely mechanically, that is to say, even if we could predict the future state of someone's brain like we can predict clock-work, it does not follow that there is no human freedom. He is not arguing that the brain is like clock-work (in fact, he believes that it is not) but that's not the point: his goal is simply to tackle the determinist argument on its own grounds, the mechanistic universe.
He argues that, though it may be possible to predict someone's actions as an onlooker, it is impossible for the subject to know his own actions in advance. In that sense, options are open to the subject: he is free to choose from among them. It is only because options are open to us that we choose a certain one.
To illustrate his point, Mackay gives an example such as this. A determinist knows all the factors acting on a particular person who is not sure whether to take the 7 o'clock train or the 9. This person decides at 6:30 to take the 7. The determinist then shows that this person could not have done otherwise (that is, he could not have taken the 9), and he is absolutely correct. But, Mackay argues, that does not mean that the person was not free in choosing. For, on the determinist theory, actions that are taken are reflected in the agent's brain-state. The determinist foretells, based someone's brain-state and a number of given factors, what his future brain-state will be (what action he will take).
But it would be impossible to tell a person at 4 o'clock that he will decide at 6:30 to take the 7. He would then not in fact be choosing at 6:30; the choice would already be made. In other words, his brain-state would be other than the determinist predicted that it would be.
I frankly do not remember if this Mackay uses this terminology, but this is an argument of the type '(non-)predictability from within'. That is, an agent cannot predict his own action. Only an outsider can predict the action of another. The agent is, then, free in that he cannot know his own future.
I do not buy this argument. It may be true that if the determinist shows his prediction to the agent, his prediction will not be valid (for the agent's brain-state will be different than he predicted it would be before he intervened); but there is a very clear flaw here. The prediction of the determinist is invalid only because the prediction did not take into account his telling the agent of his prediction. If he were going to tell the agent, he would have to have taken his telling into account in his prediction.
A complete prediction might look like this: "According to his brain-state and all other relevant factors, he will decide at 6:30 to take the 7 o'clock train. But if I tell him of my findings at 4 o'clock, he will see the truth of my findings, and will not have to decide at 6:30; he will see that the 7 o'clock is the one that he will take." Then, in telling the agent of the action, the determinist's prediction is not only kept from being falsified, but it is in fact brought about.
Further, it is irrelevant to one's freedom whether or not one can know how one will choose in the future. Whether or not one thinks that there are several options open to him, there is (in a mechanistic universe) only one action that he ever could take. So he cannot be held morally responsible for his actions the way he could if he had made a choice from among several actual possibilities. For that would simply be a case of holding responsible the son for the sins of the father.
I hope I have made myself sufficiently understood.