(See /fisking for a chapter-by-chapter breakdown.)
The key problem with Harris's argument in this book is that he never defines its topic, "free will". This leaves him free to continually move the goalposts of what "free will" allows and requires (possibly without realizing he is doing so) as he identifies arguments which support it -- until he is left with a definition which is not falsifiable. Worse, he conflates different possible usages of the term, and then is forced to gloss over the inconsistencies between the conclusions he reaches from each one.
Harris's basic conclusion is that "free will is an illusion".
He would have done much better, I think, if he had said something like this:
Free will is a very elusive concept. It is both less than we think it is, and more complex than we generally imagine. At the edges, it blends smoothly into that which we consider to be "outside" of us, with no clear boundary; at its core, it operates (and sometimes malfunctions) by means beyond our direct control or understanding. This has profound implications for society, which tends to view "free will" as being much more of a discrete component in our cognition; our view of what is and is not voluntary tends therefore to be very all-or-nothing: either we are responsible for any given action, or we are not -- but this is a false dichotomy, and one we would do well to address with all the scientific rigor we can bring to bear.
Instead, he leaves himself to defend one end of another false dichotomy -- the idea that free will doesn't really exist at all -- and consequently is unable to make any meaningful statements about how we should proceed, his (often very sensible) suggestions for modifying social mores (with regard to personal achievement and criminal rehabilitation) supported only by hand-waving.
If, instead, he had started by defining his terminology, perhaps he would not have led himself down that dead-end path, and we would see a clear argument for some very badly-needed reforms.
For convenience, I'll use the following shorthand to refer to specific meanings of "free will":
Tentatively, there are two main usages of the term, which I'll refer to as academic (FW!aca) and lay (FW!lay).
However vague Harris may be about exactly what he thinks "free will" is (even as he argues that it doesn't really exist), it does seem clear that he is talking about it in the sense that most people understand it (FW!lay): the idea that we make choices to optimize our own experience, and that sometimes we need to set up deterrents* in order to prevent people from acting in ways that benefit themselves while harming others. (I have attempted to more precisely define this concept here.)
(* actually, it's not clear that this is why he thinks FW!lay calls for "punishment"; investigating this further.)
From all the clues Harris has left scattered through his book, here is what we know about FW!SH (my understanding of the assertions contained within each quote is written in boldface):
- "...most of what is distinctly human about our lives seems to depend upon our viewing one another as autonomous persons, capable of free choice."
- Society bases its rules on the idea that people have free will.
- "Without free will, sinners and criminals would be nothing more than poorly calibrated clockwork, and any conception of justice that emphasized punishing them (rather than deterring, rehabilitating, or merely containing them) would appear utterly incongruous. And those of us who work hard and follow the rules would not “deserve” our success in any deep sense."
- Free will is required in order for people to be something other than clockwork.
- Free will is required in order for punishment to be
an effective deterrentnecessary in some way other than as deterrent, rehabilitation, or containment.
- Free will is required in order for individuals to deserve the fruits of their efforts in any meaningful way.
- "Thoughts and intentions emerge from background causes of which we are unaware and over which we exert no conscious control."
- Free will requires conscious awareness of our decisionmaking process.
- "Consider what it would take to actually have free will. You would need to be aware of all the factors that determine your thoughts and actions, and you would need to have complete control over those factors."
- Free will requires not just conscious awareness of our decisionmaking process, but conscious awareness of every detail of that process.
(Note: I'm unclear what use he thinks punishment is within FW!lay if it's not as a deterrent.)
 awareness of decisionmaking
One argument Harris returns to repeatedly is the idea that if we are not aware of every detail of the process by which we make a decision, then we are not truly the author of that decision. The decision was made by the atoms, chemical processes, and neurons that our brains are made of, and which science has demonstrated pretty conclusively to be "all we are".
This is a scope error. It's like saying that if we cannot enumerate and explain every muscle movement we make while riding a bicycle, and the process by which we decided to make each movement, then we aren't really in control of the bicycle. It's like saying I didn't really write the words you're reading now, because all I did was move my fingers up and down a lot; my computer actually did the work of noting the pattern of keystrokes caused by those finger-movements, recording them for later recall, and sending them through the internet to be available on the web.
 acting solely on past experience
Another one is the idea that we act solely on past experience rather than on some spontaneous inner drive, which somehow (to Harris and apparently to others) implies the nonexistence of free will.
I think perhaps the error here is one of scale. We have a hard time imagining, in a systematic way, just how complex "acting on past experience" can be:
We project those experiences into the future, and attempt to simulate the consequences of actions we are considering -- even if we have never taken those actions before.
Yes, this is still acting on experience -- but so are the moves in a chess game. We now have computers powerful enough to beat the best human players -- but we still can't say for sure who will win, much less name the exact moves that will be played. The algorithms used are run like clockwork, yet they aren't exactly what one would call "deterministic". Due to limitations on computing due to limitations of the size of the universe, there no purely deterministic way to know the best countermove for any given move in chess. The real-world constraints on processing and storage make it impossible to ever calculate all possible chess games, even though in theory it could be done. (Perhaps quantum computing will make it possible, but that's a separate discussion.)
We communicate our understanding of reality via symbolic and representational means.
Person A can warn Person B about an otherwise-invisible danger, causing Person B to avoid it. Is Person B "acting on the basis of one's past experience", "acting on the basis of someone else's past experience", or something more complex? In any case, it's not the simple stimulus-and-response image that "acting on one's past experience" summons. It is complex and dynamic and unrepeatable and (at some level of precision) unpredictable.
I can only wonder if I'm misunderstanding Harris's argument in some significant way; hopefully someone will set me straight if this is the case.
My more detailed fisking of the book is here.