No plot spoilers here, just wanted to flag a bit of poor reasoning that shows up in Chapter 39:
I shouldn't have believed it even for all of thirty seconds! Because if people had souls there wouldn't be any such thing as brain damage, if your soul could go on speaking after your whole brain was gone, how could damage to the left cerebral hemisphere take away your ability to talk?
This is a surprisingly common fallacy. Just because X depends on Y, it doesn't follow that X depends on nothing but Y. A phenomenon may involve more than just its most obvious failure point.
To illustrate: Suppose I'm trapped in a box, and my only way to communicate with the outside world is via radio communication. Someone on the other end argues that I don't really exist -- "There's no person beyond the radio receiver, for if there was then there wouldn't be any such thing as damaged radios!" Pretty silly, huh? But people say this kind of thing in defense of physicalism all the time.
(N.B. This is not to defend the existence of souls. It's just to point out that this particular argument against them is invalid.)
(shrug) Sure. It's not a logical impossibility that all the valuable stuff resides in the soul, and that the brain serves only relatively valueless functions (akin to a radio), such that when we eliminate the brain the bulk of the value is preserved.
One question worth asking is how much attention I want to pay to "is X logically impossible?" vs "does there exist enough evidence for X to make it worth considering?"
Or, in the context you raise this in, "should we understand Harry to be talking about whether the soul hypothesis is logically possible, or talking about whether the soul hypothesis is worth considering?"
The utterance you quote is consistent with both readings.
I agree that the soul hypothesis is not generally worth taking seriously. What I'm denying is that the existence of brain damage is good evidence for this.