The following is an outline of Chalmers' argument:
In this paper, I am only going to briefly discuss Chalmers' argument for premise 1, that zombies are logically possible. I wholeheartedly agree with the entirety of Chalmers' argument---in fact, it seems to me very similar to my own argument for the possibility of zombies that I have developed independently---but our first premise is quite seriously challenged by Harman.
When Chalmers says that zombies are ``logically possible'' he means that the existence of zombies violates no laws of math or logic (where such laws are not constrained to those of any particular formal system). Chalmers' primary reason to believe in the logical possibility of zombies is very simple and quite compelling to many people, including me: It just seems obvious that they are.
Unfortunately, no matter how convincing such an argument is to the group of people who already agree with it, this form of argument usually does little to convince those who don't. I would add (though I don't think Chalmers does) that phenomenal properties seem to have a quality to them that is just not in the domain of math and logic. Thus, logic cannot predict the existence of phenomenal properties, because they are outside of logic's domain. Chalmers does try to do a bit of intuition massaging, which I won't go into here, and he does point out that arguing for the logical possibility of something that you can't demonstrate physically is the same thing as arguing for its conceptual coherence; that is, arguing that there is no internal contradiction in its description. Showing that a position is self-consistent is notoriously difficult, so Chalmers throws the ball into the other court and challenges the opposition to find the contradiction that would make zombies logically impossible.
Harman purports to be up to this challenge. He claims that a properly programmed computer would necessarily have phenomenal properties. Zombies are not logically possible because any physical or intentional duplicate of me would necessarily be such a properly programmed computer.