Those considerations aside, the main way in which conceivability arguments can go wrong is by subtle conceptual confusion: if we are insufficiently reflective we can overlook an incoherence in a purported possibility, by taking a conceived-of situation and misdescribing it. For example, one might think that one can conceive of a situation in which Fermat’s last theorem is false, by imagining a situation in which leading mathematicians declare that they have found a counterexample. But given that the theorem is actually true, this situation is being misdescribed: it is really a scenario in which Fermat’s last theorem is true, and in which some mathematicians make a mistake. Importantly, though, this kind of mistake always lies in the a priori domain, as it arises from the incorrect application of the primary intensions of our concepts to a conceived situation. Sufficient reflection will reveal that the concepts are being incorrectly applied, and that the claim of logical possibility is not justified.
So the only route available to an opponent here is to claim that in describing the zombie world as a zombie world, we are misapplying the concepts, and that in fact there is a conceptual contradiction lurking in the description. Perhaps if we thought about it clearly enough we would realize that by imagining a physically identical world we are thereby automatically imagining a world in which there is conscious experience. But then the burden is on the opponent to give us some idea of where the contradiction might lie in the apparently quite coherent description. If no internal incoherence can be revealed, then there is a very strong case that the zombie world is logically possible.
As before, I can detect no internal incoherence; I have a clear picture of what I am conceiving when I conceive of a zombie. Still, some people find conceivability arguments difficult to adjudicate, particularly where strange ideas such as this one are concerned. It is therefore fortunate that every point made using zombies can also be made in other ways, for example by considering epistemology and analysis. To many, arguments of the latter sort (such as arguments 3-5 below) are more straightforward and therefore make a stronger foundation in the argument against logical supervenience. But zombies at least provide a vivid illustration of important issues in the vicinity.
(II.7, “Argument 1: The logical possibility of zombies”. Pg. 98).
You may find it helpful to read the relevant sections of The Conscious Mind by David Chalmers, the original thorough examination of his view:
(II.7, “Argument 1: The logical possibility of zombies”. Pg. 98).