The Zombie Argument provides, in its modern forms, a very effective attack on minimal materialism. Zombies are creatures without phenomenal consciousness. Minimal materialism says that all mental facts are entailed by the totality of physical facts. So, take a universe identical to this one in respect of all its physical properties, and without any extra non-physical stuff like ectoplasm (in technical terms, a minimal physical duplicate). Minimal materialism requires that it is impossible that that universe could differ from the actual universe in terms of its mental properties.

The zombie argument says, conversely, that it is possible that a universe could be a minimal physical duplicate of this one and have no mental properties at all.

Though it could take the form given by Nyte, above, there is an extra premise that can be unpacked:

1) Zombies are conceivable
2) x is conceivable -> x is possible.
C) Therefore zombies are possible

We can question both (1) and (2). Nyte's example of my being able to imagine "a triangle such that the length of one side is longer than the length of the other two combined" could be taken as showing two things - either, we can't really conceive what we think we can conceive of, or what we can conceive of is no guide to what is possible.

In his 2002 article "Does Conceivability Entail Possiblity", David Chalmers unpacks the notion of conceivability into a variety of forms. The most important distinction he derives, for present purposes, is that of ideal vs prima facie conceivability. Basically, I can ideally conceive of something if I can keep on enriching my conception of it without ever finding a contradiction. Conversely, I can prima facie conceive of something if it doesn't seem obviously contradictory.

Thus I can prima facie conceive of the falsity of Fermat's Last Theorem, but no-one could ideally conceive of it - in filling out my conception of it, I'd have to consider various propositions which must be true if Fermat's Last Theorem is false, and one of these would be the negation of Andrew Wiles' proof. Given that his proof works, I'd realise that I can't actually conceive of the falsity of Fermat's Last Theorem. The same is true of Nyte's example: I might think I can conceive of the kind of triangle he describes, but when I imagine it in richer detail, I'll find that it's mathematically impossible (well, in Euclidean Space, anyway).

Given that we're all finite intellects, we can't ideally conceive of the possibility of making a cup of tea. So why should we think zombies are ideally conceivable? Chalmers uses the analogy of a mile high unicycle. We can't ideally conceive that THAT is possible, since we can't imagine every last atom that would compose it, but we're totally confident that it IS possible. Why? Well, it just seems totally unbelievable that the universe is going to enforce a maximum height for unicycles; no metaphysical laws are going to kick in once I get about 1400ft.

The same, he says, is true for zombies. I can imagine one neuron at work and coherently conceive that it's not generating any mental states. Add another, and another. It seems ridiculous to imagine that there will be an n+1 neuron that makes me go, "oh, wait, I SEE, obviously this thing must be conscious". (This is indirectly related to the Sorites Paradox)

Chalmers therefore suggests that we have good reason to believe that zombies ARE ideally conceivable, and that ideal conceivability does entail possibility. Therefore, minimal physicalism is false. From this, it follows that God exists, all are one and there is no death, and pets go to heaven.

His arguments have been fairly influential, though physicalist hardliners like Daniel Dennett have remained unimpressed. The second premise has now become the battleground for the zombie argument. More moderate Physicalists have sought to challenge the zombie argument on the second premise, that the fact that X is (even ideally) conceivable does not entail that it is possible

It might seem obviously false that something that is even ideally conceivable is thereby possible. Superman is probably ideally conceivable, as is travelling faster than the speed of light, but both are impossible. The problem is that they are physically impossible. If, per impossibile, we travelled to another dimension with different physical laws, perhaps we could find Superman travelling faster than a slow beam of light. That universe would have to be fundamentally different from our own, but there's no reason we should think it's a logical impossibility that physical laws could have been different. If minimal physicalism is true, and mental states JUST ARE higher-order properties of physical states, then zombies are logically impossible: the idea that a physical duplicate of me might lack mental states is impossible in the same way that it is impossible that the number 7 might not be prime.

Despite its prima facie implausibility, then, ideal conceivability ->logical possibility is quite hard to challenge. Most attacks on this premise make use of Saul Kripke's highly complicated architecture of necessary a posteriori truths.

It's probably fair to say, however, that since Frank Jackson's repudiation of the Mary's Room argument, zombies are the biggest threat to Physicalism out there, and the fact that Physicalism still hasn't come up with a decent answer to the problem (which arguably goes back to Descartes' "I can conceive of existing without my body, so I am not identical to my body") means that if anything, it's looking increasingly dangerous.