The p-zombie theory holds that being able to conceive of something makes it possible; and because p-zombies are possible, therefore dualism or something very like it. That is: proponents hold that because they can imagine p-zombies, therefore such a thing is possible.
(I'm not going to explain it further than linking to Wikipedia. But trust me when I say there are people who actually take the idea seriously.)
The tricky bit appears to be "conceive of" in a sense that implies possibility. Consider these statements:
- I can conceive of 2+2=4 being true (in conventional everyday Peano arithmetic as we commonly know it).
- I can conceive of 2+2=5 being true (in conventional Peano arithmetic).
- I can conceive of P being equal to NP.
- I can conceive of P not being equal to NP.
- I can conceive of p-zombies, therefore dualism.
- If I can conceive of p-zombies then dualism, which is a confused idea, therefore p-zombies is a confused idea by reductio ad absurdum.
With the second, I am claiming to "conceive of" something trivially false. I arguably haven't conceived of anything actually possible; I've just shuffled some words together.
With the third and fourth, I'm claiming to have conceived of something no-one knows (though many suspect 3 is false and 4 is true). To what extent have I actually thought it through? At some point I will hit a contradiction with one of them, though no-one has yet. Both are "conceivable" in some sense; certainly that the speaker has formed a sentence in their head that they can try out for its logical implications. But one of those statements is as wrong as 2+2=5 nevertheless. Thus, conceiving of something in this sense does not imply it can possibly be true.
When someone claims that p-zombies are a conceivable thing at all, and that they have conceived of them (first part of statement five), this doesn't actually say anything about the world or what is even possible; it just says they've formed a sentence in their head they think they can try out for its logical implications. Which is fine, but the world doesn't care what philosophers think they think.
Statement six is my own view. P-zombies is like creationism for smart people. The main argument for dualism remains its advocates really really wanting it to be true.
And now, a movie.