Imagine a world in which there are nothing but two atoms. If you are not a believer in mereological sums, you will think that you are thereby imagining a situation in which (1) is true.
(1) ExEy (~x=y & Az (z=x v z=y))
The conceiving of this imagined non-believer is positive and primary, as far as I can tell. Is it ideal conception? Well in one sense no, because it isn’t really a situation in which (1) is true. But that’s a sense in which conceivability trivially entails possibility. In Dave Chalmers’ preferred sense,
S is ideally conceivable when there is a possible subject for whom S is prima facie conceivable, with justification that is undefeatable by better reasoning.
In that sense, I think (1) is ideally conceivable. Some people believe that the 2 atom room verifies (1), and some of them won’t be talked out of this by better reasoning. There’s an important point about the epistemology of metaphysics here. Ultimately getting to the right answer involves a matter of judgement. There’s no valid argument from premises everyone accepts to the impossibility of (1). At least, there’s no such argument now, and there’s no reason to think that there ever will be one.
Could we restrict the ideal conceivers to those with appropriately good judgement? Yes, but only at the cost of trivialising the conceivability/possibility thesis.
In the paper I linked, Dave briefly discusses this case, and suggests somewhat tentatively “that there is no fact of the matter about the issue, or that it can only be settled by terminological refinement”. But it’s wildly unclear how either of these strategies will help. (Here I’m following some arguments Ted Sider has made really closely.) It’s implausible that there might be no fact of the matter about claims stated using only logical vocabulary, like (1). It’s also implausible that extra refinement of the terms will help. If the logical terms aren’t refined enough to use in stating propositions, who knows what is?
So I’m inclined to conclude that this is a case of conceivability without possibility. I don’t know if this case has been discussed widely, or indeed if it’s been discussed at all outside of a small note in Dave’s paper. So this is something I should do a little more research on.
One quick note about (1) to end with. Lots of people actually take (1) to be prima facie conceivable. The fact that they do so is, to those of us who are otherwise inclined to believe it is impossible, no evidence whatsoever that (1) really is possible. So why should the prima facie conceivability of zombies be any different?