What is conceivable is not necessarily ultimately logically possible.
We can conceive of a moon made of Green Cheese, but since humans invented Cheese, this is not logically possible, since it seems obvious that the moon is older than we are. We didn't have any problem creating the mental image in order to analyze it, but we cannot believe in it either.
So belief is a state between conceivability and logical possibility. I would suggest there are at least two more that are actually important: (scientifically) creditable' and '(intuitively) appealing'.
(I hope what follows is not just belaboring the obvious as an excuse to describe a metaphor I find compelling.)
One way to capture the range of definitions involved would be to reify the language game of logic. Imagine that everyone who negotiates our culture's belief system made up a small number of people who could actually take turns talking, each one of them trying to get the group's approval of their thoughts, and that approval or disapproval was communicated directly via a visible emotional response. This is a simplified metaphor for what Wittgenstien describes as a cultural language-game. Logic is one such language game: one such topic this group might agree to focus on regularly under certain conditions.
One can conceive of anything that one might think could motivate a move -- that could suggest an utterance that captures an image and one might hope could contribute to a consensus it has meaning.
But something is only believable if there is not a defeating series of countermoves that reduces it to a contradiction quickly enough that the contradiction becomes apparent before the ideas consequences can be elaborated. In order to believe it, it has to set in and have meaning and effects on your other thoughts before it is dissolved by noticing its weaknesses.
And it is only 'scientifically' creditable if someone who has all the known material related to it as an expert, is still likely to find it believable. It can only gain credit as an idea in a domain of knowledge if it can be communicated around that domain without falling prey to an easy dismissal.
If there is still a contradiction, but it takes a lot of moves to prove it, and the proof is unlikely to occur to a random well-informed expert, so that the idea is actually likely to persist in most people if introduced, we might call such a thing an illusion.
If the illusion is hard to dispell, and the resulting deserved disbelief is often temporary, or has to be rehearsed in order to become acceptable, we sometimes call such a state 'counterintuitive'. This is a word you often hear describe results in mathematics related to infinities or complex structures. And we call the illusion itself 'intuitively appealing'.
But even the most intuitively appealing illusion is still not logically possible. What is logically possible is not disprovable with our most complete analysis. Disbelief in it cannot be stable over time even as something captured with difficulty.
We might develop conceivability in its raw state, but it does not seem like that would be an improvement.
Many would prefer to develop 'intuition' in the sense of being able to accept that illusions are illusions even when that is counterintuitive, and therefore to also accept counterintuitive truths -- things whose opposites are illusions that would be intuitively appealing.
This might need a broader range of concepts and therefore an ability to conceive of things and get to the point of believability, but that is not what it seems to be mostly 'made of'. We seem to be able to trust other humans enough to acquire a belief from them of which we would not ourselves readily conceive, stepping past the point of conceiving of it and 'falling backward' into that state from the arguments that led those other people into belief.
