David Lewis famously endorsed the possibility of "mad pain" in his article "Mad Pain and Martian Pain":
There might be a strange main who sometimes feels pain, just as we do, but whose pain differs greatly from ours in its causes and effects. Our pain is typically caused by cuts, burns, pressure, and the like; his is caused by moderate exercise on an empty stomach. Our pain is generally distracting; his turns his mind to mathematics, facilitating concentration on that but distracting him from anything else. Intense pain has no tendency whatever to cause him to groan or writhe, but does cause him to cross his legs and snap his fingers. He is not in the least motivated to prevent pain or get rid of it. In short, he feels pain but his pain does not at all occupy the typical causal role of pain.Mad pain in this sense seems to me conceivable. My question is: Could there be a parallel case for belief? Let's try to imagine such a case.
Daiyu, say, is a woman who believes that most pearls are white. However, this belief was not caused in the normal way. It was not caused by having seen white pearls nor by hearing testimony to the effect that most pearls are white or inferring that pearls are white from some other facts about pearls or whiteness. It was caused, say, by having looked for 4 seconds at the sun setting over the Pacific Ocean. And, for her, this is just the kind of event that would cause that belief: It's not the case that she would ever form that belief by any of the normal means such as those described above; rather the kinds of things that cause that belief in her in all "nearby possible worlds", or across the relevant range of counterfactual circumstances, are perception of setting-sun events of a certain sort, and maybe also eating a certain sort of salad. Furthermore, Daiyu's belief that most pearls are white has atypical effects. It does not cause her to say anything like "most pearls are white" (which she'd deny; she is actually disposed to say "most pearls are black") or to think to herself in inner speech "most pearls are white". She would feel surprise were she to see a white pearl. If a friend were to say she was looking for white jewelry to go with a dress, Daiyu would not be at all inclined to recommend a pearl necklace. She is not at all disposed to infer from her belief that most pearls are white that there is a type of precious object used in jewelry that is both round and white.
Now I'm inclined to think that this case is incoherent. If Daiyu in fact has that sort of causal/functional structure, it's not correct to say that she really does believe that most pearls are white. In this respect, belief is different from pain. If you agree with me about this, that would seem to rule out a certain class of views about belief, namely, those views that characterize belief in terms of a mental state (maybe a brain state) of the sort that, in humans, typically has certain sorts of causes and typically has certain sorts of effects but which may, in some particular individuals, be not at all apt to have been brought about those causes and be not at all apt to have those effects. It's hard to know exactly how to read "representationalists" about belief (like Fodor, Dretske, Millikan) on this point, but a certain way of reading the representationalist view would imply no incoherence in the idea of mad belief: If an individual possesses an internal representation of the right sort, held in such a way that if everything were functioning normally it would have the normal effects, that person believes -- even if everything is not functioning normally.
Compare: having a heart. Hearts might be defined in terms of their normal functional role (to pump blood), but a being can still have a heart even if that heart fails utterly to fill that normal functional role (in which case the being will presumably either not be viable or have its life sustained somehow without a functioning heart). I'm a type functionalist about hearts: To have a heart is to have the type of organ that normally fills the causal role of hearts even if in one's own case the organ does not fill that causal role. Lewis is a type functionalist about pain. But if the Daiyu case is incoherent, we should not be type functionalists about belief. Closer to the truth, I suspect, would be token functionalism: To believe is to be in a state that actually, for you, plays the functional/causal role characteristic of belief. I'm not sure how readily representationalists about belief, especially those who think of mental representations as biological types or real in-the-head entities, can take token functionalism on board. Perhaps they are committed to the possibility of mad belief.