In some of the Smullyan books he extends the knights and knaves puzzles to incorporate beliefs with sane and insane variants. This is common in his Transylvania puzzles, where vampires always lie, humans always tell the truth, the sane believe true things, and the insane believe false things.
The sane human and insane vampire always tell the truth, even though it's not the vampire's intent to tell the truth. Meanwhile, the insane human always makes false statements though their intent is to tell the truth (and they do, they tell you what they believe to be true).
One cute thing there is that the insane people (and vampires) have immediately inconsistent beliefs.
For example, they believe "the sky is red", "the sky is yellow", "the sky is green"...
Also, they believe "I am sane" but also "I am insane and 1+1=3". (Or "George Washington is dead" but also "George Washington is still alive and 1+1=3".)
I don't think Smullyan ever had the insane people try to reason from their infinite store of false beliefs (as opposed to just knowing individual isolated false assertions). That could have made the puzzles much more confusing because they might conclude true beliefs as well as false ones, although maybe they also always get immediately confused about the results of their reasoning process and invert it?
Like, insane humans in the Transylvania puzzles believe "I am sane" but they also believe "I am insane and 1+1=3"; if they could perform the valid logical inference from the conjunct they could also then conclude "I am insane" alongside "I am sane". This would make the puzzles less interesting, because then insane humans could assert anything!
The sane human and insane vampire always tell the truth, even though it's not the vampire's intent to tell the truth. Meanwhile, the insane human always makes false statements though their intent is to tell the truth (and they do, they tell you what they believe to be true).