Notes on neighborhood semantics for logics of unknown truths and false beliefs

This paper studies logics of unknown truths and false beliefs under neighborhood semantics. Intuitively, if p is true but you do not know that p, then you have an unknown truth that p; if p is false but you believe thatp, then you have a false belief thatp, or you are wrong aboutp. The notion of unknown truths is important in philosophy and formal epistemology. For instance, it is related to Verificationism, or'verification thesis' [31]. Verificationism says that all truths can be known. However, from the thesis, the unknown truth of p, formalized p Kp, gives us a consequence that all truths are actually known. In other words, the notion gives rise to a well-known counterexample to Verificationism. This is the so-called Fitch's'paradox of knowability' [13]. 1 To take another example: it gives rise to an important type of Moore sentences, which is essential to Moore's paradox, which says that one cannot claim the paradoxical sentence "p but I do not know it" [23, 18]. It is known that such a Moore sentence is unsuccessful and self-refuting (see, e.g.