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.

This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in~1995. We show that Kuhn's function is actually equal to a related function originally given by Humberstone.