Fun fact! Paradoxical propositions that are true if and only if you don't believe them are at the heart of Tarski's theorem on the undefinability of truth, and MIRI has figured out a way to make sense of them. Basically, if proposition P is true if and only if you assign less than 10% probability to P, then you ought to assign probability 10% to P, and you ought to believe that you assign probability "approximately 10%" to P.