Jump Liars and Boolean Paradoxes

The n-jump liar, being the generalization of the liar paradox, is a paradox whose sentences change their truth values every n points in any relational frame. It is proved that whenever n>1, then the n-jump liars cannot be represented by any Boolean paradox owing to the semantic closeness of the Boolean paradoxes. However, for any number n, we can construct a Boolean paradox, that is, the so-called weak n-jump liar, satisfying the condition for the n-jump liar in some weak sense. These results provide a partial solution to the definability problem of the n-jump liar.