In this paper we first develop a Dialetheic Logic with Exclusive Assumptions
and Conclusions, DLEAC. We adopt the semantics of the logic of paradox (LP)
extended with a notion of model suitable for DLEAC, and we modify its proof
theory by refining the notions of assumption and conclusion, which are understood
as speech acts. We introduce a new paradox – the rejectability paradox –
first informally, then formally. We then provide its derivation in an extension
of DLEAC contanining the rejectability predicate.