We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model M, or an axiomatization S thereof, we find a modal logic M such that a modal sentence A is a theorem of M if and only if the sentence A* obtained by translating the modal operator with the truth predicate is true in M or a theorem of S under all such translations. To this end, we introduce a novel possible world semantics featuring both classical and subclassical worlds and establish the completeness of a familiy of non-classical modal logics (in the sense of Segerberg), whose internal logic is subclassical, with respect to this semantics. In a second step we show how to emulate the models of the modal logic within the lattice of Kripkean fixed-point models.