Modal extensions of the Logics of Formal Inconsistency (LFIs) warrant new appropriate approaches to certain questions involving the Paradox of Knowability. We show that any propositional logic $L$ which extends the classical implicative fragment ImPC and which is sound and complete with respect to dyadic semantics can be, by its turn, extended to a modal logic $KL$ $\langle j,k,m,n\rangle$ (through the axiom schemes (K), $G$ $\langle j,k,m,n\rangle$ and the Necessitation) in such a way that $KL$ $\langle j,k,m,n\rangle$ is constructively characterizable by Kripke structures, independently of any properties of negation. In particular, families of modal LFIs can in this way be defined, with application to questions of knowability.