  In this paper, we investigate proof-theoretic aspects of the logics of
evidence and truth LETJ and LETF. These logics extend, respectively, Nelson's
logic N and the logic of first-degree entailment FDE, also known as Belnap-Dunn
four-valued logic, with a classicality operator that recovers classical logic
for formulas in its scope. We will present natural deduction and sequent
systems for LETJ and LETF, together with proofs of normalization and
cut-elimination theorems, respectively. As a corollary, we obtain decidability
for both logics.
