This article begins with recalling the three-valued characterizations of the cut-free and identity-free derivations in the usual Gentzen-style sequent calculus \(\mathcal {L}\mathcal {K}\) . Later we turn to a three-valued logic à la Kleene and four-valued logics in the spirit of Belnap and Priest. In each case sound and complete sequent calculi will be presented. The Priest four-valued logic plays an important role in approaching the tetralemma and the Catuskoti—prominent in Indian logic—from a Western perspective.