Alongside the sequent calculus for Classical Core Logic \({\mathbb {C}}^+\) we set forth some new sequent calculi that we call \({\mathbb {T}}\) , \({\mathbb {C}}^{++}\) , and \({\mathbb {K}}\) . \({\mathbb {T}}\) encodes truth-tabular reasoning; \({\mathbb {C}}^{++}\) classicizes Core Logic \({\mathbb {C}}\) by having multiple succedents; and \({\mathbb {K}}\) is a cut-free sequent calculus inspired by insights of Ketonen. Our aim is to establish that \({\mathbb {C}}^{++}\) is weakly complete, and to do so in a way that makes no use of standard semantic notions or the rule of Cut. All the concepts defined and applied in the course of this study will be proof-theoretic, turning on rules of inference as meaning-constitutive. It is intended to be a contribution to the program of proof-theoretic semantics, from the distinctive vantage point of the Core logician.
