A Unified Sequent Calculus for Focused Proofs

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical logic, intuitionistic logic, and multiplicative-additive linear logic are derived as fragments of LKU by increasing the sensitivity of specialized structural rules to polarity information. We develop a unified, streamlined framework for proving cut-elimination in the various fragments. Furthermore, each sublogic can interact with other fragments through cut. We also consider the possibility of introducing classical-linear hybrid logics.