We present a single sequent calculus common to classical, intuitionistic and linear logics. The main novelty is that classical, intuitionistic and linear logics appear as fragments, i.e. as particular classes of formulas and sequents. For instance, a proof of an intuitionistic formula A may use classical or linear lemmas without any restriction: but after cut-elimination the proof of A is wholly intuitionistic, what is superficially achieved by the subformula property and more deeply by a very careful treatment of structural rules. (...) This approach is radically different from the one that consists in “changing the rule of the game” when we want to change logic, e.g. pass from one style of sequent to another: here, there is only one logic, which—depending on its use—may appear classical, intuitionistic or linear. (shrink)

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative–additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut-elimination holds in such fragments. From cut-elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. (...) We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical–linear hybrid logics. (shrink)

We introduce a notion of the Kripke model for classical logic for which we constructively prove the soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications.