GIL: A Generalization of Intuitionistic Logic

We introduce GIL, a new logic inspired by linear logic, focusing and polarization. GIL is a unified logic in which connectives from intuitionistic, classical and linear logic can mix with few restrictions. Systems that resemble it include Girard’s logic LU, although GIL is best considered a generalization of intuitionistic principles. Intuitionistic logic is seen as a less-complete version of GIL. The logic is organized around three axes of polarization which correspond to three distinct layers of focalization. The need for additional polarities is explained in the context of cut-elimination. We demonstrate the strength of this approach relative to linear logic and related systems. An extensive set of structural rules allow for the uniform presentation of GIL as a focused sequent calculus.