Polarized Intuitionistic Logic

We introduce Polarized Intuitionistic Logic, which allows intuitionistic and classical logic to mix. The logic is based on a new analysis of the intuitionistic distinction between “left” and “right” as a form of polarity information. In contrast to double-negation translations, classical logic is transparently captured. The logic is given a Kripke-style semantics and is presented as a sequent calculus that admits cut elimination. We discuss the impact of this logic on traditional intuitionistic concepts such as Glivenko’s theorem and Markov’s principle.