Dualized Simple Type Theory

We propose a new bi-intuitionistic type theory called Dualized Type The-ory (DTT). It is a simple type theory with perfect intuitionistic duality, and correspondsto a single-sided polarized sequent calculus. We prove DTT strongly normalizing, andprove type preservation. DTT is based on a new propositional bi-intuitionistic logic calledDualized Intuitionistic Logic (DIL) that builds on Pinto and Uustalu’s logic L. DIL is asimplification of L by removing several admissible inference rules while maintaining consis-tency and completeness. Furthermore, DIL is defined using a dualized syntax by labelingformulas and logical connectives with polarities thus reducing the number of inference rulesneeded to define the logic. We give a direct proof of consistency, but prove completenessby reduction to L.