Extended Abstract: Reconsidering Intuitionistic Duality

This paper proposes a new syntax and proof system called Dualized Intuitionistic Logic (DIL), for intuitionistic propositional logic with the subtraction operator. Our goal is a conservative extension of standard propositional intuitionistic logic with perfect duality (symmetry) between positive and negative connectives. The proof system should satisfy the following metatheoretic properties: soundness, completeness, cut elimination, and substitution. To our knowledge, no existing system achieves these goals. Substitution is needed for cut elimination, but has posed problems for other systems; for example Crolard develops a complex dependency-tracking calculus to obtain substitution for a constructive type theory with subtraction [3].