Hypersequent Calculus for Intuitionistic Logic with Classical Atoms

Abstract

We introduce a hypersequent calculus for intuitionistic logic with classical atoms, i.e. intuitionistic logic augmented with a special class of propositional variables for which we postulate the decidability property. This system combines classical logical reasoning with constructive and computationally oriented intuitionistic logic in one system. Our main result is the cut-elimination theorem with the subformula property for this system. We show this by a semantic method, namely via proving the completeness theorem of the hypersequent calculus without the cut rule. The cut-elimination theorem gives a semantic completeness of the system, decidability, and some form of the disjunction property.