Natural Deduction with Explosion and Excluded Middle

Abstract

In this study, new natural deduction systems for intuitionistic, classical, and Gurevich logics are introduced using the new rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Gurevich logic is an extended three-valued logic obtained from intuitionistic logic by adding strong negation. Theorems for equivalence between these natural deduction systems and the corresponding previously proposed sequent calculi for the logics are proved. Normalization theorems for the proposed natural deduction systems for intuitionistic and Gurevich logics are proved as well.