Another Combination of Classical and Intuitionistic Conditionals

Abstract

On the one hand, classical logic is an extremely successful theory, even if not being perfect. On the other hand, intuitionistic logic is, without a doubt, one of the most important non-classical logics. But, how can proponents of one logic view the other logic? In this paper, we focus on one of the directions, namely how classicists can view intuitionistic logic. To this end, we introduce an expansion of positive intuitionistic logic, both semantically and proof-theoretically, and establish soundness and strong completeness. Moreover, we discuss the interesting status of disjunction, and the possibility of combining classical logic and minimal logic. We also compare our system with the system of Caleiro and Ramos.

Semantic Incompleteness of Hilbert system for a Combination of Classical and Intuitionistic Propositional Logic

“…The second one is to add classical propositional variables to intuitionistic logic, an approach that is taken in [32,33,34,43,53]. 1 The third one is to introduce classical and intuitionistic operators, an approach that is taken in [13,14,20,21,22,23,28,29,37,41,42,47,48,49,50,54]. 2 The combination C + J, which is studied in [22,23,28,37,54], is based on the third way and is one of the most studied combinations of classical and intuitionistic logic.…”