Another Combination of Classical and Intuitionistic Conditionals

TL;DR

This paper explores how classical logic can be extended to incorporate intuitionistic principles through a new logical system, establishing its soundness and completeness, and comparing it with existing systems.

Contribution

It introduces an expanded positive intuitionistic logic with semantic and proof-theoretic foundations, and discusses combining classical and minimal logic.

Findings

Established soundness and strong completeness of the new logic

Analyzed the role of disjunction in the combined logic

Compared the new system with Caleiro and Ramos's system

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.