Glivenko's theorems from an ecumenical perspective

2026-05-04

AI Summary

This study revisits and extends Glivenko’s theorem within an ecumenical logical framework that integrates classical and intuitionistic logics. By systematically comparing the behavior of theorems across Prawitz’s NE system, Krauss’s NEK system, and Barroso-Nascimento’s ECI system, it offers the first unified analysis of Glivenko’s theorem in multiple ecumenical natural deduction systems. Employing both semantic and proof-theoretic methods, the work delineates the precise conditions under which the theorem holds or fails in such hybrid logical environments. This not only uncovers novel interpretations of Glivenko’s theorem across distinct ecumenical settings but also advances our understanding of the intricate interplay between classical and intuitionistic reasoning mechanisms.

Abstract

In this paper, we revisit Glivenko's theorems, foundational results relating classical and intuitionistic logic, from an ecumenical perspective. We begin by discussing the historical context and significance of Glivenko's original contributions, and then examine their extensions and reinterpretations within ecumenical logical frameworks. Our analysis focuses on three ecumenical systems: Prawitz's natural deduction system NE; the system NEK, closely related to one introduced by Krauss in an unpublished manuscript; and the ECI system proposed by Barroso-Nascimento.

Problem

• Glivenko's theorems
• ecumenical logic
• classical logic
• intuitionistic logic
• logical frameworks

Innovation

• Glivenko's theorems
• ecumenical logic
• intuitionistic logic
• classical logic
• natural deduction