Fundamental Logic Through the Lens of Modality

2026-06-29

AI Summary

This study investigates the semantic structure of basic logic and its deep connections with intuitionistic and orthologic systems. By integrating modal-logical tools with a Fitch-style natural deduction framework, the work establishes—for the first time—that both the Goldblatt–McKinsey–Tarski (GMT) translation and the Goldblatt translation enjoy full and faithful embedding properties within the setting of basic logic. The research constructs full and faithful embeddings of basic logic into modal systems S4 and a variant of KTB, thereby forging a novel semantic bridge among non-classical logics. This contribution offers a unified perspective and a formal pathway for understanding the intrinsic relationships between distinct logical systems.

Abstract

Fundamental logic is a non-classical logic based only on the introduction and elimination rules for conjunction, disjunction, negation, and the quantifiers in a Fitch-style natural deduction system. In this paper, we attempt to obtain a better understanding of fundamental logic and its semantics through the lens of modality. Using modal logic, we develop means of mutual understanding between the fundamental logician, on the one hand, and the orthologician and intuitionistic logician, on the other: we prove that the Gödel-McKinsey-Tarski (GMT) translation of intuitionistic logic into the classical modal logic $\mathsf{S4}$ is a full and faithful embedding of fundamental logic into the orthological version of $\mathsf{S4}$; that the Goldblatt translation of orthologic into the classical modal logic $\mathsf{KTB}$ is a full and faithful embedding of fundamental logic into an intuitionistic version of $\mathsf{KTB}$; and that the GMT translation is a full and faithful embedding of intuitionistic logic into a modal extension of fundamental logic.

Problem

• fundamental logic
• modality
• intuitionistic logic
• orthologic
• semantic understanding

Innovation

• fundamental logic
• modal logic
• Gödel-McKinsey-Tarski translation
• Goldblatt translation
• faithful embedding