Fundamental Logic Through the Lens of Modality
2026-06-29 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors study fundamental logic, a system based on basic rules for how we use and combine statements, to better understand its meaning using modal logic (which deals with possibility and necessity). They show how fundamental logic connects with other types of logic, like intuitionistic and orthologic, by translating between them using known methods (GMT and Goldblatt translations). These translations demonstrate tight relationships where fundamental logic fits fully and accurately inside these other logical systems. This helps clarify how different logical approaches relate through the lens of modality.
fundamental logicmodal logicintuitionistic logicorthologicGödel-McKinsey-Tarski translationGoldblatt translationnatural deductionS4 modal logicKTB modal logicfull and faithful embedding
Authors
Wesley H. Holliday, Guillaume Massas
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.