The Geometry of Inconsistency: Modal Frames for Paraconsistent and Intuitionistic Logics

This paper explores the intricate relationship between modal logic, paraconsistent logic, and intuitionistic logic through the lens of Kripke semantics and algebraic structures. We investigate how modal frames, originally conceived for modal logics, can be adapted and extended to provide a unified geometric foundation for understanding inconsistency-tolerant reasoning and constructive proof. By developing a specialized framework of modal frames, we aim to model the distinct characteristics of paraconsistent logics, which permit contradictions without trivialization, and intuitionistic logics, which adhere to constructive principles and reject classical tenets like the law of excluded middle. The paper details the construction of such frames, elucidates the specific properties of accessibility relations required for each logical paradigm, and discusses the implications for a comprehensive "geometry of inconsistency." We examine how the structure of possible worlds and their interconnections can geometrically represent the presence of contradictory information or the incremental nature of constructive knowledge. Furthermore, we explore the duality between these relational frames and their corresponding algebraic semantics, such as Heyting algebras for intuitionistic logic and various lattices for paraconsistent systems, thereby bridging semantic and algebraic perspectives. The findings reveal a robust framework for analyzing the structural properties underlying non-classical reasoning, offering new insights into the formal treatment of inconsistency and constructive truth.