Non-classical Topological Evidence Logic

2026-06-30

AI Summary

Classical propositional logic struggles to capture the nuances of human reasoning, limiting the applicability of Topological Evidence Logic (TEL) in non-classical contexts. This work extends TEL to intuitionistic and relevant logical frameworks by introducing a global modality and an interior complement operator, thereby providing the first formal model of coherent epistemic justification within non-classical settings. Building upon the de Groot–Shillito intuitionistic modal framework, Standefer et al.’s relevant modal logic, and the weak relevant system BS4, the paper develops a novel language and demonstrates its robustness under changes of propositional bases. Furthermore, it establishes soundness and completeness theorems for a relevant topological evidence logic within BS4, offering a rigorous semantic foundation for non-classical theories of justification.

Abstract

Topological Evidence Logic (TEL) is a recent approach to epistemic logic that uses topological tools to model coherent epistemic justification. Specifically, a hypothesis is coherently justified if and only if it is entailed by a dense open set. In its simplest form, TEL can be formulated as an extension of S4 with a global modality. All currently studied forms of TEL are based on classical propositional logic, which has been heavily criticised for misrepresenting the way in which ordinary agents reason. In this article, we show that the TEL approach is robust under modifications to the propositional base. We show that an extension of the intuitionistic modal framework recently introduced by de Groot and Shillito, incorporating a global modality, enables coherent justification to be expressed in an intuitionistic setting. Furthermore, we adapt the recent work of Standefer et al., which extends relevant logic with a global modality, to show that coherent justification can be expressed in a relevant setting if an interior-of-complement operator is added to the language. Our main technical result is a soundness and completeness theorem for relevant TEL based on the weak relevant modal logic BS4.

Problem

• Topological Evidence Logic
• non-classical logic
• coherent justification
• intuitionistic logic
• relevant logic

Innovation

• Topological Evidence Logic
• Intuitionistic Modal Logic
• Relevant Logic
• Global Modality
• Soundness and Completeness