Vagueness and the Connectives

2024-11-30

arXiv.org

AI Summary

This paper addresses the non-classical logical behavior arising jointly from linguistic vagueness and epistemic modality, asking whether classical logic must be replaced as the semantic foundation—and if so, what form such a replacement should take. Systematically comparing orthologic and intuitionistic logic, the authors show that neither adequately accommodates both phenomena: orthologic fails to capture vagueness appropriately, while intuitionistic logic over-constrains modal behavior. In response, the paper proposes a novel, strictly weaker base logic that relaxes key principles—including negation introduction and disjunctive properties. This framework is the first to uniformly model vagueness (grounded in Kit Fine’s theory) and epistemic modality within a single non-classical semantic system. By doing so, it provides a more explanatory and flexible foundation for non-classical semantics, advancing the theoretical understanding of how logical structure interacts with natural language meaning.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic.

Problem

• Identify natural non-classical base logic for epistemic modals and vagueness.
• Evaluate orthologic as a potential solution for non-classical logic challenges.
• Explore weaker fundamental logic systems accommodating epistemic and vague predicates.

Innovation

• Weakening classical logic to orthologic
• Exploring intuitionistic logic for vagueness
• Integrating non-classicality from epistemic modals