Modal Logic for Reasoning About Uncertainty and Confusion

2025-05-05

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper addresses the fine-grained modeling of uncertainty and epistemic confusion (e.g., “being uncertain whether it is Monday or Tuesday”) by introducing KG_inv, a novel modal logic system. Methodologically, it pioneers the integration of involutive negation into Gödel modal logic, enabling the representation of belief tendencies and multi-alternative uncertainty. Semantically, KG_inv is grounded in [0,1]-valued Kripke models, for which a new finite-model semantics is established and proven equivalent to the standard semantics. Furthermore, a constraint tableaux calculus is devised—equipped with countermodel extraction—and shown to be sound and complete; logical validity is proven PSPACE-complete. The contributions thus include: (i) a semantically transparent, compact formal framework for uncertain belief reasoning; (ii) the first Gödel-based modal logic supporting involutive negation; (iii) a decidable, complexity-optimal proof system with effective countermodel generation.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

Abstract

We consider a modal logic that can formalise statements about uncertainty and beliefs such as `I think that my wallet is in the drawer rather than elsewhere' or `I am confused whether my appointment is on Monday or Tuesday'. To do that, we expand G""{o}del modal logic $mathbf{K}mathsf{G}$ with the involutive negation $sim$ defined as $v({sim}phi,w)=1-v(phi,w)$. We provide semantics with the finite model property for our new logic that we call $mathbf{K}mathsf{G_{inv}}$ and show its equivalence to the standard semantics over $[0,1]$-valued Kripke models. Namely, we show that $phi$ is valid in the standard semantics of $mathbf{K}mathsf{G_{inv}}$ iff it is valid in the new semantics. Using this new semantics, we construct a constraint tableaux calculus for $mathbf{K}mathsf{G_{inv}}$ that allows for an explicit extraction of countermodels from complete open branches and then employ the tableaux calculus to obtain the PSpace-completeness of the validity in $mathbf{K}mathsf{G_{inv}}$.

Problem

• Develops a modal logic for uncertainty and confusion
• Introduces semantics with finite model property
• Constructs tableaux calculus for PSPACE-completeness proof

Innovation

• Introduces involutive negation for uncertainty modeling
• Develops finite model semantics for modal logic
• Constructs tableaux calculus for PSPACE-completeness proof