From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses

2025-08-10

AI Summary

This paper addresses the limitation of classical epistemic logic in formalizing hypothetical reasoning—specifically, distinguishing factual assertions from hypothetical ones while preserving established facts and dynamically exploring their consequences. Methodologically, it abandons Axiom T and adopts a non-classical semantics without modal collapse; introduces a “settle” operator to model the dynamic transition from conjecture to fact; and constructs an incomplete semantics based on Weak Kleene logic or description logic, integrated with modal systems KC and KDC. The main contributions are: (i) the first logical separation of facts and hypotheses within a unified framework; (ii) a decidable, sound, and robust system for conjectural reasoning; and (iii) formal verification of logical consistency for multi-level hypothetical reasoning and its dynamic updating under partial knowledge.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• 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: Research challenges in human and human-AI computation

Abstract

This paper introduces a new family of cognitive modal logics designed to formalize conjectural reasoning: a modal system in which cognitive contexts extend known facts with hypothetical assumptions to explore their consequences. Unlike traditional doxastic and epistemic systems, conjectural logics rely on a principle, called Axiom C ($varphi ightarrow Boxvarphi$), that ensures that all established facts are preserved across hypothetical layers. While Axiom C was dismissed in the past due to its association with modal collapse, we show that the collapse only arises under classical and bivalent assumptions, and specifically in the presence of Axiom T. Hence we avoid Axiom T and adopt a paracomplete semantic framework, grounded in Weak Kleene logic or Description Logic, where undefined propositions coexist with modal assertions. This prevents the modal collapse and guarantees a layering to distinguish between factual and conjectural statements. Under this framework we define new modal systems, e.g., KC and KDC, and show that they are complete, decidable, and robust under partial knowledge. Finally, we introduce a dynamic operation, $mathsf{settle}(varphi)$, which formalizes the transition from conjecture to accepted fact, capturing the event of the update of a world's cognitive state through the resolution of uncertainty.

Problem

• Formalizing conjectural reasoning with cognitive modal logics
• Preventing modal collapse via paracomplete semantic frameworks
• Modeling transition from conjecture to fact dynamically

Innovation

• Cognitive modal logics formalize conjectural reasoning
• Paracomplete semantics prevent modal collapse
• Dynamic operation settles conjectures into facts