Better Understanding, Understanding Better

2026-06-30

AI Summary

Traditional epistemic logics fail to formally capture the graduality and comparability of “understanding.” This work proposes a comparative logic of understanding that introduces hierarchically indexed understanding modalities and comparative connectives. By integrating graded explanatory structures with proof-like term algebras in a multi-agent setting, the framework uniformly characterizes understanding ranging from finite to ideal degrees and enables comparisons of relative understanding strength among agents. The system provides the first formalization of both graduality and comparability of understanding, distinguishes between bounded finite and unbounded linguistic settings, and establishes a sound and strongly complete logical framework. Moreover, it proves that each finite-level fragment is decidable, thereby laying a formal foundation for modeling understanding in artificial intelligence and philosophy.

Abstract

"Any fool can know; the point is to understand." A well-known remark often attributed to Einstein captures a widely shared intuition: understanding is more than merely knowing. Yet epistemic logic has paid relatively little attention to understanding, despite its central role in contemporary epistemology, philosophy of science, and recent debates about AI. A recurring theme in the philosophical literature is that, unlike knowledge, understanding comes in degrees: one may understand something more or less well, and one's understanding may be better than another's. We introduce a comparative epistemic logic of understanding with level-indexed understanding modalities and a comparative connective for saying that one agent understands why a proposition better than another agent does. Semantically, we enrich multi-agent epistemic models with agent-indexed graded explanation structures and a justification-style term algebra. This yields a unified framework for representing minimal, ordinary, more demanding, and ideal understanding, together with comparisons between agents with respect to the same formula at issue. We distinguish a finitary bounded-level calculus from an infinitary full-language companion system. We establish soundness and strong completeness, and show that each fixed finite-level fragment is decidable.

Problem

• understanding
• epistemic logic
• graded understanding
• comparative understanding
• philosophy of science

Innovation

• comparative epistemic logic
• graded understanding
• explanation structures
• modal logic of understanding
• decidability