Better Understanding, Understanding Better

"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.