Explainable Belief Harmonization under Dynamic Epistemic Partitions

Existing approaches to multi-agent belief combination have established mature foundations for combining uncertain beliefs under common assumptions: consensus methods use iterative averaging, logic-based methods resolve conflicting knowledge bases, and epistemic logic analyzes agents' information states. Typically, these approaches assume that the structure determining what each agent can represent remains fixed. However, in many scenarios, agents gain or lose observational capacity during execution, and what was once admissible may become structurally impossible. This paper presents a formal framework for handling such runtime changes in epistemic partitions over continuous belief profiles. A hybrid approach exploits the advantages of answer set programming in elaboration tolerance, declarative integrity constraints, and explanations, with the numerical flexibility of Python. The framework applies to domains where agents operate at heterogeneous and possibly changing levels of resolution, and provides formal guarantees of admissibility preservation under refinement, unique mass-preserving repair under coarsening, and explanation completeness. Evaluation across 100 randomly generated topology changes confirms complete violation detection and explanation coverage.

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.

History of the Muddy Children Puzzle

The Muddy Children Puzzle is a puzzle about knowledge and ignorance that has been inspiring for the development of epistemic logic. Who came up with it first? This is unclear. We trace the origin of the Muddy Children Puzzle through logical and literary publications over the past two centuries. The puzzle inspired a numerous variations such as involving numbers or coloured hats. We also present a novel hats puzzle involving self-reference.