The Common Knowledge of Formula Exclusion

TL;DR

This paper investigates the structure of canonical Kripke models in epistemic logic, revealing the existence of cells without finite fanout that are surjectively mapped by all structures, challenging assumptions about possible knowledge configurations.

Contribution

It introduces the concept of cells in canonical Kripke structures that lack finite fanout yet are surjectively mapped by all related structures, expanding understanding of possible knowledge states.

Findings

Existence of cells without finite fanout in canonical structures

Surjective mapping of all structures to such cells

Implications for the structure of common knowledge

Abstract

For every set of primitive propositions and agents there is a canonical Kripke structure and a canonical map from any Kripke structure (defined with the same primitive propositions and agents) to this canonical one. A cell of the canonical Kripke structure is a set C such that if any agent considers a point x in C to be possible then all the other points considered possible by this agent are also in C. A cell C has finite fanout if at every point in C every agent considers possible only finitely many other points. We demonstrate a cell of this canonical Kripke structure such that every Kripke structure that maps to this cell does so surjectively, yet this cell does not have finite fanout.