Mathematical Foundations for Joining Only Knowing and Common Knowledge (Extended Version)

TL;DR

This paper introduces a novel logical framework using $biworlds$ to accurately model the interplay of common knowledge and only knowing, capturing essential properties and providing a three-valued semantics for multi-agent epistemic reasoning.

Contribution

It proposes a new $biworld$-based approach to encode complex interactions of common knowledge and only knowing, with a three-valued semantics and a canonical Kripke structure.

Findings

Biworlds of depth at most omega^2+1 suffice for modeling

Kripke semantics coincide with model semantics

Positive introspection can be integrated into the logic

Abstract

Common knowledge and only knowing capture two intuitive and natural notions that have proven to be useful in a variety of settings, for example to reason about coordination or agreement between agents, or to analyse the knowledge of knowledge-based agents. While these two epistemic operators have been extensively studied in isolation, the approaches made to encode their complex interplay failed to capture some essential properties of only knowing. We propose a novel solution by defining a notion of $\mu$-biworld for countable ordinals $\mu$, which approximates not only the worlds that an agent deems possible, but also those deemed impossible. This approach allows us to define a multi-agent epistemic logic with common knowledge and only knowing operators, and a three-valued model semantics for it. Moreover, we show that we only really need biworlds of depth at most $\omega^2+1$. Based on this observation, we define a Kripke semantics on a canonical Kripke structure and show that this semantics coincides with the model semantics. Finally, we discuss issues arising when combining negative introspection or truthfulness with only knowing and show how positive introspection can be integrated into our logic.