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.

翻译：常识与仅知代表了两种直观且自然的认知概念，已被证明在多种场景中具有实用价值，例如推理智能体间的协调或共识，或分析基于知识的智能体的认知状态。尽管这两种认知算子已得到广泛独立研究，但以往编码其复杂交互的方法未能捕捉仅知的一些本质属性。我们提出一种新颖解法，通过定义可数序数μ的μ-双世界概念，不仅近似于智能体所认为可能的世界，还包含了其认为不可能的世界。这一方法使我们能够构建一个包含常识与仅知算子的多智能体认知逻辑，并为其建立三值模型语义。此外，我们证明仅需深度不超过ω²+1的双世界即可实现完整刻画。基于这一发现，我们在经典克里普克结构上定义了克里普克语义，并证明该语义与模型语义等价。最后，我们讨论了将负内省或真值性与仅知结合时产生的问题，并展示了正内省如何融入我们的逻辑框架。