Lewis' account of common knowledge in Convention describes the generation of higher-order expectations between agents as hinging upon agents' inductive standards and a shared witness. This paper attempts to draw from insights in learning theory to provide a formal account of common inductive knowledge and how it can be generated by a witness. Our language has a rather rich syntax in order to capture equally rich notions central to Lewis' account of common knowledge; for instance, we speak of an agent 'having some reason to believe' a proposition and one proposition 'indicating' to an agent that another proposition holds. A similar line of work was pursued by Cubitt & Sugden 2003; however, their account was left wanting for a corresponding semantics. Our syntax affords a novel topological semantics which, following Kelly 1996's approach in The Logic of Reliable Inquiry, takes as primitives agents' information bases. In particular, we endow each agent with a 'switching tolerance' meant to represent their personal inductive standards for learning. Curiously, when all agents are truly inductive learners (not choosing to believe only those propositions which are deductively verified), we show that the set of worlds where a proposition $P$ is common inductive knowledge is invariant of agents' switching tolerances. Contrarily, the question of whether a specific witness $W$ generates common inductive knowledge of $P$ is sensitive to changing agents' switching tolerances. After establishing soundness of our proof system with respect to this semantics, we conclude by applying our logic to solve an 'inductive' variant of the coordinated attack problem.

翻译：刘易斯在《惯例》中对公共知识的论述，将主体间高阶期望的产生机制归结于主体的归纳标准与共享见证。本文试图借鉴学习理论的洞见，为公共归纳知识及其如何通过见证生成提供形式化描述。为准确捕捉刘易斯公共知识理论中的核心概念——例如“主体有理由相信”某命题，或某命题向主体“指示”另一命题成立——我们的语言设计了较为丰富的句法结构。Cubitt & Sugden 2003曾开展过类似研究，但其理论缺乏对应的语义学框架。基于Kelly 1996在《可靠探究的逻辑》中提出的方法，我们的句法系统创新性地构建了以主体信息基为原语的拓扑语义模型。特别地，我们为每个主体赋予“切换容忍度”参数，用以表征其进行归纳学习时的个体标准。有趣的是，当所有主体均为真正的归纳学习者（即不局限于仅相信经演绎验证的命题）时，命题$P$成为公共归纳知识的世界集合将不随主体切换容忍度的变化而改变。与之相对，特定见证$W$是否能生成$P$的公共归纳知识，则对主体切换容忍度的调整具有敏感性。在证明该语义框架下演绎系统的可靠性后，我们最终应用该逻辑解决了协调攻击问题的“归纳”变体。