Topological Semantics for Common Inductive Knowledge
Siddharth Namachivayam
TL;DR
This paper develops a topological semantics for generating common inductive knowledge via a witness, integrating Lewis's ideas with learning theory. It introduces a rich, non-normal logical language accompanied by information-base topologies $\mathcal{T}_i$ and agent-specific switching tolerances $n_i$, proving that when all agents learn inductively the common inductive knowledge set $\mathbf{C}(P)$ is invariant to tolerances; the witness-generating question, however, remains tolerance-sensitive. The core results show that $\mathbf{C}(P)$ equals the interior of $P$ with respect to the intersection of the $\mathcal{T}_{\mathbf{S}_i}$ topologies (Theorem 5.5), and that the inductive variant of the coordinated attack problem admits attestation protocols precisely for non-empty $(n_i+1)$-open subsets of $P$ in each $\mathcal{T}_i$ (Theorems 7.0 and 7.1). The framework yields a sound proof system (Section 6) and a constructive link between epistemic reasoning under inductive learning and distributed consensus (Section 7), with potential practical applications in robust, inductive consensus protocols.
Abstract
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.