Topological Semantics for Common Inductive Knowledge

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.

💡 Research Summary

The paper revisits David Lewis’s account of common knowledge in “Convention”, which hinges on a shared witness (W) and agents’ inductive standards, and provides a rigorous formalization using tools from learning theory and topology. The author first distinguishes two notions of belief: (i) “reason simpliciter” – a piece of evidence E that gives an agent a basic reason to believe a proposition, and (ii) “having W as a reason to believe P” – which requires that the same evidence together with W entails P. To capture the interaction between evidence, witnesses, and beliefs, a new primitive operator “A indicates to i that B” is introduced, together with a suite of inference rules that encode weak transitivity, closure under deduction, and the relationship between indication and reason.

The semantic core is a topological model inspired by Kelly’s “Logic of Reliable Inquiry”. Each agent i is associated with an information base that is represented as a collection of open sets in a topological space X_i. The author introduces a parameter called the “switching tolerance” τ_i, which quantifies how much an agent is willing to revise its belief in response to new evidence. Small τ_i corresponds to a “truly inductive learner” who changes belief only under substantial evidence; larger τ_i allows more liberal belief updates.

Common inductive knowledge (CIK) of a proposition P is defined via an infinite hierarchy of conditions (L1/L2) that mirror Lewis’s three conditions but are expressed in terms of the primitive operators. The key technical result shows that when every agent’s τ_i is sufficiently small (i.e., all agents are truly inductive learners), the set of worlds in which P is CIK is invariant under changes of τ_i. This invariance follows from the fact that, in the topological setting, the common‑knowledge operator corresponds to taking the topological closure, which does not depend on the specific tolerance thresholds as long as they are below a certain bound.

In contrast, the question of whether a particular witness W generates CIK for P is highly sensitive to the τ_i values. For W to generate CIK, each agent must have an open set that contains the intersection of W with any evidence E that gives a reason simpliciter for W, and this inclusion can fail when τ_i grows, breaking the necessary entailments. Thus the witness‑generation conditions L1/L2 become functions of the agents’ tolerances.

A Hilbert‑style proof system is presented, built around the primitive equivalence
(i has A as reason to believe B) ↔