Topological Semantics for Common Inductive Knowledge

TL;DR

This paper introduces a topological semantics-based logic for modeling how scientists can coordinate inductively to reach common conclusions without communication, ensuring convergence and avoiding false positives.

Contribution

It develops a novel logic with rich syntax and semantics based on agents' information bases and inductive standards, extending Lewis' account of common knowledge.

Findings

Proves soundness of the proof system with respect to the semantics.

Demonstrates how the logic can solve the inductive coordinated attack problem.

Provides a formal framework for reasoning about inductive common knowledge.

Abstract

Consider a community of scientists whose labs are each capable of conducting a different set of experiments. The scientists want to work together to confirm a new hypothesis, but to ensure blindness, their labs generally prohibit the scientists from communicating with each other. Further, each scientist can only make so many retractions to their lab before having to cease inquiry and suspend judgement forever. How might the scientists coordinate whether to affirm or suspend judgement on this hypothesis in light of their private experiments so that their labs are guaranteed to converge to the same conclusion and that this conclusion will not be a false positive? Call this problem 'inductive coordinated attack.' In this paper, we develop a logic for solving inductive coordinated attack by determining when and how a hypothesis can become what we call 'common inductive knowledge.' We begin by precisifying Lewis' account of common knowledge in Convention which describes the generation of higher-order expectations between agents as hinging upon agents' inductive standards and a shared witness. Our language has a rather rich syntax in order to capture equally rich notions central to Lewis' account; for instance, we speak of an agent 'having inductive reason to believe' a proposition and one proposition 'indicating' to an agent that another proposition holds. This 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. After establishing soundness of our proof system with respect to this semantics, we conclude by showing how our logic can be used to solve inductive coordinated attack.