Implicit Knowledge in Unawareness Structures

TL;DR

This paper explores the relationship between implicit and explicit knowledge in unawareness models, proposing new definitions and showing an equivalence between two prominent frameworks, enhancing understanding of subjective awareness in logic.

Contribution

It defines implicit knowledge within HMS structures, introduces a variant of HMS models with implicit knowledge and awareness as primitives, and establishes an equivalence between HMS and FH models.

Findings

Defined implicit knowledge consistent with explicit knowledge in HMS models

Introduced a variant of HMS models with implicit knowledge and awareness as primitives

Proved equivalence between HMS and FH models, ensuring soundness and completeness

Abstract

Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic awareness correspondence and accessibility relations modeling implicit knowledge. They are a flexible model of unawareness, and best interpreted from a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and Schipper (2006, 2008) (HMS) model awareness by a lattice of state spaces and explicit knowledge via possibility correspondences. Sublattices thereof can be interpreted as subjective views of agents. Open questions include (1) how implicit knowledge can be defined in HMS structures, and (2) in which way FH structures can be extended to model the agents' subjective views. In this paper, we address (1) by defining implicit knowledge such that it is consistent with explicit knowledge in HMS models. We also introduce a variant of HMS models that instead of explicit knowledge, takes implicit knowledge and awareness as primitives. Further, we address (2) by introducing a category of FH models that are modally equivalent relative to sublanguages and can be interpreted as agents' subjective views depending on their awareness. These constructions allow us to show an equivalence between HMS and FH models. As a corollary, we obtain soundness and completeness of HMS models with respect to the Logic of Propositional Awareness, based on a language featuring both implicit and explicit knowledge.