On Translating Epistemic Operators in a Logic of Awareness

TL;DR

This paper explores translating awareness-based epistemic logic into HMS models, clarifying the semantic role of awareness-induced operators and comparing implicit knowledge across models.

Contribution

It defines a model transformation from AIL to HMS models and proves truth preservation, clarifying the semantics of awareness-based epistemic operators.

Findings

Translation preserves truth between AIL and HMS models.

Clarifies the semantic role of awareness in explicit knowledge.

Highlights differences in implicit knowledge captured by the models.

Abstract

Awareness-Based Indistinguishability Logic (henceforth, AIL) is an extension of Epistemic Logic by introducing the notion of awareness, distinguishing explicit knowledge from implicit knowledge. In this framework, each of these notions is represented by a modal operator. On the other hand, HMS models, developed in the economics literature, also provide a formalization of those notions. Nevertheless, the behavior of the epistemic operators in AIL within HMS models has yet to be explored. In this paper, we define a transformation of an AIL model into an HMS model and then prove that a translation between the fragments of the language of AIL preserves truth under this transformation. As a result, we clarify the semantic role of an epistemic operator in AIL, which is induced by awareness and is essential to defining explicit knowledge, within HMS models. Furthermore, we demonstrate the differences in the implicit knowledge captured by AIL and HMS models. This work lays the groundwork for a comparative analysis between the model classes.