On Translating Epistemic Operators in a Logic of Awareness
Yudai Kubono
TL;DR
This paper addresses translating epistemic operators from Awareness-Based Indistinguishability Logic ($\text{AIL}$) into HMS-formalism to enable cross-framework analysis. It introduces the HMS-transform $t^{M\rightarrow}(M)$ that converts an $\text{AIL}$ model into an iHMS model and defines a truth-preserving translation $t:\mathcal{L}^*_{AIL}\to\mathcal{L}^*_{HMS}$, mapping $A_i^{\text{AIL}}$ to $A_i^{\text{HMS}}$ and $[\approx]_i I_i [\approx]_i$ to $I_i^{\text{HMS}}$, with a proof framework anchored by the A-condition. The results clarify that the awareness-induced epistemic operator in $\text{AIL}$ corresponds to a subjective viewpoint in iHMS, while the implicit-knowledge notion differs between $\text{AIL}$ and HMS-based models. This work lays groundwork for a formal, comparative analysis of $\text{AIL}$ and HMS-family models, including future work on comparing explicit knowledge across the two formalisms and further semantic alignment.
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.