Is “Unawareness Leads to Ignorance” Trivial?

The main theorem (we call it the Triviality Theorem) of Dekel, Lipman, and Rustichini (1998, Econometrica, 66) states that under some assumptions, standard state-space models cannot discuss non-trivial unawareness. Additionally, if the assumptions constituting this theorem are relaxed, non-trivial unawareness can be discussed in standard state-space models. This note reconsiders non-trivial unawareness and its main theorem in standard state-space models. Particularly, we focus on the “Unawareness Leads to Ignorance” property of unawareness, pointing out that in some cases, this property is not trivial. Furthermore, we propose an information function-based definition of a knowledge operator that does not satisfy the Necessitation. While previous studies have discussed knowledge operators excluding Necessitation using non-normal modal logic and non-partial state-space models, our defined knowledge operator builds on them from a set-theoretic approach.