AU Introspection and Symmetry under non-trivial unawareness

This note discusses the relationship between AU Introspection (i.e., an agent is unaware of some event, then she is unaware of that she is unaware of the event) and Symmetry (i.e., an agent is unaware of some event if and only if she is unaware of the complement set) for non-trivial unawareness (i.e., there is an event an agent is unaware of). without Negative Introspection using a set-theoretical approach in standard state-space models. Previous studies have explored the equivalence between Negative Introspection and AU Introspection, or the equivalence between Negative Introspection and Symmetry, by assuming Necessitation of the knowledge operator. As a corollary, AU Introspection is equivalent to Symmetry. However, no studies have shown the relationship between AU Introspection and Symmetry without Necessitation. Therefore, we explore this issue. Our main result shows that if the knowledge operator satisfies Monotonicity, Truth, and Positive introspection, then Modica and Rustichini’s definition of unawareness leads to the equivalence of AU Introspection and Symmetry. In other words, we show that both AU Introspection and Symmetry hold without clashing with non-trivial unawareness.