Epistemic foundations for set-algebraic representations of knowledge

On the Consistency among Prior, Posteriors, and Information Sets (Extended Abstract)

“…is the set of states at which the agent qualitatively believes E. Since ω ∈ K(P(ω)), it follows that P(ω) = E∈Σ:ω∈K(E) E. Hence, for a given qualitative belief operator K, the possibility correspondence P that induces K is unique ( [15]). Also, K satisfies:…”

“…Likewise, define the (iterative) common qualitative belief operator C : Σ → Σ by C(·) := n∈N ( i∈I K i ) n (·). 15 The common qualitative belief operator is induced by the transitive closure of (P i ) i∈I .…”

“…While an agent's beliefs satisfy Invariance within each sub-space, she exhibits unawareness in the entire enriched model. 15 Call an event E a common p-basis if E ⊆ i∈I B p i (F) for any F ∈ Σ with E ⊆ F ( [17]). In an interactive epistemic model, an event E is common p-belief at ω if there is a common p-basis F with ω ∈ F ⊆ i∈I B p i (E).…”

“…The possibility correspondence P induces the qualitative belief operator K : Σ → Σ defined by K(E) := {ω ∈ Ω | P(ω) ⊆ E} ∈ Σ for each E ∈ Σ. The event K(E) is the set of states at which the agent qualitatively believes E. Since ω ∈ K(P(ω)), it follows that P(ω) = E∈Σ:ω∈K(E) E. Hence, for a given qualitative belief operator K, the possibility correspondence P that induces K is unique ( [15]). Also, K satisfies: (i) Monotonicity:…”