A Two-Layer Partition Awareness Structure.

With the idea of representing knowledge and awareness by binary relations, we establish a two-layer partition epistemic structure, a syntax-free model, to formalize a non-trivial notion of awareness. We provide two axiomatic systems with awareness modality on our structure and prove soundness and completeness for them. Moreover, we show that our structure and the syntactic awareness structure are equivalent under certain conditions, which helps us understand the connections between knowledge and awareness.