  In this paper, we propose a new neighborhood semantics for contingency logic,
by introducing a simple property in standard neighborhood models. This
simplifies the neighborhood semantics given in (Fan and van Ditmarsch, 2015),
but does not change the set of valid formulas. Under this perspective, among
various notions of bisimulation and respective Hennessy-Milner Theorems, we
show that $c$-bisimulation is equivalent to nbh-$\Delta$-bisimulation in the
literature, which guides us to understand the essence of the latter notion.
This perspective also provides various frame definability and axiomatization
results.
