  Correspondence theory allows us to create sound and complete axiomatizations
for modal logic on frames with certain properties. For example, if we restrict
ourselves to transitive frames we should add the axiom $\square \phi
\rightarrow \square\square\phi$ which, among other things, can be interpreted
as positive introspection. One limitation of this technique is that the frame
property and the axiom are assumed to hold globally, i.e., the relation is
transitive throughout the frame, and the agent's knowledge satisfies positive
introspection in every world.
  In a modal logic with local properties, we can reason about properties that
are not global. So, for example, transitivity might hold only in certain parts
of the model and, as a result, the agent's knowledge might satisfy positive
introspection in some worlds but not in others. Van Ditmarsch et al. (2012)
introduced sound and complete axiomatizations for modal logics with certain
local properties. Unfortunately, those axiomatizations are rather complex.
Here, we introduce far simpler axiomatizations for a wide range of local
properties.
