Knowing More — from Global to Local Correspondence

Modal correspondence theory is a powerful and effective way to guarantee that adding specific syntactic axioms to a modal logic is mirrored by requiring 'corresponding' properties of the underlying Kripke models. However, such axioms not only  quantify over all formulas, but they are also global  in the sense that the corresponding semantic property is assumed to hold for all states. However, in  for instance epistemic logic one would like to have  the flexibility to say that certain properties (like "agent b knows at least what agent a knows") are  true locally in a specific state, but not necessarily  globally, in all states. This would enable one to  say "currently, b knows at least what a knows, but  this is not common knowledge," or ". . . but this is  not always true," or ". . . but this could be changed  by action α." We offer a logic for "knowing at least  as," where the (global) axiom scheme Ka ϕ → Kb ϕ  is replaced by a (local) inference rule. We give  a complete modal system, and discuss some consequences of the axiom in an epistemic setting.  Our completeness proof also suggests how achieving such local properties can be generalized to other  axioms schemes and modal logics.