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 schemeKaϕ → Kbϕ 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.

Hans van Ditmarsch, Wiebe van der Hoek, Barteld Kooi