On a Graph Approach to Modal Logics

We introduce a sound and complete graph calculus for multi-modal logics. This formalism internalizes the (Kripke) semantics of modal logics and provides uniform tools for expressing and manipulating modal formulas. We present the graph calculus for logic K and show how to extend it to handle some modalities, like the global and difference modalities, in a natural manner. We also indicate how it can be easily extended to other normal modal logics, such as T, S4, S5, etc.