A Topography of Labelled Modal Logics

Labelled Deductive Systems provide a general method for representing logics in a modular and transparent way. A Labelled Deductive System consists of two parts, a base logic and a labelling algebra, which interact through a xed interface. The labelling algebra can be viewed as an independent parameter: the base logic stays xed for a given class of related logics from which we can generate the one we want by plugging in the appropriate algebra. Our work identiies an important property of the structured presentation of logics, their combination, and extension. Namely, there is tension between modularity and extensibility: a narrow interface between the base logic and labelling algebra can limit the degree to which we can make use of extensions to the labelling algebra. We illustrate this in the case of modal logics and apply simple results from proof theory to give examples.