Labelled propositional modal logics: theory and practice

We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and well-known class of propositional modal logics (including $K$, $D$, $T$, $B$, $S4$, $S4.2$, $KD45$, $S5$). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive proof-theory (in comparision to, e.g., semantic embedding) but limits the degree to which we can make use of extensions to the labelling algebra