Labelled sequent calculi for logics of strict implication

We study the proof theory of C.I. Lewis' logics of strict conditional $\mathbf{S1}$-$\mathbf{S5}$ and we propose the first modular and uniform presentation of C.I. Lewis' systems. In particular, for each logic $\mathbf{Sn}$ we present a labelled sequent calculus $\mathbf{G3Sn}$ and we discuss its structural properties: every rule is height-preserving invertible and the structural rules of weakening, contraction and cut are admissible. Completeness of $\mathbf{G3Sn}$ is established both indirectly via the embedding in the axiomatic system $\mathbf{Sn}$ and directly via the extraction of a countermodel out of a failed proof search. Finally, the sequent calculus $\mathbf{G3S1}$ is employed to obtain a syntactic proof of decidability of $\mathbf{S1}$.