Translating Labels to Hypersequents for Intermediate Logics with Geometric Kripke Semantics

We give a procedure for translating geometric Kripke frame axioms into structural hypersequent rules for the corresponding intermediate logics in Int^*/Geo that admit weakening, contraction and in some cases, cut. We give a procedure for translating labelled sequents in the corresponding logic to hypersequents that share the same linear models (which correspond to G\"odel-Dummett logic). We prove that labelled proofs Int^*/Geo can be translated into hypersequent proofs that may use the linearity rule, which corresponds to the well-known communication rule for G\"odel-Dummett logic.