On the relationship between hypersequent calculi and labelled sequent calculi for intermediate logics with geometric Kripke semantics

Abstract

In this thesis we examine the relationship between hypersequent and some types of labelled sequent calculi for a subset of intermediate logics—logics between intuitionistic (Int), and classical logics—that have geometric Kripke semantics, which we call Int∗/Geo.

We introduce a novel calculus for a fragment of first-order classical logic, which we call partially-shielded formulae (or PSF for short), that is adequate for expressing the semantic validity of formulae in Int∗/Geo, and apply techniques from correspondence theory to provide translations of hypersequents, simply labelled sequents and relational sequents (simply labelled sequents with relational formulae) into PSF. Using these translations, we show that hypersequents and simply labelled sequents for calculi in Int∗/Geo share the same models. We also use these translations to justify various techniques that we introduce for translating simply labelled sequents into relational sequents and vice versa. In particular, we introduce a technique called "transitive unfolding" for translating relational sequents into simply labelled sequents (and by extension, hypersequents) which preserves linear models in Int∗/Geo.

We introduce syntactic translations between hypersequent calculi and simply labelled sequent calculi. We apply these translations to a novel hypersequent framework HG3ipm∗ for some logics in Int∗/Geo to obtain a corresponding simply labelled sequent framework LG3ipm∗, and to an existing simply labelled calculus for Int from the literature to obtain a novel hypersequent calculus for Int.

We introduce methods for translating a simply labelled sequent calculus into a cor- responding relational calculus, and apply these methods to LG3ipm∗ to obtain a novel relational framework RG3ipm∗ that bears similarities to existing calculi from the literature. We use transitive unfolding to translate proofs in RG3ipm∗ into proofs in LG3ipm∗ and HG3ipm∗ with the communication rule, which corresponds to the semantic restriction to linear models.