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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.

[pt] Neste trabalho propomos um framework lógico para sistemas de Dedução Natural rotulados. Sua meta-linguagem é baseada numa generalização dos esquemas de regras propostos por Prawitz, e o uso de rótulos permite a definição de lógicas intencionais como lógicas modais e de descrição, bem como a definição uniforme de quantificadores como o para um número não-renumerável de indivíduos vale a propriedade P (lógica de Keisler), ou para quase todos os indivíduos vale P (lógica de ultra-filtros), sem mencionar os quantificadores padrões de lógica de primeira-ordem. Mostramos também a implementação deste framework em um assistente de prova virtual disponível livremente na web, e comparamos a definição de sistemas lógicos nele com o mesmo feito em outros assistentes — Agda, Isabelle, Lean, Metamath. Como subproduto deste experimento comparativo, também contribuímos uma prova formal em Lean do postulado de Zolt em três dimensões usando o sistema Zp proposto por Giovaninni et al. / [en] We propose a Logical Framework for labelled Natural Deduction systems. Its meta-language is based on a generalization of the rule schemas proposed by Prawitz, and the use of labels allows the definition of intentional logics, such as Modal Logic and Description Logic, as well as some quantifiers, such as Keisler s for non-denumerable-many individuals property P, or for almost all individuals P holds, or generally P holds, not to mention standard first-order logic quantifiers, all in a uniform way. We also show an implementation of this framework as a freely-available web-based proof assistant. We then compare the definition of logical systems in our implementation and in other proof assistants — Agda, Isabelle, Lean, Metamath. As a sub-product of this comparison experiment, we contribute a formal proof (in Lean) of De Zolt s postulate for three dimensions, using the Zp system proposed by Giovaninni et al.