Multi Lingual Sequent Calculus and Coherent Spaces

Abstract

We study a Gentzen style sequent calculus where the formulas on the left and right of the turnstile need not necessarily come from the same logical system. Such a sequent can be seen as a consequence between different domains of reasoning. We discuss the ingredients needed to set up the logic generalised in this fashion. The usual cut rule does not make sense for sequents which connect different logical systems because it mixes formulas from antecedent and succedent. We propose a different cut rule which addresses this problem. The new cut rule can be used as a basis for composition in a suitable category of logical systems. As it turns out, this category is equivalent to coherent spaces with certain relations between them. Finally, cut elimination in this set-up can be employed to provide a new explanation of the domain constructions in Samson Abramsky's Domain Theory in Logical Form.