Some Analytic Systems of Rules
Timo Lang · Lecture notes in computer science · 2023
Abstract We define two simple systems of rules, i.e. calculi with a global condition on the order of rule instances in a proof, for the modal logics of shift-reflexive and Euclidean frames respectively. Cut-elimination, and therefore the subformula property, can be derived directly from the cut-elimination property of adjacent logics. We compare our system to the calculus of grafted hypersequents, which has previously been used to capture both logics. We then discuss an attempt to obtain similar ‘modular’ cut-elimination proofs in other systems of rules. This general attempt is carried out for two more logics, namely the modal logic of serial frames and the intermediate logic axiomatised by the law of the weak excluded middle.