The Method of Hypersequents in the Proof Theory of Propositional Non-classical Logics

Until not too many years ago, all logics except classical logic (and, perhaps, intuitionis-tic logic too) were considered to be things esoteric. Today this state of aaairs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers to try to put some order in the present logical jungle. Thus Cl91], Ep90] and Wo88] are three recent books in which an attempt is made to develop a general theoretical framework for the study of logics. On the more pragmatic side, several systems have been developed with the goal of providing a computerized logical framework in which many diierent logical systems can be implemented in a uniform way. An example is the Edinburgh LF((HHP91]). It is clear that there is no limit to the number of logics that logicians (and non-logicians) can produce. Logical frameworks should only be expected, therefore, to be able to handle those that are \good" or \interesting". But what is a \good" logic? One simple answer might be: a logic which has applications. This answer is not satisfactory, though. First, systems of logics are frequently introduced before they nd actual applications. Moreover: there is a tendency to choose for application exactly those that are \good" in some sense. Second: Logic is an autonomous mathematical discipline, and as such 1