Suprastructural logics

Over the last few decades, considerable attention has been paid to substructural logics—that is, logical systems that forego at least one of the structural principles sanctioned by classical logic. However, little or no attention has been paid to the dual notion of a suprastructural logic, that is, a system that sanctions at least one structural principle that classical logic rejects. In the present paper, we explore this notion and present different types of systems where at least one classically invalid structural metainference holds. We introduce two families of suprastructural logics—Boolean logics and Strong-Kleene logics—as examples of different ways to achieve suprastructurality by revising the traditional conception of inferential validity. Afterwards, we present some strictly suprastructural logics. We show that LP and K3 are strictly suprastructural from a global perspective, which also exemplifies a way to achieve suprastructurality that does not alter the usual sense of inferential validity. We then introduce metainferential logics that are strictly suprastructural from a local perspective and show that some of them are also supraclassical. This leads us to consider two different notions of suprastructurality: one that focuses on metainferences and the other that focuses on closure properties.