World Logic Day: Elia Zardini "Paradox and Substructurality"

First, I’ll introduce some of the main paradoxes in philosophy of logic: the paradoxes of selfreference, the paradoxes of vagueness and the paradoxes of relevance. Then, I’ll present the principal substructural approaches to those paradoxes and explain their advantages over structural nonclassical approaches: that they often revise classical logic without revising the fundamental principles governing logical operations, that they often provide a unified solution to the paradoxes of a certain kind; that they often afford the only way to uphold certain compelling principles concerning the original notions with their intended force. It is common to assume that such advantages accrue to substructural approaches because the structural level is more fundamental than the operational one. However, I’ll explore the opposite hypothesis according to which the operational rules of a logic ground its structural rules: a logic’s denial of a classical structural rule is due to its denial of some classical rule concerning conjunction, disjunction or the conditional. Finally, I’ll argue that such a hypothesis naturally leads to a reconceptualisation of a large class of substructural logics and of the corresponding substructural approaches to paradox as unusually centring on logical operations (conjunction and disjunction) whose arguments are all upwards monotonic—that is, in effect, logical operations of positive composition.