Super-Strict Implications

This paper introduces the logics of super-strict implications, where a super-strictimplication is a strengthening of C.I. Lewis’ strict implication that avoids not onlythe paradoxes of material implication but also those of strict implication. Thesemantics of super-strict implications is obtained by strengthening the (normal)relational semantics for strict implication. We consider all logics of super-strictimplications that are based on relational frames for modal logics in the modalcube. it is shown that all logics of super-strict implications are connexive logicsin that they validate Aristotle’s Theses and (weak) Boethius’s Theses. A proof-theoretic characterisation of logics of super-strict implications is given by meansof G3-style labelled calculi, and it is proved that the structural rules of inferenceare admissible in these calculi. It is also shown that validity in theS5-basedlogic of super-strict implications is equivalent to validity in G. Priest’s negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest’slogic.