Proof Systems for Super- Strict Implication

Abstract

This paper studies proof systems for the logics of super-strict implication $$\textsf{ST2}$$ ST 2 –$$\textsf{ST5}$$ ST 5 , which correspond to C.I. Lewis’ systems $$\textsf{S2}$$ S 2 –$$\textsf{S5}$$ S 5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating $$\textsf{STn}$$ STn in $$\textsf{Sn}$$ Sn and backsimulating $$\textsf{Sn}$$ Sn in $$\textsf{STn}$$ STn , respectively (for $${\textsf{n}} =2, \ldots , 5$$ n = 2 , … , 5 ). Next, $$\textsf{G3}$$ G 3 -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of $$\textsf{G3}$$ G 3 -style calculi, that they are sound and complete, and it is shown that the proof search for $$\mathsf {G3.ST2}$$ G 3 . ST 2 is terminating and therefore the logic is decidable.