Proof Systems for Super- Strict Implication

Abstract

AbstractThis paper studies proof systems for the logics of super-strict implication$$\textsf{ST2}$$ST2–$$\textsf{ST5}$$ST5, which correspond to C.I. Lewis’ systems$$\textsf{S2}$$S2–$$\textsf{S5}$$S5freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating$$\textsf{STn}$$STnin$$\textsf{Sn}$$Snand backsimulating$$\textsf{Sn}$$Snin$$\textsf{STn}$$STn, respectively (for$${\textsf{n}} =2, \ldots , 5$$n=2,…,5). Next,$$\textsf{G3}$$G3-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of$$\textsf{G3}$$G3-style calculi, that they are sound and complete, and it is shown that the proof search for$$\mathsf {G3.ST2}$$G3.ST2is terminating and therefore the logic is decidable.