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

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

Connexive logic has room for two pairs of universal and particular quantifiers: one pair, |$\forall $| and |$\exists $|⁠, are standard quantifiers; the other pair, |$\mathbb{A}$| and |$\mathbb{E}$|⁠, are unorthodox, but we argue, are well-motivated in the context of connexive logic. Both non-standard quantifiers have been introduced previously, but in the context of connexive logic they have a natural semantic and proof-theoretic place, and plausible natural language readings. The results are logics that are negation inconsistent but non-trivial.