Abstract
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.
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Acknowledgements
Eric Raidl’s work was funded by the Deutsche Forschungsgemeinschaft (EXC number 2064/1, project number 390727645), and by the Baden-Württemberg Stiftung (‘Verantwortliche Künstliche Intelligenz’). We are grateful to the audience of the conferences Trends in Logic XXI and NCL’22, where part of the paper was presented.
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Gherardi, G., Orlandelli, E. & Raidl, E. Proof Systems for Super- Strict Implication. Stud Logica 112, 249–294 (2024). https://doi.org/10.1007/s11225-023-10048-3
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DOI: https://doi.org/10.1007/s11225-023-10048-3