Super-Strict Implications

Authors

  • Guido Gherardi University of Bologna, Department of Philosophy and Communication Studies I-40126, Via Zamboni 38, Bologna, Italy https://orcid.org/0000-0002-3382-2292
  • Eugenio Orlandelli University of Bologna, Department of Philosophy and Communication Studies I-40126, Via Zamboni 38, Bologna, Italy https://orcid.org/0000-0002-4021-8667

DOI:

https://doi.org/10.18778/0138-0680.2021.02

Keywords:

Strict implication, paradoxes of implication, connexive implication, sequent calculi, structural rules

Abstract

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the modal cube. it is shown that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic 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's logic.

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Published

2021-01-20

How to Cite

Gherardi, G., & Orlandelli, E. (2021). Super-Strict Implications. Bulletin of the Section of Logic, 50(1), 1–34. https://doi.org/10.18778/0138-0680.2021.02

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Research Article