Geodesic
Super-Strict Implications
Bulletin of the Section of Logic, Tome 50 (2021) no. 1, pp. 1-34.

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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.
Keywords: Strict implication, paradoxes of implication, connexive implication, sequent calculi, structural rules 
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Gherardi, Guido; Orlandelli, Eugenio. Super-Strict Implications. Bulletin of the Section of Logic, Tome 50 (2021) no. 1, pp. 1-34. http://geodesic.mathdoc.fr/item/BSL_2021_50_1_a0/

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