This paper introduces the logics of super-strict implications, where a super-strictimplication is a strengthening of C.I. Lewis’ strict implication that avoids not onlythe paradoxes of material implication but also those of strict implication. Thesemantics of super-strict implications is obtained by strengthening the (normal)relational semantics for strict implication. We consider all logics of super-strictimplications that are based on relational frames for modal logics in the modalcube. it is shown that all logics of super-strict implications are connexive logicsin that they validate Aristotle’s Theses and (weak) Boethius’s Theses. A proof-theoretic characterisation of logics of super-strict implications is given by meansof G3-style labelled calculi, and it is proved that the structural rules of inferenceare admissible in these calculi. It is also shown that validity in theS5-basedlogic 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’slogic.
Orlandelli, E., Gherardi, G. (2021). Super-Strict Implications. BULLETIN OF THE SECTION OF LOGIC, 50(1), 1-34 [10.18778/0138-0680.2021.02].
Super-Strict Implications
Orlandelli, Eugenio;Gherardi, Guido
2021
Abstract
This paper introduces the logics of super-strict implications, where a super-strictimplication is a strengthening of C.I. Lewis’ strict implication that avoids not onlythe paradoxes of material implication but also those of strict implication. Thesemantics of super-strict implications is obtained by strengthening the (normal)relational semantics for strict implication. We consider all logics of super-strictimplications that are based on relational frames for modal logics in the modalcube. it is shown that all logics of super-strict implications are connexive logicsin that they validate Aristotle’s Theses and (weak) Boethius’s Theses. A proof-theoretic characterisation of logics of super-strict implications is given by meansof G3-style labelled calculi, and it is proved that the structural rules of inferenceare admissible in these calculi. It is also shown that validity in theS5-basedlogic 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’slogic.File | Dimensione | Formato | |
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