Super-Strict Implications

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 $$\mathsf{S5}$$-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.

Non-Normal Super-Strict Implications

“…What becomes valid, instead, is the negation of the first paradox: ¬(⊥ ⊲ B). Moreover, as shown in [6], all normal SSI are weakly connexive in that they validate Aristotle's Theses:…”

“…4, this can be used as a motivation to choose S2 as the best system of SSI. This paper solves the question in [6] to the affirmative by introducing the S2and S3-based logics of SSI and by showing that they are weakly connexive and well-behaved like the normal ones. 6 The rest of the paper is organised as follows.…”

“…Thus neither connexivism nor relevantism can be seen as supplementation of classical logic. In [6] we proposed an alternative solution that sides with connexivists and relevantists in claiming that there are entailment-related uses of implication where PSI are not acceptable while at the same time siding with Lewis in expressing these uses of implication in a supplementation of classical logic. This proposal is based on introducing two strengthenings of normal strict implication called, respectively, weak (⊲) and strong (◮) super-strict implication (SSI).…”

“…This proposal is based on introducing two strengthenings of normal strict implication called, respectively, weak (⊲) and strong (◮) super-strict implication (SSI). The formula A ⊲ B is defined to be true at a point w of a relational model if both A B and ♦A are true at w. 2 Intuitively, ⊲ has been introduced in [6] as expressing implications where the truth of the succedent depends, in some minimal sense, on that of the antecedent and it accords with the following amendment of Russell's essential property of implication:…”

“…4 This is a price we paid to have a paradox-free and weakly connexive implication that is a simple modal supplementation of classical logic. One question that was left open in [6] is whether non-normal logics of SSI are connexive and wellbehaved like the normal ones. This question is interesting from both a logical and a philosophical perspective.…”