Superconnexivity Reconsidered

Abstract

We reconsider the idea of superconnexivity, an idea that has not received much attention so far. We inspect more closely the problems with the proposal that are responsible for this disregard. However, we also suggest a slight modification of the idea that has a much better chance of delivering the desired results, which we call super-bot-connexivity.* The research by Andreas Kapsner has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), Project 436508789. Moreover, the research by Hitoshi Omori has been supported by a Sofja Kovalevskaja Award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research. We would like to thank the audience in Bochum for their comments on an earlier version of this paper. In particular, our special thanks goes to Francesco Paoli for extremely fruitful discussions. Finally, but not the least, we would also like to thank the anonymous referees for their supportive, careful and helpful comments.1 For an overview of connexive logic, see [11,22].

Intuitionistic Logic is a Connexive Logic

“…We provide a finite example of a connexive Heyting algebra (called L9 by Sankappanavar [42,Thm. 4.1] and also mentioned by Kapsner and Omori, see [33])) showing both that this class is nonempty and that implication, in general, fails to be symmetric therein.…”

“…Yet, these principles are dumped because they lead to triviality given a modicum of assumptions. Very recently, however, Kapsner and Omori [33] have attempted to revisit the superconnexive insight. Their goal, in a nutshell, is to salvage the spirit of superconnexivity by slightly weakening the letter of it.…”

Angell and McCall Meet Wansing

“…Once this extension is made, we obtain the following table as a result of the comparison. Though we will not argue for this here, we believe that the super-bot versions of the connexive principles are an attractive way of capturing the intuitions that might draw one towards strong connexivity (we deliver this argument and all technical details in [13]). Given that C isn't strongly connexive, it is no surprise that C ⊥ fails to obey these features, nor is it surprising that CC1 and AM3, both being strongly connexive, do.…”

“…However adding this to a system with substitutivity of logical equivalents quickly leads to triviality (cf. [13]).…”

“…Recently, the present authors revived the idea by slightly modifying it in [13]. We called our idea Super-Bot-Connexivity, because it involves a bottom constant.…”