Strong Connexivity

Abstract

Connexive logics aim to capture important logical intuitions, intuitions that can be traced back to antiquity. However, the requirements that are imposed on connexive logic are actually not enough to do justice to these intuitions, as I will argue. I will suggest how these demands should be strengthened.

Humble Connexivity

“…The same is true of Cantwell's system, which I also discussed as another example for a weakly connexive logic in [3]. It is a three valued logic which, adjusting notation, can be seen as Graham Priest's Logic of Paradox (LP) with an added conditional.…”

“…However, the interested reader should consider this piece side by side with Wolfgang Lenzen's contribution to this collection, as he argues that the charitable way of reading these authors is to take them to mean something very close to what I will develop here. 3 For the purposes of this paper, let us start with the observation that these principles certainly seem plausible at first blush. Read out with the usual natural language correspondences to the formal vocabulary, it is hard not to feel a strong pre-theoretic intuition that these principles express logical truths.…”

“…In earlier work, I have pointed out that Aristotle and Boethius by themselves might not be doing full justice to these intuitions. In [3], I suggested that in order to do so, a logic should additionally satisfy the following conditions: 4 Unsat1 In no model, A → ¬A is satisfiable (for any A). Unsat2 In no model (A → B) and (A → ¬B) are simultaneously satisfiable (for any A and B).…”

“…It seemed, and still seems, to me that whatever reason you might have to require "It is not the case that if A is the case, not-A is the 3 Our papers were developed in complete isolation, and we only found out about their surprising convergence when we were both invited to give talks at the Third Connexive Logic Workshop in Kyoto in 2017. The paper submitted to this issue by Wansing and Unterhuber, "Connexive Conditional Logic.…”

Modal Boolean Connexive Logics: Semantics and Tableau Approach

“…The first one consists of proposals of specific logical systems designed for a given intended aims, such as [12], where connexive systems of conditionals are investigated. In the second approach, such as [7] and [8], some additional conditions are proposed to distinguish the well behaving connexive logics. One could also find the third approach, such as [10], where a comparison of different connexive logics is conducted.…”