Humble Connexivity

Abstract

In this paper, I review the motivation of connexive and strongly connexive logics, and I investigate the question why it is so hard to achieve those properties in a logic with a well motivated semantic theory. My answer is that strong connexivity, and even just weak connexivity, is too stringent a requirement. I introduce the notion of humble connexivity, which in essence is the idea to restrict the connexive requirements to possible antecedents. I show that this restriction can be well motivated, while it still leaves us with a set of requirements that are far from trivial. In fact, formalizing the idea of humble connexivity is not as straightforward as one might expect, and I offer three different proposals. I examine some well known logics to determine whether they are humbly connexive or not, and I end with a more wide-focused view on the logical landscape seen through the lens of humble connexivity.1 Not only is he, by giving me this challenge, responsible for the existence of this paper, he also gave a number of suggestions that were of tremendous help to me in writing this paper; section 6 in particular owes its inclusion and form to these suggestions. Two others have had an equally great impact on this paper, and they happen to be the editors of this volume. The idea of humble connexivity originates in my joint work with Hitoshi Omori. Even if what I'll have to say is probably more opinionated than he would have put it, I would not have been able to form

Towards a bridge over two approaches in connexive logic

“…In other words, this shows that Egré-Politzer-style logics have a loose connection to humble connexivity discussed by Kapsner in [6].…”

“…To this end, we make use of a modal logic that expands the four-valued logic N4 ⊥ p (or B → 4 in the terminology of [9]) 5 , developed by Sergei Odintsov and Heinrich Wansing in [10]. 6…”

“…Connexive logics are characterized by having so-called Aristotle's theses and Boethius' theses as derivable/valid formulas: Aristotle's theses ∼(A → ∼ A), ∼(∼ A → A) Boethius' theses (A → B) → ∼(A → ∼ B), (A → ∼ B) → ∼(A → B) More recently, there are some discussions casting some doubts on the very notion of connexivity [e.g., 5,6], but in this note I will stick to the more conservative account of connexive logic.…”

“…The bottom element free fragment is known as HBe since[2] 6. See[11] for an overview of modal logics based on Belnap-Dunn logic that may also serve well for our purposes.7 WBK stands for weak BK.…”

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.…”