Connexive Logics?

Just as Hartley Slater famously questioned the meaning of paraconsistency, I want to do the same hereby with connexivity, i;e., to talk about about what "connexive logics" are taken to be the name of.The talk consists of three parts.Firstly, we consider in which sense connexive logics deal with connexivity in the four main "connexive theses" (two Aristotelian, two Boethian). Several logical concepts are related to these: implication, compatibility, consistency, soudness, contradiction, impossibility.Second, we see in what sense this special class of formulas requires a special logic: (a) as theorems (or tautologies) of the form |- A -> B; (b) as semantic consequences of the form A |- B; (c) as denied inferences of the form A |-/ B. We claim that the first reading is too strong, the second is trivial for some cases, and the third is too weak.Thirdly, we propose a logical system and its "modal" extension to validate the connexive theses. The first system is a four-valued logic of acceptance and rejection, AR4, which is a non-classical or deviant extension of Dunn-Belnap's FDE including a logical constant of strong implication. The second system is a doxastic logic AR4B, which includes at least four different sorts of epistemc agents with various criteria of justification. We show that one of these agents makes sense of connexive statements, thereby fulfilling the requirement (a) by a specific translation of the four connexive theses into AR4B. We also show that this belief operator behaves like the modal operator of necessity without falling into the net of Dugundji's Theorem (i.e., there is no finite characteristic matrix of any of the Lewis' systems between S1 and S5).The final result is an interpretation of connexivity in a fourth sense (d), namely, as a set of valid theorems which rely upon a "non-normal" constant of implication. The difference with "normal" implication lies in the relation between inference rules and consequence as a truth-preserving relation: Modus Ponens is still a truth-preserving relation from premise to consequence in AR4, whereas Modus Tollens is a non-falsity preserving relation.Two final appendices help to illustrate this added distinction, together with a square of belief oppositions that show the sound behavior of doxastic agents from a connexive perspective.