Dialogical Connexive Logic

Many of the discussions about conditionals can best be put as follows: can those conditionals that involve an entailment relation be formulated within a formal system? The reasons for the failure of the classical approach to entailment have usually been that they ignore the meaning connection between antecedent and consequent in a valid entailment. One of the first theories in the history of logic about meaning connection resulted from the stoic discussions on tightening the relation between the Ifand the Thenparts of conditionals, which in this context was called συναρτησ ις (connection). This theory gave a justification for the validity of what we today express through the formulae ¬(a → ¬a) and ¬(¬a → a). Hugh MacColl and, more recently, Storrs McCall (from 1877 to 1906 and from 1963 to 1975 respectively) searched for a formal system in which the validity of these formulae could be expressed. Unfortunately neither of the resulting systems is very satisfactory. In this paper we introduce dialogical games with the help of a new connexive If-Then (“⇒”), the structural rules of which allow the Proponent to develop (formal) winning strategies not only for the above-mentioned connexive theses but also for (a ⇒ b) ⇒ ¬(a ⇒ ¬b) and (a ⇒ b) ⇒ ¬(¬a ⇒ b). Further on, we develop the corresponding tableau systems and conclude with some remarks on possible perspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form.