Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) context was called. This theory gave a justification for the validity of what we todayexpress through the formulae ¬ and ¬. Hugh MacColl and, more recently, Storrs McCall searched for a formal system in which the validity ofthese 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 winning strategies not only for the above-mentioned connexive theses but also for ¬ and ¬. Further on, we developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form. (shrink)

Many of the discussions about conditionals can best be put as follows:can those conditionals that involve an entailment relation be formulatedwithin a formal system? The reasons for the failure of the classical approachto entailment have usually been that they ignore the meaning connectionbetween antecedent and consequent in a valid entailment. One of the firsttheories in the history of logic about meaning connection resulted from thestoic discussions on tightening the relation between the If- and the Then-parts of conditionals, which in this (...) context was called συναρτησις(connection). This theory gave a justification for the validity of what we todayexpress through the formulae ¬(a → ¬ a) and ¬(¬ a → a). Hugh MacColl and, more recently, Storrs McCall (from 1877 to 1906 and from1963 to 1975 respectively) searched for a formal system in which the validity ofthese 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 developthe corresponding tableau systems and conclude with some remarks on possibleperspectives and consequences of the dialogical approach to connexivity including the loss of uniform substitution leading to a new concept of logical form. (shrink)

The Scottish logician Hugh MacColl is well known for his innovative contributions to modal and nonclassical logics. However, until now little biographical information has been available about his academic and cultural background, his personal and professional situation, and his position in the scientific community of the Victorian era. The present article reports on a number of recent findings.

Dialogical logic is a game-theoretical approach to logic. Logic is studied with the help of certain games, which can be thought of as idealized argumentations. Two players, the Proponent, who puts forward the initial thesis and tries to defend it, and the Opponent, who tries to attack the Proponent’s thesis, alternately utter argumentative moves according to certain rules. For a long time the dialogical approach had been worked out only for classical and intuitionistic logic. The seven papers of this dissertation (...) show that this narrowness was uncalled for. The initial paper presents an overview and serves as an introduction to the other papers. Those papers are related by one central theme. As each of them presents dialogical formulations of a different non-classical logic, they show that dialogical logic constitutes a powerful and flexible general framework for the development and study of various logical formalisms and combinations thereof. As such it is especially attractive to logical pluralists that reject the idea of “the single correct logic”. The collection contains treatments of free logic, modal logic, relevance logic, connexive logic, linear logic, and multi-valued logic. (shrink)