Non-normal dialogics for a wonderful world and more

Abstract

At the end of the 19th century Hugh MacColl (1837-1909), the father of pluralism in formal logic, attempted in the north of France (Boulogne sur mer) to formulate a modal logic which would challenge the semantics of material implication of the post-Boolean wave. It seems that in some of his various attempts MacColl suggested some systems where the rule of necessitation fails. Moreover, the idea that no logical necessity has universal scope-or that no logic could be applied to any argumentative context-seems to be akin and perhaps even central to his pluralistic philosophy of logic. Some years later Clarence Irwin Lewis furnished the axiomatics for several of these logics and since then the critics on the material implication have shown an increasing interest in these modal logics called "non-normal". When Saul Kripke studied their semantics of "impossible worlds" as a way to distinguish between "necessity" and "validity" these logics reached a status of some respectability. As is well known, around the 70s non-normal logics were associated with the problem of omniscience in the epistemic interpretation of modal logic, specially in the work of Jaakko Hintikka and Veikko Rantala. Actually impossible worlds received a intensive study and development too in the context of relevant and paraconsistent logics-specially within the "Saint-Andrews-Australasian connection" in the work of such people as Graham Priest, Stephen Read, Greg Restall and Richard Routley-Sylvan. Nowadays, though the association with omniscience seems to have faded out, the study of non-normal logics has received a new impulse motivated through the study of counterlogicals. The aim of the paper is to offer a dialogical interpretation of non-normal modal dialogics which will suggest some explorations beyond the concept of non-normality. This interpretation will be connected to the discussion of two issues, namely: 1 Counterlogicals as a minimalist defense of logical pluralism (pluralism for a monist) following the path prefigured by MacColl and 2 The difficulties involved in the application of the so-called Hintikka strategy and hybrid languages while constructing tableau systems for nonnormal modal logics.