Non-normal Dialogics for a Wonderful World and More

The aim of the paper is to offer a dialogical interpretation of non-normal modal logic which will suggest some explorations beyond the concept of nonnormality. This interpretation will be connected to the discussion of two issues, one more philosophical and the second of a more technical nature: namely a minimalist defence of logical pluralism and the difficulties involved in the application of the so-called Hintikka strategy and hybrid languages while constructing tableau-systems for non-normal modal logics. At the end of the 19 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 -