Non-normal dialogics for a wonderful world and more. (English) Zbl 1123.03007

van Benthem, Johan (ed.) et al., The age of alternative logics. Assessing philosophy of logic and mathematics today. Based on the conference on philosophical insights into logic and mathematics, Nancy, France, September 30–October 4, 2002. Dordrecht: Springer (ISBN 978-1-4020-5011-4/hbk; 978-1-4020-5012-7/e-book). Logic, Epistemology, and the Unity of Science 3, 311-334 (2006).
This paper explores Proponent-Opponent dialogues in which Opponent can choose a (non-modal) logic different from that assumed by Proponent. A possible context would be an assertion of the kind (common in the formal science), “If the law of excluded middle were not valid, the one side of the de Morgan laws would fail”. Tableaux for dialogue strategies are directly connected with the more familiar tree-shaped semantic tableaux. Properties of accessibility relations can be implemented by structural rules, but more interestingly by finding object-language formulae which express the relevant frame conditions, an approach pioneered by Hintikka. Since P-O dialogues have only two force indicators (? or attack, and ! or defend), the resulting systems are more a variant on semantic tableaux than a representation of dialogues between people with different logics. Misprint: on p. 311, line 7 of the text, for “applied to any argumentative context”, read “applied to every argumentative context”.
For the entire collection see [Zbl 1110.03002].


03A05 Philosophical and critical aspects of logic and foundations
03B45 Modal logic (including the logic of norms)