Viva, Marcelo: > 2016-11-01 19:08 GMT-02:00 Carol Blasio <[email protected]>: >> >> * João Marcos: "The mystery of duality unraveled: dualizing rules, >> operators, and logics" > > Não obstante o lado místico (ou misterioso) da lógica, eu gostaria de saber > se v tem algo escrito sobre dualidade em lógica, pois é um tema que sempre > me interessou.
Há alguns anos que trabalho nestas coisas, mas o paper propriamente dito ainda é work-in-progress! Certamente partilharei assim que incorporar o feedback recebido em breve e tiver as coisas bem escritinhas. Para dar uma ideia da coisa, envio abaixo o resumo da palestra (uma encarnação anterior foi apresentada no UNILOG 2013 por Sanderson Molick, contendo resultados variados de toda a década precedente). * * * Title: The mistery of duality unraveled: dualizing rules, operators, and logics Abstract: Though one often finds in the literature the concept of "dualization" employed to describe a certain form of opposition that holds between *statement-forms*, between logical *operators*, and even between *logics* themselves, only rarely a formal definition of *duality* is actually offered by the very authors that embrace the terminology. Basic as it might seem, proper definitions of duality appear also not to have been incorporated in logic textbooks and courses beyond the level of side comments, exercises, or cheap talk. The current scarcity of basic material available addressing themes of Universal Logic has also not helped the development of firm grounds for setting up a theory of logical duality generous enough to cover non-classical territory. In the presence of a classical negation, a common syntactic approach used to produce the dual of a given logical connective is the one we might call *De Morgan Method*: Exchange each atomic argument by its negated counterpart, and add also a negation over the whole expression. Semantically, still on a classical context, this gives rise to an approach we might call *Inversion Method*: Draw a truth-table for the formula and exchange 0s and 1s both for the atoms in the input and for the complex formula in the output. A third approach we might call *Symmetry Method* proposes to read systematically from right to left any semantical clause over the set of valuations and any proof-theoretical statement that is naturally read from left to right, and vice-versa. The latter approach gets complicated when the underlying formalism is asymmetric (such as the case of Gentzen systems for intuitionistic logic), despite the lasting potential informativeness of the dualization procedure (the notion of constructive truth of intuitionistic logic, for instance, may dualize into a notion of constructive falsity, and the *verification* methodology is dualized into *falsification*, as in [3]). A straightforward abstract formal definition of duality may be found in [2]. In the present contribution we show that this definition encompasses all the above mentioned approaches, and applies equally well to logics that either extend or diverge from classical logic. The particular methods applicable to classical logic generalize in a natural way to modal logics and to many-valued logics, for instance. We shall illustrate in particular how finite-valued logics are dualized by taking advantage of their bivalent representations, following the algorithms surveyed in [1]. References: [1] Carlos Caleiro and João Marcos. Many-valuedness meets bivalence: Using logical values in an effective way. Journal of Multiple-Valued Logic and Soft Computing, 19(5–6):51–70, 2012. [2] João Marcos. Ineffable inconsistencies. In J.-Y. Béziau, W A Carnielli, and D Gabbay, editors, Handbook of Paraconsistency, volume 9 of Studies in Logic, pages 301–311. North Holland, Amsterdam, 2007. [3] Yaroslav Shramko. Dual intuitionistic logic and a variety of negations: The logic of scientific research. Studia Logica, 80(2–3):347–367, 2005. * * * -- Você está recebendo esta mensagem porque se inscreveu no grupo "LOGICA-L" dos Grupos do Google. Para cancelar inscrição nesse grupo e parar de receber e-mails dele, envie um e-mail para [email protected]. Para postar neste grupo, envie um e-mail para [email protected]. Visite este grupo em https://groups.google.com/a/dimap.ufrn.br/group/logica-l/. Para ver esta discussão na web, acesse https://groups.google.com/a/dimap.ufrn.br/d/msgid/logica-l/CAO6j_LhXbt6WFvQR2zRNgw6QtFF66gXSytDcuOwOkVWqc3stEA%40mail.gmail.com.
