Walter Carnielli

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus $ \mathcal {C}$min, stronger than $ \mathcal {C}$$\scriptstyle \omega$, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$min presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $ \mathcal {C}$Lim, the real deductive limit of da Costa's hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $ \mathcal {D}$min is proposed as the dual to $ \mathcal {C}$min.

This is the report on the XVI BRAZILIAN LOGIC CONFERENCE (EBL 2011) held in Petrópolis, Rio de Janeiro, Brazil between May 9–13, 2011 published in The Bulletin of Symbolic Logic Volume 18, Number 1, March 2012. The 16th Brazilian Logic Conference (EBL 2011) was held in Petro ́polis, from May 9th to 13th, 2011, at the Laboratório Nacional de Computação o Científica (LNCC). It was the sixteenth in a series of conferences that started in 1977 with the aim of congregating logicians from Brazil and abroad, furthering interest in logic and its applications, stimulating cooperation, and contributing to the development of this branch of science. EBL 2011 included more than one-hundred and fifty participants, all of them belonging to prominent research institutes from Brazil and abroad, especially Latin America. The conference was sponsored by the Academia Brasileira de Ciências (ABC), the As- sociation for Symbolic Logic (ASL), Universidade Estadual de Campinas (UNICAMP), Centre for Logic, Epistemology and the History of Sciences (CLE), Laboratório Nacional de Computação o Científica (LNCC), Pontif ́ıcia Universidade Cato ́lica do Rio de Janeiro (PUC- Rio), Sociedade Brasileira de Lógica (SBL), and Universidade Federal Fluminense (UFF). Funding was provided by Conselho Nacional de Desenvolvimento Cient ́ıfico e Tecnolo ́ gico (CNPq), Fundac ̧a ̃o de Amparo `a Pesquisa do Estado de São Paulo (FAPESP), Fundação Euclides da Cunha (FEC), and Universidade Federal Fluminense (UFF). The members of the Scientific Committee were: Mário Folhadela Benevides (COPPE- UFRJ), Fa ́bio Bertato (CLE-IFCH-UNICAMP), Jean-Yves Béziau (UFRJ), Ricardo Bianconi (USP), Juliana Bueno-Soler (UFABC), Xavier Caicedo (Universidad de Los An- des), Walter Carnielli (CLE-IFCH-UNICAMP), Oswaldo Chateaubriand Filho (PUC-Rio), Marcelo Esteban Coniglio (CLE-IFCH-UNICAMP), Newton da Costa (UFSC, President), Antonio Carlos da Rocha Costa (UFRG), Alexandre Costa-Leite (UnB), I ́tala M. Loffredo D’Ottaviano (CLE-IFCH-UNICAMP), Marcelo Finger (USP), Edward Hermann Haeusler (PUC-Rio), Décio Krause (UFSC), João Marcos (UFRN), Ana Teresa de Castro Martins (UFC), Maria da Paz Nunes de Medeiros (UFRN), Francisco Miraglia (USP), Luiz Car- los Pereira (PUC-Rio and UFRJ), Elaine Pimentel (UFMG), and Samuel Gomes da Silva (UFBA). The members of the Organizing Committee were: Anderson de Araujo (UNICAMP), Walter Carnielli (CLE-IFCH-UNICAMP), Oswaldo Chateaubriand Filho (PUC-Rio, Co- chair), Marcelo Correa (UFF), Renata de Freitas (UFF), Edward Hermann Haeusler (PUC- RJ), Hugo Nobrega (COPPE-UFRJ), Luiz Carlos Pereira (PUC-Rio e IFCS/UFRJ), Leandro Suguitani (UNICAMP), Rafael Testa (UNICAMP), Leonardo Bruno Vana (UFF), and Petrucio Viana (UFF, Co-chair).

his paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way. We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Lowenheim-Skolem theorem. The paper is completely self-contained and includes examples of application to particular many-valued formal systems.