Walter Carnielli

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices.

Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem.

How is it possible that beginning from the negation of rational thoughts one comes to produce knowledge? This problem, besides its intrinsic interest, acquires a great relevance when the representation of a knowledge is settled, for example, on data and automatic reasoning. Many treatment ways have been tried, as in the case of the non-monotonic logics; logics that intend to formalize an idea of reasoning by default, etc. These attempts are incomplete and are subject to failure. A possible solution would be to formulate a logic of the irrational, which offers a model for reasoning permitting to support contradictions as well as to produce knowledge from such situations. An intuition underlying the foundation of such a logic consists of the da Costa's paraconsistent logics presenting however, a different deduction theory and a whole distinct semantics, called here "the semantics of possible translations". The present proposing, following our argumentation, intends to enlight all this question, by a whole satisfactory logical point of view, being practically applicable and philosophically acceptable.Como é possível que a partir da negação do racional se possa obter conhecimento adicional? Esse problema, além de seu interesse intrínseco, adquire uma relevância adicional quando o encontramos na representação do conhecimento em bases de dados e raciocínio automático, por exemplo. Nesse caso, diversas tentativas de tratamento têm sido propostas, como as lógicas não-monotônicas, as lógicas que tentam formalizar a ideia do raciocínio por falha. Tais tentativas de solução, porém, são falhas e incompletas; proponho que uma solução possível seria formular uma lógica do irracional, que oferecesse um modelo para o raciocínio permitindo não só suportar contradições, como conseguir obter conhecimento, a partir de tais situações. A intuição subjacente à formulação de tal lógica são as lógicas paraconsistentes de da Costa, mas com uma teoria da dedução diferente e uma semântica completamente distinta. Tal proposta, como pretendo argumentar, fornece um enfoque para a questão que é ao mesmo tempo completamente satisfatório, aplicável do ponto de vista prático e aceitável do ponto de vista filosófico.

This paper studies a family of monotonic extensions of first-order logic which we call modulated logics, constructed by extending classical logic through generalized quantifiers called modulated quantifiers. This approach offers a new regard to what we call flexible reasoning. A uniform treatment of modulated logics is given here, obtaining some general results in model theory. Besides reviewing the “Logic of Ultrafilters”, which formalizes inductive assertions of the kind “almost all”, two new monotonic logical systems are proposed here, the “Logic of Many” and the “Logic of Plausibility”, that characterize assertions of the kind “many”, and “for a good number of”. Although the notion of simple majority (“more than half”) can be captured by means of a modulated quantifier semantically interpreted by cardinal measure on evidence sets, it is proven that this system, although sound, cannot be complete if checked against the intended model. This justifies the interest on a purely qualitative approach to this kind of quantification, what is guaranteed by interpreting the modulated quantifiers as notions of families of principal filters and reduced topologies, respectively. We prove that both systems are conservative extensions of classical logic that preserve important properties, such as soundness and completeness. Some additional perspectives connecting our approach to flexible reasoning through modulated logics to epistemology and social choice theory are also discussed.

This paper investigates a problem related to quantifiers which has some analogies to that of propositional completeness I give a definition of quantifier in many-valued logics generalizing the cases which already occur in first order many- valued logics. Though other definitions are possible, this particular one, which I call distribution quantifiers, generalizes the classical quantifiers in a very natural way, and occurs in finite numbers in every m-valued logic. We then call the problem of quantificationa2 completeness in m-valued logic the problem of characterizing which are the quantifiers in a given language which can generate all other quantifiers in this language, using the connectives, as is the case, for example, of the universal and exis- tential quantifiers in classical logic, using negation. We are interested, in particular, in those many-valued quantifiers which closely mimic the behavior of existential an universal quantifiers in generating all other quantifiers using negation: these I call perfect quantifiers, as defined below. The main result of this paper is the characterization of all perfect quantifiers in 3-valued logics, which are complete if the logic is functionally complete. As a byproduct, we obtain the same result for the classical logic, which we include mainly for motivation.

We present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account of paraconsistency.

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning.

EN In his “Logic, Language, and Knowledge” Chateaubriand denounces the tyranny of belief, but takes some positions on knowledge and justification which seem to be too exacting. The fact that Chateaubriand derives constraints on the notion of justification by a close parallel to the notion of proof makes it unnecessarily loaded with the individual, rather than with the collective perspective. His position seems to leave little room for common knowledge, collective knowledge and usual common-sense knowledge, and absolutely no room for explaining how people take correct decisions based on apparently faulty notions of knowledge and justification. PT Em “Logic, Language, and Knowledge” Chateaubriand denuncia a tirania da crença, mas toma posições sobre conhecimento e justificação que parecem demasiado exigentes. O fato de Chateaubriand derivar condições sobre a noção de justificação a partir de uma estreita analogia com a noção de prova torna a noção de justiticação desnecessariamente carregada com a perspectiva indivídual, em detrimento da perspectiva coletiva. Sua posição parece deixar pouco espaço para noções como conhecimento comum, conhecimento coletivo e senso comum, e absolutamente nenhum espaço para explicar como as pessoas tomam decisões corretas com base em noções aparentemente errôneas de conhecimento e de justificação.

Abduction or retroduction, as introduced by C.S. Peirce in the double sense of searching for explanatory instances and providing an explanation is a kind of complement for usual argumentation. There is, however, an inferential step from the explanandum to the abductive explanans. Whether this inferential step can be captured by logical machinery depends upon a number of assumptions, but in any case it suffers in principle from the triviality objection: any time a singular contradictory explanans occurs, the system collapses and stops working. The traditional remedies for such collapsing are the expensive mechanisms of consistency maintenance, or complicated theories of non-monotonic derivation that keep the system running at a higher cost. I intend to show that the robust logics of formal inconsistency, a particular category of paraconsistent logics which permit the internalization of the concepts of consistency and inconsistency inside the object language, provide simple yet powerful techniques for automatic abduction. Moreover, the whole procedure is capable of automatization by means of the tableau proof-procedures available for such logics. Some motivating examples are discussed in detail.

Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo contexts evolve for the sake of argumentation intoone system containing both contexts.More technically, we show a new, dialogical, way tobuild paraconsistent systems for propositional andfirst-order logic with classical and intuitionisticfeatures (i.e. paraconsistency both with and withouttertium non-datur) and present theircorresponding tableaux.

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.

A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-Gentzen-Gödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics .