Walter Alexandre Carnielli

Logics of Formal Inconsistency

Segundo a pressuposicao de consistencia classica, as contradicoes tem um cara[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocinio sensato pode entao ter lugar. Uma logica e paraconsistente se ela rejeita uma tal pressuposicao, e aceita ao inves que algumas teorias inconsistentes conquanto nao-triviais facam perfeito sentido. A? Logicas da Inconsistencia Formal, LIFs, formam uma classe de logicas paraconsistentes particularmente expressivas nas quais a nocao meta-teonca de consistencia pode ser internalizada ao nivel da linguagem obje[c]to. Como consequencia, as LIFs sao capazes de recapturar o raciocinio consistente pelo acrescimo de assuncoes de consistencia apropriadas. Assim, por exemplo, enquanto regras classicas tais como o silogismo disjuntivo (de A e {nao-,4)-ou-13, infira B) estao fadadas a falhar numa logica paraconsistente (pois A e (nao-A) poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presuncao de que estamos raciocinando em um ambiente consistente (neste caso, pelo acrescimo de (consistente-.A) como uma hipotese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustracoes destas logicas e de suas propriedades, mostrando que tais logicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocinio consistente dentro de tais sistemas inconsistentes sao tambem ilustradas Em cada caso, interpretacoes em termos de semânticas polivalentes, de traducoes possiveis ou modais sao fornecidas, e os problemas relacionados a provisao de contrapartidas algebricas para tais logicas sao examinados. Uma abordagem formal abstra[cjta e proposta para todas as definicoes relacionadas e uma extensa investigacao e feita sobre os principios logicos e as propriedades positivas e negativas da negacao Abstract

Systematization of finite many-valued logics through the method of tableaux

This 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.

Formal inconsistency and evolutionary databases

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.

A Logical Framework for Integrating Inconsistent Information in Multiple Databases

When integrating data coming from multiple different sources we are faced with the possibility of inconsistency in databases. In this paper, we use one of the paraconsistent logics introduced in [9,7] (LFI1) as a logical framework to model possibly inconsistent database instances obtained by integrating different sources. We propose a method based on the sound and complete tableau proof system of LFI1 to treat both the integration process and the evolution of the integrated database submitted to users updates. In order to treat the integrated database evolution, we introduce a kind of generalized database context, the evolutionary databases, which are databases having the capability of storing and manipulating inconsistent information and, at the same time, allowing integrity constraints to change in time. We argue that our approach is sufficiently general and can be applied in most circumstances where inconsistency may arise in databases.

Segmentation of breast tumors in mammograms by fuzzy region growing

Segmentation of tumor regions in mammograms is not easy due to the low contrast and the fuzzy nature of the boundaries of malignant tumors. General image segmentation procedures do not consider the uncertainty present around the boundaries of a tumor region. In this paper we present a segmentation method based on fuzzy region growing. The procedure starts with a seed pixel, and uses a fuzzy membership function based upon statistical measures of the region being grown. Results of testing with several mammograms indicate that the method can provide boundaries of tumors close to those drawn by an expert radiologist. The regions obtained preserve the transition information present around tumor boundaries. Statistical measures computed from the resulting regions have shown the potential to classify masses and tumors as benign or malignant.

MODEL-BASED REASONING IN SCIENCE AND TECHNOLOGY

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Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed withtruth-functional semantics. However, the techniques used so far are unableto cope with fibring of logics endowed with non-truth-functional semanticsas, for example, paraconsistent logics. The first main contribution of thepaper is the development of a suitable abstract notion of logic, that mayalso encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that thisextended notion of fibring preserves completeness under certain reasonableconditions. This completeness transfer result, the second main contributionof the paper, generalizes the one established in Zanardo et al. (2001) butis obtained using new techniques that explore the properties of a suitablemeta-logic (conditional equational logic) where the (possibly)non-truth-functional valuations are specified. The modal paraconsistentlogic of da Costa and Carnielli (1988) is studied in the context of this novel notionof fibring and its completeness is so established.

Segmentation of breast tumors in mammograms using fuzzy sets

Defining criteria to determine precisely the boundaries of masses in mammograms is a difficult task. The problem is com- pounded by the fact that most malignant tumors possess fuzzy boundaries with a slow and extended transition from a dense core region to the surrounding less-dense tissues. We propose two seg- mentation methods that incorporate fuzzy concepts. The first method determines the boundary of a mass or tumor by region growing after a preprocessing step based on fuzzy sets to enhance the region of interest (ROI). Contours provided by the method have demonstrated a good match with the contours drawn by a radiolo- gist, as indicated by good agreement between the two sets of con- tours for 47 mammograms. The second segmentation method is a fuzzy region-growing method that takes into account the uncertainty present around the boundaries of tumors. The difficult step of decid- ing on a crisp boundary is obviated in the proposed method. Mea- sures of inhomogeneity computed from the pixels present in a suit- ably defined fuzzy ribbon have indicated potential use in classifying the masses and tumors as benign or malignant, with a sensitivity of 0.8 and a specificity of 0.9.

Many-valued logics and plausible reasoning

The analysis of reasoning with inconsistent information is modeled by a formal metasystem, META, which is presented. Semantics for META is given in terms of possible translations into three-valued logics which are based on a notion of contextual negation. From the point of view of applications, it is shown that this analysis can provide better foundations for default reasoning.<<ETX>>

The Dialogical Approach to Paraconsistency

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.