Marcelo E. Coniglio

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

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.

Transfers between Logics and their Applications

In this paper, logics are conceived as two-sorted first-order structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be used to describe the relationship between logics through morphisms of structures called transfers. This leads to a formal framework for studying several properties of abstract logics and their attributes such as consequence operator, syntactical structure, and internal transformations. In particular, we treat Belief Revision Systems (BRS) as our main example, defining the Wide Belief Revision Systems (WBRSs). This generalization allows us to define BRSs in an abstract setting for classical and non-standard logics. We also show how the concept of translation between logics can be obtained as a particular case of transfers.

Combining Valuations with Society Semantics

Society Semantics, introduced by W. Carnielli and M. Lima-Marques, is a method for obtaining new logics from the combination of agents (valuations) of a given logic. The goal of this paper is to present several generalizations of this method, as well as to show some applications to many-valued logics. After a reformulation of Society Semantics in a wider setting, we develop in detail two examples of application of the new formalism, characterizing a hierarchy of paraconsistent logics called Pn (for n ? N) and a hierarchy of paracomplete logics In (for n ? N). We also propose three further generalizations, obtaining Society Semantics for several many-valued logics, including a hierarchy of logics called In Pk which are both paraconsistent and paracomplete.

Fibring Logics with Topos Semantics

The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOL-enrichments. Soundness is shown to be preserved by fibring without any further assumptions.

New Dimensions on Translations Between Logics

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: (conservative) translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.

A Graph-theoretic Account of Logics

A graph-theoretic account of logics is explored based on the general notion of m-graph (i.e; a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as multi-graphs (m-graphs). After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with non-deterministic semantics, and subsume all logics endowed with an algebraic semantics.

Modules in the category of sheaves over quantales

In this paper we develop the elementary theory of modules in the category Sh(Q) of sheaves over right-sided idempotent quantales. The main ingredient is the construction of a (first-order) logic sound for Sh(Q). As an application we prove that in Sh(Q), a finitely generated projective module is free (Theorem 7.2), a result that is relevant to the study of representation of non-commutative C∗-algebras.

Non-Commutative Topology and Quantales

The relationship between q-spaces (c.f. [9]) and quantum spaces (c.f. [5]) is studied, proving that both models coincide in the case of Spec A, the spectrum of a non-commutative C*-algebra A. It is shown that a sober T1 quantum space is a classical topological space. This difficulty is circumvented through a new definition of point in a quantale. With this new definition, it is proved that Lid A has enough points. A notion of orthogonality in quantum spaces is introduced, which permits us to express the usual topological properties of separation. The notion of stalks of sheaves over quantales is introduced, and some results in categorial model theory are obtained.

Hilbert-style Presentations of Two Logics Associated to Tetravalent Modal Algebras

We analyze the variety of A. Monteiro’s tetravalent modal algebras under the perspective of two logic systems naturally associated to it. Taking profit of the contrapositive implication introduced by A. Figallo and P. Landini, sound and complete Hilbert-style calculi for these logics are presented.