Non-deterministic algebraization of logics by swap structures1

Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics

Some modal and temporal translations of generalized basic logic

We introduce a family of modal expansions of Łukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal Łukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna’s poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Łukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter

Logics of formal inconsistency

Orientadores: Walter Alexandre Carnielli, Carlos M. C. L. CaleiroTexto em ingles e portuguesTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias HumanasTese (doutorado) - Universidade Tecnica de Lisboa, Instituto Superior TecnicoResumo: Segundo a pressuposição de consistência clássica, as contradições têm um cará[c]ter explosivo; uma vez que estejam presentes em uma teoria, tudo vale, e nenhum raciocínio sensato pode então ter lugar. Uma lógica é paraconsistente se ela rejeita uma tal pressuposição, e aceita ao invés que algumas teorias inconsistentes conquanto não-triviais façam perfeito sentido. A? Lógicas da Inconsistência Formal, LIFs, formam uma classe de lógicas paraconsistentes particularmente expressivas nas quais a noção meta-teónca de consistência pode ser internalizada ao nível da linguagem obje[c]to. Como consequência, as LIFs são capazes de recapturar o raciocínio consistente pelo acréscimo de assunções de consistência apropriadas. Assim, por exemplo, enquanto regras clássicas tais como o silogismo disjuntivo (de A e {não-,4)-ou-13, infira B) estão fadadas a falhar numa lógica 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 presunção de que estamos raciocinando em um ambiente consistente (neste caso, pelo acréscimo de (consistente-.A) como uma hipótese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustrações destas lógicas e de suas propriedades, mostrando que tais lógicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do raciocínio consistente dentro de tais sistemas inconsistentes são também ilustradas Em cada caso, interpretações em termos de semânticas polivalentes, de traduções possíveis ou modais são fornecidas, e os problemas relacionados à provisão de contrapartidas algébricas para tais lógicas são examinados. Uma abordagem formal abstra[cjta é proposta para todas as definições relacionadas e uma extensa investigação é feita sobre os princípios lógicos e as propriedades positivas e negativas da negação.Abstract: According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and (not-A)-or-B, infer B) are bound to fail in a paraconsistent logic (because A and (not-.4) could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of (consistent-/!) as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature. Several ways of performing the recapture of consistent reasoning inside such inconsistent systems are also illustrated. In each case, interpretations in terms of many-valued, possible-translations, or modal semantics are provided, and the problems related to providing algebraic counterparts to such logics are surveyed. A formal abstract approach is proposed to all related definitions and an extended investigation is carried out into the logical principles and the positive and negative properties of negation.DoutoradoFilosofiaDoutor em Filosofia e Matemátic

Universal Logic and the Geography of Thought - Reflections on logical pluralism in the light of culture

The aim of this dissertation is to provide an analysis for those involved and interested in the interdisciplinary study of logic, particularly Universal Logic. While continuing to remain aware of the importance of the central issues of logic, we hope that the factor of culture is also given serious consideration. Universal Logic provides a general theory of logic to study the most general and abstract properties of the various possible logics. As well as elucidating the basic knowledge and necessary definitions, we would especially like to address the problems of motivation concerning logical investigations in different cultures. First of all, I begin by considering Universal Logic as understood by Jean-Yves Béziau, and examine the basic ideas underlying the Universal Logic project. The basic approach, as originally employed by Universal Logicians, is introduced, after which the relationship between algebras and logics at an abstract level is discussed, i.e., Universal Algebra and Universal Logic. Secondly,I focus on a discussion of the translation paradox , which will enable readers to become more familiar with the new subject of logical translation, and subsequently comprehensively summarize its development in the literature. Besides helping readers to become more acquainted with the concept of logical translation, the discussion here will also attempt to formulate a new direction in support of logical pluralism as identified by Ruldof Carnap (1934), JC Beall and Greg Restall (2005), respectively. Thirdly, I provide a discussion of logical pluralism. Logical pluralism can be traced back to the principle of tolerance raised by Ruldof Carnap (1934), and readers will gain a comprehensive understanding of this concept from the discussion. Moreover,an attempt will be made to clarify the real and important issues in the contemporary debate between pluralism and monism within the field of logic in general. Fourthly, I study the phenomena of cultural-difference as related to the geography of thought. Two general systems in the geography of thought are distinguished, which we here call thought-analytic and thought-holistic. They are proposed to analyze and challenge the universality assumption regarding cognitive processes. People from different cultures and backgrounds have many differences in diverse areas, and these differences, if taken for granted, have proven particularly problematic in understanding logical thinking across cultures. Interestingly, the universality of cognitive processes has been challenged, especially by Richard Nisbett s research in cultural psychology. With respect to these concepts, C-UniLog can also be considered in relation to empirical evidence obtained by Richard Nisbett et al. In the final stage of this dissertation, I will propose an interpretation of the concept of logical translation, i.e., translations between formal logical mode (as cognitive processes in the case of westerners) and dialectical logical mode (as cognitive processes in the case of Asians). From this, I will formulate a new interpretation of the principle of tolerance, as well as of logical pluralism

Poset products as relational models

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics

Quantale Modules, with Applications to Logic and Image Processing

We propose a categorical and algebraic study of quantale modules. The results and constructions presented are also applied to abstract algebraic logic and to image processing tasks.Comment: 150 pages, 17 figures, 3 tables, Doctoral dissertation, Univ Salern