This work presents some basic results on a theory of translations between logics and a short revision about Łukasiewicz's logics. Then, it is shown, using facts about algebraic semantics, that there is a conservative translation from every finite Łukasiewicz's logic into classical logic. However, this is not a constructive result.

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L3, the intuitionistic system I1 and several paraconsistent logics, as for instance Sette's system P1, the D'Ottaviano and da Costa system J3 and da Costa's systems Cn, 1≤ n≤ω. (shrink)

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L₃, the intuitionistic system Iⁱ and several para-consistent logics, as for instance Sette's system Pⁱ, the D'Ottaviano and da Costa system J₃ and da Costa's systems $\text{C}_{n}$ , 1 ≤ n ≤ ω. (shrink)

The Joint Non-Trivialization Theorem, two Definability Theorems and the generalized Quantifier Elimination Theorem are proved for J 3-theories. These theories are three-valued with more than one distinguished truth-value, reflect certain aspects of model type logics and can. be paraconsistent. J 3-theories were introduced in the author's doctoral dissertation.

This work presents the concepts of translation and conservative translation between logics. By using algebraic semantics we introduce several conservative translations involving the classical propositional calculus and the many-valued calculi of Post and Lukasiewicz.

In this paper we present a new hierarchy of analytical tableaux systems TNDC n, 1≤n (...) and also prove that these systems are logically equivalent to the corresponding systems Cn, 1≤nshrink)

In the Organon Aristotle describes some deductive schemata in which inconsistencies do not entail the trivialization of the logical theory involved. This thesis is corroborated by three different theoretical topics by him discussed, which are presented in this paper. We analyse inference schema used by Aristotle in the Protrepticus and the method of indirect demonstration for categorical syllogisms. Both methods exemplify as Aristotle employs classical reductio ad absurdum strategies. Following, we discuss valid syllogisms from opposite premises (contrary and contradictory) studied (...) by the Stagerian in the Analytica Priora (B15). According to him, the following syllogisms are valid from opposite premises, in which small Latin letters stand for terms such as subject and predicate, and capital Latin letters stand for the categorical propositions such as in the traditional notation: (i) in the second figure, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), and Aba,Oba ` Oaa (Baroco); (ii) in the third one, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) and Eab, Iab ` Oaa (Ferison). Finally, we discuss the passage from the Analytica Posteriora (A11) in which Aristotle states that the Principle of Non-Contradiction is not generally presupposed in all demonstrations (scientific syllogisms), but only in those in which the conclusion must be proved from the Principle; the Stagerian states that if a syllogism of the first figure has the major term consistent, the other terms of the demonstration can be each one separately inconsistent. These results allow us to propose an interpretation of his deductive theory as a broad sense paraconsistent theory. Firstly, we proceed to a hermeneutical analysis, evaluating its logical significance and the interplay of the results with some other points of Aristotle’s philosophy. Secondly, we point to a logical interpretation of the Aristotelian syllogisms from opposite premises in the antilogisms method proposed by Christine Ladd-Franklin in 1883, and we also present a logical treatment of the Aristotelian demonstration with inconsistent material in the paraconsistent logics Cn, 1 _ n _!, introduced by da Costa in 1963. These two issues seem having not yet been analysed in detail in the literature. (shrink)

The aim of this paper is to argue about the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm that represent the human capacity of recognizing mathematical truths. More specifically, based on a direct argument from Gödel's First Incompleteness Theorem, we discuss the impossibility of constructing a complete formal theory or a complete Turing machines' algorithm to the human capacity of recognition of first-order arithmetical truths and so of mathematical truths in general.

No Órganon Aristóteles descreve alguns esquemas dedutivos nos quais a presença de inconsistências não acarreta a trivialização da teoria lógica envolvida. Esta tese é corroborada por três diferentes situações teóricas estudadas por ele, as quais são apresentadas neste trabalho. Analizamos o esquema de inferência utilizado por Aristóteles no Protrepticus e o método de demonstração indireta para os silogismos categóricos. Ambos os métodos exemplificam como Aristóteles emprega estratégias de redução ao absurdo logicamente clássicas. Na sequência, discutimos os silogismos válidos a partir (...) de premissas opostas (contrárias e contraditórias) estudadas pelo Estagirita no Analytica Priora (B15). De acordo com o autor, os seguintes silogismos são válidos a partir de premissas opostas, nos quais letras latinas minúsculas denotam termos como sujeito e predicado, enquanto que letras latinas maiúsculas denotam proposições categóricas tal como na notação tradicional: (i) na segunda figura, Eba,Aba ` Eaa (Cesare), Aba, Eba ` Eaa (Camestres), Eba, I ba ` Oaa (Festino), e Aba,Oba ` Oaa (Baroco); (ii) na terceira, Eab,Aab ` Oaa (Felapton), Oab,Aab ` Oaa (Bocardo) e Eab, Iab ` Oaa (Ferison). Por fim, discutimos a passagem do Analytica Posteriora (A11) no qual Aristóteles enuncia que o Princípio de Não-Contradição não é, em geral, pressuposto de toda demonstração (silogismo científico), mas apenas daquelas nas quais a conclusão deve ser provada a partir do Princípio; o Estagirita enuncia que se um silogismo da primeira figura tiver o termo maior consistente, os outros termos da demonstração podem ser separadamente inconsistentes. Estes resultados permitem-nos propor uma interpretação de sua teoria dedutiva como uma teoria paraconsistente lato sensu. Primeiramente, efetuamos uma análise hermenêutica, avaliando seu significado lógico e a correlação desses resultados com outros aspectos da filosofia de Aristóteles. Em segundo lugar, consignamos uma interpretação dos silogismos aristotélicos a partir de premissas opostas à luz dos antilogismos propostos por Christine Ladd-Franklin em 1883, e da demonstração aristotélica com termos inconsistentes nas lógicas paraconsistentes Cn, 1 n !, introduzidas por da Costa em 1963. Esses dois aspectos não parecem ter sido ainda detalhadamente analisados na literatura. DOI:10.5007/1808-1711.2010v14n1p71. (shrink)

The main objective o f this descriptive paper is to present the general notion of translation between logical systems as studied by the GTAL research group, as well as its main results, questions, problems and indagations. Logical systems here are defined in the most general sense, as sets endowed with consequence relations; translations between logical systems are characterized as maps which preserve consequence relations (that is, as continuous functions between those sets). In this sense, logics together with translations form a (...) bicomplete category of which topological spaces with topological continuous functions constitute a full subcategory. We also describe other uses of translations in providing new semantics for non-classical logics and in investigating duality between them. An important subclass of translations, the conservative translations, which strongly preserve consequence relations, is introduced and studied. Some specific new examples of translations involving modal logics, many-valued logics, para- consistent logics, intuitionistic and classical logics are also described. (shrink)

This volume corresponds to the Proceedings of the XIII Brazilian Logic Conference held at the CLE - Centre for Logic, Epistemology and the History of Science in Campinas, SP, Brazil from May 26-30, 2003 under the auspices of the SBL - Brazilian Logic Society and the ASL - Association for Symbolic Logic.

Inspired by the early Wittgenstein’s concept of nonsense (meaning that which lies beyond the limits of language), we define two different, yet complementary, types of nonsense: formal nonsense and pragmatic nonsense. The simpler notion of formal nonsense is initially defined within Tarski’s semantic theory of truth; the notion of pragmatic nonsense, by its turn, is formulated within the context of the theory of pragmatic truth, also known as quasi-truth, as formalized by da Costa and his collaborators. While an expression will (...) be considered formally nonsensical if the formal criteria required for the assignment of any truth-value (whether true, false, pragmatically true, or pragmatically false) to such sentence are not met, a (well-formed) formula will be considered pragmatically nonsensical if the pragmatic criteria (inscribed within the context of scientific practice) required for the assignment of any truth-value to such sentence are not met. Thus, in the context of the theory of pragmatic truth, any (well-formed) formula of a formal language interpreted on a simple pragmatic structure will be considered pragmatically nonsensical if the set of primary sentences of such structure is not well-built, that is, if it does not include the relevant observational data and/or theoretical results, or if it does include sentences that are inconsistent with such data. (shrink)

O principal objetivo deste artigo é apresentar uma identidade possível entre os conceitos de forma e de qualidade na filosofia de Charles Sanders Peirce, por meio de seus argumentos em sua Semiótica e em sua Cosmologia. Em outras palavras, nosso objetivo é mostrar que a primeiridade consiste em uma forma, parte constitutiva da natureza da terceiridade, na medida em que a tendência à generalização ou à aquisição de hábitos estava prefigurada na origem do cosmos. De natureza indutiva, o passo do (...) nada absoluto para uma unidade de qualidades prefigura um universo inteligível de natureza formal. Esta unidade já pode ser considerada uma restrição de uma potencialidade de força maior presente naquele nada germinal: a primeira categoria é configurada, então, como sendo de natureza potencial qualitativa. O adjetivo “qualitativa” apresenta uma espécie de restrição da potencialidade para a qual nos referimos: tal potencial é desta ou daquela espécie. A segunda categoria, por sua vez, surge de um caos de sentimentos: não é a interação entre tais sentimentos que traz a segunda categoria à realidade, mas a mera manifestação do sentimento que é, por sua vez, caracterizado como a aparência momentânea da qualidade. Esta aparência não é potencial, mas atual. Ela já é um fato; ela já é uma restrição da potencialidade qualitativa. Sentimento enquanto atualidade e qualidade em seu estado potencial prefiguram a origem do outro que se apresenta: este outro está, portanto, inscrito na natureza da qualidade e já apresenta a dualidade objeto e objeto representado. Por fim, a tendência à generalização é reconhecida por meio das relações que as qualidades estabelecem entre si: na medida em que o aparecimento de uma ou mais qualidades se manteve insistente, tais qualidades começaram a estabelecer relações entre si, permitindo a formação indutiva de leis e objetos mais complexos, mas sempre devedores do material disponível prefigurado na origem: qualidades. (shrink)

In this paper we introduce the concept of conservative translation between logics. We present some necessary and sufficient conditions for a translation to be conservative and study some general properties of logical systems, these properties being characterized by the existence of conservative translations between the systems. We prove that the class constituted by logics and conservative translations between them determines a co-complete subcategory of the bi-complete category constituted by logics and translations.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

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.

In a recent contribution to this journal, D. Kovacs addresses the following passage from the fourth Eclogue :o mihi tum longae maneat pars ultima uitae,spiritus et quantum sat erit tua dicere facta!Kovacs takes it for granted that the meaning of l. 54 should correspond to the Loeb translation, ‘and inspiration enough to hymn your deeds!’. Starting from this assumption, he rejects the reading spiritus, arguing that a genitive is required ; the possible solution he suggests is pectoris, used metaphorically in (...) the sense of ‘poetical ability’. That seems a clever conjecture, restoring excellent style. Nevertheless, we have to assume that spiritus replaced pectoris as a gloss, but I cannot see any reason why somebody would have wanted to explain obscurum per obscurius: other words would have been far more appropriate. On the other hand, Virgil most likely chose the word spiritus, whose basic meaning is not ‘inspiration’ but ‘breath’, in order to amplify line 53. (shrink)

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L 3, the intuitionistic system I 1 and several paraconsistent logics, as for instance Sette's system P 1, the D'Ottaviano and da Costa system J 3 and da Costa's systems C n, 1 n. (shrink)

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, (...) the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered. (shrink)

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...) ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by Löwe and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing Löwe and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF). (shrink)

We develop an algebraic study of W.S. Cooper’s three-valued propositional logic of ordinary discourse ( $$\mathcal{O}\mathcal{L}$$ O L ). This logic displays a number of unusual features: $$\mathcal{O}\mathcal{L}$$ O L is not weaker but incomparable with classical logic, it is connexive, paraconsistent and contradictory. As a non-structural logic, $$\mathcal{O}\mathcal{L}$$ O L cannot be algebraized by the standard methods. However, we show that $$\mathcal{O}\mathcal{L}$$ O L has an algebraizable structural companion, and determine its equivalent semantics, which turns out to be a (...) finitely-generated discriminator variety. We provide an equational and a twist presentation for this class of algebras, which allow us to compare it with other well-known algebras of non-classical logics. In this way we establish that $$\mathcal{O}\mathcal{L}$$ O L is definitionally equivalent to an expansion of the three-valued logic $${\mathcal {J}}3$$ J 3 of D’Ottaviano and da Costa, itself a schematic extension of paraconsistent Nelson logic. (shrink)

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). (shrink)

Sanskrit text with commentaries on the theories of error in Mīmāṃsa school in Hindu philosophy by Maṇḍanamiśra, 7th century philosopher.