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S. Jakowski introduced the discussive prepositional calculus D 2as a basis for a logic which could be used as underlying logic of inconsistent but nontrivial theories (see, for example, N. C. A. da Costa and L. Dubikajtis, On Jakowski's discussive logic, in Non-Classical Logic, Model Theory and Computability, A. I. Arruda, N. C. A da Costa and R. Chuaqui edts., North-Holland, Amsterdam, 1977, 37–56). D 2has afterwards been extended to a first-order predicate calculus and to a higher-order logic (...) (cf. the quoted paper). In this paper we present a natural version of D 2, in the sense of Jakowski and Gentzen; as a consequence, we suggest a new formulation of the discussive predicate calculus (with equality). A semantics for the new calculus is also presented. (shrink)

In his thesis Para uma Teoria Geral dos Homomorfismos (1944), the Portuguese mathematician José Sebastiāo e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein's Erlangen Program and that foreshadows some notions and results of today's model theory; an analogous theory was independently worked out by M. Krasner in 1938. In this paper, we present a version of the theory making use of tools which were not at Silva's disposal. At the same time, we (...) tried to keep in mind, so much as possible, the gist of his standpoint. (shrink)

The paper reveals some mostly unnoticed and unexpected trends in reception of novel ideas in science. The author formulates certain principles of the reception of these ideas by scientific communities and justifies them by examples from modern mathematics and non-classical logic.

We show that the P 2 0 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. da Costa on the P

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The article provides a critical-comparative analysis of selected positions lending grounds for limitation of the clasical principle of consistency of belief systems. The first part contains arguments of C. Cherniak, P. Klein, R.C. Pinto, N. Rescher and R. Brandom for the inevitability of contradictions in human belief systems. The second part presents several systems of paraconsistent doxastic logic (N.C.A. da Costa, S. French, G. Priest) formally tolerating (in various way) contradictory beliefs. The analysis especially focuses on the weaknesses of (...) the paraconsistent interpretation of the self-deception phenomenon and the contradictions resulting from the so-called Moore's paradox. (shrink)

MEDIEVAL LOGICS LAMBERT MARIE DE RIJK (ed.), Die mittelalterlichen Traktate De mod0 opponendiet respondendi, Einleitung und Ausgabe der einschlagigen Texte. (Beitrage zur Geschichte der Philosophie und Theologie des Mittelalters, Neue Folge Band 17.) Miinster: Aschendorff, 1980. 379 pp. No price stated. THE SEVENTEENTH CENTURY MARTA FATTORI, Lessico del Novum Organum di Francesco Bacone. Rome: Edizioni dell'Ateneo 1980. Two volumes, il + 543, 520 pp. Lire 65.000. VIVIAN SALMON, The study of language in 17th century England. (Amsterdam Studies in the Theory (...) and History of Linguistic Science, Series 111: Studies in theHistory of Linguistics, Volume 17.) Amsterdam: John Benjamins B.V., 1979.x + 218 pp. Dfl. 65. Theoria cum Praxi. Zum Verhaltnis von Theorie und Praxis im 17. und 18. Jahrhundert. (Akten des 111. Internationalen Leibnizkongress, Hannover, 12. bis 17.November 1977, Band 111: Logik, Erkenntnistheorie, Wissenschaftstheorie, Metaphysik, Theologie.) Wiesbaden: Franz Steiner Verlag, 1980. vii + 269 pp. DM 48. CLASSICAL AND NON-CLASSICAL LOGICS MICHAEL CLARK, The place of syllogistic in logical theory. Nottingham: University of Nottingham Press, 1980. ix + 151 pp. £3.00. A.F. PARKER-RHODES, The theory of indistinguishables. Dordrecht, Boston and London: D. Reidel Publishing Company, 1981. xvii + 216 pp. Dfl.90.00/$39.50. NICHOLAS RESCHER and ROBERT BRANDOM, The logic of inconsistency. Oxford:Basil Blackwell, 1980. x + 174 pp. f 11.50. MISCELLANEOUS J. ZELENY, The logic of Marx. Translated from the German by T. Carver. Oxford: Basil Blackwell, 1980. xcii + 247 pp. £12.50. FELIX KAUFMANN, The infinite in mathematics. Edited by Brian McGuinness. Introduction by E. Nagel. Translation from the German by Paul Foulkes. Dordrecht: Reidel, 1978. xvii + 235 pp. Dfl 85/$39.50 (cloth); Dfl 45/$19.95 (paper). PAMELA MCCORDUCK, Machines who think. San Francisco: W.H. Freeman and Company, 1979. xiv + 275 pp. $14.95. J. MITTELSTRASS (ed.), Enzyklopadie Philosophie und Wissenschaftstheorie Bd. 1 : A-G. Mannheim, Wien, Ziirich: Bibliographisches Institut, 1980. 835 pp. DM 128. (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)

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. (shrink)

The prepositional calculiC n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum''s algebra forC n . C. Mortensen settled the problem, proving that no equivalence relation forC n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC n , as well as the concept of paraconsistent (...) closure system, are introduced in this paper. (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)

Da Costa's paraconsistent systems of the series Cm (for finite m) (see [C1], [C2], and esp. [C3], pp. 237ff.) share important features with transitive logic, TL (which has been gone into in [P1] and [P2]), namely, they all coincide in that: (c1) they possess a strong negation, `¬', a conditional, `⊃', a conjunction, `∧', and a disjunction, `∨', with respect to which they are conservative extensions of CL or Classical Logic; (c2) they possess a non strong negation, `N' (notations (...) are different for systems C) which does not possess all properties of classical negation, but for which the following schemata are theorematic (I use the letters `p', `q', etc as schematic letters; my notational conventions are basically Church's: associativity leftwards; a dot stands for.. (shrink)

Trois sortes de définitions sont présentées et discutées: les définitions nominales, les définitions contextuelles et les définitions amplificatrices. On insiste sur le fait que I’elimination des definitions n’est pas forcement un procede automatique en particulier dans le cas de la logique paraconsistante. Finalement on s’int’resse à la théorie des objets de Meinong et l’on montre comment elle peut êrre considéréecomme une théorie des descripteurs.Three kinds of definitions are presented and discussed: nominal definitions, contextual definitions, amplifying definitions. It is emphasized that (...) the elimination of definitions is not necessarily straightforward in particular in the case of paraconsistent logic. Finally we have a look at Meinong’s theory objects and we show how it can be considered as a theory of descriptors. (shrink)

Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, (...) as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions. (shrink)