Skip to main content
Log in

Paraconsistent Logics and Translations

  • Published:
Synthese Aims and scope Submit manuscript

Abstract

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≤ω.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Carnielli, W. A. and I. M. L. D'Ottaviano: 1997, 'Translations between Logics: A Manifesto', Logique et Analyse 40(157), 67–81.

    Google Scholar 

  • Da Costa, N. C. A.: 1963a, Sistemas Formais Inconsistentes, Thesis, Universidade Federal do Paraná, Curitiba.

    Google Scholar 

  • Da Costa, N. C. A.: 1963b, 'Calculs propositionels pour les systèmes formels inconsistants', Comptes Rendus de l'Académie de Sciences de Paris 257, 3790–3793.

    Google Scholar 

  • Da Silva, J. J., I. M. L. D'Ottaviano and A. M. Sette: 1999, 'Translations between Logics', in X. Caicedo, C. H. Montenegro (eds), Models, Algebras and Proofs, Marcel Dekker, New York. (Lecture Notes in Pure and Applied Mathematics 203, 435–448).

    Google Scholar 

  • D'Ottaviano, I. M. L. and N. C. A. da Costa: 1970, 'Sur un problème de Jaskowski', C.R. Acad. Sc. Paris 270A, 1349–1353.

    Google Scholar 

  • D'Ottaviano, I. M. L.: 1973, Fechos caracterizados por interpretações (Closures characterized by interpretations), Master Dissertation, versidade Estadual de Campinas, Campinas.

    Google Scholar 

  • D'Ottaviano, I. M. L.: 1985, 'The Completeness and Compactness of a Three-Valued First-Order Logic', Revista Colombiana de Matemáticas 19, 77–94.

    Google Scholar 

  • D'Ottaviano, I. M. L.: 1990, 'On the Development of Paraconsistent Logic and da Costa's Work', The Journal of Non-Classical Logic 7(1/2), 9–72.

    Google Scholar 

  • D'Ottaviano, I. M. L. and H. A. Feitosa: 1999, 'Many-Valued Logics and Translations', Journal of Applied Non-Classical Logics 9(1), 121–140.

    Article  Google Scholar 

  • Epstein, R. L.: 1990, The Semantic Foundations of Logic. Propositional Logics, Vol. 1, Kluwer Academic, Dordrecht.

    Google Scholar 

  • Epstein, R. L. and I. M. L. D'Ottaviano: 1998, 'A Many-Valued Paraconsistent Logic', Reports on Mathematical Logic 22, 89–103.

    Google Scholar 

  • Feitosa, H. A.: 1997, Traduções conservatives (Conservative translations), Doctoral Thesis, Universidade Estadual de Campinas, Campinas.

    Google Scholar 

  • Feitosa, H. A. and I. M. L. D'Ottaviano: 2000, 'Conservative Translations', to appear in Annals of Pure and Applied Logic.

  • Gentzen, G.: 1969, 'On the Relation between Intuitionistic and Classical Arithmetic', in M. E. Szabo (ed.), The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam, pp. 53–67.

    Google Scholar 

  • Glivenko, V.: 1929, 'FrSur quelques points de la logique de M. Brouwer.', Académie Royale de Belgique. Bulletins de la Classe de Sciences Série 5, 15, 183–188.

    Google Scholar 

  • Gödel, K.: 1986, 'On Intuitionistic Arithmetic and Number Theory (1933a)', in S. Feferman et al. (eds), Collected Works, Oxford University Press, Oxford, pp. 287–295.

    Google Scholar 

  • Gödel, K.: 1986, 'An Interpretation of the Intuitionistic Propositional Calculus (1933b)', in S. Feferman et al. (eds), Collected Works, Oxford University Press, Oxford, pp. 301–303.

    Google Scholar 

  • Heyting, A.: 1930, 'Die formalen Regeln der intuitionistschen Logik Sitzungsberichte der Preussischen (Berlin) Akademie der Wissenschaften, Phys. Math. K1. 42–56.

  • Hoppmann, A. G.: 1973, Fecho e imersão (Closure and embedding), Doctoral Thesis, Universidade Estadual Paulista, Rio Claro.

    Google Scholar 

  • Kolmogoroff, A. N.: 1977, 'On the Principle of Excluded Middle (1925)', in J. Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic 1879–1931, Harvard University Press, Cambridge, MA, pp. 414–437.

    Google Scholar 

  • Lewis, C. I. and C. H. Langford: 1932, Symbolic Logic, 2nd edn with corrections, Century, Dover, 1959.

    Google Scholar 

  • Prawitz, D. and P. E. Malmnäs: 1968, 'A Survey of Some Connections between Classical, Intuitionistic and Minimal Logic', in H. Schmidt et al. (eds), Contributions to Mathematical Logic, North-Holland, Amsterdam, pp. 215–229.

    Google Scholar 

  • Queiroz, G. S.: 1997, Sobre a dualidade entre intuicionismo e paraconsistência (On the duality between intuitionism and paraconsistency), Doctoral Thesis, Universidade Estadual de Campinas, Campinas.

    Google Scholar 

  • Rasiowa, G.: 1974, An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam.

    Google Scholar 

  • Sette, A. M.: 1973, 'On the Propositional Calculus P 1', Mathematica Japonica 18, 173–180.

    Google Scholar 

  • Sette, A. M. and W. A. Carnielli (1995), 'Maximal Weakly-Intuitionistic Logics', Studia Logica 55, 181–203.

    Article  Google Scholar 

  • Sette, A. M., E. H. Alves and G. S. Queiroz: 199?, 'Brouwerian Algebras and Paraconsistent Logic, 14 pp. Submitted for publication.

  • Szabo, M. E. (ed.): 1969, The Collected Papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam.

    Google Scholar 

  • Wójcicki, R.: 1988, Theory of Logical Calculi: Basic Theory of Consequence Operations, Vol. 199, Synthese Library, Kluwer Academic, Dordrecht.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

D’Ottaviano, I.L., Feitosa, H.A. Paraconsistent Logics and Translations. Synthese 125, 77–95 (2000). https://doi.org/10.1023/A:1005298624839

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005298624839

Keywords

Navigation