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Paraconsistent algebras

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Abstract

The prepositional calculiC n , 1 ⩽n ⩽ ω 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.

We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent.

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References

  1. [1]

    A. I. Arruda,A survey of paraconsistent logic, inMathematical Logic in Latin America North-Holland, Amsterdam, 1980, pp. 1–41.

  2. [2]

    A. I. Arruda,Aspects of the historical development of paraconsistent logic, to appear.

  3. [3]

    D. Batens,Material propositional logics for inconsistent worlds, to appear.

  4. [4]

    S. L. Bloom andD. J. Brown,Classical abstract logics,Dissertationes Mathematical CII (1973).

  5. [5]

    M. W. Bunder,A new hierarchy of paraconsistent logics,Proceedings of the Third Brasilian Conference on Mathematical Logic, Sociedade Brasileira de Lógica, São Paulo, 1981, 13–22.

  6. [6]

    N. C. A. da Costa,Calculs propositionnels pour les systèmes formels inconsistants,Compte Rendue des Sciences de l'Academie des Sciences de Paris t. 257, 3790–3792 (1963).

  7. [7]

    N. C. A. da Costa,Opérations non-monotones dans le treillis,Compte Rendue des Sciences de l'Academie des Sciences de Paris, t. 263, (1966), pp. 429–432.

  8. [8]

    N. C. A. da Costa,Filtres et idéaux d'une algèbre C n .Compte Rendue des Sciences de l'Academie des Sciences de Paris, t. 264 (1967), pp. 549–552.

  9. [9]

    M. Fidel,The decidability of the calculi C n ,Reports on Mathematical Logic, 8 (1977), pp. 31–40.

  10. [10]

    L. Henkin,La structure algébrique des théories mathématiques, Gauthier-Villars, Paris, 1956.

  11. [11]

    L. Henkin, J. D. Monk andA. Tarski,Cylindric Algebras, North-Holland, Amsterdam, 1971.

  12. [12]

    G. J. Logan,Closure algebras and boolean algebras,Zeitschrift für Mathemaische Logic und Grundlagen der Mathematik, 23 (1967), pp. 93–96.

  13. [13]

    C. Mortensen,Every quotient algebra for C 1 is trivial, to appear.

  14. [14]

    C. Mortensen,Paraconsistency and C 1 Notre Dame Journal of Formal Logic, Vol. 21, No 4 (1980), pp. 694–700.

  15. [15]

    A. M. Sette,Sobre as algebras e hiper. retioulados C ω , (Master's Thesis — UNICAMP), 1971.

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Carnielli, W.A., de Alcantara, L.P. Paraconsistent algebras. Stud Logica 43, 79–88 (1984). https://doi.org/10.1007/BF00935742

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Keywords

  • Equivalence Relation
  • Mathematical Logic
  • Closure System
  • Special Kind
  • Computational Linguistic