Historical and Computational Aspects of Paraconsistency in View of the Logic Foundation of Databases

  • Hendrik Decker
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2582)


We expose paraconsistent logic with regard to its potential to contribute to the foundations of databases. We do so from a historical perspective, starting at the ancient inception and arriving at the contemporary use of logic as a computational device. We show that an understanding of the logic foundations of databases in terms of paraconsistency is adequate. It avoids absurd connotations of the ex contradictione quodlibet principle, which in fact never applies in databases. We interpret datalog, its origins and some of its extensions by negation and abduction, in terms of paraconsistency. We propose a procedural definition of paraconsistency and show that many well-known query answering procedures comply with it.


Logic Program Inference Rule Classical Logic Integrity Constraint Intuitionistic Logic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [ABK]
    M. Arenas, L. Bertossi, M. Kifer: Applications of Annotated Predicate Calculus to Querying Inconsistent Databases. Proc. Computational Logic 2000, Springer LNCS 1861, 926–941, 2000.Google Scholar
  2. [Av]
    J. Avigad: Classical and Constructive Logic., 2000.
  3. [Ba]
    D. Batens: A survey of inconsistency-adaptive logics. In D. Batens et al (eds): Frontiers of Paraconsistent Logic. King’s College Publications, 49–73, 2000.Google Scholar
  4. [Bb]
    A. Bobenrieth: Inconsistencias ¿por qué no? Un estudio filosófico sobre la lógica paraconsistente. Colcultura, 1996.Google Scholar
  5. [Bc]
    I. M. Bocheński: Formale Logik, 2nd edition. Verlag Karl Arber, Freiburg, München, 1956.zbMATHGoogle Scholar
  6. [BS]
    H. A. Blair, V. S. Subrahmanian. Paraconsistent Logic Programming. Theoretical Computer Science 68(2), 135–154, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [Bz]
    V. A. Bazhanov: Toward the Reconstruction of the Early History of Paraconsistent Logic: The Prerequisites of N. A. Vasiliev’s Imaginary Logic. Logique et Analyse 41(161–163), 17–20, 1998.zbMATHMathSciNetGoogle Scholar
  8. [Ch]
    A. Church: A Note on the Entscheidungsproblem. J. Symbolic Logic 1(40–41, 101-102), 1936.Google Scholar
  9. [Cl]
    K. Clark: Negation as Failure. In H. Gallaire, J. Minker (eds): Logic and Data Bases, Plenum Press, 293–322, 1978.Google Scholar
  10. [dA]
    N. da Costa, E. Alves: On a Paraconsistent Predicate Calculus. In O. T. Alas, N. da Costa, C. Hönig (eds): Collected papers dedicated to Professor Edison Farah on the occasion of his retirement. Instituto de Matematica e Estatistica, Universidade de São Paulo, 83–90, 1982.Google Scholar
  11. [dBB]
    N. da Costa, J.-Y. Béziau, O. Bueno: Aspects of Paraconsistent Logic. IGLP Bulletin 3(4), 597–614, 1995.zbMATHGoogle Scholar
  12. [dC1]
    N. da Costa: Nota sobre o conceito de contradicao. Anuario da Sociedade Paranaense de Mathematica (2a. Serie) 1, 6–8, 1958.Google Scholar
  13. [dC2]
    N. da Costa: Sistemas Formais Inconsistentes. Universidade do Paraná, 1963.Google Scholar
  14. [De1]
    H. Decker: Integrity Enforcement on Deductive Databases. In L. Kerschberg: Expert Database Systems. Benjamin Cummings, 381–395, 1987.Google Scholar
  15. [De2]
    H. Decker: Drawing Updates From Derivations. Proc. 3rd ICDT, Springer LNCS 470, 437–451, 1990.Google Scholar
  16. [De3]
    H. Decker: Some Notes on Knowledge Assimilation in Deductive Databases. In Transactions and Change in Logic Databases, Springer LNCS 1472, 249–286, 1998.CrossRefGoogle Scholar
  17. [De4]
    H. Decker: An Extension of SLD by Abduction and Integrity Maintenance for View Updating in Deductive Databases. Proc. JICSLP. MIT Press, 157–169, 1996.Google Scholar
  18. [DP]
    C. Damasio, L. Pereira: A Survey of Paraconsistent Semantics for Logic Programs. In D. Gabbay, P. Smets (eds): Handbook of Defeasible Reasoning and Uncertainty Management Systems, Vol. 2, Kluwer, 241–320, 1998.Google Scholar
  19. [Du]
    P. Dung: An argumentation-theoretic foundation for logic programming. J. Logic Programming 22(2), 151–171, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [DVW]
    H. Decker, J. Villadsen, T. Waragai (eds): Paraconsistent Computational Logic (Proc. FLoC’02 Workshop). Datalogiske Skrifter, Vol. 95, Roskilde University, 2002. Electronic proceedings, edited by D. Goldin et al, available at
  21. [EK]
    K. Eshghi, R. Kowalski: Abduction compared with negation by failure. Proc. 6th ICLP, MIT Press, 234–254, 1989.Google Scholar
  22. [Fi]
    M. Fitting: A Kripke-Kleene semantics for logic programs. J. Logic Programming 2(4), 295–312, 1985.CrossRefMathSciNetzbMATHGoogle Scholar
  23. [Ge]
    G. Gentzen: Untersuchungen über das logische Schlieβen. Mathematische Zeitschrift 39, 176–210, 405-431, 1934, 1935.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Goe]
    K. Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198, 1931. Reprinted in [Hj].zbMATHCrossRefGoogle Scholar
  25. [HA]
    D. Hilbert, W. Ackermann: Grundzüge der theoretischen Logik. Springer, 1928.Google Scholar
  26. [Hi]
    D. Hilbert: Die logischen Grundlagen der Mathematik. Mathematische Annalen 88, 151–165, 1923.CrossRefMathSciNetGoogle Scholar
  27. [HB]
    D. Hilbert, P. Bernays: Grundlagen der Mathematik I, II. Springer, 1934, 1939.Google Scholar
  28. [Hj]
    J. van Heijenoort: From Frege to Gödel. Harvard University Press, 1967.Google Scholar
  29. [Hy]
    A. Heyting, Die intuitionistische Grundlegung der Mathematik. Erkenntnis 2, 106–115, 1931.CrossRefzbMATHGoogle Scholar
  30. [Ja1]
    S. Jaśkowski: The theory of deduction based on the method of suppositions. Studia Logica 1, 5–32, 1934. Reprinted in St. McCall (ed): Polish Logic 1920-1939, North-Holland, 232-258, 1963.Google Scholar
  31. [Ja2]
    S. Jaśkowski: Rachunek zdan dla systemow dedukcyjnych sprzecznych. Studia Societatis Scientiarun Torunesis, Sectio A, 1(5), 55–77, 1948. Translated as: Propositional Calculus for Contradictory Deductive Systems. Studia Logica 24, 143-157, 1969.Google Scholar
  32. [Jo]
    I. Johansson: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compositio Mathematica 4(1), 119–136, 1936.zbMATHMathSciNetGoogle Scholar
  33. [Kg]
    A. N. Kolmogorov: O principie tertium non datur. Matematiceskij Sbornik (Recueil Mathématique) 32, 1924/25. Translated as: On the principle of excluded middle, in [Hj].Google Scholar
  34. [KK]
    W. Kneale, M. Kneale: The Development of Logic. Clarendon Press, 1962.Google Scholar
  35. [KKT]
    A. Kakas, R. Kowalski, F. Toni: The Role of Abduction in Logic Programming. Handbook of Logic in Artificial Intelligence and Logic Programming, Oxford University Press, 235–324, 1995.Google Scholar
  36. [KL]
    M. Kifer, E. Lozinskii: RI: A Logic for Reasoning with Inconsistency. Proc. 4th LICS, 253–262, 1989.Google Scholar
  37. [Kl]
    S. C. Kleene: Introduction to Metamathematics. North-Holland, 1952.Google Scholar
  38. [Ku]
    K. Kunen: Negation in Logic Programming. J. Logic Programming 4(4), 289–308, 1987.CrossRefMathSciNetzbMATHGoogle Scholar
  39. [KuK]
    R. Kowalski, D. Kuehner: Linear Resolution with Selection Function. Artificial Intelligence 2(3/4), 227–260, 1971.zbMATHCrossRefMathSciNetGoogle Scholar
  40. [Kw1]
    R. Kowalski: Logic for Problem Solving. North-Holland, 1979.Google Scholar
  41. [Kw2]
    R. Kowalski: Logic without Model Theory. In D. Gabbay (ed): What is a logical system? Oxford University Press 1994, pp 35–71.Google Scholar
  42. [LL]
    C. Lewis, C. Langford: Symbolic Logic. Century Co., 1932.Google Scholar
  43. [Ll]
    J. Lloyd: Foundations of Logic Programming, 2nd edition. Springer, 1987.Google Scholar
  44. [Lo]
    P. Lorenzen: Einführung in die operative Logik und Mathematik, 2nd edition. Springer, 1969.Google Scholar
  45. [Lu]
    J. Łukasiewicz: O zasadzie sprzecznosci u Arystotelesa: Studium Krytyczne (On Aristotle’s Principle of Contradiction: A Critical Study), 1910. Translated as: On the Principle of Contradiction in Aristotle, Review of Metaphysics 24, 485–509, 1971.Google Scholar
  46. [My]
    M. Minsky: A Framework for Representing Knowledge. Tech. Report 306, MIT AI Lab, 1974. Reprinted (sans appendix) in P. Winston (ed): The Psychology of Computer Vision, McGraw-Hill, 177–211, 1975.Google Scholar
  47. [MZ]
    J. Minker, G. Zanon: An Extension to Linear Resolution with Selection Function. Information Processing Letters 14(4), 191–194, 1982.zbMATHCrossRefMathSciNetGoogle Scholar
  48. [M+]
    F. Muñoz, L. Irún, P. Galdámez, J. Bernabéu, J. Bataller, C. Bañuls, H. Decker: Flexible Management of Consistency and Availability of Networked Data Replications. Proc. 5th FQAS, Springer LNCS 2522, 289–300, 2002.Google Scholar
  49. [Pl]
    D. Poole: Logic Programming, Abduction and Probability. Proc. Fifth Generation Computer Systems’ 92, 530–538, 1992.Google Scholar
  50. [Pt]
    E. Post: Introduction to a General Theory of Propositions. American Journal of Mathematics 43, 1921. Reprinted in [Hj].Google Scholar
  51. [Ro]
    J. A. Robinson: Logic: Form and Function-The Mechanization of Deductive Reasoning. Edinburgh University Press, 1979.Google Scholar
  52. [rel]
  53. [Ru]
    B. Russell: Letter to Frege, 1902, in [Hj].Google Scholar
  54. [RW]
    B. Russell, A. Whitehead: Principia Mathematica. Cambridge University Press, 1910–1913, reprinted 1962.Google Scholar
  55. [Se]
    A. Sette: On the Propositional Calculus P1. Mathematica Japonicae 18, 173–180, 1973.zbMATHMathSciNetGoogle Scholar
  56. [Sl]
    J. Słupecki: Der volle dreiwertige Aussagenkalkül. Comptes Rendus Séances Société des Sciences et Lettres Varsovie 29, 9–11, 1936.zbMATHGoogle Scholar
  57. [Ta]
    A. Tarski: Der Wahrheitsbegri. in den formalisierten Sprachen. Studia Philosophica 1, 261–405, 1935.Google Scholar
  58. [Ur]
    A. Urquhart: Many-Valued Logic. In Gabbay, Guenthner (eds): Handbook of Philosophical Logic. Kluwer, 71–116, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Hendrik Decker
    • 1
  1. 1.Instituto Tecnológico de InformáticaUniversidad Politécnica de ValenciaSpain

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