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Tableaux with Four Signs as a Unified Framework

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2003)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2796))

Abstract

We show that the use of tableaux with four types of signed formulas (the signs intuitively corresponding to positive/negative information concerning truth/falsity) provides a framework in which a diversity of logics can be handled in a uniform way. The logics for which we provide sound and complete tableau systems of this type are classical logic, the most important three-valued logics, the four-valued logic of logical bilattices (an extension of Belnap’s four-valued logic), Nelson’s logics for constructive negation, and da Costa’s paraconsistent logic C ω (together with some of its extensions). For the latter we provide new, simple semantics for which our tableau systems are sound and complete.

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References

  1. Arieli, O., Avron, A.: Reasoning with logical bilattices. J. of Logic, Language and Information 5(1), 25–63 (1996)

    MATH  MathSciNet  Google Scholar 

  2. Arieli, O., Avron, A.: The value of four values. Artificial Intelligence 102(1), 97–141 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Anderson, A.R., Belnap, N.D.: Entailment, vol. I. Princeton University Press, Princeton (1975)

    MATH  Google Scholar 

  4. Anderson, A.R., Belnap, N.D.: Entailment, vol. II. Princeton University Press, Princeton (1992)

    MATH  Google Scholar 

  5. Almukdad, A., Nelson, D.: Constructible falsity and inexact predicates. Journal of Symbolic Logic 49, 231–333 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Avron, A.: On an implication connective of RM. Notre Dame Journal of Formal Logic 27, 201–209 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  7. Avron, A.: Natural 3-valued logics: characterization and proof theory. J. of Symbolic Logic 56(1), 276–294 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Avron, A.: On the expressive power of three-valued and four-valued languages. Journal of Logic and Computation 9, 977–994 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Avron, A.: Classical Gentzen-type methods in propositional many-valued logics. In: Fitting, M., Orlowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic. Studies in Fuzziness and Soft Computing, vol. 114, pp. 117–155. Physica Verlag, Heidelberg (2003)

    Google Scholar 

  10. Baaz, M.: Kripke-type semantics for da Costa’s paraconsistent logic c ω . Notre Dame Journal of Formal Logic 27, 523–527 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  11. Belnap, N.D.: How computers should think. In: Ryle, G. (ed.) Contemporary Aspects of Philosophy, pp. 30–56. Oriel Press, Stocksfield (1977)

    Google Scholar 

  12. Belnap, N.D.: A useful four-valued logic. In: Epstein, G., Dunn, J.M. (eds.) Modern Uses of Multiple-Valued Logic, pp. 7–37. Reidel, Dordrecht (1977)

    Google Scholar 

  13. Baaz, M., Fermüller, C.G., Salzer, G.: Automated deduction for many-valued logics. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning. Elsevier Science Publishers, Amsterdam (2000)

    Google Scholar 

  14. Busch, D.: Sequent formalizations of three-valued logic. In: Partiality, Modality, and Nonmonotonicity. Studies in Logic, Language and Information, pp. 45–75. CSLI Publications, Stanford (1996)

    Google Scholar 

  15. Carnielli, W.A., Marcos, J.: A taxonomy of c-systems. In: Coniglio, M.E., Carnielli, W.A., D’ottaviano, I.L.M. (eds.) AAECC 1984. Lecture notes in pure and applied Mathematics, Marcell Dekker, New York, vol. 228. Springer, Heidelberg (2002)

    Google Scholar 

  16. da Costa, N.C.A.: On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic 15, 497–510 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  17. D’ottaviano, I.L.M., da Costa, N.C.A.: Sur un problḿe de Jaskowski. C. R.Acad. Sc. Paris, Série A 270, 1349–1353 (1970)

    MATH  Google Scholar 

  18. D’ottaviano, I.L.M.: The completeness and compactness of a three-valued firstorder logic. Revista Colombiana de Matematicas XIX(1-2), 31–42 (1985)

    MathSciNet  Google Scholar 

  19. Dunn, J.M.: Relevance logic and entailment, in [26], vol. III, ch. 3, pp. 117–224 (1986)

    Google Scholar 

  20. Epstein, R.L.: The semantic foundation of logic. In: Propositional logics, vol. I ch. IX. Kluwer Academic Publisher, Dordrecht (1990)

    Google Scholar 

  21. Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. Reidel, Dordrecht (1983)

    Google Scholar 

  22. Fitting, M.: Bilattices in logic programming. In: Epstein, G. (ed.) 20th Int. Symp. on Multiple- Valued Logic, pp. 238–246. IEEE Press, Los Alamitos (1990)

    Google Scholar 

  23. Fitting, M.: Kleene’s logic, generalized. Journal of Logic and Computation 1, 797–810 (1990)

    Article  Google Scholar 

  24. Fitting, M.: Bilattices and the semantics of logic programming. Journal of Logic Programming 11(2), 91–116 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  25. Fitting, M.: Kleene’s three-valued logics and their children. Fundamenta Informaticae 20, 113–131 (1994)

    MATH  MathSciNet  Google Scholar 

  26. Gabbay, D.M., Guenthner, F.: Handbook of Philosophical Logic. D. Reidel Publishing company, Dordrecht (1986)

    Google Scholar 

  27. Ginsberg, M.L.: Multiple-valued logics. In: Ginsberg, M.L. (ed.) Readings in Non-Monotonic Reasoning, Los-Altos, CA, pp. 251–258 (1987)

    Google Scholar 

  28. Ginsberg, M.L.: Multivalued logics: a uniform approach to reasoning in AI. Computer Intelligence 4, 256–316 (1988)

    Google Scholar 

  29. Hähnle, R.: Tableaux for multiple-valued logics. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 529–580. Kluwer Publishing Company, Dordrecht (1999)

    Google Scholar 

  30. Jones, C.B.: Systematic Software Development Using VDM. Prentice-Hall International, U.K (1986)

    MATH  Google Scholar 

  31. Łukasiewicz, J.: On 3-valued logic. In: McCall, S. (ed.) Polish Logic. Oxford University Press, Oxford (1967)

    Google Scholar 

  32. Monteiro, A.: Construction des algebres de Łukasiewicz trivalentes dans les algebres de Boole monadiques, i. Mat. Jap. 12, 1–23 (1967)

    MATH  MathSciNet  Google Scholar 

  33. Raggio, A.R.: Propositional sequence-calculi for inconsistent systems. Notre Dame Journal of Formal logic 9, 359–366 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  34. Rozoner, L.I.: On interpretation of inconsistent theories. Information Sciences 47, 243–266 (1989)

    Article  MathSciNet  Google Scholar 

  35. Schmitt, P.H.: Computational aspects of three-valued logic. In: Siekmann, J.H. (ed.) CADE 1986. LNCS, vol. 230, pp. 190–198. Springer, Heidelberg (1986)

    Google Scholar 

  36. Slupecki, J.: Der volle dreiwertige aussagenkalkül. Com. rend. Soc. Sci. Lett. de Varsovie 29, 9–11 (1936)

    MATH  Google Scholar 

  37. von Kutschera, F.: Ein verallgemeinerter widerlegungsbegriff für Gentzenkalküle. Archiv fur Mathematische Logik und Grundlagenforschung 12, 104–118 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wansing, H. (ed.): The Logic of Information Structures. LNCS (LNAI), vol. 681. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  39. Wójcicki, R.: Lectures on Propositional Calculi. Ossolineum, Warsaw (1984)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Computer Science, Aviv University, Ramat Aviv, 69978, Israel

    Arnon Avron

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  1. Arnon Avron
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Automazione, Università di Roma Tre, via Vasca Navale 79, 00146, Roma, Italia

    Marta Cialdea Mayer

  2. Dipartimento di Informatica e Sistemistica (DIS), Sapienza, University of Rome “Sapienza”, via Ariosto 25, 00185, Roma

    Fiora Pirri

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Avron, A. (2003). Tableaux with Four Signs as a Unified Framework. In: Cialdea Mayer, M., Pirri, F. (eds) Automated Reasoning with Analytic Tableaux and Related Methods . TABLEAUX 2003. Lecture Notes in Computer Science(), vol 2796. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45206-5_4

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  • DOI: https://doi.org/10.1007/978-3-540-45206-5_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40787-4

  • Online ISBN: 978-3-540-45206-5

  • eBook Packages: Springer Book Archive

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