Journal of Philosophical Logic

, Volume 44, Issue 5, pp 473–487 | Cite as

Analytic Tableaux for all of SIXTEEN3

Article

Abstract

In this paper we give an analytic tableau calculus PL16 for a functionally complete extension of Shramko and Wansing’s logic. The calculus is based on signed formulas and a single set of tableau rules is involved in axiomatising each of the four entailment relations ⊧t, ⊧f, ⊧i, and ⊧ under consideration—the differences only residing in initial assignments of signs to formulas. Proving that two sets of formulas are in one of the first three entailment relations will in general require developing four tableaux, while proving that they are in the ⊧ relation may require six.

Keywords

Trilattice SIXTEEN3 Tableau calculi Functional completeness Truth entailment Falsity entailment Information entailment 

References

  1. 1.
    Baaz, M., Fermüller, C., Zach, R. (1994). Elimination of cuts in first-order finite-valued logics. Journal of Information Processing and Cybernetics, 29, 333–355.Google Scholar
  2. 2.
    Beckert, B., & Posegga, J. (1995). lean T A P: Lean Tableau-based deduction. Journal of Automated Reasoning, 15 (3), 339–358. http://web.sec.uni-passau.de/papers/LeanTaP.pdf.CrossRefGoogle Scholar
  3. 3.
    Belnap, N.D. (1976). How a computer should think. In G. Ryle (Ed.), Contemporary aspects of philosophy (pp. 30–56). Stocksfield: Oriel Press.Google Scholar
  4. 4.
    Belnap, N.D. (1977). A useful four-valued logic. In J. Dunn, & G. Epstein (Eds.), Modern uses of multiple-valued logic (pp. 8–37). Dordrecht: Reidel.Google Scholar
  5. 5.
    Kamide, N., & Wansing, H. (2009). Sequent calculi for some trilattice logics. The Review of Symbolic Logic, 2(2), 374–395.CrossRefGoogle Scholar
  6. 6.
    Langholm, T. (1996). How different is partial logic? In P. Doherty (Ed.), Partiality, modality, and nonmonotonicity (pp. 3–43). Stanford: CSLI.Google Scholar
  7. 7.
    Muskens, R.A. (1995). Meaning and partiality. Stanford: CSLI.Google Scholar
  8. 8.
    Muskens, R.A. (1999). On partial and paraconsistent logics. Notre Dame Journal of Formal Logic, 40(3), 352–374.CrossRefGoogle Scholar
  9. 9.
    Odintsov, S. (2009). On Axiomatizing Shramko-Wansing’s logic. Studia Logica, 91, 407–428.CrossRefGoogle Scholar
  10. 10.
    Odintsov, S., & Wansing, H. (2014). The logic of generalized truth values and the logic of bilattices. In Studia Logica.Google Scholar
  11. 11.
    Rivieccio, U. (2013). Representation of interlaced trilattices. Journal of Applied Logic, 11, 174–189.CrossRefGoogle Scholar
  12. 12.
    Rousseau, G. (1967). Sequents in many-valued logic I. Fundamenta Mathematicae, 60, 23–33.Google Scholar
  13. 13.
    Schröter, K. (1955). Methoden zur Axiomatisierung beliebiger Aussagen- und Prädikatenkalküle. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 1, 241–251.CrossRefGoogle Scholar
  14. 14.
    Shramko, Y., & Wansing, H. (2005). Some Useful 16-Valued Logics: How a Computer Network Should Think. Journal of Philosophical Logic, 34, 121–153.CrossRefGoogle Scholar
  15. 15.
    Shramko, Y., & Wansing, H. (2011). Truth and falsehood: An inquiry into generalized logical values. In Trends in logic Vol. 36: Springer.Google Scholar
  16. 16.
    Wansing, H. (2009). The power of Belnap: Sequent systems for S I X T E E N 3. Journal of Philosophical Logic, 39, 369–393.CrossRefGoogle Scholar
  17. 17.
    Wansing, H. (2012). A non-inferentialist, anti-realistic conception of logical truth and falsity. Topoi, 31, 93–100.CrossRefGoogle Scholar
  18. 18.
    Wansing, H., & Kamide, N. (2010). Intuitionistic Trilattice Logics. Journal of Logic and Computation, 20 (6), 1201–1229.CrossRefGoogle Scholar
  19. 19.
    Wintein, S., & Muskens, R.A. (2012). A calculus for Belnap’s logic in which each proof consists of two trees. Logique & Analyse, 220, 643–656.Google Scholar
  20. 20.
    Wintein, S., & Muskens, R.A. (2014). From bi-facial truth to bi-facial proofs. Studia Logica. Online First.Google Scholar
  21. 21.
    Zach, R. (1993). Proof theory of finite-valued logics. Technical Report TUW-E185.2-Z.1-93. Institut für Computersprachen, Technische Universität Wien.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Tilburg Center for Logic, Ethics, and Philosophy of Science (TiLPS)Tilburg UniversityTilburgNetherlands
  2. 2.Faculty of PhilosophyErasmus University RotterdamRotterdamNetherlands

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