Studia Logica

, Volume 91, Issue 2, pp 217–238 | Cite as

Proof Systems Combining Classical and Paraconsistent Negations

  • Norihiro KamideEmail author


New propositional and first-order paraconsistent logics (called L ω and FL ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L ω and FL ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L ω and FL ω are proved. The cut-elimination theorems for L ω and FL ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems.


Paraconsistent negation sequent calculus cut-elimination completeness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Almukdad A., Nelson D. (1984) ‘Constructible falsity and inexact predicates’. Journal of Symbolic Logic 49(1): 231–233CrossRefGoogle Scholar
  2. 2.
    Anderson, A.R., and N.D. Belnap, Entailment: The logic of relevance and necessity, Vol. I, Princeton University Press, Princeton, NJ, 1975.Google Scholar
  3. 3.
    Arieli O., Avron A. (1996) ‘Reasoning with logical bilattices’. Journal of Logic, Language and Information 5: 25–63CrossRefGoogle Scholar
  4. 4.
    Belnap, N.D., ‘A useful four-valued logic’, in J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 5–37.Google Scholar
  5. 5.
    Dunn J.M. (1976) ‘Intuitive semantics for first-degree entailment and ‘coupled trees’’. Philosophical Studies 29(3): 149–168CrossRefGoogle Scholar
  6. 6.
    Gargov G. (1999) ‘Knowledge, uncertainty and ignorance in logic: bilattices and beyond’. Journal of Applied Non-Classical Logics 9: 195–283Google Scholar
  7. 7.
    Kamide N. (2004) ‘A relationship between Rauszer’s H-B logic and Nelson’s logic’. Bulletin of the Section of Logic 33(4): 237–249Google Scholar
  8. 8.
    Kamide N. (2005) ‘Gentzen-type methods for bilattice negation’. Studia Logica 80(2-3): 265–289CrossRefGoogle Scholar
  9. 9.
    Kamide N. (2005) ‘A cut-free system for 16-valued reasoning’. Bulletin of the Section of Logic 34(4): 213–226Google Scholar
  10. 10.
    Shramko Y., Dunn J.M., Takenaka T. (2001) ‘The trilattice of constructive truth values’. Journal of Logic and Computation 11(6): 761–788CrossRefGoogle Scholar
  11. 11.
    Shramko Y., Wansing H. (2005) ‘Some useful 16-valued logics: how a computer network should think’. Journal of Philosophical Logic 34(2): 121–153CrossRefGoogle Scholar
  12. 12.
    Shramko Y., Wansing H. (2006) ‘Hyper-contradictions, generalized truth values and logics of truth and falsehood’. Journal of Logic, Language and Information 15(4): 403–424CrossRefGoogle Scholar
  13. 13.
    Takeuti, G., Proof theory, North-Holland Pub. Co., 1975.Google Scholar
  14. 14.
    Wansing, H., ‘The logic of information structures’, Lecture Notes in Artificial Intelligence 681, Springer-Verlag, 1993, pp. 1–163.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Waseda Institute for Advanced StudyWaseda UniversityTokyoJapan

Personalised recommendations