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Proof Systems Combining Classical and Paraconsistent Negations

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Abstract

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.

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References

  1. Almukdad A., Nelson D. (1984) ‘Constructible falsity and inexact predicates’. Journal of Symbolic Logic 49(1): 231–233

    Article  Google Scholar 

  2. Anderson, A.R., and N.D. Belnap, Entailment: The logic of relevance and necessity, Vol. I, Princeton University Press, Princeton, NJ, 1975.

  3. Arieli O., Avron A. (1996) ‘Reasoning with logical bilattices’. Journal of Logic, Language and Information 5: 25–63

    Article  Google Scholar 

  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.

  5. Dunn J.M. (1976) ‘Intuitive semantics for first-degree entailment and ‘coupled trees’’. Philosophical Studies 29(3): 149–168

    Article  Google Scholar 

  6. Gargov G. (1999) ‘Knowledge, uncertainty and ignorance in logic: bilattices and beyond’. Journal of Applied Non-Classical Logics 9: 195–283

    Google Scholar 

  7. Kamide N. (2004) ‘A relationship between Rauszer’s H-B logic and Nelson’s logic’. Bulletin of the Section of Logic 33(4): 237–249

    Google Scholar 

  8. Kamide N. (2005) ‘Gentzen-type methods for bilattice negation’. Studia Logica 80(2-3): 265–289

    Article  Google Scholar 

  9. Kamide N. (2005) ‘A cut-free system for 16-valued reasoning’. Bulletin of the Section of Logic 34(4): 213–226

    Google Scholar 

  10. Shramko Y., Dunn J.M., Takenaka T. (2001) ‘The trilattice of constructive truth values’. Journal of Logic and Computation 11(6): 761–788

    Article  Google Scholar 

  11. Shramko Y., Wansing H. (2005) ‘Some useful 16-valued logics: how a computer network should think’. Journal of Philosophical Logic 34(2): 121–153

    Article  Google Scholar 

  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): 403–424

    Article  Google Scholar 

  13. Takeuti, G., Proof theory, North-Holland Pub. Co., 1975.

  14. Wansing, H., ‘The logic of information structures’, Lecture Notes in Artificial Intelligence 681, Springer-Verlag, 1993, pp. 1–163.

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Authors and Affiliations

  1. Waseda Institute for Advanced Study, Waseda University, 1-6-1 Nishi Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Norihiro Kamide

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  1. Norihiro Kamide
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Correspondence to Norihiro Kamide.

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Presented by Yaroslav Shramko and Heinrich Wansing

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Kamide, N. Proof Systems Combining Classical and Paraconsistent Negations. Stud Logica 91, 217–238 (2009). https://doi.org/10.1007/s11225-009-9173-6

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  • DOI: https://doi.org/10.1007/s11225-009-9173-6

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