The main objective of the paper is to axiomatize operators − ¬ and − □′ in logics N* and HK□′, respectively. The result is formulated in terms of normal extensions HKN□′ and HKNR of the logic HK□, which are embedded in corresponding logics via a natural translation. In addition, for the logic HKNR, the finite model property and decidability are established using a hybrid calculus.
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References
P. Cabalar, S. P. Odintsov, and D. Pearce, “Logical foundations of well-founded semantics,” in Principles of Knowledge Representation and Reasoning, Proc. 10th Int. Conf. (KR2006), P. Doherty et al. (eds.), AAAI Press, Menlo Park, California (2006), pp. 25–36.
K. Dosen, “Negation as a modal operator,” Rep. Math. Log., 20, 15–28 (1986).
V. H. Sotirov, “Modal theories with intuitionistic logic,” in Proc. Conf. Math. Logic, Sofia, 1980, Bulgarian Academy of Sciences (1984), pp. 139–171.
M. Božić and K. Došen, “Models for normal intuitionistic logics,” Stud. Log., 43, 217–245 (1984).
K. Dosen, “Negative modal operators in intuitionistic logic,” Publ. Inst. Math., Nouv. Ser., 35(49), 3–14 (1984).
D. Vakarelov, “Consistency, completeness and negation,” in Paraconsistent Logics. Essays on the Inconsistent. Analytica, G. Priest, R. Routley, and J. Norman (eds.), Philosophia Verlag, München (1989), pp. 328–363.
D. Vakarelov, “The non-classical negation in the works of Helena Rasiowa and their impact on the theory of negation,” Stud. Log., 84, No. 1, 105–127 (2006).
R. Routley and V. Routley, “The semantics of first degree entailment,” Nous, 6, 335–359 (1972).
S. P. Odintsov, “Combining intuitionistic connectives and Routley negation,” Sib. El. Mat. Izv., 7, 21–41 (2010).
S. A. Drobyshevich, “A hybrid calculus for logic N*: Residual finiteness and decidability,” Algebra Logika, 50, No. 3, 351–367 (2011).
S. P. Odintsov and H. Wansing, “Inconsistency-tolerant description logic. II: A tableau algorithm for CALCC,” J. Appl. Log., 6, No. 3, 343–360 (2008).
R. Dyckhoff, “Contraction-free sequent calculi for intuitionistic logic,” J. Symb. Log., 57, No. 3, 795–807 (1992).
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Translated from Algebra i Logika, Vol. 52, No. 3, pp. 305-331, May-June, 2013.
*Supported by RFBR (project No. 12-01-00168-a) and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-276.2012.1).
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Drobyshevich, S.A. Composition of an intuitionistic negation and negative modalities as a necessity operator. Algebra Logic 52, 203–221 (2013). https://doi.org/10.1007/s10469-013-9235-8
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DOI: https://doi.org/10.1007/s10469-013-9235-8