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Inference rules in Nelson’s logics, admissibility and weak admissibility. (English) Zbl 1336.03036
Summary: Our paper aims to investigate inference rules for Nelson’s logics and to discuss possible ways to determine admissibility of inference rules in such logics. We will use the technique offered originally for intuitionistic logic and paraconsistent minimal Johannson’s logic. However, the adaptation is not an easy and evident task since Nelson’s logics do not enjoy replacement of equivalences rule. Therefore we consider and compare standard admissibility and weak admissibility. Our paper founds algorithms for recognizing weak admissibility and admissibility itself – for restricted cases, to show the problems arising in the course of study.

MSC:
03B53 Paraconsistent logics
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[1] Almukdad, A.; Nelson, D., Constructible falsity and inexact predicates, J. Symbo. Log., 49, 231-233, (1984) · Zbl 0575.03016
[2] Babenyshev, S.; Rybakov, V., Linear temporal logic LTL: basis for admissible rules, J. Log. Comput., 21, 157-177, (2011) · Zbl 1233.03026
[3] Babenyshev, S.; Rybakov, V., Unification in linear temporal logic LTL, Ann. Pure Appl. Logic, 162, 991-1000, (2011) · Zbl 1241.03014
[4] Cignoli, R.; Busaniche, M., Constructive logic with strong negation as a substructural logic, J. Log. Comput, 20, 761-793, (2010) · Zbl 1205.03040
[5] Iemhoff, R., On the admissible rules of intuitionistic propositional logic, J. Symb. Log., 66, 281-294, (2001) · Zbl 0986.03013
[6] Iemhoff, R.: Towards a proof system for admissibility. CSL-2003 255-270 (2003) · Zbl 1116.03304
[7] Iemhoff, R.; Metcalfe, G., Proof theory for admissible rules, Ann. Pure Appl. Logic, 159, 171-186, (2009) · Zbl 1174.03024
[8] Jerabek, E., Bases of admissible rules of lukasiewicz logic, J. Log. Comput., 20, 1149-1163, (2010) · Zbl 1216.03043
[9] Jerabek, E., Admissible rules of lukasiewicz logic, J. Log. Comput., 20, 425-447, (2010) · Zbl 1216.03042
[10] Jerabek, E., Independent bases of admissible rules, Log. J. IGPL, 16, 249-267, (2008) · Zbl 1146.03008
[11] Jerabek, E., Complexity of admissible rules, Arch. Math. Log., 46, 73-92, (2007) · Zbl 1115.03010
[12] Jerabek, E., Admissible rules of modal logics, J. Log. Comput., 15, 411-431, (2005) · Zbl 1077.03011
[13] Ghilardi, S., Unification in intuitionistic logic, J. Symb. Log., 64, 859-880, (1999) · Zbl 0930.03009
[14] Gurevich, Y., Intuitionistic logic with strong negation, Stud. Log., 36, 49-59, (1977) · Zbl 0366.02015
[15] Friedman, H.: One hundred and two problems in mathematical logic. J. Symb. Log. 40(3), 113-130 (1975) · Zbl 0318.02002
[16] Johansson, I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compos. Math. 4, 119-136 (1937) · Zbl 0015.24102
[17] Latkin, E.I.: Generalized Kripke semantics for Nelson’s logic. Algebra Log. 49(5), 426-443 (2010) · Zbl 1272.03119
[18] López-Escobar, E.G.K.: Refutability and elementary number theory. Indag. Math. 34, 362-374 (1972) · Zbl 0262.02027
[19] Metcalfe, G., A sequent calculus for constructive logic with strong negation as a substructural logic, Bull. Sect. Log., 38, 5-11, (2009) · Zbl 1286.03154
[20] Nelson, D., Constructible falsity, J. Symb. Log., 14, 16-26, (1949) · Zbl 0033.24304
[21] Odintsov, S.P.: Algebraic semantics for paraconsistent Nelson’s logic. J. Log. Comput. 13(4), 453-468 (2003) · Zbl 1034.03029
[22] Odintsov, S.P., The class of extensions of nelson’s paraconsistent logic, Stud. Log., 80, 293-322, (2005)
[23] Odintsov, S., Pearce, D.: Routley semantics for answer sets. In: Proceedings of 8th International Conference on Logic Programming and Nonmonotonic Reasoning, LNCS 3662, pp. 343-355 (2005) · Zbl 1152.68416
[24] Odintsov, S.P., Rybakov, V.V.: Unification and admissible rules for paraconsistent minimal Johannson’s logic J and positive intuitionistic logic IPC\^{+}. Ann. Pure Appl. Log. 164, 771-784 (2013) · Zbl 1323.03029
[25] Priest, G., Tanaka, K., Weber, Z.: Paraconsistent logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Fall 2013 Edition). (2013) · Zbl 1205.03040
[26] Rasiowa, H.: \(N\)-Lattices and constructive logic with strong negation. Fund. Math. 46, 61-80 (1958) · Zbl 0087.00905
[27] Rasiowa, H.: An Algebraic Approach to Non-Classical Logic. PWN, Warsaw and North-Holland, Amsterdam (1974) · Zbl 1233.03026
[28] Rivieccio, U., Implicative twist-structures, Algebra Univ., 71, 155-185, (2014) · Zbl 1386.06011
[29] Routley, R., Semantical analyses of propositional systems of Fitch and Nelson, Stud. Log., 33, 283-298, (1974) · Zbl 0356.02022
[30] Rybakov, V.: A criterion for admissibility of rules in the modal system S4 and the intuitionistic logic. Algebra Log. 23(5), 369-384 (1984) · Zbl 0598.03013
[31] Rybakov, V.V., Problems of substitution and admissibility in the modal system grz and in intuitionistic propositional calculus, Ann. Pure Appl. Log., 50, 71-106, (1990) · Zbl 0709.03009
[32] Rybakov, V.V.: Rules of inference with parameters for intuitionistic logic. J. Symb. Logic 57(3), 912-923 (1992) · Zbl 0788.03007
[33] Rybakov, V.V.: Construction of an explicit basis for rules admissible in modal system S4. Math. Log. Q. 47(4), 441-446 (2001) · Zbl 0992.03027
[34] Rybakov, V.V.: Logics with the universal modality and admissible consecutions. J. Appl. Non-Class. Log. 17(3), 383-396 (2007) · Zbl 1186.03048
[35] Rybakov, V.V., Linear temporal logic with until and next, Logical consecutions. Ann. Pure Appl. Logic, 155, 32-45, (2008) · Zbl 1147.03008
[36] Rybakov, V.V.: Multi-modal and temporal logics with universal formula—reduction of admissibility to validity and unification. J. Log. Comput. 18(4), 509-519 (2008) · Zbl 1149.03017
[37] Rybakov, V.V., Rules admissible in transitive temporal logic TS4, sufficient condition, Theor. Comput. Sci., 411, 4323-4332, (2010) · Zbl 1209.03011
[38] Spinks, M., Veroff, R.: Constructive logic with strong negation is a substructural logic I. Stud. Log. 88(3), 325-348 (2008) · Zbl 1145.03013
[39] Spinks, M., Veroff, R.: Constructive logic with strong negation is a substructural logic II. Stud. Log. 89(3), 401-425 (2008) · Zbl 1166.03010
[40] Thomason, R.H., A semantical study of constructible falsity, Z. Math. Logik Grundl. Math., 15, 247-257, (1969) · Zbl 0181.00901
[41] Vorob’ev, N.N.: A constructive propositional calculus with strong negation. Doklady Akademii Nauk SSSR 85, 465-468 (1952)
[42] Vorob’ev, N.N., The problem of deducibility in constructive propositional calculus with strong negation, Doklady Akademii Nauk SSSR, 85, 689-692, (1952)
[43] Wansing, H.: Negation. In: Goble, L.: (ed.) The Blackwell Guide to Philosophical Logic. Basil Blackwell Publishers, Cambridge, pp. 415-436 (2001)
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