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On one generalization of the principle reductio ad absurdum
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 3, pp. 62-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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On the base of comparing of the Curry logic of classical refutability and the Łukasiewicz modal logic we suggest a generalization of the notion of negation as reducibility to a unary absurdity operator, $\lnot \varphi:=\varphi\supset A(\varphi)$. We study the possibility to represent in this form the negation in such well known systems of paraconsistent logic as the logic of Batens $\mathbf{CLuN}$ and the maximal paraconsistent logic of Sette $P^1$. 
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S. P. Odintsov. On one generalization of the principle reductio ad absurdum. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 3, pp. 62-87. http://geodesic.mathdoc.fr/item/VNGU_2006_6_3_a5/

[1] A. N. Kolmogorov, “O printsipe tercium no datur”, Mat. sbornik, 32 (1925), 646–667 | Zbl

[2] S. P. Odintsov, “Algebraicheskaya semantika i semantika Kripke dlya rasshirenii minimalnoi logiki”, Logicheskie issledovaniya, 1999, no. 2 http://www.logic.ru

[3] S. P. Odintsov, “Negativno ekvivalentnye rasshireniya minimalnoi logiki i ikh logiki protivorechii”, Logicheskie issledovaniya, 2000, no. 7, 117–129

[4] E. H. Alves, A. M. Sette, “On the equivalence between some systems of non-classical logic”, Bull. Sect. Logic, 25:2 (1996), 68–72 | MR | Zbl

[5] D. Batens, “Paraconsistent extensional propositional logics”, Logique et Analyse, 23 (1980), 195–234 | MR | Zbl

[6] D. Batens, K. De Clercq, N. Kurtonina, “Embedding and interpolation for some paralogics. The propositional case”, Rep. on Math. Logic, 33 (1999), 29–44 | DOI | MR | Zbl

[7] P. Bernays, “Review of [9] and [10]”, J. Symb. Logic, 18 (1953), 266–268 | DOI

[8] N. C. A. da Costa, “On the theory of inconsistent formal systems”, Notre Dame J. Form. Logic, 15 (1974), 497–510 | DOI | MR | Zbl

[9] H. B. Curry, “On the definition of negation by a fixed proposition in the inferential calculus”, J. Symb. Logic, 17 (1952), 98–104 | DOI | MR | Zbl

[10] H. B. Curry, “The system LD”, J. Symb. Logic, 17 (1952), 35–42 | DOI | MR | Zbl

[11] H. B. Curry, Foundations of mathematical logic, McGrow-Hill Book Company, New York, 1963 | MR | Zbl

[12] K. Dosen, “Negative modal operators in intuitionistic logic”, Publ. Inst. Math., Nouv. Ser., 35(49) (1984), 15–20 | MR | Zbl

[13] K. Dosen, “Negation as a modal operator”, Rep. Math. Logic, 20 (1986), 15–28 | MR | Zbl

[14] K. Dosen, “Negation in the light of modal logic”, What is Negation?, eds. D. Gabbay, H. Wansing, Kluwer, Dordrecht, 1999, 77–86 | DOI | MR | Zbl

[15] J. M. Font, P. Hajek, Łukasiewicz and modal logic, Math. Preprint Ser., No 278, Univ. de Barselona, 2000 | Zbl

[16] I. Johansson, “Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus”, Compositio Mathematika, 4 (1937), 119–136 | MR

[17] A. Loparić, N. C. A. da Costa, “Paraconsistency, paracompleteness and induction”, Logique et Analyse, 113 (1986), 73–80 | MR

[18] J. Łukasiewicz, “A system of modal logic”, The Journal of Computing Systems, 1 (1953), 111–149 ; Reprinted in [20], 352–390 | MR

[19] J. Łukasiewicz, Aristotle's syllogistic from the standpoint of modern formal logic, 2nd enlarged edition, Clarendon Press, Oxford, 1957

[20] J. Łukasiewicz, Selected works, Stud. in Logic and the Found. of Math., ed. L. Borkowski, North-Holland, Amsterdam, 1970 | MR

[21] S. P. Odintsov, “Maximal paraconsistent extension of Johansson logic”, Logique et Analyse, 161/163 (1998), 107–120 | MR | Zbl

[22] S. P. Odintsov, “Logic of classical refutability and class of extensions of minimal logic”, Logic and Logical Philosophy, 9 (2002), 91–107 | DOI | MR

[23] S. P. Odintsov, “Representation of $j$-algebras and Segerberg's logics”, Logique et analyse, 165/166 (1999), 81–106 | MR | Zbl

[24] S. P. Odintsov, “Reductio ad absurdum and Łukasiewicz modalities”, Logic and Logical Philosophy, 11/12 (2003), 149–166 | DOI | MR | Zbl

[25] J. Porte, “The $\Omega$-system and the $L$-system of modal logic”, Notre Dame J. formal Logic, 20 (1979), 915–920 | DOI | MR | Zbl

[26] J. Porte, “Lukasiewicz's L-modal system and classical refutability”, Logique Anal. Now. Ser., 27 (1984), 87–92 | MR | Zbl

[27] G. Priest, R. Routley, J. Norman (eds.), Paraconsistent logic. Essays on the inconsistent, Philosophia Verlag, München, 1989 | MR

[28] A. P. Pynko, “Algebraic study of Sette's maximal paraconsistent logic”, Studia Logica, 54 (1995), 89–128 | DOI | MR | Zbl

[29] A. M. Sette, “On the propositional calculus $P^1$”, Math. Japon., 18:3 (1973), 173–180 | MR | Zbl