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Dual intuitionistic logic and a variety of negations: the logic of scientific research. (English) Zbl 1085.03022
Summary: We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic logic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson’s logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to some other logical systems.

03B60 Other nonclassical logic
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[1] CZERMAK, J., ’A remark on Gentzen’s calculus of sequents’, Notre Dame Journal of Formal Logic 18:471–474, 1977. · Zbl 0352.02014
[2] DOŠEN, K., ’Negation as a modal operator’, Reports on Mathematical Logic 20:15–27, 1986. · Zbl 0626.03006
[3] DOŠEN, K., ’Negation in the light of modal logic’, in D.M. Gabbay and H. Wansing, (eds.), What is Negation?, Applied Logic Series, 13, pp. 77–86, Kluwer Academic Publishers, Dordrecht, 1999. · Zbl 0974.03019
[4] DUNN, J.M., ’Generalized ortho negation’, in H. Wansing, (ed.), Negation. A Notion in Focus, pp. 3–26, De Gruyter, Berlin, 1995. · Zbl 0979.03027
[5] DUNN, J.M., ’A Comparative study of various model-theoretic treatments of negation: a history of formal negation’, in D.M. Gabbay and H. Wansing, (eds.), What is Negation?, Applied Logic Series, 13, pp. 23–51, Kluwer Academic Publishers, Dordrecht, 1999. · Zbl 0972.03028
[6] DUNN, J.M., ’Partiality and its dual’, Studia Logica 65:5–40, 2000. · Zbl 0988.03012
[7] HAZEN, A., Subminimal Negation. Unpublished Manuscript, 1992.
[8] JOHANSSON, I., ’Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus’, Compositio Mathematika 4:119–136, 1937. · Zbl 0015.24102
[9] FITTING, M., Proof Methods for Modal and Intuitionistic Logics. (Synthese Library; v. 169) D. Reidel Publishing Company, Dordrecht, Boston, Lancaster, 1983.
[10] GOODMAN, N., ’The logic of contradiction’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 27:119–126, 1981. · Zbl 0467.03019
[11] GORÉ, R., ’Dual intuitionistic logic revisited’, in R. Dyckhoff, (ed.), Proceedings Tableaux 2000 (Lecture Notes in Artificial Intelligence 1847), pp. 252–267, Springer Verlag, Berlin, 2000. · Zbl 0963.03042
[12] GRZEGORCZYK, A., ’A philosophically plausible formal interpretation of intuitionistic Logic’, Indagationes Matimaticae 26:596–601, 1964. · Zbl 0131.00701
[13] KAMIDE, N., ’A note on dual-intuitionistic logic’, Mathematical Logic Quarterly, 49:519–524, 2003. · Zbl 1036.03006
[14] KRIPKE, S., ’Semantical analysis of intuitionistic logic I’, in J.N. Crossley and M.A.E. Dumment, (eds.), Formal Systems and Recursive Functions, pp. 92–130, North-Holland Publishing Company, Amsterdam, 1965. · Zbl 0137.00702
[15] LUKOWSKI, P., ’A deductive-reductive form of logic: general theory and intuitionistic case’, Logic and Logical Philosophy 10:59–78, 2002.
[16] NELSON, D., ’Constructible falsity’, Journal of Symbolic Logic 14:16–26, 1949. · Zbl 0033.24304
[17] POPPER, K., The Logic of Scientific Discovery, Routledge, London and New York, 1992. · Zbl 0083.24104
[18] POPPER, K., Conjectures and Refutations. The Growth of Scientific Knowledge, Routlege, London, 1989.
[19] RABINOWICZ, W., ’Intuitionistic truth’, Journal of Philosophical Logic 14:191–228, 1985. · Zbl 0574.03047
[20] RAUSZER, C., ’Semi-Boolean algebras and their applications to intuitionistic logic with dual operations’, Fundamenta Mathematicae LXXXIII:219–249, 1974. · Zbl 0298.02064
[21] RAUSZER, C., ’A formalization of the propositional calculus of H-B-logic’, Studia Logica XXXIII:23–34, 1974. · Zbl 0289.02015
[22] RAUSZER, C., ’An algebraic and Kripke-style approach to a certain extension of intuitionistic logic’, Dissertationes Mathematicae, CLXVII, PWN, Warszawa, 1980. · Zbl 0442.03024
[23] SHRAMKO, Y., J.M. DUNN, and T. TAKENAKA, ’The trilaticce of constructive truth values’, Journal of Logic and Computation 11:761–788, 2001. · Zbl 0996.03014
[24] SHRAMKO, Y., ’Semantics for constructive negations’, in H. Wansing (ed.) Essays on Non-Classical Logic, pp. 187–217, World Scientific, New Jersey – London – Singapore – Hong Kong, 2001. · Zbl 0997.03004
[25] SHRAMKO, Y., ’The logic of scientific research’ (In Russian), in Logic and V.E.K.: Festschrift dedicated to 90th birthday of Prof. E.K. Voishvillo, pp. 237–249, Sovremennye tetradi, Moscow, 2003. (The On-line version see: Logical Studies, Online Journal, No. 10, 2003, http://www.logic.ru/Russian/LogStud/10/No10-20.html, ISBN 5-85593-140-4).
[26] SMIRNOV, V.A., ’On a system of paraconsistent logic’ (in Russian), in V.A. Smirnov and A.S. Karpenko, (eds.), Many-valued, Relevant ans Paraconsistent Logics, pp. 129–133, Moscow, 1984, (Reprinted in V.A. Smirnov, A Theory of Logical Inference, pp. 293–298, ROSSPEN, Moscow, 1999).
[27] URBAS, I., ’Dual-intuitionistic logic’, Notre Dame Journal of Formal Logic 37:440–451, 1996. · Zbl 0869.03008
[28] WANSING, H., The Logic of Information Structures, Springer-Verlag, Berlin Heidelberg, 1993. · Zbl 0788.03001
[29] WOLTER, F., ’On logics with coimplication’, Journal of Philosophical Logic 27:353–387, 1998. · Zbl 0976.03020
[30] ZASLAVSKIJ, I.D., Symmetrical Constructive Logic (in Russian), Erevan, 1978.
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