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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
3
Pages
28-39
 Section Mathematics Title Axiomatics of P.S. Novikov complete extensions of the superintuitionistic logic $L2$ in the language containing an additional constant Author(-s) Koshcheeva A.K.a Affiliations Udmurt State Universitya Abstract The Novikov problem for a superintuitionistic logic $L$ is to describe the class of all maximal conservative (i.e. P.S. Novikov complete) extensions of $L$ in the language with additional logical connectives and logical constants. Since the family of all superintuitionistic logics has the power of the continuum, it is sensible to apply the P.S. Novikov problem to superintuitionistic logics which for one reason or other have already come to researchers' attention. In particular, there are three so-called pretabular superintuitionistic logics (i.e. non-tabular, but all their own extensions are tabular). One of them - the logic $L2$ - is characterized by the class of finite rooted linearly ordered sets of depth 2. It is established that for superintuitionistic logic $L2$ in the language with one additional constant there are exactly five P.S. Novikov complete extensions; their semantic description is given. In this paper we propose an explicit axiomatics for each of the five existing P.S. Novikov complete extensions of the superintuitionistic logic $L2$ in the language containing an additional constant. Keywords the superintuitionistic logic $L2$, a new logical constant, an explicit axiomatics of Novikov-complete extensions UDC 510.64 MSC 03B55, 03B60 DOI 10.20537/vm140303 Received 27 August 2014 Language Russian Citation Koshcheeva A.K. Axiomatics of P.S. Novikov complete extensions of the superintuitionistic logic $L2$ in the language containing an additional constant, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 28-39. References Ershov Yu.L., Palyutin E.A. Matematicheskaya logika (Mathematical logic), Moscow: Nauka, 1987, 336 p. Skvortsov D.P. On intuitionistic propositional calculus with an additional logical connective, Issledovaniya po neklassicheskim logikam i formal'nym sistemam (Studies in nonclassical logics and formal systems), Moscow: Nauka, 1983, pp. 154-173 (in Russian). Smetanich Ya.S. On statement calculi with an additional operation, Soviet Math. Doklady, 1961, vol. 2, pp. 937-939. Smetanich Ya.S. On the completeness of the propositional calculus with additional operations in one argiment, Tr. Mosk. Mat. Obs., 1960, vol. 9, pp. 357-371 (in Russian). Yankov V.A. Constructing a sequence of strongly independent superintuitionistic propositional calculi, Soviet Math. Dokl., 1968, vol. 9, pp. 806-807. Yashin A.D. On a new constant in intuitionistic propositional logic, Fundam. Prikl. Mat., 1999, vol. 5, no. 3, pp. 903-926 (in Russian). Yashin A.D. New intuitionistic logical constants and Novikov completeness, Stud. Log., 1999, vol. 63, no. 2, pp. 151-180. Yashin A.D. Classification of Novikov complete logics with extra logical constants, Algebra and Logic, 2003, vol. 42, no. 3, pp. 207-216. Yashin A.D. A new regular constant in intuitionistic propositional logic, Siberian Math. Journal, 1996, vol. 37, no. 6, pp. 1242-1258. Maksimova L.L. Pretabular superintuitionistic logics, Algebra and Logic, 1972, vol. 11, no. 5, pp. 308-314. Yashin A.D. New constants in two pretabular superintuitionistic logics, Algebra and Logic, 2011, vol. 50, no. 2, pp. 171-186. Rasiowa H., Sikorski R. The mathematics of metamathematics, Warszawa: PWN, 1963. Translated under the title Matematika metamatematiki, Moscow: Nauka, 1972, 592 p. Zakharyaschev M.V. Syntax and semantics of intermediate logics, Algebra and Logic, 1989, vol. 28, no. 4, pp. 262-282. Chagrov A., Zakharyaschev M. Modal logic, New York: Oxford University Press, 1997, 605 р. Grigoliya R.Sh. Free S4.3-algebra with a finite number of generators, Issledovaniya po neklassicheskim logikam i formal'nym sistemam (Studies in nonclassical logics and formal systems), Moscow: Nauka, 1983, pp. 281-287 (in Russian). Lavrov I.A., Maksimova L.L. Zadachi po teorii mnozhestv, matematicheskoi logike i teorii algoritmov (Tasks in set theory, mathematical logic and the theory of algorithms), Moscow: Fizmatlit, 2004, 256 p. Kleene S.C. Introduction to metamathematics, New York: D. Van Nostrand Company, 1952. Translated under the title Vvedenie v metamatematiku, Moscow: Inostrannaya literatura, 1957, 526 p. Schutte K. Complete systems of modal and intuitionistic logic, Modal'naya logika (Modal Logic), Ed.: Feys R., Moscow: Nauka, 1974, pp. 324-421 (in Russian). Gabbay D.M. On some new intuitionistic propositional connectives. I, Stud. Log., 1977, vol. 36, no. 1-2, pp. 127-139. Full text