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On superintuitionistic logics as fragments of proof logic extensions

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Abstract

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign ο to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI Δ set up correspondingly by the calculi

For any calculusи we denote byяи the set of all formulae of the calculusи and byℒи the lattice of all logics that are the extensions of the logic of the calculusи, i.e. sets ofяи formulae containing the axioms ofи and closed with respect to its rules of inference. In the logiclɛℒG the sign □ is decoded as follows: □A = (A & ΔA). The result of placing □ in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and λ the decoding of □ is meant), by virtue of which the diagram is constructed

In this diagram the mapsσ, x andλ are isomorphisms, thereforex −1 = λ; and the maps △ andμ are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place:σ −1 =μ −1 λ△ and σ = △−1 xμ. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.

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References

  1. A. V. Bessonov,On new operations in intuitionistic calculus,Mathematical Notes, 22, No. 1 (1977), pp. 23–28.

    Google Scholar 

  2. V. Ia. Gerchiu,On finite approximability of superintuitionistic logics,Mathematical Research, 7, No. 1, (1972), pp 186–192.

    Google Scholar 

  3. V. Ia. Gerchiu, A. V. Kuznetsov,On varieties of pseudoboolean algebras, defined by an identity of a bounded length, In:IX All-Union Algebraical Colloquium (abstracts), Homel, 1968, pp. 54–56.

  4. A. V. Kuznetsov,On undecidability of general completeness, decidability and equivalence problems for propositional calculae,Algebra and Logics, 2, No. 4, (1963), pp. 47–66.

    Google Scholar 

  5. A. V. Kuznetsov,The analogs of Sheffer operation in constructive logic,Doklady Akademii Nauk SSSR, 160, No. 2, (1965), pp. 274–277.

    Google Scholar 

  6. A. V. Kuznetsov,Some properties of pseudoboolean algebras varieties lattice, In:XI All-Union Algebraical Colloquium (abstracts), Kishinev, 1971, pp. 255–256.

  7. A. V. Kuznetsov,Proof intuitionistic logic, In:Modal and Intentional Logics (abstracts of coordinative conference), Moscow, 1978, pp. 75–79.

  8. A. V. Kuznetsov,The deltable elements of pseudoboolean algebras, In:VI All-Union Conference on Logic (abstracts), Tbilisi, 1983, p. 93.

  9. A. V. Kuznetsov,On proof-intuitionistic propositional calculus,Doklady Akademii Nauk SSSR, 283, No. 1, (1985), pp. 27–29

    Google Scholar 

  10. A. V. Kuznetsov, V. Ia. Gerchiu,On superintuitionistic logics and finite approximability,Doklady Akademii Nauk SSSR, 195, No. 5, (1970), pp. 1029–1032.

    Google Scholar 

  11. A. V. Kuznetsov, A. Yu. Muravitsky,Proof logic, In:IV All-Union Confe rence on Mathematical Logic (abstracts), Kishinev, 1976, No. 3.

  12. A. V. Kuznetsov, A. Yu. Muravitsky,Magari's algebras, In:XIV All-Union Algebraical Conference (abstracts), part 2, Novosibirsk, 1977, pp. 105–106.

  13. A. V. Kuznetsov, A. Yu. Muravitsky,Provability as modality, In:Actual Problems on Logic and Methodology of Science, Kiev, 1980, pp. 193–230.

  14. A. V. Kuznetsov, M. F. Ratsa,Functional completeness criterium, for first-order predicate classical logic,Doklady Akademii Nauk SSSR, 249, No 3, (1979), pp. 540–544.

    Google Scholar 

  15. L. L. Maksimova,Pretabular superintuitionistic logics,Algebra and Logics, 11, No. 5, (1972), pp. 558–570.

    Google Scholar 

  16. L. L. Maksimova,Pretabular extensions of Lewis S4, logic,Algebra and Logics, 14, No. 1, (1975), pp. 28–55.

    Google Scholar 

  17. L. L. Maksimova,Modal logics of finite slices,Algebra and Logics, 14, No. 3, (1975), pp. 304–314.

    Google Scholar 

  18. L. L. Maksimova, V. V. Rybakov,On normal modal logics lattice,Algebra and Logics, 13, No. 2, (1974), pp. 188–216.

    Google Scholar 

  19. V. Yu. Meskhi, L. L. Esakia,On five “critical” modal systems, In:Logical Deduction Theory (abstracts of All-Union symposium reports), Moskow, 1974.

  20. A. Yu. Muravitsky,Strong equivalence on intuitionistic Kripke models and assertorically equivoluminous logics,Algebra and Logics, 20, No. 2, (1981), pp. 165–182.

    Google Scholar 

  21. A. Yu. Muravitsky, On finite approximability of the calculusI Δ and non-modelability of some of its extensions,Mathematical Notes, 29, No. 6, (1981), pp. 907–916.

    Google Scholar 

  22. A. Yu. Muravitsky,On proof logic extensions,Mathematical Notes, 33, No. 6, (1983), pp. 915–927.

    Google Scholar 

  23. A. Yu. Muravitsky,On superintuitionistic logics, approximable by algebras with descending chain condition,Mathematical Notes, 35, No. 2, (1984), pp. 273–276.

    Google Scholar 

  24. A. Yu. Muravitsky,Correspondence of proof-intuitionistic logic extensions to proof logic extensions,Doklady Akademii Nauk SSSR, 281, No. 4, (1985), pp. 789–793.

    Google Scholar 

  25. P. S. Novikov,Constructive mathematical logic from the point of view of the classical one, Moscow, 1977.

  26. V. V. Rybakov,Hereditary finite-axiomatizable extensions of the logic S4,Algebra and Logics, 15, No. 2, (1976), pp. 185–204.

    Google Scholar 

  27. Ia. S. Smetanich,On the completeness of the propositional calculus with an extra one-argument operation,Proceedings of the Moscow Mathematical Society, 9, (1960), pp. 357–371.

    Google Scholar 

  28. V. B. Shehtman,On incomplete propositional logics,Doklady Akademii Nauk SSSR, 235, No. 3, (1977), pp. 542–545.

    Google Scholar 

  29. V. B. Shehtman,Topological models of propositional logics,Semiotics and Informatics, 15, (1980), pp. 74–98.

    Google Scholar 

  30. L. L. Esakia,On some new results in modal and superintuitionistic systems theory, In:Logical Deduction Theory (abstracts of the All-Union symposium reports), pert 1, Moscow, 1974, pp. 173–184.

  31. L. L. Esakia,On modal “counterparts” of superintuitionistic logic, In:VII All-Union Symposium on Logic and Methodology of Science (abstracts), Kiev 1976, pp. 135–136.

  32. L. L. Esakia,On varieties of Grzegorczyk's algebras, In:Investigation on Non-Classical Logics and Set Theory, Moscow, 1979, pp. 257–287.

  33. V. A. Yankov,On some superconstructive prepositional calculus,Doklady Akademii Nauk SSSR, 151, No. 4, (1963), pp. 796–798.

    Google Scholar 

  34. V. A. Yankov,Constructing a sequence of strongly independent superintuitionistic propositional calculi,Doklady Akademii Nauk SSSR, 181, No. 1, (1968), pp. 33–34.

    Google Scholar 

  35. C. Bernardi,On the equational class of diagonalizable algebras,Studia Logica, 34, No. 4, (1975), pp. 321–331.

    Google Scholar 

  36. W. J. Blok,Varieties of Interior Algebras, dissertation, University of Amsterdam, 1976.

  37. W. J. Blok,Pretabular varieties of modal algebras,Studia Logica, 39, No. 2/3, (1980), pp. 101–124.

    Google Scholar 

  38. G. Boolos,The unprovability of consistency (an assay in modal logic), Cambrige, 1979.

  39. G. Boolos,Provability in arithmetic and schema of Grzegorczyk,Fundamental Mathematicae, 106, No. 1, (1980), pp. 41–45.

    Google Scholar 

  40. N. Bourbaki,Topologie Generale, Paris, 1953.

  41. M. A. Dummett,A propositional calculus with denumerable matrex,Journal of Symbolic Logic, vol. 24, No. 2, (1959), pp. 97–106.

    Google Scholar 

  42. M. A. Dummett, E. J. Lemmon,Modal logics between S4 and S5,Zeitschrift für Mathematische Logic and Grundlagen der Mathematic, 5, (1959), pp. 250–264.

    Google Scholar 

  43. M. C. Fitting,Intuitionistic logic, model theory and forcing, Amsterdam, 1969.

  44. K. Gödel,Eine interpretation des intuitionistische Aussagenkalkulus,Ergeb nisse Math. Colloq., 4, 1933, pp. 39–40.

    Google Scholar 

  45. R. Goldblatt,Arithmetical necessity, provability and intuitionistic logic,Theoria, vol. 54, No. 1, (1978), pp. 38–46.

    Google Scholar 

  46. R. Goldblatt,Topoi (The categorial analysis of logic), Amsterdam, 1979.

  47. A. Grzegorczyk,Some relational systems and the associated topological spaces,Fundamenta Mathematicae, 60, (1967), pp. 223–231.

    Google Scholar 

  48. T. Hosoi,On intermediate logics I,J. Fac. Sci., Univ. Tokyo, Sec. 1, 14, (1967), pp. 293–312.

    Google Scholar 

  49. T. Hosoi, H. Ono,The intermediate logics on the second slice,J. Fac. Sci., Univ. Tokyo, Sec. 1A, 17, (1970), pp. 457–461.

    Google Scholar 

  50. T. Hosoi, H. Ono,Intermediate Propositional Logics (A Survey),J. Tsuda College, 5, (1973), pp. 67–82.

    Google Scholar 

  51. S. C. Kleene,Introduction to Metamathematics, New York, 1952.

  52. A. Kolmogoroff,Zur Deutung der intuitionistischen Logic,Math. Z., 35, (1932), pp. 58–65.

    Google Scholar 

  53. S. A. Kripke,Semantical analysis of intuitionistic logic I, In:Formal Systems and Recursive Functions, Amsterdam, 1965, pp. 92–129.

  54. A. V. Kuznetsov,On superintuitionistic logics,Proceedings of the International Congress of Mathematicians, August 1974, Vancouver, 1, 1975, pp. 243–249.

    Google Scholar 

  55. A. V. Kuznetsov,Proof-intuitionistic propositional calculus, In:Logic, Methodology and Philosophy of Science (Papers of Soviet National Organization Committee for the VII International Congress on Logic, Methodology and Philosophy of Science, Austria, Salzburg, 11–16 July, 1983, sections 1–5 and 7, Moscow, 1983, pp. 21–24.

  56. E. J. Lemmon,The “Lemmon notes”: an introduction to modal logic (in collabo ration with D. Scott),American Philosophical Quartely Monograph Series, No. 11, Blackwell, Oxford, 1977.

    Google Scholar 

  57. R. Magari,The diagonalisable algebras,Boll. Unione Mat. Ital., 12, suppl. fasc. 3, (1975), pp. 117–125.

    Google Scholar 

  58. J. C. C. McKinsey, A. Tarski,Some theorems about the sentential calculi of Lewis and Heyting,Journal of Symbolic Logic, 13, (1948), pp. 1–15.

    Google Scholar 

  59. S. Nagata,A series of successive modifications of Peirce's rule,Proc. Japan Acad., 42, (1966), pp. 859–861.

    Google Scholar 

  60. H. Ono,Kripke models and intermediate logics,Publ. Res. Inst. Math. Sci. Kyoto Univ., 6, (1970), pp. 461–476.

    Google Scholar 

  61. H. Rasiowa, R. Sikorski,The Mathematics of Metamathematics, Warszawa, 1963.

  62. G. Rose,Propositional calculus and realizability,Tans. Amer. Math. Soc., vol. 75, (1953), pp. 1–19.

    Google Scholar 

  63. G. F. Schumm,Solution to four modal problems of Sobocinski,Notre Dame Journal of Formal Logic, 12, No. 3, (1971), pp. 335–340.

    Google Scholar 

  64. K. Segerberg,Modal logics with linear alternative relations,Theoria, 36, No. 3, (1970), pp. 301–322.

    Google Scholar 

  65. K. Segerbebg,An Eassay in Classical Modal Logic, Filosofiska Studier, Uppsala, 1971.

    Google Scholar 

  66. K. Segerberg,On some extensions of K4,Journal of Symbolic Logic, 36, No. 4, (1971), p. 697.

    Google Scholar 

  67. B. Sobocincki,Certain extensions of modal system S4,Notre Dame Journal of Formal Logic, 11, No. 3, (1970), pp. 347–368.

    Google Scholar 

  68. R. M. Solovay,Provability interpretations of modal logic,Israel Journal of Mathematics, vol. 25, (1976), pp. 287–304.

    Google Scholar 

  69. A. Tarski,Der Aussagenkalkül und die Topologie,Fundamenta Mathematicae, 31, (1938), pp. 103–134.

    Google Scholar 

  70. T. Umezawa,Über Zwischensysteme der Aussagenlogik,Nagoya Math. J., 9, (1955), pp. 181–189.

    Google Scholar 

  71. T. Umezawa,On intermediate prepositional logics,Journal of Symbolic Logic, vol. 24, (1959), pp. 20–36.

    Google Scholar 

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  1. Dept. of Algebra and Mathematical Logic, Mathematical Institute of Computer Center of Moldavian SSR, Kishiniev, USSR

    A. V. Kuznetsov & A. Yu. Muravitsky

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  1. A. V. Kuznetsov
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Kuznetsov, A.V., Muravitsky, A.Y. On superintuitionistic logics as fragments of proof logic extensions. Stud Logica 45, 77–99 (1986). https://doi.org/10.1007/BF01881551

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