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References

  1. M. A. Abashidze, Ordinal completeness of the Gödel-Löb modal system [in Russian], In: Intensional Logics and Logical Structure of Theories: Material from the Fourth Soviet-Finnish Symposium on Logic, Telavi, May 20–24, 1985, Tbilisi, Metsniereba, 1988, pp. 49–73.

    Google Scholar 

  2. S. Abramsky, Algorithmic game semantics. A tutorial introduction, In: H. Schwichtenberg (ed.) et al, Proof and System-Reliability. Proc. NATO Advanced Study Institute, Marktoberdorf, Germany, July 24–August 5, 2001, Dordrecht, Kluwer Academic Publishers, 2002, pp. 21–47.

    Google Scholar 

  3. S. Artemov, Explicit provability and constructive semantics, Bull. Symbolic Log. 7 (2001), no. 1, 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. N. Artemov, Applications of modal logic in proof theory [in Russian], In: Questions of Cybernetics: Nonclassical Logics and Their Applications, Moscow, Nauka, 1982, pp. 3–20.

    Google Scholar 

  5. S. N. Artemov and L. D. Beklemishev, On propositional quantifiers in provability logic, Notre Dame J. Formal Logic 34 (1993), no. 3, 401–419.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. N. Artemov and L.D. Beklemishev, Provability logic, In: D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, 2nd ed., Vol. 13, Dordrecht, Kluwer, 2004, pp. 229–403.

    Google Scholar 

  7. H. Barendregt, Towards an interactive mathematical proof language, In: F. Kamareddine (ed.), Thirty Five Years of Automath, Dordrecht, Kluwer, 2003, pp. 25–36.

    Google Scholar 

  8. J. Barwise and J. Etchemendy, Visual information and valid reasoning, In: W. Zimmerman and S. Cunningham (eds.), Visualization in Teaching and Learning Mathematics, Washington, Math. Ass. Amer. 1990, pp. 9–24.

    Google Scholar 

  9. L. D. Beklemishev, On the classification of propositional provability logics [in Russian], Izv. AN SSSR, Ser. Mat. 53 (1989), no. 5, 915–943; English transl., Math. USSR Izv. 35 (1990), 247-275.

    MathSciNet  Google Scholar 

  10. L. D. Beklemishev, On bimodal logics of provability, Ann. Pure Appl. Logic 68 (1994), no. 2, 115–160.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. D. Beklemishev, Bimodal logics for extensions of arithmetical theories, J. Symbolic Logic 61 (1996), no. 1, 91–124.

    Article  MATH  MathSciNet  Google Scholar 

  12. L. D. Beklemishev, Provability algebras and proof-theoretic ordinals I, Ann. Pure Appl. Logic 128 (2004), 103–123.

    Article  MATH  MathSciNet  Google Scholar 

  13. L. D. Beklemishev, The Worm Principle, Logic Group Preprint Ser. no. 219, Univ. Utrecht, March 2003. http://preprints.phil.uu.nl/lgps/.

    Google Scholar 

  14. L. D. Beklemishev, M. Pentus, and N. Vereshchagin, Provability, Complexity, Grammars, Am. Math. Soc. Transl. Series 2, 192, 1999.

    Google Scholar 

  15. L. D. Beklemishev and A. Visser, On the Limit Existence Principles in Elementary Arithmetic and Related Topics, Tech. Report LGPS no. 224, Dept. Phil., Univ. Utrecht, 2004.

    Google Scholar 

  16. A. Berarducci, The interpretability logic of Peano arithmetic, J. Symbolic Logic 55 (1990), 1059–1089.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Berarducci and R. Verbrugge, On the provability logic of bounded arithmetic, Ann. Pure Appl. Logic 61 (1993), 75–93.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Blass, Infinitary combinatorics and modal logic, J. Symbolic Logic 55 (1990), 7611–778.

    MathSciNet  Google Scholar 

  19. A. Blass and Yu. Gurevich, Algorithms vs. Machines, Bull. Europ. Ass. Theoret. Comput. Sci. 77(200), June, 96–118.

    Google Scholar 

  20. A. Blass and Yu. Gurevich, Algorithms: A Quest for Absolute Definitions, Bull. Europ. Ass. Theoret. Comput. Sci. 81 (2003), Oct., 195–225.

    MathSciNet  MATH  Google Scholar 

  21. G. Boolos, The Logic of Provability, Cambridge, Cambridge Univ. Press, 1993.

    MATH  Google Scholar 

  22. G. Boolos and G. Sambin, Provability: the emergence of a mathematical modality, Studia Logica 50 (1991), no. 1, 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  23. W. Burr, Fragments of Heyting arithmetic, J. Symbolic Logic 65 (2000), no. 3, 1223–1240.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Buss, Bounded arithmetic, Napoli, Bibliopolis, 1986.

    MATH  Google Scholar 

  25. T. Carlson, Modal logics with several operators and provability interpretations, Israel J. Math. 54 (1986), 14–24.

    MATH  MathSciNet  Google Scholar 

  26. D. de Jongh and G. Japaridze, The logic of provability, In: S. R. Buss (ed.), Handbook of Proof Theory, Stud. Logic Found. Math. 137 Amsterdam, Elsevier, 1998, pp. 475–546.

    Google Scholar 

  27. D. H. J. de Jongh, The maximality of the intuitionistic predicate calculus with respect to Heyting’s Arithmetic, J. Symbolic Logic 36 (1970), 606.

    Google Scholar 

  28. D. H. J. de Jongh and F. Veltman, Provability logics for relative interpretability, In: [64, pp. 31–42].

    Google Scholar 

  29. D. H. J. de Jongh and A. Visser, Embeddings of Heyting algebras, In: M. Hyland, C. Steinhorn, and J. Truss (eds.), Logic: from Foundations to Applications, Oxford, Clarendon Press, 1996 [49, pp. 187–213].

    Google Scholar 

  30. K. Došen, Idedntity of proofs based on normalization and generality, Bull. Symbolic Log. 9 (2003), 477–503.

    Article  MATH  Google Scholar 

  31. S. Feferman, Arithmetization of metamathematics in a general setting, Fundamen. Math. 49 (1960), 35–92.

    MATH  MathSciNet  Google Scholar 

  32. S. Feferman, Transfinite recursive progressions of axiomatic theories, J. Symbolic Logic 27 (1962), 259–316.

    Article  MathSciNet  Google Scholar 

  33. H. Friedman, Some applications of Kleene’s methods for intuitionistic systems, In: A. R. D. Mathias and H. Rogers (eds.), Cambridge Summerschool in Mathematical Logic, Berlin etc., Springer-Verlag, 1973, pp. 113–170.

    Google Scholar 

  34. H. Friedman, The disjunction property implies the numerical existence property, Proc. Nat. Acad. USA, 72 (1975), 2877–2878.

    Article  MATH  Google Scholar 

  35. Yu. V. Gavrilenko, Recursive realizability from the inuitionistic point of view, Soviet. Math. Dokl. 23 (1981), 9–14.

    MATH  Google Scholar 

  36. G. Gentzen, Die Wiederspruchsfreiheit der reinen Zahlentheorie, Math. Ann. 112 (1936), no. 4, 493–565.

    Article  MATH  MathSciNet  Google Scholar 

  37. G. Gentzen, Neue Fassung des Wiederspruchsfreiheitsbeweises für die reine Zahlentheorie, Forsch. Logik Grund. Wiss. 4 (1938), 19–44.

    MATH  Google Scholar 

  38. S. Ghilardi, Unification in intuitionistic logic, J. Symbolic Logic 64 (1990), 859–880.

    MathSciNet  Google Scholar 

  39. J.-Y. Girard, Linear logic, Theoret. Comput. Sci. 50 (1987), 1–102.

    Article  MATH  MathSciNet  Google Scholar 

  40. J.-Y. Girard, Proof Theory and Logical Complexity. Studies in Proof Theory. Monographs 1, Napoli, Bibliopolis, 1987.

    Google Scholar 

  41. J.-Y. Girard, Locus solum: From the rules of logic to the logic of rules, Math. Structures Comput. Sci. 11 (2001), no. 3, 301–506.

    Article  MATH  MathSciNet  Google Scholar 

  42. J.-Y. Girard, Y. Lafont, and P. Taylor, Proofs and Types, Cambridge, Cambridge Univ. Press, 1989.

    MATH  Google Scholar 

  43. K. Gödel, Eine Interpretation des intuitionistischen Aussagenkalkuls Ergebnisse Math. Kolloq. (1933), 39–40.

    Google Scholar 

  44. O. Goldreich, Zero-knowledge twenty years after its invention, Tutorial, URL: http://www.wisdom.weizmann.ac.il/oded/zk-tut02.html, 2004.

    Google Scholar 

  45. S. Goldwasser, S. Micali, and C. Rako, The knowledge complexity of interactive proof systems, SIAM J. Comput. 18 (1989), 186–208.

    Article  MATH  MathSciNet  Google Scholar 

  46. S. V. Goryachev, On interpretability of some extensions of arithmetic [in Russian], Mat. Zametki, 40 (1986), 561–572.

    MATH  MathSciNet  Google Scholar 

  47. P. Hájek and F. Montagna, The logic of Π1-conservativity, Arch. Math. Logik 30 (1990), 113–123.

    Article  MATH  Google Scholar 

  48. P. Hájek and P. Pudlák, Metamathematics of First Order Arithmetic, Berlin etc., Springer-Verlag, 1993.

    MATH  Google Scholar 

  49. W. Hodges, M. Hyland, C. Steinhorn, and J. Truss (eds.), Logic: from Foundations to Applications, Oxford, Clarendon Press, 1996.

    MATH  Google Scholar 

  50. R. Iemhof, On the admissible rules of intuitionistic propositional logic, J. Symbolic Logic 66 (2001), no. 1, 281–294.

    Article  MathSciNet  Google Scholar 

  51. R. Iemhof, Provability Logic and Admissible Rules, PhD Thesis, Univ. Amsterdam, 2001.

    Google Scholar 

  52. K. N. Ignatiev, Partial Conservativity and Modal logics, ITLI Prepublication Ser. X-91-04, Univ. Amsterdam, 1991.

    Google Scholar 

  53. K. N. Ignatiev, On strong provability predicates and the associated modal logics, J. Symbolic Logic 58 (1993), 249–290.

    Article  MATH  MathSciNet  Google Scholar 

  54. G. K. Japaridze, The modal logical means of investigation of provability [in Russian], Thesis in Phil., Moscow, 1986.

    Google Scholar 

  55. J. J. Joosten, Interpretability Formalized, PhD Thesis, Univ. Utrecht, 2004.

    Google Scholar 

  56. J. J. Joosten and A. Visser, The interpretability logic of all reasonable arithmetical theories, Erkenntnis, 53 (2000), no. 1-2, 3–26.

    Article  MATH  MathSciNet  Google Scholar 

  57. J. J. Joosten and A. Visser, How to derive principles of interpretability logic. A toolkit, In: J. van Benthem, A. Troelstra, F. Veltman, and A. Visser (eds.), Liber Amicorum for Dick de Jongh. ILLC, hhttp://www.illc.uva.nl/D65/i, 2004.

    Google Scholar 

  58. A. Kolmogorov and V. Uspensky, On the definition of algorithm [in Russian], Uspekhi Mat. Nauk 13 (1958), no. 4, 3–28; English transl.: Am. Math. Soc. Transl. 29 (1963), 217-245.

    MATH  Google Scholar 

  59. J. Krajíček, Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge, Cambridge Univ. Press, 1995.

    MATH  Google Scholar 

  60. G. Kreisel, A survey of proof theory. II, In: Proc. 2nd Scandinav. Logic Symposium (Univ. Oslo, Oslo, 1970), Stud. Logic Found. Math. 63 Amsterdam, Elsevier, 1971, pp. 109–170.

    Google Scholar 

  61. G. Kreisel and A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems, Z. Math. Logik 14 (1968), 97–142.

    MATH  Google Scholar 

  62. R. Magari, The diagonalizable algebras (the algebraization of the theories which express Theor.:II), Boll. Unione Mat. Ital., Ser. 4, 12 (1975). Suppl. fasc. 3, 117–125.

    MATH  MathSciNet  Google Scholar 

  63. Y. Moschovakis, What is an algorithm? In: B. Engquist and W. Schmid (eds.), Mathematics Unlimited, Berlin, Springer-Verlag, 2001, pp. 919–936.

    Google Scholar 

  64. P. P. Petkov (ed.), Mathematical Logic, Proc. the Heyting 1988 summer school in Varna, Bulgaria, Plenum Press, 1990.

    Google Scholar 

  65. W. Pohlers, Subsystems of set theory and second order number theory, In: S. R. Buss (ed.), Handbook of Proof Theory, Stud. Logic Found. Math. 137 Amsterdam, Elsevier, 1998, pp. 210–335.

    Google Scholar 

  66. H. Prakken, Logical Tools for Modelling Legal Argument, A Study of Defeasible Reasoning in Law, Dordrecht, 1997.

    Google Scholar 

  67. D. Prawitz, Ideas and results in proof theory, In: Proc. 2nd Scandinav. Logic Symposium (Univ. Oslo, Oslo, 1970), Stud. Logic Found. Math. 63 Amsterdam, Elsevier, 1971, pp. 235–307.

    Google Scholar 

  68. N. Preining, Sketch-as-proof, In: G. Gottlob, A. Leitsch, and D. Mundici (eds.), Computational Logic and Proof Theory, Lect. Notes Comput. Sci. 1289 (1997), pp. 264–277.

    Google Scholar 

  69. M. Rathjen, Recent advances in ordinal analysis: Π 21 -CA and related systems, Bull. Symbolic Log. 1 (1995) no. 4, 468–485.

    Article  MATH  MathSciNet  Google Scholar 

  70. G. F. Rose, Propositional calculus and realizability, Trans. Am. Math. Soc. 61 (1953), 1–19.

    Article  Google Scholar 

  71. V. V. Rybakov, A criterion for admissibility of rules in the modal system S4 and intuitionistic logic, Algebra Logic 23 (1984), 369–384.

    Article  MATH  Google Scholar 

  72. V. V. Rybakov, Admissibility of Logical Inference Rules, Amsterdam, Elsevier, 1997.

    MATH  Google Scholar 

  73. A. Schönhage, Storage modification machines, SIAM J. Comput. 9 (1980), 490–508.

    Article  MATH  MathSciNet  Google Scholar 

  74. V. Yu. Shavrukov, The logic of relative interpretability over Peano arithmetic [in Russian], Tech. Report No. 5, Steklov Math. Institute, Moscow, 1988.

    Google Scholar 

  75. V. Yu. Shavrukov, A note on the diagonalizable algebras of PA and ZF, Ann. Pure Appl. Logic 61 (1993), 161–173.

    Article  MATH  MathSciNet  Google Scholar 

  76. V. Yu. Shavrukov, Subalgebras of Diagonalizable Algebras of Theories Containing Arithmetic, Disser. Math., no. 323, 1993.

    Google Scholar 

  77. V. Yu. Shavrukov, Isomorphisms of diagonalizable algebras, Theoria 63 (1997), no. 3, 210–221.

    Article  MathSciNet  Google Scholar 

  78. V. Yu. Shavrukov, Undecidability in diagonalizable algebras, J. Symbolic Logic 62 (1997), no. 1, 79–116.

    Article  MATH  MathSciNet  Google Scholar 

  79. T. Smiley, The logical basis of ethics, Act. Phil. Fennica 16 (1963), 237–246.

    MathSciNet  Google Scholar 

  80. C. Smoryński, Applications of Kripke models, In: [87], pp. 324–391.

    Google Scholar 

  81. C. Smoryński, Fifty years of self-reference, Notre Dame J. Formal Logic 22 (1981), 357–374.

    Article  MathSciNet  MATH  Google Scholar 

  82. C. Smoryński, Self-Reference and Modal Logic, Berlin, Springer-Verlag, 1985.

    MATH  Google Scholar 

  83. R. M. Solovay, Provability interpretations of modal logic, Israel J. Math. 28 (1976), 33–71.

    MathSciNet  Google Scholar 

  84. G. Sundholm, Proofs as acts versus proofs as objects: Some questions for Dag Prawitz, Theoria 64 (1998), 187–216.

    Article  MathSciNet  Google Scholar 

  85. V. Švejdar, Modal analysis of generalized Rosser sentences, J. Symbolic Logic 48 (1983), 986–999.

    Article  MathSciNet  MATH  Google Scholar 

  86. A. Tarski, A. Mostowski, and R. M. Robinson, Undecidable Theories, Amsterdam, North-Holland, 1953.

    MATH  Google Scholar 

  87. A. Troelstra, Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, Berlin, Springer-Verlag, Lect. Notes 344, 1973.

    Google Scholar 

  88. A. Troelstra and D. van Dalen, Constructivism in Mathematics, Vols. 1, 2, Amsterdam, North-Holland, 1988.

    Google Scholar 

  89. A. M. Turing, System of logics based on ordinals, Proc. London Math. Soc., Ser. 2 45 (1939), 161–228.

    MATH  Google Scholar 

  90. R. Verbrugge, Efficient Metamathematics, PhD Thesis, Univ. Amsterdam, 1993.

    Google Scholar 

  91. A. Visser, Aspects of Diagonalization and Provability, PhD Thesis, Univ. Utrecht, 1981.

    Google Scholar 

  92. A. Visser, On the completeness principle, Ann. Math. Logic 22(1982), 263–295.

    Article  MATH  MathSciNet  Google Scholar 

  93. A. Visser, Evaluation, provably deductive equivalence in Heyting’s Arithmetic of substitution instances of propositional formulas, Logic Group Preprint Ser. 4, Dept. Phil., Univ. Utrecht, 1985.

    Google Scholar 

  94. A. Visser, Interpretability logic, In: [64, pp. 175–209].

    Google Scholar 

  95. A. Visser, The formalization of interpretability, Studia Logica 51(1991), 81–105.

    Article  MathSciNet  Google Scholar 

  96. A. Visser, An overview of interpretability logic, In: M. Kracht, M. de Rijke, H. Wansing, and M. Zakhariaschev (eds.), Advances in Modal Logic, Vol. 1, CSLI Lect. Notes, 87 (1998), 307–359.

    Google Scholar 

  97. A. Visser, Rules and arithmetics, Notre Dame J. Formal Logic 40 (1999), no. 1, 116–140.

    Article  MATH  MathSciNet  Google Scholar 

  98. A. Visser, Substitutions of Σ 10 -sentences: Explorations between intuitionistic propositional logic and intuitionistic arithmetic, Ann. Pure Appl. Logic 114 (2002),no. 1–3, 227–271.

    Article  MATH  MathSciNet  Google Scholar 

  99. A. Visser, Faith and falsity, Ann. Pure Appl. Logic, 131 (2005),103–131.

    Article  MATH  MathSciNet  Google Scholar 

  100. A. Visser, J. van Benthem, D. de Jongh, and G. R. de Lavalette, NNIL, a study in intuitionistic propositional logic In: A. Ponse, M. de Rijke, and Y. Venema (eds.), Modal Logic and Process Algebra, a Bisimulation Perspective, CSLI Lect. Notes 53 (1995), 289–326.

    Google Scholar 

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Beklemishev, L., Visser, A. (2006). Problems in the Logic of Provability. In: Gabbay, D.M., Goncharov, S.S., Zakharyaschev, M. (eds) Mathematical Problems from Applied Logic I. International Mathematical Series, vol 4. Springer, New York, NY. https://doi.org/10.1007/0-387-31072-X_2

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