Handbook of Philosophical Logic, 2nd Edition pp 189-360

Part of the Handbook of Philosophical Logic book series (HALO, volume 13)

Provability Logic

  • Sergei N. Artemov
  • Lev D. Beklemishev

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [Ackermann, 1940]
    W. Ackermann. Zur Wiederspruchsfreiheit der reinen Zahlentheorie. Math. Ann., 117:162–194, 1940.CrossRefGoogle Scholar
  2. [Adamovicz and Bigorajska, 1989]
    Z. Adamovicz and T. Bigorajska. Functions provably total in I Σ1. Fundamenta mathematicae, 132:189–194, 1989.Google Scholar
  3. [Allen et al., 1990] S. Allen, R. Constable, D. Howe, and W. Aitken. The semantics of refected proofs. In Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, pages 95–107, Los Alamitos, CA, USA, 1990. IEEE Computer Society Press.Google Scholar
  4. [Alt and Artemov, 2001]
    J. Alt and S. Artemov. Refective λ-calculus. In Proceedings of the Dagstuhl-Seminar on Proof Theory in Computer Science, volume 2183 of Lecture Notes in Computer Science, pages 22–37. Springer, 2001.Google Scholar
  5. [Artemov and Beklemishev, 1993]
    S.N. Artemov and L.D. Beklemishev. On propositional quantifiers in provability logic. Notre Dame Journal of Formal Logic, 34:401–419, 1993.Google Scholar
  6. [Artemov and Krupski, 1996]
    S. Artemov and V. Krupski. Data storage interpretation of labeled modal logic. Annals of Pure and Applied Logic, 78(1):57–71, 1996.Google Scholar
  7. [Artemov and Nogina, 2004]
    S. Artemov and E. Nogina. Logic of knowledge with justifications from the provability perspective. Technical Report TR-2004011, CUNY Ph.D. Program in Computer Science, 2004.Google Scholar
  8. [Artemov and Sidon-Yavorskaya, 2001]
    S. Artemov and T. Sidon-Yavorskaya. On the first order logic of proofs. Moscow Mathematical Journal, 1:475–490, 2001.Google Scholar
  9. [Artemov and Strassen, 1992a]
    S. Artemov and T. Strassen. The basic logic of proofs. In E. Börger, G. Jäger, H. Kleine Büning, S. Martini, and M. M. Richter, editors, Computer Science Logic. 6th Workshop, CSL’92. San Miniato, Italy, September/October 1992. Selected Papers, volume 702 of Lecture Notes in Computer Science, pages 14–28. Springer, 1992.Google Scholar
  10. [Artemov and Strassen, 1992b]
    S. Artemov and T. Strassen. Functionality in the basic logic of proofs. Technical Report IAM 92-004, Department of Computer Science, University of Bern, Switzerland, 1992.Google Scholar
  11. [Artemov and Strassen, 1993]
    S. Artemov and T. Strassen. The logic of the Gödel proof predicate. In G. Gottlob, A. Leitsch, and D. Mundici, editors, Computational Logic and Proof Theory. Third Kurt Gödel Colloquium, KGC’93. Brno, Chech Republic, August 1993. Proceedings, volume 713 of Lecture Notes in Computer Science, pages 71–82. Springer, 1993.Google Scholar
  12. [Artemov et al., 1999] S. Artemov, E. Kazakov, and D. Shapiro. Epistemic logic with justifications. Technical Report CFIS 99-12, Cornell University, 1999.Google Scholar
  13. [Artemov, 1979]
    S.N. Artemov. Extensions of arithmetic and modal logics. PhD thesis, Steklov Mathematical Insitute, Moscow, 1979. In Russian.Google Scholar
  14. [Artemov, 1980]
    S.N. Artemov. Arithmetically complete modal theories. In Semiotika i Informatika, 14, pages 115–133. VINITI, Moscow, 1980. In Russian. English translation in: Amer. Math. Soc. Transl. (2), 135: 39–54, 1987.Google Scholar
  15. [Artemov, 1982]
    S.N. Artemov. Applications of modal logic in proof theory. In Problems of Cyberneticsr: Nonclassical logics and their applications, pages 3–20. Nauka, Moscow, 1982. In Russian.Google Scholar
  16. [Artemov, 1985a]
    S.N. Artemov. Nonarithmeticity of truth predicate logics of provability. Doklady Akad. Nauk SSSR, 284(2):270–271, 1985. In Russian. English translation in Soviet Mathematics Doklady 33:403–405, 1985.Google Scholar
  17. [Artemov, 1985b]
    S.N. Artemov. On modal logics axiomatizing provability. Izvestiya Akad. pmNauk SSSR, ser. mat., 49(6):1123–1154, 1985. In Russian. English translation in: Math. USSR Izvestiya 27(3).Google Scholar
  18. [Artemov, 1986]
    S.N. Artemov. Numerically correct provability logics. Doklady Akad. Nauk SSSR, 290(6):1289–1292, 1986. In Russian. English translation in Soviet Mathematics Doklady 34:384–387, 1987.Google Scholar
  19. [Artemov, 1990]
    S. Artemov. Kolmogorov logic of problems and a provability interpretation of intuitionistic logic. In Theoretical Aspects of Reasoning about Knowledge — III Proceedings, pages 257–272. Morgan Kaufman Pbl., 1990.Google Scholar
  20. [Artemov, 1994]
    S. Artemov. Logic of proofs. Annals of Pure and Applied Logic, 67(1):29–59, 1994.Google Scholar
  21. [Artemov, 1995]
    S. Artemov. Operational modal logic. Technical Report MSI 95-29, Cornell University, 1995.Google Scholar
  22. [Artemov, 1998]
    S. Artemov. Logic of proofs: a unified semantics for modality and λ-terms. Technical Report CFIS 98-06, Cornell University, 1998.Google Scholar
  23. [Artemov, 1999]
    S. Artemov. On explicit reflection in theorem proving and formal verification. In Automated Deduction — CADE-16. Proceedings of the 16th International Conference on Automated Deduction, Trento, Italy, July 1999, volume 1632 of Lecture Notes in Artificial Intelligence, pages 267–281. Springer, 1999.Google Scholar
  24. [Artemov, 2000]
    S. Artemov. Operations on proofs that can be specified by means of modal logic. In Advances in Modal Logic. Volume 2. CSLI Publications, Stanford University, 2000.Google Scholar
  25. [Artemov, 2001]
    S. Artemov. Explicit provability and constructive semantics. Bulletin of Symbolic Logic, 7(1):1–36, 2001.CrossRefGoogle Scholar
  26. [Artemov, 2002]
    S. Artemov. Unified semantics for modality and λ-terms via proof polynomials. In K. Vermeulen and A. Copestake, editors, Algebras, Diagrams and Decisions in Language, Logic and Computation, pages 89–119. CSLI Publications, Stanford University, 2002.Google Scholar
  27. [Artemov, 2004]
    S. Artemov. Kolmogorov and Gödel’s approach to intuitionistic logic: current developments. Russian Mathematical Surveys, 59(2):203–229, 2004.CrossRefGoogle Scholar
  28. [Avigad and Feferman, 1998]
    J. Avigad and S. Feferman. Gödel’s functional (”Dialectica“) interpretation. In S. Buss, editor, Handbook of Proof Theory, pages 337–406. Elsevier, 1998.Google Scholar
  29. [Avron, 1984]
    A. Avron. On modal systems having arithmetical interpretations. The Journal of Symbolic Logic, 49:935–942, 1984.Google Scholar
  30. [Beeson, 1975]
    M. Beeson. The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations. The Journal of Symbolic Logic, 40:321–346, 1975.Google Scholar
  31. [Beeson, 1980]
    M. Beeson. Foundations of Constructive Mathematics. Springer-Verlag, 1980.Google Scholar
  32. [Beklemishev et al., 1999] L.D. Beklemishev, M. Pentus, and N. Vereshchagin. Provability, complexity, grammars. American Mathematical Society Translations, Series 2, 192, 1999.Google Scholar
  33. [Beklemishev, 1989a]
    L.D. Beklemishev. On the classification of propositional provability logics. Izvestiya Akademii Nauk SSSR, ser. mat., 53(5):915–943, 1989. In Russian. English translation in Math.USSR Izvestiya 35 (1990) 247–275.Google Scholar
  34. [Beklemishev, 1989b]
    L.D. Beklemishev. A provability logic without Craig’s interpolation property. Matematicheskie Zametki, 45(6):12–22, 1989. In Russian. English translation in Math. Notes 45 (1989).Google Scholar
  35. [Beklemishev, 1991]
    L.D. Beklemishev. Provability logics for natural Turing progressions of arithmetical theories. Studia Logica, 50(1):107–128, 1991.CrossRefGoogle Scholar
  36. [Beklemishev, 1992]
    L.D. Beklemishev. Independent numerations of theories and recursive progressions. Sibirskii Matematichskii Zhurnal, 33(5):22–46, 1992. In Russian. English translation in Siberian Math. Journal, 33 (1992).).Google Scholar
  37. [Beklemishev, 1994]
    L.D. Beklemishev. On bimodal logics of provability. Annals of Pure and Applied Logic, 68(2):115–160, 1994.CrossRefGoogle Scholar
  38. [Beklemishev, 1996]
    L.D. Beklemishev. Bimodal logics for extensions of arithmetical theories. The Journal of Symbolic Logic, 61(1):91–124, 1996.Google Scholar
  39. [Beklemishev, 1997a]
    L.D. Beklemishev. Induction rules, reflection principles, and provably recursive functions. Annals of Pure and Applied Logic, 85:193–242, 1997.CrossRefGoogle Scholar
  40. [Beklemishev, 1997b]
    L.D. Beklemishev. Notes on local reflection principles. Theoria, 63(3):139–146, 1997.Google Scholar
  41. [Beklemishev, 1997c]
    L.D. Beklemishev. Parameter free induction and reflection. In G. Gottlob, A. Leitsch, and D. Mundici, editors, Lecture Notes in Computer Science 1289. Computational Logic and Proof Theory, 5-th K.Gödel Colloquium KGC’97, Proceedings, pages 103–113. Springer-Verlag, Berlin, 1997.Google Scholar
  42. [Beklemishev, 1998a]
    L.D. Beklemishev. A proof-theoretic analysis of collection. Archive for Mathematical Logic, 37:275–296, 1998.CrossRefGoogle Scholar
  43. [Beklemishev, 1998b]
    L.D. Beklemishev. Reflection principles in formal arithmetic. Doctor of Sciences Dissertation, Steklov Math. Institute, Moscow. In Russian, 1998.Google Scholar
  44. [Beklemishev, 1999a]
    L.D. Beklemishev. Open least element principle and bounded query computation. In J. Flum and M. Rodrigues-Artalejo, editors, Lecture Notes in Computer Science 1683. Computer Science Logic, 13-th international workshop, CSL’99. Madrid, Spain, September 20–25, 1999. Proceedings, pages 389–404. Springer-Verlag, Berlin, 1999.Google Scholar
  45. [Beklemishev, 1999b]
    L.D. Beklemishev. Parameter free induction and provably total computable functions. Theoretical Computer Science, 224(1-2):13–33, 1999.CrossRefGoogle Scholar
  46. [Beklemishev, 2001]
    L.D. Beklemishev. Provability algebras and proof-theoretic ordinals, I. Logic Group Preprint Series 208, University of Utrecht, 2001. http://preprints.phil.uu.nl/lgps/.Google Scholar
  47. [Beklemishev, 2003a]
    L.D. Beklemishev. Proof-theoretic analysis by iterated reflection. Archive for Mathematical Logic, 42:515–552, 2003. DOI: 10.1007/s00153-002-0158-7.CrossRefGoogle Scholar
  48. [Beklemishev, 2003b]
    L.D. Beklemishev. The Worm principle. Logic Group Preprint Series 219, University of Utrecht, 2003. http://preprints.phil.uu.nl/lgps/.Google Scholar
  49. [Beklemishev, 2004]
    L.D. Beklemishev. Provability algebras and proof-theoretic ordinals, I. Annals of Pure and Applied Logic, 128:103–123, 2004.CrossRefGoogle Scholar
  50. [Bellin, 1985]
    G. Bellin. A system of natural deduction for GL. Theoria, 51:89–114, 1985.Google Scholar
  51. [Berarducci and Verbrugge, 1993]
    A. Berarducci and R. Verbrugge. On the provability logic of bounded arithmetic. Annals of Pure and Applied Logic, 61:75–93, 1993.CrossRefGoogle Scholar
  52. [Berarducci, 1990]
    A. Berarducci. The interpretability logic of Peano Arithmetic. The Journal of Symbolic Logic, 55:1059–1089, 1990.Google Scholar
  53. [Bernardi, 1976]
    C. Bernardi. The uniqueness of the fixed point in every diagonalizable algebra. Studia Logica, 35:335–343, 1976.CrossRefGoogle Scholar
  54. [Boolos and McGee, 1987]
    G. Boolos and V. McGee. The degree of the set of sentences of predicate provability logic that are true under every interpretation. The Journal of Symbolic Logic, 52(1):165–171, 1987.Google Scholar
  55. [Boolos and |Sambin, 1991]
    G. Boolos and G. Sambin. Provability: the emergence of a mathematical modality. Studia Logica, 50(1):1–23, 1991.CrossRefGoogle Scholar
  56. [Boolos, 1976]
    G. Boolos. On deciding the truth of certain statements involving the notion of consistency. Journal of Symbolic Logic, 41:779–781, 1976.Google Scholar
  57. [Boolos, 1979a]
    G. Boolos. Reflection principles and iterated consistency assertions. The Journal of Symbolic Logic, 44:33–35, 1979.Google Scholar
  58. [Boolos, 1979b]
    G. Boolos. The Unprovability of Consistency: An Essay in Modal Logic. Cambridge University Press, Cambridge, 1979.Google Scholar
  59. [Boolos, 1980]
    G. Boolos. Omega-consistency and the diamond. Studia Logica, 39:237–243, 1980.CrossRefGoogle Scholar
  60. [Boolos, 1982]
    G. Boolos. Extremely undecidable sentences. The Journal of Symbolic Logic, 47:191–196, 1982.Google Scholar
  61. [Boolos, 1993]
    G. Boolos. The Logic of Provability. Cambridge University Press, Cambridge, 1993.Google Scholar
  62. [Brezhnev, 2000]
    V. Brezhnev. On explicit counterparts of modal logics. Technical Report CFIS 2000-05, Cornell University, 2000.Google Scholar
  63. [Brezhnev, 2001]
    V. Brezhnev. On the logic of proofs. In Proceedings of the Sixth ESSLLI Student Session, Helsinki, pages 35–46, 2001. http://www.helsinki.fi/esslli/.Google Scholar
  64. [Bull and Segerberg, 2001]
    R. Bull and K. Segerberg. Basic modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, 2nd ed., volume 3, pages 1–83. Kluwer, Dordrecht, 2001.Google Scholar
  65. [Burr, 2000]
    W. Burr. Fragments of Heyting arithmetic. The Journal of Symbolic Logic, 65(3):1223–1240, 2000.Google Scholar
  66. [Buss, 1986]
    S. Buss. Bounded arithmetic. Bibliopolis, Napoli, 1986.Google Scholar
  67. [Buss, 1990]
    S. Buss. The modal logic of pure provability. Notre Dame Journal of Formal Logic, 31(2):225–231, 1990.Google Scholar
  68. [Buss, 1998]
    S.R. Buss. Introduction to Proof Theory. In S.R. Buss, editor, Handbook of Proof Theory, pages 1–78, Amsterdam, 1998. Elsevier, North-Holland.Google Scholar
  69. [Carbone and Montagna, 1989]
    A. Carbone and F. Montagna. Rosser orderings in bimodal logics. Zeitschrift f. math. Logik und Grundlagen d. Math., 35:343–358, 1989.Google Scholar
  70. [Carbone and Montagna, 1990]
    A. Carbone and F. Montagna. Much shorter proofs: A bimodal investigation. Zeitschrift f. math. Logik und Grundlagen d. Math., 36:47–66, 1990.Google Scholar
  71. [Carlson, 1986]
    T. Carlson. Modal logics with several operators and provability interpretations. Israel Journal of Mathematics, 54:14–24, 1986.Google Scholar
  72. [Chagrov and Zakharyaschev, 1997]
    A. Chagrov and M. Zakharyaschev. Modal Logic. Oxford Science Publications, 1997.Google Scholar
  73. [Chagrov et al., 2001]
    [Chagrov et al., 2001] A.V. Chagrov, F. Wolter, and M. Zakhariashchev. Advanced modal logic. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, 2nd ed., volume 3, pages 83–266. Kluwer, Dordrecht, 2001.Google Scholar
  74. [Chagrov, 1985]
    A.V. Chagrov. On the complexity of propositional logics. In Complexity problems in Mathematical Logic, pages 80–90. Kalinin State University, Kalinin, 1985. In Russian.Google Scholar
  75. [Constable, 1994]
    Robert L. Constable. Using reflection to explain and enhance type theory. In Helmut Schwichtenberg, editor, Proof and Computation, volume 139 of NATO Advanced Study Institute, International Summer School held in Marktoberdorf, Germany, July 20–August 1, NATO Series F, pages 65–100. Springer, Berlin, 1994.Google Scholar
  76. [Constable, 1998]
    R. Constable. Types in logic, mathematics and programming. In S. Buss, editor, Handbook of Proof Theory, pages 683–786. Elsevier, 1998.Google Scholar
  77. [Cutland, 1980]
    N. Cutland. Computability. An introduction to recursive function theory. Cambridge University Press, Cambridge, etc., 1980.Google Scholar
  78. [Davis and Schwartz, 1979]
    M. Davis and J. Schwartz. Metamathematical extensibility for theorem verifiers and proof checkers. Computers and Mathematics with Applications, 5:217–230, 1979.CrossRefGoogle Scholar
  79. [de Jongh and Japaridze, 1998]
    D. de Jongh and G. Japaridze. The Logic of Provability. In S.R. Buss, editor, Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, Vol.137., pages 475–546. Elsevier, Amsterdam, 1998.Google Scholar
  80. [de Jongh and Montagna, 1989]
    D. de Jongh and F. Montagna. Much shorter proofs. Z. Math. Logik Grundlagen Math., 35(3):247–260, 1989.Google Scholar
  81. [de Jongh and Visser, 1996]
    D. de Jongh and A. Visser. Embeddings of Heyting algebras. In W. et al. Hodges, editor, Logic: from foundations to applications. European Logic Colloquium, Keele, UK, July 20–29, 1993, pages 187–213. Clarendon Press, Oxford, 1996.Google Scholar
  82. [de Jongh, 1970]
    D. de Jongh. The maximality of the intuitionistic predicate calculus with respect to Heyting’s Arithmetic. The Journal of Symbolic Logic, 36:606, 1970.Google Scholar
  83. [Dzhaparidze, 1992]
    G. Dzhaparidze. The logic of linear tolerance. Studia Logica, 51(2):249–277, 1992.CrossRefGoogle Scholar
  84. [Dzhaparidze, 1993]
    G. Dzhaparidze. A generalized notion of weak interpretability and the corresponding modal logic. Annals of Pure and Applied Logic, 61(1-2):113–160, 1993.CrossRefGoogle Scholar
  85. [Fagin et al., 1995] R. Fagin, J. Halpern, Y. Moses, and M. Vardi. Reasoning About Knowledge. MIT Press, 1995.Google Scholar
  86. [Feferman, 1960]
    S. Feferman. Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae, 49:35–92, 1960.Google Scholar
  87. [Feferman, 1962]
    S. Feferman. Transfinite recursive progressions of axiomatic theories. The Journal of Symbolic Logic, 27:259–316, 1962.Google Scholar
  88. [Fitting, 2003a]
    M. Fitting. A semantic proof of the realizability of modal logic in the logic of proofs. Technical Report TR-2003010, CUNY Ph.D. Program in Computer Science, 2003.Google Scholar
  89. [Fitting, 2003b]
    M. Fitting. A semantics for the logic of proofs. Technical Report TR-2003012, CUNY Ph.D. Program in Computer Science, 2003.Google Scholar
  90. [Fitting, 2004]
    M. Fitting. Semantics and tableaus for LPS4. Technical Report TR-2004016, CUNY Ph.D. Program in Computer Science, 2004.Google Scholar
  91. [Fitting, 2005]
    M. Fitting. The logic of proofs, semantically. To appear in Annals of Pure and Applied Logic. Available on http://comet.lehman.cuny.edu/fitting., 2005.Google Scholar
  92. [Friedman, 1975a]
    H. Friedman. 102 problems in mathematical logic. The Journal of Symbolic Logic, 40:113–129, 1975.Google Scholar
  93. [Friedman, 1975b]
    H. Friedman. The disjunction property implies the numerical existence property. Proc. Nat. Acad. USA, 72:2877–2878, 1975.Google Scholar
  94. [Friedman, 1975c]
    H. Friedman. Some applications of Kleene’s methods for intuitionistic systems. In A.R.D. Mathias and H. Rogers, editors, Cambridge Summerschool in Mathematical Logic, pages 113–170. Springer-Verlag, Berlin, 1975.Google Scholar
  95. [Gabbay, 1996]
    D.M. Gabbay. Labelled Deductive Systems. Oxford University Press, 1996.Google Scholar
  96. [Gavrilenko, 1981]
    Yu.V. Gavrilenko. Recursive realizability from the intuitionistic point of view. Soviet Math. Doklady, 23:9–14, 1981.Google Scholar
  97. [Gentzen, 1936]
    G. Gentzen. Die Wiederspruchsfreiheit der reinen Zahlentheorie. Math. Ann., 112:493–565, 1936.CrossRefGoogle Scholar
  98. [Gentzen, 1938]
    G. Gentzen. Neue Fassung des Wiederspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik ung Grundlegung der exakten Wissenschaften, 4:19–44, 1938.Google Scholar
  99. [Ghilardi and Zawadowski, 1995]
    S. Ghilardi and M. Zawadowski. A sheaf representation and duality for finitely presented Heyting algebras. The Journal of Symbolic Logic, 60:911–939, 1995.Google Scholar
  100. [Ghilardi, 1999]
    S. Ghilardi. Unification in intuitionistic logic. The Journal of Symbolic Logic, 64:859–880, 1999.Google Scholar
  101. [Ghilardi, 2000]
    S. Ghilardi. Best solving modal equations. Annals of Pure and Applied Logic, 102:183–198, 2000.CrossRefGoogle Scholar
  102. [Girard et al., 1989] J.-Y. Girard, Y. Lafont, and P. Taylor. Proofs and Types. Cambridge University Press, 1989.Google Scholar
  103. [Gödel, 1933]
    K. Gödel. Eine Interpretation des intuitionistischen Aussagenkalkuls. Ergebnisse Math. Kolloq., 4:39–40, 1933. English translation in: S. Feferman et al., editors, Kurt Gödel Collected Works, Vol. 1, pages 301–303. Oxford University Press, Oxford, Clarendon Press, New York, 1986.Google Scholar
  104. [Gödel, 1995]
    K. Gödel. Vortrag bei Zilsel, 1938. In S. Feferman, editor, Kurt Gödel Collected Works. Volume III, pages 86–113. Oxford University Press, 1995.Google Scholar
  105. [Goldblatt, 1978]
    R. Goldblatt. Arithmetical necessity, provability and intuitionistic logic. Theoria, 44:38–46, 1978.Google Scholar
  106. [Goncharov, 1997]
    S.S. Goncharov. Countable Boolean algebras and decidability, Siberian School of Algebra and Logic. Plenum Press, New York, 1997. Russian original: Schetnye bulevy algebry i razreshimost’. Sibirskaya Shkola Algebry i Logiki. Novosibirsk: Nauchnaya Kniga. xii, 362 p. (1996).Google Scholar
  107. [Goodman, 1970]
    N.D. Goodman. A theory of constructions is equivalent to arithmetic. In J. Myhill, A. Kino, and R.E. Vesley, editors, Intuitionism and Proof Theory, pages 101–120. North-Holland, 1970.Google Scholar
  108. [Goryachev, 1986]
    S. Goryachev. On interpretability of some extensions of arithmetic. Mat. Zametki, 40:561–572, 1986. In Russian. English translation in Math. Notes, 40.Google Scholar
  109. [Grigolia, 1987]
    R.Sh. Grigolia. Free algebras of non-classical logics. Metzniereba, Tbilissi, 1987. In Russian.Google Scholar
  110. [Grzegorczyk, 1953]
    A. Grzegorczyk. Some classes of recursive functions. In Rozprawy Matematiczne, IV. Warszawa, 1953.Google Scholar
  111. [Guaspari and Solovay, 1979]
    D. Guaspari and R. Solovay. Rosser sentences. Annals of Math. Logic, 16:81–99, 1979.Google Scholar
  112. [Guaspari, 1979]
    D. Guaspari. Partially conservative sentences and interpretability. Transactions of AMS, 254:47–68, 1979.Google Scholar
  113. [Hájek and Montagna, 1990]
    P. Hájek and F. Montagna. The logic of Π1-conservativity. Archive for Mathematical Logic, 30(2):113–123, 1990.Google Scholar
  114. [Hájek and Montagna, 1992]
    P. Hájek and F. Montagna. The logic of Π1-conservativity continued. Archive for Mathematical Logic, 32:57–63, 1992.Google Scholar
  115. [Hájek and Pudlák, 1993]
    P. Hájek and P. Pudlák. Metamathematics of First Order Arithmetic. Springer-Verlag, Berlin, Heidelberg, New York, 1993.Google Scholar
  116. [Harrison, 1995]
    J. Harrison. Methatheory and reflection in theorem proving: A survey and critique. Technical report, University of Cambridge, 1995. URL http://www.dcs.glasgow.ac.uk/ tfm/hol-bib.html#H.Google Scholar
  117. [Heyting, 1931]
    A. Heyting. Die intuitionistische grundlegung der mathematik. Erkenntnis, 2:106–115, 1931.CrossRefGoogle Scholar
  118. [Heyting, 1934]
    A. Heyting. Mathematische Grundlagenforschung. Intuitionismus. Beweistheorie. Springer, Berlin, 1934.Google Scholar
  119. [Hilbert and Bernays, 1968]
    D. Hilbert and P. Bernays. Grundlagen der Mathematik, Vols. I and II, 2d ed. Springer-Verlag, Berlin, 1968.Google Scholar
  120. [Iemhoff, 2001a]
    R. Iemhoff. A modal analysis of some principles of the provability logic of Heyting arithmetic. In Advances in modal logic. Vol. 2. Selected papers from the 2nd international workshop (AiML’ 98), Uppsala, Sweden, October 16–18, 1998. CSLI Lecture Notes 119, pages 301–336. CSLI Publications, Stanford, 2001.Google Scholar
  121. [Iemhoff, 2001b]
    R. Iemhoff. On the admissible rules of intuitionistic propositional logic. The Journal of Symbolic Logic, 66(1):281–294, 2001.Google Scholar
  122. [Iemhoff, 2001c]
    R. Iemhoff. Provability logic and admissible rules. PhD thesis, University of Amsterdam, Amsterdam, 2001.Google Scholar
  123. [Ignatiev, 1991]
    K.N. Ignatiev. Partial conservativity and modal logics. ITLI Prepublication Series X-91-C04, University of Amsterdam, 1991.Google Scholar
  124. [Ignatiev, 1993a]
    K.N. Ignatiev. On strong provability predicates and the associated modal logics. The Journal of Symbolic Logic, 58:249–290, 1993.Google Scholar
  125. [Ignatiev, 1993b]
    K.N. Ignatiev. The provability logic for Σ1-interpolability. Annals of Pure and Applied Logic, 64:1–25, 1993.CrossRefGoogle Scholar
  126. [Japaridze, 1986]
    G.K. Japaridze. The modal logical means of investigation of provability. Thesis in Philosophy, in Russian, Moscow, 1986.Google Scholar
  127. [Japaridze, 1988]
    G.K. Japaridze. The polymodal logic of provability. In Intensional Logics and Logical Structure of Theories: Material from the fourth Soviet-Finnish Symposium on Logic, Telavi, May 20–24, 1985, pages 16–48. Metsniereba, Tbilisi, 1988. In Russian.Google Scholar
  128. [Japaridze, 1994]
    G. Japaridze. A simple proof of arithmetical completeness for Π1-conservativity logic. Notre Dame Journal of Formal Logic, 35:346–354, 1994.CrossRefGoogle Scholar
  129. [Kaye et al., 1988]
    [Kaye et al., 1988] R. Kaye, J. Paris, and C. Dimitracopoulos. On parameter free induction schemas. The Journal of Symbolic Logic, 53(4):1082–1097, 1988.Google Scholar
  130. [Kirby and Paris, 1982]
    L.A.S. Kirby and J.B. Paris. Accessible independence results for Peano arithmetic. Bull. London Math. Soc., 14:285–293, 1982.Google Scholar
  131. [Kleene, 1945]
    S. Kleene. On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, 10(4):109–124, 1945.Google Scholar
  132. [Kleene, 1952]
    S. Kleene. Introduction to Metamathematics. Van Norstrand, 1952.Google Scholar
  133. [Kolmogoro, 1932]
    A. Kolmogoro. Zur Deutung der intuitionistischen logik. Mathematische Zeitschrift, 35:58–65, 1932. In German. English translation in V.M. Tikhomirov, editor, Selected works of A.N. Kolmogorov. Volume I: Mathematics and Mechanics, pages 151–158. Kluwer, Dordrecht 1991.Google Scholar
  134. [Kolmogorov, 1985]
    A.N. Kolmogorov. About my papers on intuitionistic logic. In V.M. Tikhomirov, editor, Selected works of A.N. Kolmogorov. Volume I: Mathematics and Mechanics, page 393. Nauka, Moscow, 1985. In Russian, English translation in V.M. Tikhomirov, editor, Selected works of A.N. Kolmogorov. Volume I: Mathematics and Mechanics, pages 451–452. Kluwer, Dordrecht 1991.Google Scholar
  135. [Kozen and Tiuryn, 1990]
    D. Kozen and J. Tiuryn. Logic of programs. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. Volume B, Formal Models and Semantics, pages 789–840. Elsevier, 1990.Google Scholar
  136. [Kreisel and Lévy, 1968]
    G. Kreisel and A. Lévy. Reflection principles and their use for establishing the complexity of axiomatic systems. Zeitschrift f. math. Logik und Grundlagen d. Math., 14:97–142, 1968.Google Scholar
  137. [Kreisel, 1952]
    G. Kreisel. On the interpretation of non-finitist proofs, II. The Journal of Symbolic Logic, 17:43–58, 1952.Google Scholar
  138. [Kreisel, 1962a]
    G. Kreisel. Foundations of intuitionistic logic. In E. Nagel, P. Suppes, and A. Tarski, editors, Logic, Methodology and Philosophy of Science. Proceedings of the 1960 International Congress, pages 198–210. Stanford University Press, 1962.Google Scholar
  139. [Kreisel, 1962b]
    G. Kreisel. On weak completeness of intuitionistic predicate logic. The Journal of Symbolic Logic, 27:139–158, 1962.Google Scholar
  140. [Kreisel, 1965]
    G. Kreisel. Mathematical logic. In T.L. Saaty, editor, Lectures in Modern Mathematics III, pages 95–195. Wiley and Sons, New York, 1965.Google Scholar
  141. [Kripke, 1963]
    S. Kripke. Semantical considerations on modal logic. Acta Philosophica Fennica, 16:83–94, 1963.Google Scholar
  142. [Krupski, 1997]
    V. Krupski. Operational logic of proofs with functionality condition on proof predicate. In S. Adian and A. Nerode, editors, Logical Foundations of Computer Science’ 97, Yaroslavl’, volume 1234 of Lecture Notes in Computer Science, pages 167–177. Springer, 1997.Google Scholar
  143. [Krupski, 2002]
    V. Krupski. The single-conclusion proof logic and inference rules specification. Annals of Pure and Applied Logic, 113(1-3):181–201, 2002.Google Scholar
  144. [Krupski, 2005]
    V. Krupski. Referential logic of proofs. Theoretical Computer Science, (accepted), 2005.Google Scholar
  145. [Krupski (jr.), 2003]
    N.V. Krupski (jr.). On the complexity of the reflected logic of proofs. Technical Report TR-2003007, CUNY Ph. D. Program in Computer Science, 2003.Google Scholar
  146. [Kuznets, 2000]
    R. Kuznets. On the complexity of explicit modal logics. In Computer Science Logic 2000, volume 1862 of Lecture Notes in Computer Science, pages 371–383. Springer-Verlag, 2000.Google Scholar
  147. [Kuznetsov and Muravitsky, 1976]
    A. Kuznetsov and A. Muravitsky. The logic of provability. In Abstracts of the 4-th All-Union Conference on Mathematical Logic, page 73, Kishinev, 1976. In Russian.Google Scholar
  148. [Kuznetsov and Muravitsky, 1977]
    A.V. Kuznetsov and A.Yu. Muravitsky. Magari algebras. In Fourteenth All-Union Algebra Conf., Abstract part 2: Rings, Algebraic Structures, pages 105–106, 1977. In Russian.Google Scholar
  149. [Kuznetsov and Muravitsky, 1986]
    A.V. Kuznetsov and A.Yu. Muravitsky. On superintuitionistic logics as fragments of proof logic. Studia Logica, XLV:76–99, 1986.Google Scholar
  150. [Läuchli, 1970]
    H. Läuchli. An abstract notion of realizability for which intuitionistic predicate logic is complete. In J. Myhill, A. Kino, and R.E. Vesley, editors, Intuitionism and Proof Theory, pages 227–234. North-Holland, 1970.Google Scholar
  151. [Leivant, 1981]
    D. Leivant. On the proof theory of the modal logic for arithmetic provability. The Journal of Symbolic Logic, 46:531–538, 1981.Google Scholar
  152. [Leivant, 1983]
    D. Leivant. The optimality of induction as an axiomatization of arithmetic. The Journal of Symbolic Logic, 48:182–184, 1983.Google Scholar
  153. [Lemmon, 1957]
    E. Lemmon. New foundations for Lewis’s modal systems. The Journal of Symbolic Logic, 22:176–186, 1957.Google Scholar
  154. [Lindström, 1984]
    P. Lindström. On partially conservative sentences and interpretability. Proceedings of the AMS, 91(3):436–443, 1984.Google Scholar
  155. [Lindström, 1994]
    P. Lindström. The modal logic of Parikh provability. Tech. Rep. Filosofiska Meddelanden, Gröna Serien 5, Univ. Göteborg, 1994.Google Scholar
  156. [Lindström, 1996]
    P. Lindström. Provability logic — a short introduction. Theoria, 62(1-2):19–61, 1996.Google Scholar
  157. [Löb, 1955]
    M.H. Löb. Solution of a problem of Leon Henkin. The Journal of Symbolic Logic, 20:115–118, 1955.Google Scholar
  158. [Magari, 1975a]
    R. Magari. The diagonalizable algebras (the algebraization of the theories which express Theor.: II). Bollettino della Unione Matematica Italiana, Serie 4, 12, 1975. Suppl. fasc. 3, 117–125.Google Scholar
  159. [Magari, 1975b]
    R. Magari. Representation and duality theory for diagonalizable algebras (the algebraization of theories which express Theor.:IV). Studia Logica, 34:305–313, 1975.CrossRefGoogle Scholar
  160. [McCarthy, 2004]
    J. McCarthy. Notes on self-awareness. Internet posting by URL http://www-formal.stanford.edu/jmc/selfaware/selfaware.html, April 2004.Google Scholar
  161. [McKinsey and Tarski, 1946]
    J.C.C. McKinsey and A. Tarski. On closed elements of closure algebras. Annals of Mathematics, 47:122–162, 1946.Google Scholar
  162. [McKinsey and Tarski, 1948]
    J.C.C. McKinsey and A. Tarski. Some theorems about the sentential calculi of Lewis and Heyting. The Journal of Symbolic Logic, 13:1–15, 1948.Google Scholar
  163. [Medvedev, 1962]
    Yu. Medvedev. Finite problems. Soviet Mathematics Doklady, 3:227–230, 1962.Google Scholar
  164. [Meyer, 1975]
    A.R. Meyer. The inherent complexity of theories of ordered sets. In Proceedings of the international congress of math., Vancouver, 1974, pages 477–482. Canadian Math. Congress, 1975.Google Scholar
  165. [Mints, 1971]
    G.E. Mints. Quantifier free and one quantifier systems. Zapiski nauchnyh seminarov LOMI, 20:115–133, 1971. In Russian.Google Scholar
  166. [Mints, 1974]
    G. Mints. Lewis’ systems and system T (a survey 1965–1973). In Feys. Modal Logic (Russian translation), pages 422–509. Nauka, Moscow, 1974. In Russian, English translation in G. Mints, Selected papers in proof theory, Bibliopolis, Napoli, 1992.Google Scholar
  167. [Mkrtychev, 1997]
    A. Mkrtychev. Models for the logic of proofs. In S. Adian and A. Nerode, editors, Logical Foundations of Computer Science’ 97, Yaroslavl’, volume 1234 of Lecture Notes in Computer Science, pages 266–275. Springer, 1997.Google Scholar
  168. [Montagna, 1978]
    F. Montagna. On the algebraization of a Feferman’s predicate (the algebraiztion of theories which express Theor; X). Studia Logica, 37:221–236, 1978.CrossRefGoogle Scholar
  169. [Montagna, 1979]
    F. Montagna. On the diagonalizable algebra of Peano arithmetic. Bollettino della Unione Matematica Italiana, B (5), 16:795–812, 1979.Google Scholar
  170. [Montagna, 1980]
    F. Montagna. Undecidability of the first order theory of diagonalizable algebras. Studia Logica, 39:347–354, 1980.Google Scholar
  171. [Montagna, 1987a]
    F. Montagna. The predicate modal logic of provability. Notre Dame Journal of Formal Logic, 25:179–189, 1987.Google Scholar
  172. [Montagna, 1987b]
    F. Montagna. Provability in. nite subtheories of PA. The Journal of Symbolic Logic, 52(2):494–511, 1987.Google Scholar
  173. [Montague, 1963]
    R. Montague. Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability. Acta Philosophica Fennica, 16:153–168, 1963.Google Scholar
  174. [Moses, 1988]
    Y. Moses. Resource-bounded knowledge. In M. Vardi, editor, Theoretical Aspects of Reasoning about Knowledge, pages 261–276. Morgan Kaufman Pbl., 1988.Google Scholar
  175. [Mostowski, 1953]
    A. Mostowski. On models of axiomatic systems. Fundamenta Mathematicae, 39:133–158, 1953.Google Scholar
  176. [Myhill, 1960]
    J. Myhill. Some remarks on the notion of proof. Journal of Philosophy, 57:461–471, 1960.Google Scholar
  177. [Myhill, 1985]
    J. Myhill. Intensional set theory. In S. Shapiro, editor, Intensional Mathematics, pages 47–61. North-Holland, 1985.Google Scholar
  178. [Nogina, 1994]
    E. Nogina. Logic of proofs with the strong provability operator. Technical Report ILLC Prepublication Series ML-94-10, Institute for Logic, Language and Computation, University of Amsterdam, 1994.Google Scholar
  179. [Nogina, 1996]
    E. Nogina. Grzegorczyk logic with arithmetical proof operators. Fundamental and Applied Mathematics, 2(2):483–499, 1996. In Russian, URL http://mech.math.msu.su/∼fpm/eng/96/962/96206.htm.Google Scholar
  180. [Ono, 1987]
    H. Ono. Reflection principles in fragments of Peano Arithmetic. Zeitschrift f. math. Logik und Grundlagen d. Math., 33(4):317–333, 1987.Google Scholar
  181. [Orevkov, 1979]
    V.P. Orevkov. Lower bounds for lengthening of proofs after cut-elimination. Zapiski Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova, 88:137–162, 1979. In Russian. English translation in: Journal of Soviet Mathematics 20, 2337–2350 (1982).Google Scholar
  182. [Orlov, 1928]
    I.E. Orlov. The calculus of compatibility of propositions. Matematicheskii Sbornik, 35:263–286, 1928. In Russian.Google Scholar
  183. [Parikh, 1971]
    R. Parikh. Existence and feasibility in arithmetic. The Journal of Symbolic Logic, 36:494–508, 1971.Google Scholar
  184. [Parikh, 1987]
    R. Parikh. Knowledge and the problem of logical omniscience. In Z. Ras and M. Zemankova, editors, ISMIS-87 (International Symposium on Methodolody for Intellectual Systems), pages 432–439. North-Holland, 1987.Google Scholar
  185. [Parikh, 1995]
    R. Parikh. Logical omniscience. In D. Leivant, editor, Logic and Computational Complexity, pages 22–29. Springer Springer Lecture Notes in Computer Science No. 960, 1995.Google Scholar
  186. [Parsons, 1970]
    C. Parsons. On a number-theoretic choice schema and its relation to induction. In A. Kino, J. Myhill, and R.E. Vessley, editors, Intuitionism and Proof Theory, pages 459–473. North Holland, Amsterdam, 1970.Google Scholar
  187. [Parsons, 1972]
    C. Parsons. On n-quantifier induction. The Journal of Symbolic Logic, 37(3):466–482, 1972.Google Scholar
  188. [Pitts, 1992]
    A. Pitts. On an interpretation of second-order quantification in first order intuitionistic propositional logic. The Journal of Symbolic Logic, 57:33–52, 1992.Google Scholar
  189. [Plisko, 1977]
    V. Plisko. The nonarithmeticity of the class of realizable predicate formulas. Soviet Mathematics Izvestia, 11:453–471, 1977.Google Scholar
  190. [Pohlers, 1998]
    W. Pohlers. Subsystems of set theory and second order number theory. In S.R. Buss, editor, Handbook of Proof Theory, pages 210–335. Elsevier, North-Holland, Amsterdam, 1998.Google Scholar
  191. [Pour-El and Kripke, 1967]
    M.B. Pour-El and S. Kripke. Deduction-preserving “recursive isomorphisms” between theories. Fundamenta Mathematicae, 61:141–163, 1967.Google Scholar
  192. [Rabin, 1961]
    M.O. Rabin. Non-standard models and independence of the induction axiom. In Essays on the Foundations of Mathematics: Dedicated to A. Fraenkel on his 70th anniversary, pages 287–299. North-Holland, Amsterdam, 1961.Google Scholar
  193. [Rasiowa and Sikorski, 1963]
    H. Rasiowa and R. Sikorski. The Mathematics of Metamathematics. Polish Scientific Publishers, 1963.Google Scholar
  194. [Ratajczyk, 1989]
    Z. Ratajczyk. Functions provably total in I Σn. Fundamenta Mathematicae, 133:81–95, 1989.Google Scholar
  195. [Rathjen, 1994]
    M. Rathjen. Proof theory of reflection. Annals of Pure and Applied Logic, 68(2):181–224, 1994.CrossRefGoogle Scholar
  196. [Rathjen, 1999]
    M. Rathjen. The realm of ordinal analysis. In S.B. Cooper and J.K. Truss, editors, Sets and proofs. London Math. Soc. Lect. Note Series 258, pages 219–279. Cambridge University Press, Cambridge, 1999.Google Scholar
  197. [Renne, 2004]
    B. Renne. Tableaux for the logic of proofs. Technical Report TR-2004001, CUNY Ph.D. Program in Computer Science, 2004.Google Scholar
  198. [Rose, 1984]
    H.E. Rose. Subrecursion: Functions and Hierarchies. Clarendon Press, Oxford, 1984.Google Scholar
  199. [Rosser, 1936]
    J.B. Rosser. Extensions of some theorems of Gödel and Church. The Journal of Symbolic Logic, 1:87–91, 1936.Google Scholar
  200. [Rybakov, 1984]
    V.V. Rybakov. A criterion for admissibility of rules in the modal system S4 and intuitionistic logic. Algebra and Logic, 23:369–384, 1984.Google Scholar
  201. [Rybakov, 1989]
    V.V. Rybakov. On admissibility of the inference rules in the modal system G. In Yu.L. Ershov, editor, Trudy instituta matematiki, volume 12, pages 120–138. Nauka, Novosibirsk, 1989. In Russian.Google Scholar
  202. [Rybakov, 1997]
    V.V. Rybakov. Admissibility of logical inference rules. Elsevier, Amsterdam, 1997.Google Scholar
  203. [Ryll-Nardzewski, 1953]
    C. Ryll-Nardzewski. The role of the axiom of induction in elementary arithmetic. Fundamenta Mathematicae, 39:239–263, 1953.Google Scholar
  204. [Sambin and Valentini, 1982]
    G. Sambin and S. Valentini. The modal logic of provability, the sequential approach. Journal of Philosophic Logic, 11:311–342, 1982.Google Scholar
  205. [Sambin and Valentini, 1983]
    G. Sambin and S. Valentini. The modal logic of provability: cut-elimination. Journal of Philosophic Logic, 12:471–476, 1983.Google Scholar
  206. [Schmerl, 1979]
    U.R. Schmerl. A fine structure generated by reflection formulas over Primitive Recursive Arithmetic. In M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium’78, pages 335–350. North Holland, Amsterdam, 1979.Google Scholar
  207. [Segerberg, 1971]
    K. Segerberg. An essay in classical modal logic. Filosofiska Föreningen och Filosofiska Insitutionen vid Uppsala Universitet, Uppsala, 1971.Google Scholar
  208. [Shapiro, 1985a]
    S. Shapiro. Epistemic and intuitionistic arithmetic. In S. Shapiro, editor, Intensional Mathematics, pages 11–46. North-Holland, 1985.Google Scholar
  209. [Shapiro, 1985b]
    S. Shapiro. Intensional mathematics and constructive mathematics. In S. Shapiro, editor, Intensional Mathematics, pages 1–10. North-Holland, 1985.Google Scholar
  210. [Shavrukov, 1988]
    V.Yu. Shavrukov. The logic of relative interpretability over Peano arithmetic. Preprint, Steklov Mathematical Institute, Moscow, 1988. In Russian.Google Scholar
  211. [Shavrukov, 1991]
    V.Yu. Shavrukov. On Rosser’s provability predicate. Zeitschrift f. math. Logik und Grundlagen d. Math., 37:317–330, 1991.Google Scholar
  212. [Shavrukov, 1993a]
    V.Yu. Shavrukov. A note on the diagonalizable algebras of PA and ZF. Annals of Pure and Applied Logic, 61:161–173, 1993.CrossRefGoogle Scholar
  213. [Shavrukov, 1993b]
    V.Yu. Shavrukov. Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes Mathematicae, 323, 1993.Google Scholar
  214. [Shavrukov, 1994]
    V.Yu. Shavrukov. A smart child of Peano’s. Notre Dame Journal of Formal Logic, 35:161–185, 1994.Google Scholar
  215. [Shavrukov, 1997a]
    V.Yu. Shavrukov. Isomorphisms of diagonalizable algebras. Theoria, 63(3):210–221, 1997.Google Scholar
  216. [Shavrukov, 1997b]
    V.Yu. Shavrukov. Undecidability in diagonalizable algebras. The Journal of Symbolic Logic, 62(1):79–116, 1997.Google Scholar
  217. [Sidon, 1997]
    T. Sidon. Provability logic with operations on proofs. In S. Adian and A. Nerode, editors, Logical Foundations of Computer Science’ 97, Yaroslavl’, volume 1234 of Lecture Notes in Computer Science, pages 342–353. Springer, 1997.Google Scholar
  218. [Sidon, 1998]
    T.L. Sidon. Craig interpolation property for operational logics of proofs. Vestnik Moskovskogo Universiteta. Ser. 1 Mat., Mech., (2):34–38, 1998. In Russian. English translation in: Moscow University Mathematics Bulletin, v.53, n.2, pp.37–41, 1999.Google Scholar
  219. [Simmons, 1988]
    H. Simmons. Large discrete parts of the E-tree. The Journal of Symbolic Logic, 53:980–984, 1988.Google Scholar
  220. [Smiley, 1963]
    T. Smiley. The logical basis of ethics. Acta Philosophica Fennica, 16:237–246, 1963.Google Scholar
  221. [Smoryński, 1973]
    C. Smoryński. Applications of Kripke models. In A. Troelstra, editor, Metamathematical investigations of intuitionistic arithmetic and analysis. Springer Lecture Notes 344, pages 324–391. Springer, Berlin, 1973.Google Scholar
  222. [Smoryński, 1977a]
    C. Smoryński. ω-consistency and reflection. In Colloque International de Logique (Colloq. Int. CNRS), pages 167–181. CNRS Inst. B. Pascal, Paris, 1977.Google Scholar
  223. [Smoryński, 1977b]
    C. Smoryński. The incompleteness theorems. In J. Barwise, editor, Handbook of Mathematical Logic, pages 821–865. North Holland, Amsterdam, 1977.Google Scholar
  224. [Smoryński, 1978]
    C. Smoryński. Beth’s theorem and self-referential sentences. In A. Macintyre et al., editor, Logic Colloquium’77. North Holland, Amsterdam, 1978.Google Scholar
  225. [Smoryński, 1981]
    C. Smoryński. Fifty years of self-reference. Notre Dame Journal of Formal Logic, 22:357–374, 1981.Google Scholar
  226. [Smoryński, 1982]
    C. Smoryński. The finite inseparability of the first order theory of diagonalizable algebras. Studia Logica, 41:347–349, 1982.Google Scholar
  227. [Smoryński, 1985]
    C. Smoryński. Self-Reference and Modal Logic. Springer-Verlag, Berlin, 1985.Google Scholar
  228. [Smoryński, 1989]
    C. Smoryński. Arithmetical analogues of McAloon’s unique Rosser sentences. Archive for Mathematical Logic, 28:1–21, 1989.Google Scholar
  229. [Smoryński, 2004]
    C. Smoryński. Modal logic and self-reference. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, 2nd ed., volume 11, pages 1–53. Springer, Berlin, 2004.Google Scholar
  230. [Solovay, 1976]
    R.M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 28:33–71, 1976.Google Scholar
  231. [Statman, 1978]
    R. Statman. Bounds for proof-search and speed-up in the predicate calculus. Annals of Mathematical Logic, 15:225–287, 1978.CrossRefGoogle Scholar
  232. [Takeuti, 1975]
    G. Takeuti. Proof Theory. North-Holland, 1975.Google Scholar
  233. [Troelstra and Schwichtenberg, 1996]
    A. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge University Press, Amsterdam, 1996.Google Scholar
  234. [Troelstra and van Dalen, 1988]
    A. Troelstra and D. van Dalen. Constructivism in Mathematics, vols 1, 2. North-Holland, Amsterdam, 1988.Google Scholar
  235. [Troelstra, 1973]
    A. Troelstra. Metamathematical investigations of intuitionistic arithmetic and analysis. Springer Lecture Notes 344. Springer-Verlag, Berlin, 1973.Google Scholar
  236. [Troelstra, 1998]
    A.S. Troelstra. Realizability. In S. Buss, editor, Handbook of Proof Theory, pages 407–474. Elsevier, 1998.Google Scholar
  237. [Turing, 1939]
    A.M. Turing. System of logics based on ordinals. Proc. London Math. Soc., ser. 2, 45:161–228, 1939.Google Scholar
  238. [Uspensky and Plisko, 1985]
    V. Uspensky and V. Plisko. Intuitionistic Logic. Commentary on [Kolmogoroff, 1932] and [Kolmogorov, 1985]. In V.M. Tikhomirov, editor, Selected works of A.N. Kolmogorov. Volume I: Mathematics and Mechanics, pages 394–404. Nauka, Moscow, 1985. In Russian, English translation in V.M. Tikhomirov, editor, Selected works of A.N. Kolmogorov. Volume I: Mathematics and Mechanics, pages 452–466. Kluwer, Dordrecht 1991.Google Scholar
  239. [Uspensky, 1992]
    V.A. Uspensky. Kolmogorov and mathematical logic. Journal of Symbolic Logic, 57(2):385–412, 1992.Google Scholar
  240. [van Benthem, 1991]
    J. van Benthem. Reflections on epistemic logic. Logique & Analyse, 133-134:5–14, 1991.Google Scholar
  241. [van Benthem, 2001]
    J. van Benthem. Correspondence theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, 2nd ed., volume 3, pages 325–408. Kluwer, Dordrecht, 2001.Google Scholar
  242. [van Dalen, 1986]
    D. van Dalen. Intuitionistic logic. In D. Gabbay and F. Guenther, editors, Handbook of Philosophical Logic. Volume 3, pages 225–340. Reidel, 1986.Google Scholar
  243. [van Dalen, 1994]
    D. van Dalen. Logic and Structure. Springer-Verlag, 1994.Google Scholar
  244. [Vardanyan, 1986]
    V.A. Vardanyan. Arithmetic comlexity of predicate logics of provability and their fragments. Doklady Akad. Nauk SSSR, 288(1):11–14, 1986. In Russian. English translation in Soviet Mathematics Doklady 33:569–572, 1986.Google Scholar
  245. [Visser et al., 1995]
    [Visser et al., 1995] A. Visser, J. van Benthem, D. de Jongh, and G. Renardel de Lavalette. NNIL, a study in intuitionistic propositional logic. In A. Ponse, M. de Rijke, and Y. Venema, editors, Modal Logic and Process Algebra, a bisimulation perspective. CSLI Lecture Notes, 53, pages 289–326. CSLI Publications, Stanford, 1995.Google Scholar
  246. [Visser, 1980]
    A. Visser. Numerations, λ-calculus and arithmetic. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry. Essays on combinatory logic, lambda-calculus and formalism, pages 259–284. Academic Press, London, 1980.Google Scholar
  247. [Visser, 1981]
    A. Visser. Aspects of Diagonalization and Provability. PhD thesis, University of Utrecht, Utrecht, The Netherlands, 1981.Google Scholar
  248. [Visser, 1982]
    A. Visser. On the completeness principle. Annals of Mathematical Logic, 22:263–295, 1982.CrossRefGoogle Scholar
  249. [Visser, 1984]
    A. Visser. The provability logics of recursively enumerable theories extending Peano Arithmetic at arbitrary theories extending Peano Arithmetic. Journal of Philosophic Logic, 13:97–113, 1984.Google Scholar
  250. [Visser, 1985]
    A. Visser. Evaluation, provably deductive equivalence in Heyting arithmetic of substitution instances of propositional formulas. Logic Group Preprint Series 4, Department of Philosophy, University of Utrecht, 1985.Google Scholar
  251. [Visser, 1989]
    A. Visser. Peano’s smart children. A provability logical study of systems with built-in consistency. Notre Dame Journal of Formal Logic, 30:161–196, 1989.Google Scholar
  252. [Visser, 1990]
    A. Visser. Interpretability logic. In P.P. Petkov, editor, Mathematical Logic, pages 175–208. Plenum Press, New York, 1990.Google Scholar
  253. [Visser, 1991]
    A. Visser. The formalization of interpretability. Studia Logica, 50(1):81–106, 1991.CrossRefGoogle Scholar
  254. [Visser, 1992]
    A. Visser. An inside view of EXP. the closed fragment of the provability logic of IΔ0 + Ω1 with a propositional constant for EXP. The Journal of Symbolic Logic, 57(1):131–165, 1992.Google Scholar
  255. [Visser, 1994]
    A. Visser. Propositional combinations of Σ1-sentences in Heyting’s arithmetic. Logic Group Preprint Series 117, Department of Philosophy, University of Utrecht, 1994.Google Scholar
  256. [Visser, 1995]
    A. Visser. A course in bimodal provability logic. Annals of Pure and Applied Logic, 73:109–142, 1995.CrossRefGoogle Scholar
  257. [Visser, 1996]
    A. Visser. Uniform interpolation and layered bisimulation. In P. Hájek, editor, Lecture Notes in Logic 6. Logical foundations of Mathematics, Computer Science and Physics — Kurt Gödel’s Legacy, Gödel’ 96, Brno, Chech Republic, Proceedings, pages 139–164. Springer-Verlag, Berlin, 1996.Google Scholar
  258. [Visser, 1998]
    A. Visser. An overview of interpretability logic. In M. Kracht, M. de Rijke, H. Wansing, and M. Zakhariaschev, editors, Advances in Modal Logic, v.1, CSLI Lecture Notes, No. 87, pages 307–359. CSLI Publications, Stanford, 1998.Google Scholar
  259. [Visser, 1999]
    A. Visser. Rules and arithmetics. Notre Dame Journal of Formal Logic, 40(1):116–140, 1999.Google Scholar
  260. [Visser, 2002a]
    A. Visser. Faith and falsity. Logic Group Preprint Series 216, Department of Philosophy, University of Utrecht, 2002.Google Scholar
  261. [Visser, 2002b]
    A. Visser. Substitutions of Σ10-sentences: Explorations between intuitionistic propositional logic and intuitionistic arithmetic. Annals of Pure and Applied Logic, 114(1-3):227–271, 2002.CrossRefGoogle Scholar
  262. [Švejdar, 2003]
    V. Švejdar. The decision problem of provability logic with only one atom. Archive for Mathematical Logic, 42(8):763–768, 2003.Google Scholar
  263. [Weinstein, 1983]
    S. Weinstein. The intended interpretation of intuitionistic logic. Journal of Philosophical Logic, 12:261–270, 1983.CrossRefGoogle Scholar
  264. [Wickline et al., 1998] P. Wickline, P. Lee, F. Pfenning, and R. Davies. Modal types as staging specifications for run-time code generation. ACM Computing Surveys, 30(3es), 1998.Google Scholar
  265. [Wilkie and Paris, 1987]
    A. Wilkie and J. Paris. On the scheme of induction for bounded arithmetic formulas. Annals of Pure and Applied Logic, 35:261–302, 1987.CrossRefGoogle Scholar
  266. [Wolter, 1998]
    F. Wolter. All finitely axiomatizable subframe logics containing CSM are decidable. Archive for Mathematical Logic, 37:167–182, 1998.CrossRefGoogle Scholar
  267. [Yavorskaya (Sidon), 2002]
    T. Yavorskaya (Sidon). Logic of proofs and provability. Annals of Pure and Applied Logic, 113(1-3):345–372, 2002.Google Scholar
  268. [Yavorsky, 2000]
    R. Yavorsky. On the logic of the standard proof predicate. In Computer Science Logic 2000, volume 1862 of Lecture Notes in Computer Science, pages 527–541. Springer, 2000.Google Scholar
  269. [Yavorsky, 2002]
    R. Yavorsky. Provability logics with quantifiers on proofs. Annals of Pure and Applied Logic, 113(1-3):373–387, 2002.Google Scholar
  270. [Zambella, 1994]
    D. Zambella. Shavrukov’s theorem on the subalgebras of diagonalizable algebras for theories containing IΔ0 + exp. Notre Dame Journal of Formal Logic, 35:147–157, 1994.CrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Sergei N. Artemov
  • Lev D. Beklemishev

There are no affiliations available

Personalised recommendations