Studia Logica

, Volume 104, Issue 1, pp 1–46 | Cite as

Franco Montagna’s Work on Provability Logic and Many-valued Logic

  • Lev Beklemishev
  • Tommaso Flaminio


Franco Montagna, a prominent logician and one of the leaders of the Italian school on Mathematical Logic, passed away on February 18, 2015. We survey some of his results and ideas in the two disciplines he greatly contributed along his career: provability logic and many-valued logic.


Provability Logic Many-valued Logic Franco Montagna 


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  1. 1.
    Aglianò, P., I. M. A. Ferreirim, and F. Montagna, Basic Hoops: an Algebraic Study of Continuous t-norms. Studia Logica 87(1): 73–98, 2007.Google Scholar
  2. 2.
    Aglianò P., Montagna F.: Varieties of BL-Algebras I: General Properties. Journal of Pure and Applied Algebra 181, 105–129 (2003)CrossRefGoogle Scholar
  3. 3.
    Aguzzoli S., Bova S.: The free \({n}\)-generated BL-algebra. Annals of Pure and Applied Logic. 161(9), 1144–1170 (2010)CrossRefGoogle Scholar
  4. 4.
    Artemov, S. N., Extensions of arithmetic and modal logics. PhD thesis, Steklov Mathematical Insitute, Moscow, 1979. In Russian.Google Scholar
  5. 5.
    Artemov, S.N., Arithmetically complete modal theories. Semiotika i Informatika (14): 115–133. VINITI, Moscow, 1980. In Russian. English translation in: American Mathematical Society Translations 135(2): 39–54, 1987.Google Scholar
  6. 6.
    Artemov, S. N., Nonarithmeticity of truth predicate logics of provability. Doklady Akademii Nauk SSSR 284(2): 270–271, 1985. In Russian. English translation in Soviet Mathematics Doklady 33: 403–405, 1985.Google Scholar
  7. 7.
    Artemov, S. N., Numerically correct provability logics. Doklady Akademii Nauk SSSR, 290(6): 1289–1292, 1986. In Russian. English translation in Soviet Mathematics Doklady 34: 384–387, 1987.Google Scholar
  8. 8.
    Artemov S. N., Beklemishev L. D.: On propositional quantifiers in provability logic. Notre Dame Journal of Formal Logic 34, 401–419 (1993)CrossRefGoogle Scholar
  9. 9.
    Baaz M., Hájek P., Montagna F., Veith H.: Complexity of t-tautologies. Annals of Pure and Applied Logic 113(1-3), 3–11 (2001)CrossRefGoogle Scholar
  10. 10.
    Beklemishev, L. D., 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: 247–275, 1990.Google Scholar
  11. 11.
    Beklemishev, L. D., M. Pentus, and N. Vereshchagin, Provability, complexity, grammars. American Mathematical Society Translations, Series 2, 192, 1999.Google Scholar
  12. 12.
    Berarducci A.: The interpretability logic of Peano Arithmetic. The Journal of Symbolic Logic 55, 1059–1089 (1990)CrossRefGoogle Scholar
  13. 13.
    Bianchi M., Montagna F.: \({n}\)-Contractive BL-logics. Archive for Mathematical Logic 50(3-4), 257–285 (2011)CrossRefGoogle Scholar
  14. 14.
    Blok, W., and D. Pigozzi, Algebraizable Logics. Memoirs of the American Mathematical Society, 396 (77). Amer. Math Soc. Providence, 1989.Google Scholar
  15. 15.
    Boolos G., McGee V.: 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)CrossRefGoogle Scholar
  16. 16.
    Bova S., Flaminio T.: The coherence of Łukasiewicz assessment is NPcomplete. International Journal of Approximate Reasoning 51, 294–304 (2010)CrossRefGoogle Scholar
  17. 17.
    Bova S., Montagna F.: The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete. Theoretical Computer Science 410(12-13), 1143–1158 (2009)CrossRefGoogle Scholar
  18. 18.
    Busaniche, M., and F. Montagna, Hájek Logic BL and BL-algebras. In P. Cintula, P. Hájek, and C. Noguera, (eds.), Handbook of Mathematical Fuzzy Logic - Volume 1: 355–447. Volume 37 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011.Google Scholar
  19. 19.
    Carlson T.: Modal logics with several operators and provability interpretations. Israel Journal of Mathematics 54, 14–24 (1986)CrossRefGoogle Scholar
  20. 20.
    Chang C. C.: Algebraic Analysis of Many-valued Logics. Transactions of the American Mathematical Society 88, 467–490 (1958)CrossRefGoogle Scholar
  21. 21.
    Ciabattoni A., Metcalfe G., Montagna F.: Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. Fuzzy Sets and Systems 161(3), 369–389 (2010)CrossRefGoogle Scholar
  22. 22.
    Ciabattoni A., Montagna F.: Proof theory for locally finite many-valued logics: Semi-projective logics. Theoretical Computer Science 480, 26–42 (2013)CrossRefGoogle Scholar
  23. 23.
    Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht, 2000.Google Scholar
  24. 24.
    Cignoli R., Esteva F., Godo L., Torrens A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Computing 4(2), 106–112 (2000)CrossRefGoogle Scholar
  25. 25.
    Cintula, P., P. Hájek, and C. Noguera (eds.). Handbook of Mathematical Fuzzy Logic (Vol 1). volume 37 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011.Google Scholar
  26. 26.
    Cintula, P., Hájek P., and C. Noguera (eds.). Handbook of Mathematical Fuzzy Logic (Vol 2). volume 38 of Studies in Logic, Mathematical Logic and Foundations. College Publications, London, 2011.Google Scholar
  27. 27.
    Corsi, E. A., and F. Montagna, The Rényi-Ulam games and many-valued logics. Fuzzy Sets and Systems, in print. DOI: 10.1016/j.fss.2015.09.006.
  28. 28.
    de Finetti, B., Sul significato soggettivo della probabilità. Fundamenta Mathematicae 17: 298–329, 1931. Translated into English as “On the subjective meaning of probability”, in P. Monari and D. Cocchi (eds.), Probabilità e Induzione, Clueb, Bologna: 291–321, 1993.Google Scholar
  29. 29.
    de Jongh D., Jumelet M., Montagna F.: On the proof of Solovay’s theorem. Studia Logica 50(1), 51–70 (1991)CrossRefGoogle Scholar
  30. 30.
    de Jongh, D., and F. Veltman, Provability logics for relative interpretability. In P.P. Petkov (ed.), Mathematical Logic: 175–208. Plenum Press, New York, 1990.Google Scholar
  31. 31.
    de Jongh D., Visser A.: Explicit fixed points in interpretability logic. Studia Logica 50(1), 39–49 (1991)CrossRefGoogle Scholar
  32. 32.
    Di Nola A.: Representation and reticulation by quotients of MV-algebras. Ricerche di Matematica (Naples) 40, 291–297 (1991)Google Scholar
  33. 33.
    Di Nola A., Dvurečenskij A.: Product MV-algebras. Multiple-Valued Logics 5, 193–215 (2001)Google Scholar
  34. 34.
    Di Nola, A., and A. Dvurečenskij, State-morphism MV-algebras, Annals of Pure and Applied Logic, Festschrift on the occasion of Franco Montagna’s 60th birthday 161: 161–173, 2009.Google Scholar
  35. 35.
    Dvurečenskij A., Kowalski T., Montagna F.: State morphism MV-algebras. International Journal of Approximate Reasoning 52, 1215–1228 (2011)CrossRefGoogle Scholar
  36. 36.
    Esteva F., Gispert J., Godo L., Montagna F.: On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2), 199–226 (2002)CrossRefGoogle Scholar
  37. 37.
    Esteva F., Godo L.: Putting Together Łukasiewicz and Product Logics. Mathware and Soft Computing 6, 219–234 (1999)Google Scholar
  38. 38.
    Esteva F., Godo L.: Monoidal t-norm based logic: towards a logic for leftcontinuous t-norms. Fuzzy Sets and Systems 124, 271–288 (2001)CrossRefGoogle Scholar
  39. 39.
    Esteva, F., L. Godo, and F. Montagna, The Ł\({\prod}\) and Ł\({\prod}\) \({\frac{1}{2}}\) logics: two complete fuzzy systems joining Łukasiewicz and Product Logics. Archive for Mathematical Logic 40(1): 39–67, 2001.Google Scholar
  40. 40.
    Fedel M., Hosni H., Montagna F.: A logical characterization of coherence for imprecise probabilities. International Journal of Approximate Reasoning 52(8), 1147–1170 (2011)CrossRefGoogle Scholar
  41. 41.
    Fedel, M., K. Keimel, F. Montagna, and W. Roth, Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Forum Mathematicum 25: 405–441, 2013.Google Scholar
  42. 42.
    Feferman S.: Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae 49, 35–92 (1960)Google Scholar
  43. 43.
    Flaminio T.: NP-Containment for the Coherence Tests of Assessment of Conditional Probability: a Fuzzy-Logical Approach. Archive for Mathematical Logic 46(3-4), 301–319 (2007)CrossRefGoogle Scholar
  44. 44.
    Flaminio T., Godo L.: A logic for reasoning on the probability of fuzzy events. Fuzzy Sets and Systems 158, 625–638 (2007)CrossRefGoogle Scholar
  45. 45.
    Flaminio, T., H. Hosni, and F. Montagna, A characterization of strict coherence for infinite-valued events. Manuscript.Google Scholar
  46. 46.
    Flaminio T., Montagna F.: A logical and algebraic treatment of conditional probability. Archive for Mathematical Logic 44, 245–262 (2005)CrossRefGoogle Scholar
  47. 47.
    Flaminio T., Montagna F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50(1), 138–152 (2009)CrossRefGoogle Scholar
  48. 48.
    Flaminio T., Montagna F.: Models for many-valued probabilistic reasoning. Journal of Logic and Computation 21(3), 447–464 (2011)CrossRefGoogle Scholar
  49. 49.
    Friedman S.-D., Rathjen M., Weiermann A.: Slow consistency. Annals of Pure and Applied Logic 164(3), 382–393 (2013)CrossRefGoogle Scholar
  50. 50.
    Gerla B.: MV-algebras, multiple bets and subjective states. International Journal of Approximate Reasoning 25, 1–13 (2000)CrossRefGoogle Scholar
  51. 51.
    Gottwald, S., A Treatise on Many-valued Logics, Studies in Logic and Computation 9, Research Studies Press Ltd., Baldock, UK, 2001.Google Scholar
  52. 52.
    Goris E., Joosten J. J.: A new principle in the interpretability logic of all reasonable arithmetical theories. Logic Journal of the IGPL 19, 1–17 (2011)CrossRefGoogle Scholar
  53. 53.
    Hájek, P., Metamathematics of Fuzzy Logic, Kluwer, 1998.Google Scholar
  54. 54.
    Hájek P., Montagna F.: The logic of \({\prod_1}\)-conservativity. Archive for Mathematical Logic 30(2), 113–123 (1990)CrossRefGoogle Scholar
  55. 55.
    Hájek P., Montagna F.: The logic of \({\prod_1}\)-conservativity continued. Archive for Mathematical Logic 32, 57–63 (1992)CrossRefGoogle Scholar
  56. 56.
    Hájek P., Montagna F.: A note on the first-order logic of complete BL-chains. Mathematical Logic Quarterly 54(4), 435–446 (2008)CrossRefGoogle Scholar
  57. 57.
    Hčrcík R.: Standard completeness theorem for \({\prod}\)MTL. Archive for Mathematical Logic 44(4), 413–424 (2005)CrossRefGoogle Scholar
  58. 58.
    Hosni, H., and F. Montagna, Stable Non-standard Imprecise Probabilities. Proceedings of IPMU2014 (A. Laurent, et al. eds.), Communication in Computer and Information Science 444: 436–445, 2014.Google Scholar
  59. 59.
    Japaridze, G. K., Modal-logical means of investigation of provability. PhD Thesis (Diss. kand. philos. nauk), Moscow, Moscow State University, 1986.Google Scholar
  60. 60.
    Jenei S., Montagna F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica 70, 183–192 (2002)CrossRefGoogle Scholar
  61. 61.
    Jipsen P., Montagna F.: On the structure of generalized BL-algebras. Algebra Universalis 55, 226–237 (2006)CrossRefGoogle Scholar
  62. 62.
    Kemeny J. G.: Fair Bets and Inductive Probabilities. The Journal of Symbolic Logic 20(3), 263–273 (1955)CrossRefGoogle Scholar
  63. 63.
    Kihara H., Ono H.: Interpolation properties, Beth definability properties and amalgamation properties for substructural logics. Journal of Logic Computation 20(4), 823–875 (2010)CrossRefGoogle Scholar
  64. 64.
    Klement, E.P., R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
  65. 65.
    Krauss P. H.: Representation of conditional probability measures on boolean algebras. Acta Mathematica Academiae Scientiarum Hungaricae Tomus 19((3-4), 229–241 (1968)CrossRefGoogle Scholar
  66. 66.
    Kroupa T.: Conditional probability over MV algebras. Fuzzy Sets and Systems 149(2), 369–384 (2005)CrossRefGoogle Scholar
  67. 67.
    Kroupa T.: Representation and extension of states on MV-algebras. Archive for Mathematical Logic 45, 381–392 (2006)CrossRefGoogle Scholar
  68. 68.
    Kühr J.: MundiciD., De Finetti Theorem and Borel states in [0, 1]-valued Algebraic Logic. International Journal of Approximate Reasoning 46(3), 605–616 (2007)CrossRefGoogle Scholar
  69. 69.
    Lapenta, S., MV-algebras with product: connecting the Pierce-Birkhoff conjecture with Łukasiewicz logic. Ph. D. Thesis, Università deli Studi della Basilicata, 2015.Google Scholar
  70. 70.
    Lindström, P., and V. Shavrukov, The \({\forall \exists}\)-theory of Peano \({\sum_1}\)-sentences. Journal of Mathematical Logic 8(2): 251–280, 2008.Google Scholar
  71. 71.
    Löb M. H.: Solution of a problem of Leon Henkin. The Journal of Symbolic Logic 20, 115–118 (1955)CrossRefGoogle Scholar
  72. 72.
    Łukasiewicz, J., and A. Tarski, Untersuchungen über den Aussagenkalkül. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie cl. III, 23(iii): 30–50, 1930.Google Scholar
  73. 73.
    Macintyre A., Simmons H.: Gödel’s diagonalization technique and related properties of theories. Colloquium Mathematicae 28, 165–180 (1973)Google Scholar
  74. 74.
    Magari, R., The diagonalizable algebras (the algebraization of the theories which express Theor.; II). Bollettino della Unione Matematica Italiana, Serie 4 12:117–125, 1975.Google Scholar
  75. 75.
    Marchioni, E., and L. Godo, A logic for reasoning about coherent conditional probability: a fuzzy modal logic approach. Lecture Notes in Artificial Intelligence, 9th European Conference on Logics in Artificial Intelligence JELIA’04. J. J. Alferes and J. Leite (eds.), 3229: 213–225, 2004.Google Scholar
  76. 76.
    Marchioni E., Metcalfe G.: Craig interpolation for semilinear substructural logics. Mathematical Logic Quarterly 58(6), 468–481 (2012)CrossRefGoogle Scholar
  77. 77.
    Marchioni E., Spada L.: Advances in the theory of \({\mu}\)Ł\({\prod}\) algebras. Logic Journal of the IGPL 19(3), 476–489 (2011)CrossRefGoogle Scholar
  78. 78.
    Metcalfe G., Montagna F., Tsinakis C.: Amalgamation and interpolation in ordered algebras. Journal of Algebra 402, 21–82 (2014)CrossRefGoogle Scholar
  79. 79.
    Montagna F.: On the algebraization of a Feferman’s predicate. Studia Logica 37, 221–236 (1978)CrossRefGoogle Scholar
  80. 80.
    Montagna, F., On the diagonalizable algebra of Peano arithmetic. Bollettino della Unione Matematica Italiana, B (5), 16: 795–812, 1979.Google Scholar
  81. 81.
    Montagna F.: Interpretations of the first-order theory of diagonalizable algebras in Peano arithmetic. Studia Logica 39, 347–354 (1980)CrossRefGoogle Scholar
  82. 82.
    Montagna F.: Undecidability of the first-order theory of diagonalizable algebras. Studia Logica 39, 355–359 (1980)CrossRefGoogle Scholar
  83. 83.
    Montagna F.: The predicate modal logic of provability. Notre Dame Journal of Formal Logic 25, 179–189 (1987)CrossRefGoogle Scholar
  84. 84.
    Montagna F.: Provability in finite subtheories of PA. The Journal of Symbolic Logic 52(2), 494–511 (1987)CrossRefGoogle Scholar
  85. 85.
    Montagna F.: An algebraic approach to propositional fuzzy logic. Journal of Logic, Language, and Information 9, 91–124 (2000)CrossRefGoogle Scholar
  86. 86.
    Montagna F.: The Free BL-Algebra on One Generator. Neural Network World 5, 837–844 (2000)Google Scholar
  87. 87.
    Montagna F.: Storage Operators and Multiplicative Quantifiers in Many-valued Logics. Journal of Logic and Computation 14(2), 299–322 (2004)CrossRefGoogle Scholar
  88. 88.
    Montagna F.: On the predicate logics of continuous t-norm BL-algebras. Archive for Mathematical Logic 44(1), 97–114 (2005)CrossRefGoogle Scholar
  89. 89.
    Montagna F.: Subreducts of MV-algebras with product and product residuation. Algebra Universalis 53, 109–137 (2005)CrossRefGoogle Scholar
  90. 90.
    Montagna F.: Interpolation and Beth’s property in propositional many-valued logics: A semantic investigation. Annals of Pure and Applied Logic 141(1-2), 148–179 (2006)CrossRefGoogle Scholar
  91. 91.
    Montagna F.: A notion of coherence for books on conditional events in manyvalued logic. Journal of Logic and Computation 21(5), 829–850 (2011)CrossRefGoogle Scholar
  92. 92.
    Montagna F.: Partially Undetermined Many-Valued Events and Their Conditional Probability. Journal of Philosophical Logic 41(3), 563–593 (2012)CrossRefGoogle Scholar
  93. 93.
    Montagna F., Fedel M., Scianna G.: Non-standard probability, coherence and conditional probability on many-valued events. International Journal of Approximate Reasoning 54, 573–589 (2013)CrossRefGoogle Scholar
  94. 94.
    Montagna F., Marini C., Simi G.: Product logic and probabilistic Ulam games. Fuzzy Sets and Systems 158(6), 639–651 (2007)CrossRefGoogle Scholar
  95. 95.
    Montagna F., Noguera C.: Arithmetical complexity of first order predicate fuzzy logics over distinguished semantics. Journal of Logic and Computation 20, 399–424 (2010)CrossRefGoogle Scholar
  96. 96.
    Montagna F., Noguera C., HǕrcík R.: On Weakly Cancellative Fuzzy Logics. Journal of Logic and Computation 16(4), 423–450 (2006)CrossRefGoogle Scholar
  97. 97.
    Montagna F., Panti G.: Adding structure to MV-algebras. Journal of Pure and Applied Algebra 164(3), 365–387 (2001)CrossRefGoogle Scholar
  98. 98.
    Montagna F., Spada L.: Continuous approximations of product implication in MV-algebras with product. Soft Computing 9(3), 149–154 (2005)CrossRefGoogle Scholar
  99. 99.
    Montagna F., Tsinakis C.: Ordered groups with a conucleus. Journal of Pure and Applied Algebra 214(1), 71–88 (2010)CrossRefGoogle Scholar
  100. 100.
    Montagna F., Ugolini S.: A Categorical Equivalence for Product Algebras. Studia Logica 103(2), 345–373 (2015)CrossRefGoogle Scholar
  101. 101.
    Mundici D.: Averaging the Truth-value in Łukasiewicz Logic. Studia Logica 55(1), 113–127 (1995)CrossRefGoogle Scholar
  102. 102.
    Mundici D.: Tensor Products and the Loomis-Sikorski Theorem for MV-Algebras. Advances in Applied Mathematics 22, 227–248 (1999)CrossRefGoogle Scholar
  103. 103.
    Mundici D.: Bookmaking over Infinite-valued Events. International Journal of Approximate Reasoning 43, 223–240 (2006)CrossRefGoogle Scholar
  104. 104.
    Mundici, D., Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic 35, Springer 2011.Google Scholar
  105. 105.
    Nelson, E., Radically Elementary Probability Theory. Annals of Mathematics Studies, Princeton University Press, 1988.Google Scholar
  106. 106.
    Noguera, C., Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph. D. Thesis, University of Barcelona, 2006.Google Scholar
  107. 107.
    Pakhomov F. N.: Undecidability of the elementary theory of the semilattice of GLP-words. Sbornik Mathematics 203(8), 1211–1229 (2012)CrossRefGoogle Scholar
  108. 108.
    Pakhomov, F.N., On elementary theories of GLP-algebras. Technical report, 2014. ArXiv:1412.4439 [math.LO].
  109. 109.
    Pakhomov F.N.: On elementary theories of ordinal notation systems based on reflection principles. Proceedings of the Steklov Institute of Mathematics 289, 194–212 (2015)CrossRefGoogle Scholar
  110. 110.
    Paris, J. B., The uncertain reasoner’s companion. A Mathematical Perspective. Cambridge University Press, 1994.Google Scholar
  111. 111.
    Paris, J. B., A Note on the Dutch Book Method. In Proceedings of the Second International Symposium on Imprecise Probabilities and their Applications (ISIPTA), 2001, pp. 301–306.Google Scholar
  112. 112.
    Segerberg, K., An essay in classical modal logic. Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet, Uppsala, 1971.Google Scholar
  113. 113.
    Shavrukov, V.Yu, The logic of relative interpretability over Peano arithmetic. Preprint, Steklov Mathematical Institute, Moscow, 1988. In Russian.Google Scholar
  114. 114.
    Shavrukov, V.Yu, Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes Mathematicae 323, 1993.Google Scholar
  115. 115.
    Shavrukov, V.Yu, A smart child of Peano’s. Notre Dame Journal of Formal Logic 35: 161–185, 1994.Google Scholar
  116. 116.
    Shavrukov, V.Yu, Undecidability in diagonalizable algebras. The Journal of Symbolic Logic 62(1): 79–116.Google Scholar
  117. 117.
    Shavrukov, V.Yu, Effectively inseparable Boolean algebras in lattices of sentences. Archive for Mathematical Logic 49(1): 69–89, 2010.Google Scholar
  118. 118.
    Shimony A.: Coherence and the Axioms of Confirmation. The Journal of Symbolic Logic 20(1), 1–28 (1955)CrossRefGoogle Scholar
  119. 119.
    Smoryński C.: The finite inseparability of the first order theory of diagonalizable algebras. Studia Logica 41, 347–349 (1982)CrossRefGoogle Scholar
  120. 120.
    Smoryński C.: Arithmetical analogues of McAloon’s unique Rosser sentences. Archive for Mathematical Logic 28, 1–21 (1989)CrossRefGoogle Scholar
  121. 121.
    Solovay R. M.: Provability interpretations of modal logic. Israel Journal of Mathematics 28, 33–71 (1976)Google Scholar
  122. 122.
    Spada L.: \({\mu}\)MV-algebras: An approach to fixed points in Łukasiewicz logic. Fuzzy Sets and Systems 159(10), 1260–1267 (2008)CrossRefGoogle Scholar
  123. 123.
    Spada L.: Ł\({\prod}\) logic with fixed points. Archive for Mathematical Logic 47(7-8), 741–763 (2008)CrossRefGoogle Scholar
  124. 124.
    Švejdar V.: Modal analysis of generalized Rosser sentences. The Journal of Symbolic Logic 48, 986–999 (1983)CrossRefGoogle Scholar
  125. 125.
    Vardanyan, V.A., Arithmetic complexity of predicate logics of provability and their fragments. Doklady Akademii Nauk SSSR 288(1): 11–14, 1986. In Russian. English translation in Soviet Mathematics Doklady 33: 569–572, 1986.Google Scholar
  126. 126.
    Visser A.: Peano’s smart children. A provability logical study of systems with built-in consistency. Notre Dame Journal of Formal Logic 30, 161–196 (1989)CrossRefGoogle Scholar
  127. 127.
    Visser, A., Interpretability logic. In P.P. Petkov, (ed.), Mathematical Logic, Plenum Press, New York, 1990, pp. 175–208.Google Scholar
  128. 128.
    Visser, A., An overview of interpretability logic. In M. Kracht, M. de Rijke, H. Wansing, and M. Zakhariaschev, (eds.), Advances in Modal Logic, v.1, CSLI Lecture Notes, No. 87: 307–359. CSLI Publications, Stanford, 1998.Google Scholar
  129. 129.
    Visser A.: The Second Incompleteness Theorem and bounded interpretations. Studia Logica 100(1), 399–418 (2012)CrossRefGoogle Scholar
  130. 130.
    Visser A., de Jonge M.: No escape from Vardanyan’s theorem. Archive for Mathematical Logic 45(5), 539–554 (2006)CrossRefGoogle Scholar
  131. 131.
    Walley, P., Statistical Reasoning with Imprecise Probabilities, Volume 42 of Monographs on Statistics and Applied Probability, Chapman and Hall, London 1991.Google Scholar
  132. 132.
    Zambella D.: Shavrukov’s theorem on the subalgebras of diagonalizable algebras for theories containing \({I\Delta_0}\)+exp. Notre Dame Journal of Formal Logic 35, 147–157 (1994)CrossRefGoogle Scholar
  133. 133.
    Zambella, D., Chapters on Bounded Arithmetic & Provability Logic. PhD thesis, Universiteit van Amsterdam, September 1994.Google Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.Department of Theoretical and Applied SciencesUniversity of InsubriaVareseItaly

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