Skip to main content
Log in

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

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  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.

  2. Aglianò P., Montagna F.: Varieties of BL-Algebras I: General Properties. Journal of Pure and Applied Algebra 181, 105–129 (2003)

    Article  Google Scholar 

  3. Aguzzoli S., Bova S.: The free \({n}\)-generated BL-algebra. Annals of Pure and Applied Logic. 161(9), 1144–1170 (2010)

    Article  Google Scholar 

  4. Artemov, S. N., Extensions of arithmetic and modal logics. PhD thesis, Steklov Mathematical Insitute, Moscow, 1979. In Russian.

  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.

  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.

  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.

  8. Artemov S. N., Beklemishev L. D.: On propositional quantifiers in provability logic. Notre Dame Journal of Formal Logic 34, 401–419 (1993)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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.

  11. Beklemishev, L. D., M. Pentus, and N. Vereshchagin, Provability, complexity, grammars. American Mathematical Society Translations, Series 2, 192, 1999.

  12. Berarducci A.: The interpretability logic of Peano Arithmetic. The Journal of Symbolic Logic 55, 1059–1089 (1990)

    Article  Google Scholar 

  13. Bianchi M., Montagna F.: \({n}\)-Contractive BL-logics. Archive for Mathematical Logic 50(3-4), 257–285 (2011)

    Article  Google Scholar 

  14. Blok, W., and D. Pigozzi, Algebraizable Logics. Memoirs of the American Mathematical Society, 396 (77). Amer. Math Soc. Providence, 1989.

  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)

    Article  Google Scholar 

  16. Bova S., Flaminio T.: The coherence of Łukasiewicz assessment is NPcomplete. International Journal of Approximate Reasoning 51, 294–304 (2010)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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.

  19. Carlson T.: Modal logics with several operators and provability interpretations. Israel Journal of Mathematics 54, 14–24 (1986)

    Article  Google Scholar 

  20. Chang C. C.: Algebraic Analysis of Many-valued Logics. Transactions of the American Mathematical Society 88, 467–490 (1958)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  22. Ciabattoni A., Montagna F.: Proof theory for locally finite many-valued logics: Semi-projective logics. Theoretical Computer Science 480, 26–42 (2013)

    Article  Google Scholar 

  23. Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht, 2000.

  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)

    Article  Google Scholar 

  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.

  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.

  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. 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.

  29. de Jongh D., Jumelet M., Montagna F.: On the proof of Solovay’s theorem. Studia Logica 50(1), 51–70 (1991)

    Article  Google Scholar 

  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.

  31. de Jongh D., Visser A.: Explicit fixed points in interpretability logic. Studia Logica 50(1), 39–49 (1991)

    Article  Google Scholar 

  32. Di Nola A.: Representation and reticulation by quotients of MV-algebras. Ricerche di Matematica (Naples) 40, 291–297 (1991)

    Google Scholar 

  33. Di Nola A., Dvurečenskij A.: Product MV-algebras. Multiple-Valued Logics 5, 193–215 (2001)

    Google Scholar 

  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.

  35. Dvurečenskij A., Kowalski T., Montagna F.: State morphism MV-algebras. International Journal of Approximate Reasoning 52, 1215–1228 (2011)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  37. Esteva F., Godo L.: Putting Together Łukasiewicz and Product Logics. Mathware and Soft Computing 6, 219–234 (1999)

    Google Scholar 

  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)

    Article  Google Scholar 

  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.

  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)

    Article  Google Scholar 

  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.

  42. Feferman S.: Arithmetization of metamathematics in a general setting. Fundamenta Mathematicae 49, 35–92 (1960)

    Google Scholar 

  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)

    Article  Google Scholar 

  44. Flaminio T., Godo L.: A logic for reasoning on the probability of fuzzy events. Fuzzy Sets and Systems 158, 625–638 (2007)

    Article  Google Scholar 

  45. Flaminio, T., H. Hosni, and F. Montagna, A characterization of strict coherence for infinite-valued events. Manuscript.

  46. Flaminio T., Montagna F.: A logical and algebraic treatment of conditional probability. Archive for Mathematical Logic 44, 245–262 (2005)

    Article  Google Scholar 

  47. Flaminio T., Montagna F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50(1), 138–152 (2009)

    Article  Google Scholar 

  48. Flaminio T., Montagna F.: Models for many-valued probabilistic reasoning. Journal of Logic and Computation 21(3), 447–464 (2011)

    Article  Google Scholar 

  49. Friedman S.-D., Rathjen M., Weiermann A.: Slow consistency. Annals of Pure and Applied Logic 164(3), 382–393 (2013)

    Article  Google Scholar 

  50. Gerla B.: MV-algebras, multiple bets and subjective states. International Journal of Approximate Reasoning 25, 1–13 (2000)

    Article  Google Scholar 

  51. Gottwald, S., A Treatise on Many-valued Logics, Studies in Logic and Computation 9, Research Studies Press Ltd., Baldock, UK, 2001.

  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)

    Article  Google Scholar 

  53. Hájek, P., Metamathematics of Fuzzy Logic, Kluwer, 1998.

  54. Hájek P., Montagna F.: The logic of \({\prod_1}\)-conservativity. Archive for Mathematical Logic 30(2), 113–123 (1990)

    Article  Google Scholar 

  55. Hájek P., Montagna F.: The logic of \({\prod_1}\)-conservativity continued. Archive for Mathematical Logic 32, 57–63 (1992)

    Article  Google Scholar 

  56. Hájek P., Montagna F.: A note on the first-order logic of complete BL-chains. Mathematical Logic Quarterly 54(4), 435–446 (2008)

    Article  Google Scholar 

  57. Hčrcík R.: Standard completeness theorem for \({\prod}\)MTL. Archive for Mathematical Logic 44(4), 413–424 (2005)

    Article  Google Scholar 

  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.

  59. Japaridze, G. K., Modal-logical means of investigation of provability. PhD Thesis (Diss. kand. philos. nauk), Moscow, Moscow State University, 1986.

  60. Jenei S., Montagna F.: A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica 70, 183–192 (2002)

    Article  Google Scholar 

  61. Jipsen P., Montagna F.: On the structure of generalized BL-algebras. Algebra Universalis 55, 226–237 (2006)

    Article  Google Scholar 

  62. Kemeny J. G.: Fair Bets and Inductive Probabilities. The Journal of Symbolic Logic 20(3), 263–273 (1955)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  64. Klement, E.P., R. Mesiar, and E. Pap, Triangular Norms, Kluwer Academic Publishers, Dordrecht, 2000.

  65. Krauss P. H.: Representation of conditional probability measures on boolean algebras. Acta Mathematica Academiae Scientiarum Hungaricae Tomus 19((3-4), 229–241 (1968)

    Article  Google Scholar 

  66. Kroupa T.: Conditional probability over MV algebras. Fuzzy Sets and Systems 149(2), 369–384 (2005)

    Article  Google Scholar 

  67. Kroupa T.: Representation and extension of states on MV-algebras. Archive for Mathematical Logic 45, 381–392 (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  69. Lapenta, S., MV-algebras with product: connecting the Pierce-Birkhoff conjecture with Łukasiewicz logic. Ph. D. Thesis, Università deli Studi della Basilicata, 2015.

  70. Lindström, P., and V. Shavrukov, The \({\forall \exists}\)-theory of Peano \({\sum_1}\)-sentences. Journal of Mathematical Logic 8(2): 251–280, 2008.

  71. Löb M. H.: Solution of a problem of Leon Henkin. The Journal of Symbolic Logic 20, 115–118 (1955)

    Article  Google Scholar 

  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.

  73. Macintyre A., Simmons H.: Gödel’s diagonalization technique and related properties of theories. Colloquium Mathematicae 28, 165–180 (1973)

    Google Scholar 

  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.

  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.

  76. Marchioni E., Metcalfe G.: Craig interpolation for semilinear substructural logics. Mathematical Logic Quarterly 58(6), 468–481 (2012)

    Article  Google Scholar 

  77. Marchioni E., Spada L.: Advances in the theory of \({\mu}\)Ł\({\prod}\) algebras. Logic Journal of the IGPL 19(3), 476–489 (2011)

    Article  Google Scholar 

  78. Metcalfe G., Montagna F., Tsinakis C.: Amalgamation and interpolation in ordered algebras. Journal of Algebra 402, 21–82 (2014)

    Article  Google Scholar 

  79. Montagna F.: On the algebraization of a Feferman’s predicate. Studia Logica 37, 221–236 (1978)

    Article  Google Scholar 

  80. Montagna, F., On the diagonalizable algebra of Peano arithmetic. Bollettino della Unione Matematica Italiana, B (5), 16: 795–812, 1979.

  81. Montagna F.: Interpretations of the first-order theory of diagonalizable algebras in Peano arithmetic. Studia Logica 39, 347–354 (1980)

    Article  Google Scholar 

  82. Montagna F.: Undecidability of the first-order theory of diagonalizable algebras. Studia Logica 39, 355–359 (1980)

    Article  Google Scholar 

  83. Montagna F.: The predicate modal logic of provability. Notre Dame Journal of Formal Logic 25, 179–189 (1987)

    Article  Google Scholar 

  84. Montagna F.: Provability in finite subtheories of PA. The Journal of Symbolic Logic 52(2), 494–511 (1987)

    Article  Google Scholar 

  85. Montagna F.: An algebraic approach to propositional fuzzy logic. Journal of Logic, Language, and Information 9, 91–124 (2000)

    Article  Google Scholar 

  86. Montagna F.: The Free BL-Algebra on One Generator. Neural Network World 5, 837–844 (2000)

    Google Scholar 

  87. Montagna F.: Storage Operators and Multiplicative Quantifiers in Many-valued Logics. Journal of Logic and Computation 14(2), 299–322 (2004)

    Article  Google Scholar 

  88. Montagna F.: On the predicate logics of continuous t-norm BL-algebras. Archive for Mathematical Logic 44(1), 97–114 (2005)

    Article  Google Scholar 

  89. Montagna F.: Subreducts of MV-algebras with product and product residuation. Algebra Universalis 53, 109–137 (2005)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  92. Montagna F.: Partially Undetermined Many-Valued Events and Their Conditional Probability. Journal of Philosophical Logic 41(3), 563–593 (2012)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  94. Montagna F., Marini C., Simi G.: Product logic and probabilistic Ulam games. Fuzzy Sets and Systems 158(6), 639–651 (2007)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  96. Montagna F., Noguera C., HǕrcík R.: On Weakly Cancellative Fuzzy Logics. Journal of Logic and Computation 16(4), 423–450 (2006)

    Article  Google Scholar 

  97. Montagna F., Panti G.: Adding structure to MV-algebras. Journal of Pure and Applied Algebra 164(3), 365–387 (2001)

    Article  Google Scholar 

  98. Montagna F., Spada L.: Continuous approximations of product implication in MV-algebras with product. Soft Computing 9(3), 149–154 (2005)

    Article  Google Scholar 

  99. Montagna F., Tsinakis C.: Ordered groups with a conucleus. Journal of Pure and Applied Algebra 214(1), 71–88 (2010)

    Article  Google Scholar 

  100. Montagna F., Ugolini S.: A Categorical Equivalence for Product Algebras. Studia Logica 103(2), 345–373 (2015)

    Article  Google Scholar 

  101. Mundici D.: Averaging the Truth-value in Łukasiewicz Logic. Studia Logica 55(1), 113–127 (1995)

    Article  Google Scholar 

  102. Mundici D.: Tensor Products and the Loomis-Sikorski Theorem for MV-Algebras. Advances in Applied Mathematics 22, 227–248 (1999)

    Article  Google Scholar 

  103. Mundici D.: Bookmaking over Infinite-valued Events. International Journal of Approximate Reasoning 43, 223–240 (2006)

    Article  Google Scholar 

  104. Mundici, D., Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic 35, Springer 2011.

  105. Nelson, E., Radically Elementary Probability Theory. Annals of Mathematics Studies, Princeton University Press, 1988.

  106. Noguera, C., Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph. D. Thesis, University of Barcelona, 2006.

  107. Pakhomov F. N.: Undecidability of the elementary theory of the semilattice of GLP-words. Sbornik Mathematics 203(8), 1211–1229 (2012)

    Article  Google Scholar 

  108. Pakhomov, F.N., On elementary theories of GLP-algebras. Technical report, 2014. ArXiv:1412.4439 [math.LO].

  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)

    Article  Google Scholar 

  110. Paris, J. B., The uncertain reasoner’s companion. A Mathematical Perspective. Cambridge University Press, 1994.

  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.

  112. Segerberg, K., An essay in classical modal logic. Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet, Uppsala, 1971.

  113. Shavrukov, V.Yu, The logic of relative interpretability over Peano arithmetic. Preprint, Steklov Mathematical Institute, Moscow, 1988. In Russian.

  114. Shavrukov, V.Yu, Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes Mathematicae 323, 1993.

  115. Shavrukov, V.Yu, A smart child of Peano’s. Notre Dame Journal of Formal Logic 35: 161–185, 1994.

  116. Shavrukov, V.Yu, Undecidability in diagonalizable algebras. The Journal of Symbolic Logic 62(1): 79–116.

  117. Shavrukov, V.Yu, Effectively inseparable Boolean algebras in lattices of sentences. Archive for Mathematical Logic 49(1): 69–89, 2010.

  118. Shimony A.: Coherence and the Axioms of Confirmation. The Journal of Symbolic Logic 20(1), 1–28 (1955)

    Article  Google Scholar 

  119. Smoryński C.: The finite inseparability of the first order theory of diagonalizable algebras. Studia Logica 41, 347–349 (1982)

    Article  Google Scholar 

  120. Smoryński C.: Arithmetical analogues of McAloon’s unique Rosser sentences. Archive for Mathematical Logic 28, 1–21 (1989)

    Article  Google Scholar 

  121. Solovay R. M.: Provability interpretations of modal logic. Israel Journal of Mathematics 28, 33–71 (1976)

    Google Scholar 

  122. Spada L.: \({\mu}\)MV-algebras: An approach to fixed points in Łukasiewicz logic. Fuzzy Sets and Systems 159(10), 1260–1267 (2008)

    Article  Google Scholar 

  123. Spada L.: Ł\({\prod}\) logic with fixed points. Archive for Mathematical Logic 47(7-8), 741–763 (2008)

    Article  Google Scholar 

  124. Švejdar V.: Modal analysis of generalized Rosser sentences. The Journal of Symbolic Logic 48, 986–999 (1983)

    Article  Google Scholar 

  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.

  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)

    Article  Google Scholar 

  127. Visser, A., Interpretability logic. In P.P. Petkov, (ed.), Mathematical Logic, Plenum Press, New York, 1990, pp. 175–208.

  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.

  129. Visser A.: The Second Incompleteness Theorem and bounded interpretations. Studia Logica 100(1), 399–418 (2012)

    Article  Google Scholar 

  130. Visser A., de Jonge M.: No escape from Vardanyan’s theorem. Archive for Mathematical Logic 45(5), 539–554 (2006)

    Article  Google Scholar 

  131. Walley, P., Statistical Reasoning with Imprecise Probabilities, Volume 42 of Monographs on Statistics and Applied Probability, Chapman and Hall, London 1991.

  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)

    Article  Google Scholar 

  133. Zambella, D., Chapters on Bounded Arithmetic & Provability Logic. PhD thesis, Universiteit van Amsterdam, September 1994.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lev Beklemishev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beklemishev, L., Flaminio, T. Franco Montagna’s Work on Provability Logic and Many-valued Logic. Stud Logica 104, 1–46 (2016). https://doi.org/10.1007/s11225-016-9654-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-016-9654-3

Keywords

Navigation