Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
153 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Lewis and Brouwer meet Strong Löb (2404.11969v1)

Published 18 Apr 2024 in math.LO and cs.LO

Abstract: We study the principle phi implies box phi, known as Strength' orthe Completeness Principle', over the constructive version of L\"ob's Logic. We consider this principle both for the modal language with the necessity operator and for the modal language with the Lewis arrow, where L\"ob's Logic is suitably adapted. Central insights of provability logic, like the de Jongh-Sambin Theorem and the de Jongh-Sambin-Bernardi Theorem, take a simple form in the presence of Strength. We present these simple versions. We discuss the semantics of two salient systems and prove uniform interpolation for both. In addition, we sketch arithmetical interpretations of our systems. Finally, we describe the various connections of our subject with Computer Science.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (102)
  1. Provability logic. In Dov Gabbay and Franz Guenthner, editors, Handbook of Philosophical Logic, 2nd ed., volume 13, pages 229–403. Springer, Dordrecht, 2004.
  2. On modal μ𝜇\muitalic_μ-calculus and Gödel-Löb logic. Studia Logica, 91(2):145–169, 2009.
  3. Productive coprogramming with guarded recursion. In Greg Morrisett and Tarmo Uustalu, editors, International Conference on Functional Programming, (ICFP), pages 197–208. ACM SIGPLAN, 2013.
  4. The Σ1subscriptΣ1{\Sigma}_{1}roman_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-provability logic of HA. Annals of Pure and Applied Logic, 169(10):997–1043, 2018.
  5. Well-pointed coalgebras (extended abstract). In Lars Birkedal, editor, Proceedings of FoSSaCS, volume 7213 of LNCS, pages 89–103, 2012.
  6. A very modal model of a modern, major, general type system. In Martin Hofmann and Matthias Felleisen, editors, Proceedings of POPL, pages 109–122. ACM SIGPLAN-SIGACT, 2007.
  7. A formalized proof of strong normalization for guarded recursive types. In Jacques Garrigue, editor, Proceedings of APLAS, volume 8858 of LNCS, pages 140–158. Springer International Publishing, 2014.
  8. Guarded cubical type theory. Journal of Automated Reasoning, 63(2):211–253, Aug 2019.
  9. Computational types from a logical perspective. Journal of Functional Programming, 8(2):177–193, 1998.
  10. A filter lambda model and the completeness of type assignment. Journal of Symbolic Logic, 48(4):931–940, 1983.
  11. Extended Curry-Howard correspondence for a basic constructive modal logic. In Proceedings of Methods for Modalities, 2001.
  12. Michael Beeson. The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations. Journal of Symbolic Logic, 40:321–346, 1975.
  13. Claudio Bernardi. The uniqueness of the fixed-point in every diagonalizable algebra. Studia Logica, 35(4):335–343, 1976.
  14. Guarded dependent type theory with coinductive types. In Bart Jacobs and Christof Löding, editors, Foundations of Software Science and Computation Structures, pages 20–35, Berlin, Heidelberg, 2016. Springer Berlin Heidelberg.
  15. Intensional type theory with guarded recursive types qua fixed points on universes. In Proceedings of LiCS, pages 213–222. ACM/IEEE, 2013.
  16. First steps in synthetic guarded domain theory: Step-indexing in the topos of trees. Logical Methods in Computer Science, 8:1–45, 2012.
  17. George Boolos. The Logic of Provability. Cambridge University Press, 1993.
  18. On an intuitionistic modal logic. Studia Logica, 65(3):383–416, 2000.
  19. Compiling functional types to relational specifications for low level imperative code. In Andrew Kennedy and Amal Ahmed, editors, Types in Languages Design and Implementation (TLDI), pages 3–14. ACM SIGPLAN, 2009.
  20. Samuel R. Buss. Bounded Arithmetic. Bibliopolis, Napoli, 1986.
  21. The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types. Logical Methods in Computer Science, Volume 12, Issue 3, April 2017.
  22. New foundations for fixpoint computations: Fix-hyperdoctrines and fix-logic. Information and Computation, 98(2):171–210, 1992.
  23. Roy L. Crole. Programming Metalogics with a Fixpoint Type. PhD thesis, Computer Laboratory, University of Cambridge, 1991.
  24. Logical step-indexed logical relations. Logical Methods in Computer Science, 7(2), 2011.
  25. The relevance of semantic subtyping. Electronic Notes in Theoretical Computer Science, 70(1):88–105, 2003. ITRS ’02, Intersection Types and Related Systems (FLoC Satellite Event).
  26. The semantics of entailment omega. Notre Dame J. Formal Logic, 43(3):129–145, 07 2002.
  27. Explicit fixed points in interpretability logic. Studia Logica, 50:39–50, 1991.
  28. Embeddings of Heyting algebras. In [HHST96], pages 187–213, 1996.
  29. Positive formulas in intuitionistic and minimal logic. In Martin Aher, Daniel Hole, Emil Jeřábek, and Kupke Clemens, editors, International Tbilisi Symposium on Logic, Language, and Computation 2013, pages 175–189. Springer-Verlag, 2015.
  30. Basic constructive modality. In Jean-Yves Beziau and Marcelo Esteban Coniglio, editors, Logic Without Frontiers- Festschrift for Walter Alexandre Carnielli on the occasion of his 60th birthday, pages 411–428. College Publications, 2011.
  31. Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57:795–807, 1992.
  32. On the algebraic structure of rooted trees. J. Comput. System Sci., 16:362–399, 1978.
  33. Scattered toposes. Annals of Pure and Applied Logic, 103(1-3):97–107, 2000.
  34. Calvin C. Elgot. Monadic computation and iterative algebraic theories. In H. E. Rose and John C. Shepherdson, editors, Logic Colloquium ’73, volume 80, pages 175–230, Amsterdam, 1975. North-Holland Publishers.
  35. Solomon Feferman. Arithmetization of metamathematics in a general setting. Fundamenta Mathematicæ, 49:35–92, 1960.
  36. Slow consistency. Annals of Pure and Applied Logic, 164(3):382–393, 2013.
  37. Mechanised uniform interpolation for modal logics K, GL and iSL. arXiv 2206.00445, 2024.
  38. Characters and Fixed Points in Provability Logic. Notre Dame Journal of Formal Logic, 31:26–36, 1990.
  39. Robert I. Goldblatt. Grothendieck topology as geometric modality. Mathematical Logic Quarterly, 27(31–35):495–529, 1981.
  40. Robert I. Goldblatt. Topoi: the categorial analysis of logic. Dover Books on Mathematics. Dover Publications, 2006.
  41. Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond. Mathematical Structures in Computer Science, 30(6):572–596, 2020.
  42. A sheaf representation and duality for finitely presented Heyting algebras. Journal of Symbolic Logic, 60:911–939, 1995.
  43. Masihito Hasegawa. Recursion from cyclic sharing: Traced monoidal categories and models of cyclic lambda calculi. In Proc. 3rd International Conference on Typed Lambda Calculi and Applications, volume 1210 of Lecture Notes Comput. Sci., pages 196–213. Springer-Verlag, 1997.
  44. Masahito Hasegawa. Models of Sharing Graphs: A Categorical Semantics of let and letrec. Distinguished Dissertation Series. Springer, 1999.
  45. Logic: from foundations to applications. Clarendon Press, Oxford, 1996.
  46. Eva Hoogland. Definability and interpolation: Model-theoretic investigations. Institute for Logic, Language and Computation, 2001.
  47. Jörg Hudelmaier. Bounds for cut elimination in intuitionistic propositional logic. Ph.D. Thesis. University of Tübingen, Tübingen, 1989.
  48. Properties of intuitionistic provability and preservativity logics. Logic Journal of IGPL, 13(6):615–636, 2005.
  49. Rosalie Iemhoff. A modal analysis of some principles of the provability logic of Heyting Arithmetic. In Proceedings of AiML’98, volume 2, Uppsala, 2001.
  50. Rosalie Iemhoff. On the admissible rules of intuitionistic propositional logic. Journal of Symbolic Logic, pages 281–294, 2001.
  51. Rosalie Iemhoff. Provability Logic and Admissible Rules. PhD thesis, University of Amsterdam, 2001.
  52. Rosalie Iemhoff. Preservativity logic: An analogue of interpretability logic for constructive theories. Mathematical Logic Quarterly, 49(3):230–249, 2003.
  53. The logic of provability. In S. Buss, editor, Handbook of proof theory, pages 475–546. North-Holland Publishing Co., Amsterdam, 1998.
  54. P.T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides. Clarendon Press, 2002.
  55. Extending type theory with forcing. In Proceedings of LiCS, pages 395–404. IEEE, 2012.
  56. A semantic model for graphical user interfaces. In Manuel M. T. Chakravarty, Zhenjiang Hu, and Olivier Danvy, editors, Proceedings of ICFP, pages 45–57. ACM SIGPLAN, ACM, 2011.
  57. Ultrametric semantics of reactive programs. In Proceedings of LiCS, pages 257–266. IEEE, 2011.
  58. On superintuitionistic logics as fragments of proof logic extensions. Studia Logica, 45(1):77–99, 1986.
  59. Taishi Kurahashi. Arithmetical completeness theorem for modal logic K. Studia Logica, 106(2):219–235, 2018.
  60. Taishi Kurahashi. Arithmetical soundness and completeness for Σ2subscriptΣ2\Sigma_{2}roman_Σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-numerations. Studia Logica, 106(6):1181–1196, 2018.
  61. Tadeusz Litak. Constructive modalities with provability smack. In Guram Bezhanishvili, editor, Leo Esakia on duality in modal and intuitionistic logics, volume 4 of Outstanding Contributions to Logic, pages 179–208. Springer, 2014.
  62. Martin H. Löb. Solution of a problem of Leon Henkin. Journal of Symbolic Logic, 20:115–118, 1955.
  63. Negative Translations and Normal Modality. In Dale Miller, editor, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017), volume 84 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1–27:18, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
  64. J. Lambek and P. J. Scott. Introduction to Higher Order Categorical Logic. Number 7 in Cambridge studies in advanced mathematics. Cambridge University Press, 1986.
  65. Lewis meets Brouwer: Constructive strict implication. Indagationes Mathematicæ, 29(1):36–90, February 2018. Special Issue: L.E.J. Brouwer, 50 years later, editors: van Dalen, Dirk and Jongbloed, Geurt and Klop, Jan Willem and van Mill, Jan. URL: https://arxiv.org/abs/1708.02143.
  66. Lewisian fixed points I: two incomparable constructions. CoRR, abs/1905.09450, 2019. To appear in ”Dick de Jongh on Intuitionistic and Provability Logics”, Outstanding Contributions to Logic, 2024, eds. N. Bezhanishvili, R. Iemhoff, Fan Yang. URL: http://arxiv.org/abs/1905.09450.
  67. Larisa Maksimova. Definability theorems in normal extensions of the probability logic. Studia Logica, 48(4):495–507, 1989.
  68. Stefan Milius. Completely iterative algebras and completely iterative monads. Information and Computation, 196:1–41, 2005.
  69. Borja Sierra Miranda. On the provability logic of constructive arithmetic. The Σ1subscriptΣ1\Sigma_{1}roman_Σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-provability logics of fragments of Heyting Arithmetic. MSc Thesis. ILLC, https://eprints.illc.uva.nl/id/eprint/2266/1/MoL-2023-17.text.pdf, 2023.
  70. Guard your daggers and traces: Properties of guarded (co-)recursion. Fundamenta Informaticæ, 150:407–449, 2017. special issue FiCS’13 edited by David Baelde, Arnaud Carayol, Ralph Matthes and Igor Walukiewicz.
  71. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer, corrected edition, May 1992.
  72. Mojtaba Mojtahedi. On provability logic of HA. ArXiv 2206.00445, 2022.
  73. Mojtaba Mojtahedi. Relative unification in intuitionistic logic: Towards provability logic of HA. ArXiv 2206.00446, 2022.
  74. Denotational semantics of recursive types in synthetic guarded domain theory. Mathematical Structures in Computer Science, 29(3):465–510, 2019.
  75. Hiroshi Nakano. A modality for recursion. In Proceedings of LiCS, pages 255–266. IEEE, 2000.
  76. Hiroshi Nakano. Fixed-point logic with the approximation modality and its Kripke completeness. In Naoki Kobayashi and Benjamin C. Pierce, editors, Proceedings of TACS, volume 2215 of LNCS, pages 165–182. Springer, 2001.
  77. Andrew M. Pitts. On an interpretation of second order quantification in first order intuitionistic propositional logic. Journal of Symbolic Logic, 57:33–52, 1992.
  78. Andrew M. Pitts. Categorical logic. In S. Abramsky, D. M. Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures, chapter 2, pages 39–128. Oxford University Press, 2000.
  79. Lisa Reidhaar-Olson. A new proof of the fixed-point theorem of provability logic. Notre Dame journal of formal logic, 31(1):37–43, 1990.
  80. Wim Ruitenburg. On the period of sequences (An⁢(p))superscript𝐴𝑛𝑝(A^{n}(p))( italic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_p ) ) in intuitionistic propositional calculus. Journal of Symbolic Logic, 49:892–899, 1984.
  81. Giovanni Sambin. An effective fixed-point theorem in intuitionistic diagonalizable algebras. Studia Logica, 35:345–361, 1976.
  82. Helmut Schwichtenberg. Eine klassifikation der ϵ0subscriptitalic-ϵ0\epsilon_{0}italic_ϵ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT-rekursiven funktionen. Zeitschrift für mathematische Logik und Grundlagenforschung, 17(1):61–74, 1971.
  83. Vladimir Yu. Shavrukov. Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes mathematicae (Rozprawy matematyczne), CCCXXIII, 1993.
  84. Craig Smoryński. Applications of Kripke Models. In A.S. Troelstra, editor, Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, Springer Lecture Notes 344, pages 324–391. Springer, Berlin, 1973.
  85. Craig Smoryński. Beth’s theorem and self-referential sentences. In Agnus Macintyre, Leszek Pacholski, and Jeff Paris, editors, Logic Colloquium ’77, Studies in Logic. North Holland, Amsterdam, 1978.
  86. Complete axioms for categorical fixed-point operators. In Proc. 15th Symposium on Logic in Computer Science (LICS’00), pages 30–41. IEEE Computer Society, 2000.
  87. The modal logic of provability. the sequential approach. Journal of Philosophical Logic, 11(3):311–342, 1982.
  88. Johan Van Benthem. Modal frame correspondences and fixed-points. Studia Logica, 83(1-3):133–155, 2006.
  89. Albert Visser. Aspects of diagonalization and provability. Ph.D. Thesis, Department of Philosophy, Utrecht University, 1981. URL: https://eprints.illc.uva.nl/id/eprint/1854/.
  90. Albert Visser. On the completenes principle: A study of provability in Heyting’s Arithmetic and extensions. Annals of Mathematical Logic, 22(3):263–295, 1982.
  91. Albert Visser. Evaluation, provably deductive equivalence in Heyting’s Arithmetic of substitution instances of propositional formulas. Logic Group Preprint Series 4, Faculty of Humanities, Philosophy, Utrecht University, Janskerkhof 13, 3512 BL Utrecht, 1985.
  92. Albert Visser. Uniform interpolation and layered bisimulation. In P. Hájek, editor, Gödel ’96, Logical Foundations of Mathematics, Computer Science and Physics —Kurt Gödel’s Legacy, pages 139–164, Berlin, 1996. Springer. reprinted as Lecture Notes in Logic 6, Association of Symbolic Logic.
  93. Albert Visser. Interpretations over Heyting’s Arithmetic. In Eva Orłowska, editor, Logic at Work, Studies in Fuzziness and Soft Computing, pages 255–284. Physica-Verlag, Heidelberg/New York, 1998.
  94. A. Visser. Rules and Arithmetics. Notre Dame Journal of Formal Logic, 40(1):116–140, 1999.
  95. Albert Visser. Substitutions of Σ10subscriptsuperscriptΣ01{\Sigma}^{0}_{1}roman_Σ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic. Annals of Pure and Applied Logic, 114:227–271, 2002.
  96. Albert Visser. Löb’s Logic Meets the μ𝜇\muitalic_μ-Calculus. In Aart Middeldorp, Vincent van Oostrom, Femke van Raamsdonk, and Roel de Vrijer, editors, Processes, Terms and Cycles, Steps on the Road to Infinity, Essays Dedicated to Jan Willem Klop on the Occasion of His 60th Birthday, LNCS 3838, pages 14–25. Springer, Berlin, 2005.
  97. Albert Visser. Closed fragments of provability logics of constructive theories. Journal of Symbolic Logic, 73(3):1081–1096, 2008.
  98. Albert Visser. The Second Incompleteness Theorem and bounded interpretations. Studia Logica, 100(1-2):399–418, 2012. doi: 10.1007/s11225-012-9385-z.
  99. Albert Visser. The absorption law, or: how to Kreisel a Hilbert-Bernays-Löb. Archive for Mathematical Logic, 60(3-4):441–468, 2020.
  100. Albert Visser. Cyclic Henkin Logic. ArXiv 2101.11462, 2021.
  101. Provability logic and the completeness principle. Annals of Pure and Applied Logic, 170(6):718–753, 2019.
  102. Stan Wainer. A classification of the ordinal recursive functions. Archive for Mathematical Logic, 13(3):136–153, 1970.
Citations (1)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now