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On the Blok-Esakia Theorem

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Leo Esakia on Duality in Modal and Intuitionistic Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 4))

Abstract

We discuss the celebrated Blok-Esakia theorem on the isomorphism between the lattices of extensions of intuitionistic propositional logic and the Grzegorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimplication.

In memory of Leo Esakia

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Notes

  1. 1.

    Topological Boolean algebras [60] are also known as closure algebras [48], interior algebras [6] and S4-algebras. Heyting algebras are called pseudo-Boolean algebras in [60].

  2. 2.

    Actually, Orlov [57] considered a somewhat weaker logic, which can be regarded as the first relevant system.

  3. 3.

    There are different variants of the translation \(T\); in fact, it is enough to prefix \(\Box \) to propositional variables, implications and negations only.

  4. 4.

    That every si-logic \(L\) has a greatest modal companion was first established by Maksimova and Rybakov [47], who gave an answer to an open question by R. Bull; however, they did not observe that greatest modal companion is actually \(\tau L \oplus \mathbf{Grz }\).

References

  1. Artemov S (2001) Explicit provability and constructive semantics. Bull Symbolic Logic 7:1–36

    Article  Google Scholar 

  2. Bezhanishvili G (2001) Glivenko type theorems for intuitionistic modal logics. Studia Logica 67(1):89–109

    Google Scholar 

  3. Bezhanishvili G (2009) The universal modality, the center of a Heyting algebra, and the Blok-Esakia theorem. Ann Pure Appl Logic 161(3):253–267

    Article  Google Scholar 

  4. Bezhanishvili G, Bezhanishvili N (2011) An algebraic approach to canonical formulas: modal case. Studia Logica 99(1–3):93–125

    Google Scholar 

  5. Blok W, Dwinger P (1975) Equational classes of closure algebras. Indagationes Math 37:189–198

    Article  Google Scholar 

  6. Blok W (1976) Varieties of interior algebras. PhD thesis, University of Amsterdam

    Google Scholar 

  7. Boolos G (1980) On systems of modal logic with provability interpretations. Theoria 46:7–18

    Article  Google Scholar 

  8. Buisman L, Goré R (2007) A cut-free sequent calculus for bi-intuitionistic logic. In: TABLEAUX, Springer, Berlin, pp 90–106

    Google Scholar 

  9. Chagrov A, Zakharyaschev M (1997) Modal logic, volume 35 of Oxford logic guides. Clarendon Press, Oxford

    Google Scholar 

  10. Chagrov A, Zakharyaschev M (1992) Modal companions of intermediate propositional logics. Studia Logica 51:49–82

    Google Scholar 

  11. Dummett M, Lemmon E (1959) Modal logics between S4 and S5. Zeitschrift für Math Logik und Grundlagen der Mathematik 5:250–264

    Article  Google Scholar 

  12. Esakia L (1975) The problem of dualism in the intuitionistic logic and Brouwerian lattices. In: 5th international congress of logic, methodology and philosophy of science, Canada, pp 7–8

    Google Scholar 

  13. Esakia L (1976) On modal companions of superintuitionistic logics. In: VII Soviet Symposium on logic. Kiev (Russian)

    Google Scholar 

  14. Esakia L (1978) Semantical analysis of bimodal (tense) systems. In: Logic, semantics and methodology, Metsniereba Press, Tbilisi, pp 87–99 (Russian)

    Google Scholar 

  15. Esakia L (1979) On varieties of Grzegorczyk algebras. In: Mikhailov A (ed) Studies in non-classical logics and set theory, Moscow, Nauka, pp 257–287 (Russian)

    Google Scholar 

  16. Esakia L (1979) To the theory of modal and superintuitionistic systems. In: Smirnov V (ed) Proceedings of the USSR symposium on the theory of logical inference, Nauka, Moscow, pp 147–172 (Russian)

    Google Scholar 

  17. Fine K (1985) Logics containing K4. Part II. J Symbolic Logic 50:619–651

    Article  Google Scholar 

  18. Fischer Servi G (1977) On modal logics with an intuitionistic base. Studia Logica 36:141–149

    Google Scholar 

  19. Fischer Servi G (1980) Semantics for a class of intuitionistic modal calculi. In: M Dalla Chiara (ed) Italian studies in the philosophy of science, Reidel, Dordrecht, pp 59–72

    Google Scholar 

  20. Fischer Servi G (1984) Axiomatizations for some intuitionistic modal logics. Rendiconti di Matematica di Torino 42:179–194

    Google Scholar 

  21. Font J (1986) Modality and possibility in some intuitionistic modal logics. Notre Dame J Formal Logic 27:533–546

    Google Scholar 

  22. Gabbay D, Kurucz A, Wolter F, Zakharyaschev M (2003) Many-dimensional modal logics: theory and applications. Elsevier, Amsterdam

    Google Scholar 

  23. Gabbay D, Shehtman V, Skvortsov D (2009) Quantification in nonclassical logic. Elsevier, Amsterdam

    Google Scholar 

  24. Gödel K (1932) Zum intuitionistischen Aussagenkalkül. Anzeiger der Akademie der Wissenschaften in Wien 69:65–66

    Google Scholar 

  25. Gödel K (1933) Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums 4:39–40

    Google Scholar 

  26. Goldblatt R (1978) Arithmetical necessity, provability and intuitionistic logic. Theoria 44:38–46

    Article  Google Scholar 

  27. Goldblatt R (2003) Mathematical modal logic: a view of its evolution. J. Appl Logic 1(5–6):309–392

    Article  Google Scholar 

  28. Goranko V, Passy S (1992) Using the universal modality: gains and questions. J Logic Comput 2:5–30

    Article  Google Scholar 

  29. Goré R, Postniece L, Tiu A (2010) Cut-elimination and proof search for bi-intuitionistic tense logic. In: Advances in modal logic, pp 156–177

    Google Scholar 

  30. Grädel E, Kolaitis P, Vardi M (1997) On the decision problem for two-variable first order logic. Bull Symbolic Logic 3:53–69

    Article  Google Scholar 

  31. Grätzer G (1971) Lattice theory: first concepts and distributive lattices. Freeman Co, San Francisco

    Google Scholar 

  32. Grefe C (1998) Fischer Servi’s intuitionistic modal logic has the finite model property. In: Kracht M, de Rijke M, Wansing H, Zakharyaschev M (eds) Advances in modal logic. CSLI Publications, Stanford, pp 85–98

    Google Scholar 

  33. Grzegorczyk A (1967) Some relational systems and the associated topological spaces. Fundamenta Math 60:223–231

    Google Scholar 

  34. Hacking I (1963) What is strict implication? J Symbolic Logic 28:51–71

    Article  Google Scholar 

  35. Heyting A (1930) Die formalen Regeln der intuitionistischen Logik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp 42–56

    Google Scholar 

  36. Jankov V (1968) The construction of a sequence of strongly independent superintuitionistic propositional calculi. Sov Math Doklady 9:806–807

    Google Scholar 

  37. Jerábek E (2009) Canonical rules. J Symbolic Logic 74(4):1171–1205

    Article  Google Scholar 

  38. Kleene S (1945) On the interpretation of intuitionistic number theory. J Symbolic Logic 10:109–124

    Article  Google Scholar 

  39. Köhler R (1980) A subdirectly irreducible double Heyting algebra which is not simple. Algebra Univers 10:189–194

    Article  Google Scholar 

  40. Kontchakov R, Kurucz A, Zakharyaschev M (2005) Undecidability of first-order intuitionistic and modal logics with two variables. Bull Symbolic Logic 11(3):428–438

    Article  Google Scholar 

  41. Kuznetsov A (1985) Proof-intuitionistic propositional calculus. Doklady Academii Nauk SSSR 283:27–30 (Russian)

    Google Scholar 

  42. Kuznetsov A, Muravitskij A (1980) Provability as modality. In: Actual problems of logic and methodology of science, Naukova Dumka, Kiev, pp 193–230 (Russian)

    Google Scholar 

  43. Lewis C, Langford C (1932) Symbolic logic. Appleton-Century-Crofts, New York

    Google Scholar 

  44. Maehara S (1954) Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya Math J 7:45–64

    Google Scholar 

  45. Makkai M, Reyes G (1995) Completeness results for intuitionistic and modal logic in a categorical setting. Ann Pure Appl Logic 72(1):25–101

    Google Scholar 

  46. Maksimova L (1982) Failure of the interpolation property in modal companions of Dummett’s logic. Algebra Logic 21:690–694

    Article  Google Scholar 

  47. Maksimova L, Rybakov V (1974) Lattices of modal logics. Algebra Logic 13:105–122

    Article  Google Scholar 

  48. McKinsey J, Tarski A (1944) The algebra of topology. Ann Math 45:141–191

    Google Scholar 

  49. McKinsey J, Tarski A (1948) Some theorems about the sentential calculi of Lewis and Heyting. J Symbolic Logic 13:1–15

    Google Scholar 

  50. Medvedev Y (1966) Interpretation of logical formulas by means of finite problems. Sov Math Doklady 7:857–860

    Google Scholar 

  51. Muravitskij A (1985) Correspondence of proof-intuitionistic logic extensions to provability logic extensions. Sov Math Doklady 31:345–348

    Google Scholar 

  52. Muravitsky A (1989) The correspondence of proof-intuitionistic logic extensions to proof-logic extensions. Mat Logika i Algoritm Probl Trudy Inst Mat 12:104–120

    Google Scholar 

  53. Muravitsky A (2006) The embedding theorem: its further developments and consequences. Part 1. Notre Dame J Formal Logic 47(4):525–540

    Article  Google Scholar 

  54. Nelson D (1947) Recursive functions and intuitionistic number theory. Trans Am Math Soc 61:307–368

    Article  Google Scholar 

  55. Novikov P (1977) Constructive mathematical logic from the point of view of classical logic. Nauka, Moscow (Russian)

    Google Scholar 

  56. Ono H (1977) On some intuitionistic modal logics. Publ Res Inst Math Sci Kyoto Univ 13:55–67

    Article  Google Scholar 

  57. Orlov I (1928) The calculus of compatibility of propositions. Math USSR Sbornik 35:263–286 (Russian)

    Google Scholar 

  58. Pinto L, Uustalu T (2010) Relating sequent calculi for bi-intuitionistic propositional logic. In: CL&C, pp 57–72

    Google Scholar 

  59. Prior A (1957) Time and modality. Clarendon Press, Oxford

    Google Scholar 

  60. Rasiowa H, Sikorski R (1963) The mathematics of metamathematics. Polish Scientific Publishers, Warsaw

    Google Scholar 

  61. Rauszer C (1974) A formalization of the propositional calculus of H-B logic. Studia Logica 33:23–34

    Google Scholar 

  62. Rauszer C (1974) Semi-boolean algebras and their applications to intuitionistic logic with dual operators. Fundam Math 83:219–249

    Google Scholar 

  63. Rauszer C (1977) Applications of Kripke models to Heyting-Brouwer logic. Studia Logica 36:61–71

    Google Scholar 

  64. Reyes G, Zawadowski M (1993) Formal systems for modal operators on locales. Studia Logica 52(4):595–614

    Google Scholar 

  65. Rose G (1953) Propositional calculus and realizability. Trans Am Math Soc 75:1–19

    Article  Google Scholar 

  66. Rybakov V (1976) Hereditarily finitely axiomatizable extensions of the logic \({S}4\). Algebra Logic 15:115–128

    Google Scholar 

  67. Schütte K (1968) Vollständige systeme modaler und intuitionistischer logik. Springer, Berlin

    Book  Google Scholar 

  68. Shehtman V (1980) Topological models of propositional logics. Semiot Inf Sci 15:74–98 (Russian)

    Google Scholar 

  69. Shehtman V (1999) “Everywhere” and “here”. J Appl Non-Classical Logics 9(2–3)

    Google Scholar 

  70. Simpson A (1994) The proof theory and semantics of intuitionistic modal logic. PhD thesis, University of Edinburgh

    Google Scholar 

  71. Sotirov V (1984) Modal theories with intuitionistic logic. In: Proceedings of the conference on mathematical logic, Sofia, Bulgarian Academy of Sciences, pp 139–171

    Google Scholar 

  72. Umezawa T (1955) Über die Zwischensysteme der Aussagenlogik. Nagoya Math J 9:181–189

    Google Scholar 

  73. Umezawa T (1959) On intermediate propositional logics. J Symbolic Logic 24:20–36

    Article  Google Scholar 

  74. Wolter F (1995) The finite model property in tense logic. J Symbolic Logic 60:757–774

    Article  Google Scholar 

  75. Wolter F (1998) On logics with coimplication. J Philos Logic 27:387

    Article  Google Scholar 

  76. Wolter F, Zakharyaschev M (1997) On the relation between intuitionistic and classical modal logics. Algebra Logic 36:121–155

    Article  Google Scholar 

  77. Wolter F, Zakharyaschev M (1999) Intuitionistic modal logics as fragments of classical bimodal logics. In: Orlowska E (ed) Logic at work. Springer, Berlin, pp 168–186

    Google Scholar 

  78. Zakharyaschev M (1989) Syntax and semantics of intermediate logics. Algebra Logic 28:262–282

    Google Scholar 

  79. Zakharyaschev M (1991) Modal companions of superintuitionistic logics: syntax, semantics and preservation theorems. Math USSR Sbornik 68:277–289

    Article  Google Scholar 

  80. Zakharyaschev M (1992) Canonical formulas for \({K}4\). Part I: basic results. J Symbolic Logic 57:1377–1402

    Article  Google Scholar 

  81. Zakharyaschev M (1996) Canonical formulas for \({K}4\). Part II cofinal subframe logics. J Symbolic Logic 61:421–449

    Article  Google Scholar 

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Acknowledgments

We are grateful to Guram Bezhanishvili, Alexander Chagrov, Alex Citkin, Tadeusz Litak, Larisa Maksimova, Aleksei Muravitsky and Vladimir Rybakov for their help and comments.

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Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    Frank Wolter

  2. Department of Computer Science and Information Systems, Birkbeck, University of London, London, UK

    Michael Zakharyaschev

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Correspondence to Frank Wolter .

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  1. New Mexico State University, Las Cruces, New Mexico, USA

    Guram Bezhanishvili

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Wolter, F., Zakharyaschev, M. (2014). On the Blok-Esakia Theorem. In: Bezhanishvili, G. (eds) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8860-1_5

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