A Universal Algebra for the Variable-Free Fragment of \({\mathrm {RC}^\nabla }\)

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10703)

Abstract

The language of Reflection Calculus \(\mathrm {RC}\) consists of implications between formulas built up from propositional variables and the constant ‘true’ using only conjunction and the diamond modalities which are interpreted in Peano arithmetic as restricted uniform reflection principles. In [6] we introduced \({\mathrm {RC}^\nabla }\), an extension of \(\mathrm {RC}\) by a series of modalities representing the operators associating with a given arithmetical theory T its fragment axiomatized by all theorems of T of arithmetical complexity \(\varPi ^0_n\), for all \(n>0\). In this paper we continue the study of the variable-free fragment of \({\mathrm {RC}^\nabla }\) and characterize its Lindenbaum–Tarski algebra in several natural ways.

Keywords

Strictly positive logics Reflection principle Provability GLP 

References

  1. 1.
    Beklemishev, L.D., Onoprienko, A.A.: On some slowly terminating term rewriting systems. Sbornik Math. 206, 1173–1190 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Beklemishev, L.D.: Provability algebras and proof-theoretic ordinals. Ann. Pure Appl. Logic 128, 103–123 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Beklemishev, L.D.: Reflection principles and provability algebras in formal arithmetic. Russ. Math. Surv. 60(2), 197–268 (2005). Russian original: Uspekhi Matematicheskikh Nauk, 60(2): 3–78 (2005)CrossRefMATHGoogle Scholar
  4. 4.
    Beklemishev, L.D.: The Worm principle. In: Chatzidakis, Z., Koepke, P., Pohlers, W. (eds.) Logic Colloquium 2002. Lecture Notes in Logic, vol. 27, pp. 75–95. AK Peters (2006). Preprint: Logic Group Preprint Series 219, Utrecht University, March 2003Google Scholar
  5. 5.
    Beklemishev, L.D.: Calibrating provability logic: from modal logic to reflection calculus. In: Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (eds.) Advances in Modal Logic, vol. 9, pp. 89–94. College Publications, London (2012)Google Scholar
  6. 6.
    Beklemishev, L.D.: On the reflection calculus with partial conservativity operators. In: Kennedy, J., de Queiroz, R.J.G.B. (eds.) WoLLIC 2017. LNCS, vol. 10388, pp. 48–67. Springer, Heidelberg (2017).  https://doi.org/10.1007/978-3-662-55386-2_4 CrossRefGoogle Scholar
  7. 7.
    Beklemishev, L.D., Joosten, J., Vervoort, M.: A finitary treatment of the closed fragment of Japaridze’s provability logic. J. Logic Comput. 15(4), 447–463 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Boolos, G.: The Logic of Provability. Cambridge University Press, Cambridge (1993)MATHGoogle Scholar
  9. 9.
    Dashkov, E.V.: On the positive fragment of the polymodal provability logic GLP. Matematicheskie Zametki 91(3), 331–346 (2012). English translation: Mathematical Notes 91(3):318–333, 2012MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fernández-Duque, D., Joosten, J.: Models of transfinite provability logic. J. Symbol. Logic 78(2), 543–561 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Reyes, E.H., Joosten, J.J.: The logic of Turing progressions. arXiv:1604.08705v2 [math.LO] (2016)
  12. 12.
    Reyes, E.H., Joosten, J.J.: Relational semantics for the Turing Schmerl calculus. arXiv:1709.04715 [math.LO] (2017)
  13. 13.
    Icard III, T.F.: A topological study of the closed fragment of GLP. J. Logic Comput. 21(4), 683–696 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ignatiev, K.N.: On strong provability predicates and the associated modal logics. J. Symbol. Logic 58, 249–290 (1993)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jackson, M.: Semilattices with closure. Algebra Universalis 52, 1–37 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Japaridze, G.K.: The modal logical means of investigation of provability. Thesis in Philosophy, in Russian, Moscow (1986)Google Scholar
  17. 17.
    Joosten, J.J.: Turing–Taylor expansions of arithmetical theories. Studia Logica 104, 1225–1243 (2015).  https://doi.org/10.1007/s11225-016-9674-z CrossRefMATHGoogle Scholar
  18. 18.
    Kikot, S., Kurucz, A., Tanaka, Y., Wolter, F., Zakharyaschev, M.: On the completeness of EL-equations: first results. In: 11th International Conference on Advances in Modal Logic, Short Papers (Budapest, 30 August – 2 September, 2016), pp. 82–87 (2016)Google Scholar
  19. 19.
    Kikot, S., Kurucz, A., Tanaka, Y., Wolter, F., Zakharyaschev, M.: Kripke Completeness of strictly positive modal logics over meet-semilattices with operators. ArXiv e-prints, August 2017Google Scholar
  20. 20.
    Kurucz, A., Tanaka, Y., Wolter, F., Zakharyaschev, M.: Conservativity of Boolean algebras with operators over semi lattices with operators. In: Proceedings of TACL 2011, pp. 49–52 (2011)Google Scholar
  21. 21.
    Pakhomov, F.: On the complexity of the closed fragment of Japaridze’s provability logic. Archive Math. Logic 53(7), 949–967 (2014)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pakhomov, F.: On elementary theories of ordinal notation systems based on reflection principles. Proc. Steklov Inst. Math. 289, 194–212 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Shamkanov, D.: Nested sequents for provability logic GLP. Logic J. IGPL 23(5), 789–815 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sofronie-Stokkermans, V.: Locality and subsumption testing in EL and some of its extensions. In: Areces, C., Goldblatt, R. (eds.) Advances in Modal Logic, vol. 7, pp. 315–339. College Publications, London (2008)Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

Personalised recommendations