Constructive Embedding from Extensions of Logics of Strict Implication into Modal Logics

Part of the Logic in Asia: Studia Logica Library book series (LIAA)


Dyckhoff and Negri (Arch Math Logic 51:71–92 (2012), [8]) give a constructive proof of Gödel–Mckinsey–Tarski embedding from intermediate logics to modal logics via labelled sequent calculi. Then, they regard a monotonicity of atomic propositions in intuitionistic logic as an initial sequent, i.e., an axiom. However, we regard the monotonicity as an additional inference rule and employ a modified translation sending an atomic variable P to \( P \& \Box P\) to generalize their result to an embedding from extensions of Corsi’s \(\mathbf {F}\) of logic of strict implication to normal extensions of modal logics \(\mathbf {K}\). In this process, we provide a \(\mathbf {G3}\)-style labelled sequent calculi for extensions of \(\mathbf {F}\) and show that our calculi admit the cut rule and enjoy soundness and completeness for Kripke semantics.


Labelled sequent calculus Modal logic Intermediate logic Gödel–Mckinsey–Tarski embedding Cut elimination Completeness Kripke semantics Strict implication 



We would like to thank an anonymous reviewer for his/her invaluable comments. We also would like to thank Sara Negri for her sharing her draft [23] on a similar topic to our paper. We are grateful to Ryo Kashima for setting opportunities for the first author to give presentations on this topic at Tokyo Institute of Technology for giving helpful suggestions to us. The first author wishes to thank her supervisor Kengo Okamoto for a regular weekly discussion. The authors have presented material related to this paper at several occasions. We would like to thank the audiences of these events, including 2014 annual meetings of the Japan Association for Philosophy of Science in Japan, Trends in Logic XIII in Poland, the Second Taiwan Philosophical Logic Colloquium (TPLC 2014) in Taiwan, and the 49th MLG meeting at Kaga, Japan. The first author’s visit to Taiwan for attending TPLC 2014 was supported by the grant from Tokyo Metropolitan University for graduate students. The work of the second author was partially supported by JSPS Core-to-Core Program (A. Advanced Research Networks) and JSPS KAKENHI, Grant-in-Aid for Young Scientists (B) 24700146 and 15K21025.


  1. 1.
    Aghaei, M., Ardeshir, M.: A bounded translation of intuitionistic propositional logic into basic propositional logic. Math. Log. Q. 46, 199–206 (2000)CrossRefGoogle Scholar
  2. 2.
    Ardeshir, M., Ruitenburg, W.: Basic propositional calculus I. Math. Log. Q. 44, 317–343 (1998)CrossRefGoogle Scholar
  3. 3.
    Cerrato, C.: Natural deduction based upon strict implication for normal modal logics. Notre Dome J. Form. Log. 35, 471–495 (1994)CrossRefGoogle Scholar
  4. 4.
    Chagrov, A., Zakharyaschev, N.: Modal Logic. Oxford University Press (1997)Google Scholar
  5. 5.
    Corsi, G.: Weak logics with strict implication. Math. Log. Q. 33, 389–406 (1987)CrossRefGoogle Scholar
  6. 6.
    Došen, K.: Modal translation in K and D. Diamond and Defaults, pp. 103–127 (1993)Google Scholar
  7. 7.
    Dragalin, A.: Mathmatical Intuitionism: Introduction to Proof Theory. American Mathematics Society (1988)Google Scholar
  8. 8.
    Dyckhoff, R., Negri, S.: Proof analysis in intermediate logics. Arch. Math. Log. 51, 71–92 (2012)CrossRefGoogle Scholar
  9. 9.
    Gödel, K.: Eine interpretation des intuitionistischen Aussagenkalküls. Ergebnisse Eines Mathematischen Kolloquiums 4, 39–40 (1933)Google Scholar
  10. 10.
    Hughes, G., Cresswell, M.: A New Introduction to Modal Logic. Routledge, London (1996)CrossRefGoogle Scholar
  11. 11.
    Ishigaki, R., Kashima, R.: Sequent calculi for some strict implication logics. Log. J. IGPL 16(2), 155–174 (2008)CrossRefGoogle Scholar
  12. 12.
    Ishii, K., Kashima, R., Kikuchi, K.: Sequent calculi for Visser’s propositional logics. Notre Dame J. Form. Log. 42(1), 1–22 (2001)CrossRefGoogle Scholar
  13. 13.
    Kikuchi, K.: Relationships between basic propositional calculus and substructural logics. Bull. Sect. Log. 30(1), 15–20 (2001)Google Scholar
  14. 14.
    Kikuchi, K., Sasaki, K.: A cut-free Gentzen formulation of basic propositional calculus. J. Log. Lang. Inf. 12, 213–225 (2003)CrossRefGoogle Scholar
  15. 15.
    Kleene, S.C.: Introduction to Metamathematics. North-Holland Public Co. (1952)Google Scholar
  16. 16.
    Lewis, C.I.: Implication and the algebra of logic. Mind 21, 522–531 (1912)CrossRefGoogle Scholar
  17. 17.
    Lewis, C.I.: A new algebra of strict implications and some consequents. J. Philos. Psychol. Sci. Methods 10, 428–438 (1913)Google Scholar
  18. 18.
    Ma, M., Sano, K.: On extensions of basic propositional logic. In: Proceedings of the 13th Asian Logic Conference, pp. 170–200 (2015)Google Scholar
  19. 19.
    Mckinsey, J.C.C., Tarski, A.: Some theorems about the sentential calculi of Lewis and Heyting. J. Symbol. Log. 13, 1–15 (1948)CrossRefGoogle Scholar
  20. 20.
    Mints, G.: The Gödel-Tarski translations of intuitionistic propositional formulas. Correct Reason. 487–491 (2012)Google Scholar
  21. 21.
    Negri, S.: Proof analysis in modal logic. J. Philos. Log. 34, 507–544 (2005)CrossRefGoogle Scholar
  22. 22.
    Negri, S.: Proof analysis in non-classical logics. Logic Colloquium’ 05,ASL Lecture Notes in Logic, vol. 28, pp. 107–128 (2008)Google Scholar
  23. 23.
    Negri, S.: The intensional side of algebraic-topological representation theorems. SubmittedGoogle Scholar
  24. 24.
    Negri, S., Von Plato, J.: Structural Proof Theory. Cambridge University Press (2001)Google Scholar
  25. 25.
    Negri, S., Von Plato, J.: Proof Analysis. Cambridge University Press (2011)Google Scholar
  26. 26.
    Restall, G.: Subintuitionistic logics. Notre Dome J. Form. Log. 35, 116–129 (1994)CrossRefGoogle Scholar
  27. 27.
    Ruitenburg, W.: Constructive logic and the paradoxes. Modern Log. 1, 271–301 (1991)Google Scholar
  28. 28.
    Sano, K., Ma, M.: Alternative semantics for Visser’s propositional logics. In: Logic, Language, and Computation, volume 8984 of Lecture Notes in Computer Science, pp. 257–275 (2015)Google Scholar
  29. 29.
    Suzuki, Y., Ono, H.: Hilbert-style proof system for BPL. Technical Report IS-RR-97-0040F, Japan Advanced Institute of Science and Technology (1997)Google Scholar
  30. 30.
    Suzuki, Y., Wolter, F., Zakharyaschev, M.: Speaking about transitive frames in propositional languages. J. Log. Lang. Inf. 7, 317–339 (1998)CrossRefGoogle Scholar
  31. 31.
    Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press (2000)Google Scholar
  32. 32.
    Visser, A.: A propositional logic with explicit fixed points. Studia Logica 40, 155–175 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Graduate School of HumanitiesTokyo Metropolitan UniversityTokyoJapan
  2. 2.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan

Personalised recommendations