Skip to main content

Positive Formulas in Intuitionistic and Minimal Logic

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8984)

Abstract

In this article we investigate the positive, i.e. \(\lnot ,\bot \)-free formulas of intuitionistic propositional and predicate logic, IPC and IQC, and minimal logic, MPC and MQC. For each formula \(\varphi \) of IQC we define the positive formula \(\varphi ^+\) that represents the positive content of \(\varphi \). The formulas \(\varphi \) and \(\varphi ^+\) exhibit the same behavior on top models, models with a largest world that makes all atomic sentences true. We characterize the positive formulas of IPC and IQC as the formulas that are immune to the operation of turning a model into a top model. With the +-operation on formulas we show, using the uniform interpolation theorem for IPC, that both the positive fragment of IPC and MPC respect a revised version of uniform interpolation. In propositional logic the well-known theorem that KC is conservative over the positive fragment of IPC is shown to generalize to many logics with positive axioms. In first-order logic, we show that IQC + DNS (double negation shift) + KC is conservative over the positive fragment of IQC and similar results as for IPC.

Keywords

  • Intuitionistic logic
  • Minimal logic
  • Jankov’s logic
  • Intermediate logics
  • Positive formulas
  • Interpolation
  • Conservativity

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-662-46906-4_11
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-662-46906-4
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)

Notes

  1. 1.

    We do not consider identity and functional symbols, but our results will surely hold for the extension with such symbols.

  2. 2.

    A Kripke frame is of depth \(n\) if the largest chain contains \(n\) nodes.

References

  1. Brouwer, L.E.J.: Essentieel negatieve eigenschappen. Indag. Math. 10, 322–323 (1948)

    Google Scholar 

  2. Chagrov, A.V., Zakharyaschev, M.: Modal Logic. Oxford Logic Guides. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  3. de Jongh, D., Yang, F.: Jankov’s theorems for intermediate logics in the setting of universal models. In: Bezhanishvili, N., Löbner, S., Schwabe, K., Spada, L. (eds.) TbiLLC 2009. LNCS, vol. 6618, pp. 53–76. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  4. Diego, A.: Sur les Algèbres de Hilbert, vol. 21. E. Nauwelaerts, Gauthier-Villars, Louvain (1966)

    MATH  Google Scholar 

  5. Enderton, H.B.: A Mathematical Introduction to Logic. Harcourt/Academic Press, Burlington (2001)

    MATH  Google Scholar 

  6. Gabbay, D.M., de Jongh, D.H.J.: A sequence of decidable finitely axiomatizable: intermediate logics with the disjunction property. J. Symbolic Logic 39(1), 67–78 (1974)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Gabbay, D.M., Shehtman, V., Skvortsov, D.: Quantification in Nonclassical Logic I. Studies in Logic and the Foundations of Mathematics. Clarendon Press, Oxford (2009)

    MATH  Google Scholar 

  8. Ghilardi, S., Zawadowski, M.W.: Undefinability of propositional quantifiers in the modal system S4. Stud. Logica. 55(2), 259–271 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Griss, G.F.C.: Negationless intuitionistic mathematics I. Indag. Math. 8, 675–681 (1946)

    Google Scholar 

  10. Heyting, A.: Die formalen Regeln der intuitionistischen Logik, pp. 42–56 (1930)

    Google Scholar 

  11. Jankov, V.A.: Calculus of the weak law of the excluded middle (in russian). Izv. Akad. Nauk SSSR Ser. Mat. 32(5), 1044–1051 (1968)

    MATH  MathSciNet  Google Scholar 

  12. Johansson, I.: Der Minimalkalkül Ein Reduzierter Intuitionistischer Formalismus. Compos. Math. 4, 119–136 (1937)

    Google Scholar 

  13. Kolmogorov, A.: Zur Deutung der intuitionistischen Logik. Math. Z. 35(1), 58–65 (1932)

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Pitts, A.M.: On an interpretation of second order quantification in first order intuitionistic propositional logic. J. Symb. Logic 57, 33–52 (1992)

    CrossRef  MATH  MathSciNet  Google Scholar 

  15. Renardel de Lavalette, G.R., Hendriks, A., de Jongh, D.: Intuitionistic implication without disjunction. J. Log. Comput. 22(3), 375–404 (2012)

    CrossRef  MATH  Google Scholar 

  16. Troelstra, A., van Dalen, D.: Constructivism in Mathematics, vols. 2. North-Holland, Amsterdam (1988)

    Google Scholar 

  17. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2000)

    CrossRef  MATH  Google Scholar 

  18. Tzimoulis, A., Zhao, Z.: The Universal Model for the Negation-free Fragment of IPC. Technical Notes (X) Series X-2013-01, ILLC, University of Amsterdam (2013)

    Google Scholar 

  19. van Dalen, D.: Logic and Structure. Universitext. Springer, London (2012)

    Google Scholar 

  20. Visser, A.: Uniform interpolation and layered bisimulation. In: Hájek, P. (ed.) Proceedings of Gödel 1996: Logical Foundations of Mathematics, Computer Science and Physics - Kurt Gödel’s Legacy, Brno, Czech Republic. Lecture Notes Logic, vol. 6, pp. 139–164. Springer Verlag, Berlin (1996)

    Google Scholar 

  21. Zhao, Z.: An Investigation of Jankov’s Logic (2012) (Unpublished paper)

    Google Scholar 

Download references

Acknowledgement

We thank Albert Visser, Nick Bezhanishvili, Rosalie Iemhoff, Grisha Mints and Anne Troelstra for informative discussions on the subject. We thank the referees for their corrections and Linde Frölke for her preparatory work.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dick de Jongh .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

de Jongh, D., Zhao, Z. (2015). Positive Formulas in Intuitionistic and Minimal Logic. In: Aher, M., Hole, D., Jeřábek, E., Kupke, C. (eds) Logic, Language, and Computation. TbiLLC 2013. Lecture Notes in Computer Science(), vol 8984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46906-4_11

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-46906-4_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46905-7

  • Online ISBN: 978-3-662-46906-4

  • eBook Packages: Computer ScienceComputer Science (R0)