Skip to main content
SpringerLink
Log in

On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

By using algebraic-categorical tools, we establish four criteria in order to disprove canonicity, strong completeness, w-canonicity and strong w-completeness, respectively, of an intermediate propositional logic. We then apply the second criterion in order to get the following result: all the logics defined by extra-intuitionistic one-variable schemata, except four of them, are not strongly complete. We also apply the fourth criterion in order to prove that the Gabbay-de Jongh logic D1 is not strongly w-complete.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chagrov, A., and M. Zakharyaschev, Modal Logic, Clarendon Press, Oxford 1997.

    Google Scholar 

  2. Ferrari, M., and P. Miglioli, ‘A method to single out maximal propositional logics with the disjunction property I’, Annals of Pure and Applied Logic 76 (1995), 1–46.

    Article  Google Scholar 

  3. Ferrari, M., and P. Miglioli, ‘A method to single out maximal propositional logics with the disjunction property II’, Annals of Pure and Applied Logic 76 (1995), 117–168.

    Article  Google Scholar 

  4. Fine, K., ‘Normal forms in modal logic’, Notre Dame Journal of Formal Logic 16 (1975), 31–42.

    Article  Google Scholar 

  5. Gabbay, D. M., and D. de Jongh, ‘A Sequence of decidable finitely axiomatizable intermediate logics with the disjunction property’, Journal of Symbolic Logic 39 (1974), 67–78.

    Article  Google Scholar 

  6. Ghilardi, S., ‘Free Heyting algebras as bi-Heyting algebras’, Math. Rep. Acad. Sci. of Canada XVI (1992), 6, 240–244.

    Google Scholar 

  7. Ghilardi, S., and G. Meloni, ‘Constructive canonicity in non classical logics’, Annals of Pure and Applied Logic 86 (1997), 1–32.

    Article  Google Scholar 

  8. Goldblatt, R. I., ‘The McKinsey axiom is not canonical’, Journal of Symbolic Logic 56 (1991), 554–562.

    Article  Google Scholar 

  9. Miglioli, P., ‘An infinite class of maximal intermediate propositional logics with the disjunction property’, Archive for Mathematical Logic 31 (1991), 6, 415–432.

    Article  Google Scholar 

  10. Mc Lane, S., Categories for the Working Mathematician, Springer, Berlin 1971.

    Google Scholar 

  11. Shimura, T., ‘On completeness of intermediate predicate logics with respect to Kripke semantics’, Bulletin of the Section of Logic 24 (1995), 41–45.

    Google Scholar 

  12. Wang, X., ‘The McKinsey axiom is not compact’, Journal of Symbolic Logic 57 (1992), 1230–1238.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze della Informazione, Università degli Studi di Milano, via Comelico 39/41, 20135, Milano, Italy

    Silvio Ghilardi & Pierangelo Miglioli

Authors
  1. Silvio Ghilardi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierangelo Miglioli
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ghilardi, S., Miglioli, P. On Canonicity and Strong Completeness Conditions in Intermediate Propositional Logics. Studia Logica 63, 353–385 (1999). https://doi.org/10.1023/A:1005203020752

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1005203020752

Search

Navigation