Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-17T18:56:52.671Z Has data issue: false hasContentIssue false
  • English
  • Français

All intermediate logics with extra axioms in one variable, except eight, are not strongly ω-complete

Published online by Cambridge University Press:  12 March 2014

Camillo Fiorentini
Camillo Fiorentini*
Affiliation:
Dipartimento di Scienze dell'Informazione, Università Degli Studi di Milano, Via Comelico 39, 20135 Milano, Italy, E-mail:fiorenti@dsi.unimi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

In [8] it is proved that all the intermediate logics axiomatizable by formulas in one variable, except four of them, are not strongly complete. We considerably improve this result by showing that all the intermediate logics axiomatizable by formulas in one variable, except eight of them, are not strongly ω-complete. Thus, a definitive classification of such logics with respect to the notions of canonicity, strong completeness, ω-canonicity and strong ω-completeness is given.

Keywords

ω-canonicityextensive ω-canonicitystrong ω-completeness
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1576 - 1604
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Anderson, J. G., Superconstructive propositional calculi with extra schemes containing one variable, Zeitschrift fÜr Mathematische Logik und Grundlagen der Mathematik, vol. 18 (1972), pp. 113–130.CrossRefGoogle Scholar
[2]Chagrov, A. and Zakharyaschev, M., The disjunction property of intermediate propositional logics, Studia Logica, vol. 50 (1991), no. 2, pp. 189–216.CrossRefGoogle Scholar
[3], Modal logic, Oxford University Press, 1997.Google Scholar
[4]Cresswell, M. J. and Hughes, G. E., An introduction to modal logic, Methuen, London, 1968.Google Scholar
[5]Ferrari, M. and Miglioli, P., A method to single out maximal propositional logics with the disjunction property I, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 1–46.CrossRefGoogle Scholar
[6]Ferrari, M. and Miglioli, P.A method to single out maximal propositional logics with the disjunction property II, Annals of Pure and Applied Logic, vol. 76 (1995), pp. 117–168.CrossRefGoogle Scholar
[7]Fiorentini, C., Kripke completeness for intermediate logics, PhD thesis, Dipartimento di Scienze dell'Informazione. UniversitÀ degli Studi di Milano, Italy, 1999, available at http://gongolo.usr.dsi.unimi.it/~fiorenti.Google Scholar
[8]Ghilardi, S. and Miglioli, P., On canonicity and strong completeness conditions in intermediate propositional logics, Studia Logica, vol. 63 (1999), no. 3, pp. 353–385.CrossRefGoogle Scholar
[9]Ghilardi, S. and Zawadowski, M., A sheaf representation and duality for finitely presented Heyting algebras, this Journal, vol. 60 (1995), no. 3, pp. 911–939.Google Scholar
[10]Levy, A., Basic set theory, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[11]Nishimura, I., On formulas of one variable in intuitionistic propositional calculus, this Journal, vol. 25 (1960), pp. 327–331.Google Scholar
[12]Shimura, T., On completeness of intermediate predicate logics with respect to Kripke semantics, Bulletin of the Section of Logic, vol. 24 (1995), pp. 41–45.Google Scholar
[13]Sobolev, S. K., On the finite approximability of superintuitionistk logics, Mathematics of USSR, Sbornik, vol. 31 (1977), pp. 257–268.CrossRefGoogle Scholar
[14]Zakharyaschev, M., Canonical formulas for K4. III. The finite model property, this Journal, vol. 62 (1997), no. 3, pp. 950–975.Google Scholar