Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T06:13:33.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Counting the maximal intermediate constructive logics

Published online by Cambridge University Press:  12 March 2014

Mauro Ferrari and
Pierangelo Miglioli
Mauro Ferrari
Affiliation:
Universita degli Studi di Milano, Dipartimento di Scienze Dell'Informazione, 20135 Milano, Italia, E-mail: ferrari@hermes.dsi.unimi.it
Pierangelo Miglioli
Affiliation:
Universita degli Studi di Milano, Dipartimento di Scienze Dell'Informazione, 20135 Milano, Italia, E-mail: miglioli@hermes.dsi.unimi.it
Get accessRights & Permissions [Opens in a new window]

Abstract

A proof is given that the set of maximal intermediate propositional logics with the disjunction property and the set of maximal intermediate predicate logics with the disjunction property and the explicit definability property have the power of continuum. To prove our results, we introduce various notions which might be interesting by themselves. In particular, we illustrate a method to generate wide sets of pairwise “constructively incompatible constructive logics”. We use a notion of “semiconstructive” logic and define wide sets of “constructive” logics by representing the “constructive” logics as “limits” of decreasing sequences of “semiconstructive” logics. Also, we introduce some generalizations of the usual filtration techniques for propositional logics. For instance, “fitrations over rank formulas” are used to show that any two different logics belonging to a suitable uncountable set of “constructive” logics are “constructively incompatible”.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 4 , December 1993, pp. 1365 - 1401
Copyright
Copyright © Association for Symbolic Logic 1993

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.)

Article purchase

Temporarily unavailable

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. V., The cardinality of the set of maximal intermediate logics with the disjunction property is of continuum, Matematicheskie Zametki, vol. 51, No. 2 (1992), pp. 117–123. (Russian)Google Scholar
[3]Chagrov, A. V. and Zacharyaschev, M. V., The disjunction property of intermediate propositional logics, The ITLI prepuhlication series, X-91-01, Department of Mathematics and Computer Science, University of Amsterdam, Amsterdam, 1991.Google Scholar
[4]Doets, K., Definability in intensional and high order logic, Ph.D. Thesis, University of Amsterdam, Amsterdam, 1986.Google Scholar
[5]Fine, K., Logics containing K4, part II, this Journal, vol. 50 (1985), pp. 619–651.Google Scholar
[6]Gabbay, D. M., The decidability of the Kreisel-Putnam system, this Journal, vol. 35 (1970), pp. 431–437.Google Scholar
[7]Gabbay, D. M., Semantical investigations in Heyting's intuitionistic logic, Reidel, Dordrecht, 1981.CrossRefGoogle Scholar
[8]Gabbay, D. M. and de Jongh, D. H. J., A sequence of decidable finitely axiomatizable intermediate logics with the disjunction property, this Journal, vol. 39 (1974), pp. 67–78.Google Scholar
[9]Galanter, G. I., A continuum of intermediate logics which are maximal among the logics having the intuitionistic disjunctionless fragment, Proceedings of 10th USSR Conference for Mathematical Logic, Alma Ata, 1990, p. 41. (Russian)Google Scholar
[10]Jankov, V. A., Constructing a sequence of strongly independent superintuilionistic propositional calculi, Soviet Mathematics-Doklady, vol. 9 (1968), pp. 806–807.Google Scholar
[11]Kirk, R. E., A result on propositional logics having the disjunction properly, Notre Dame Journal of Formal Logic, vol. 23, No. 1 (1982), pp. 71–74.CrossRefGoogle Scholar
[12]Kracht, M., Intensional definability and completeness in modal logic, Ph.D. Thesis, Freie Universität, Berlin, 1990.Google Scholar
[13]Kreisel, G. and Putnam, H., Eine Unableitbarkeitsbeweismethode für den intuit ionistischen Aussagenkalkül, Archiv für Mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 74–78.CrossRefGoogle Scholar
[14]Maksimova, L. L., On maximal intermediate logics with the disjunction property, Studia Logica, vol. 45 (1986), pp. 69–75.CrossRefGoogle Scholar
[15]Maksimova, L. L., The number of maximal intermediate logics with the disjunction property, Proceedings of the 7th All-Union Conference for Mathematical Logic, Novosibirsk, 1984. (Russian)Google Scholar
[16]Miglioli, P., An infinite class of maximal intermediate propositional logics with the disjunction property. Archive for Mathematical Logic, vol. 31, No. 6 (1992), pp. 415–432.CrossRefGoogle Scholar
[17]Minari, P., Indagini semantiche sulle logiche intermedie proposizionali, Bibliopolis, Naples, 1989. (Italian)Google Scholar
[18]Nagata, S., A series of successive modifications of Peirce's rule, Proceedings of the Japan Academy, Mathematical Sciences, vol. 42 (1966), pp. 859–861.Google Scholar
[19]Ono, H., A study of intermediate predicate logics, Publications of the Research Institute for Mathematical Sciences, vol. 8 (1972), pp. 619–649.CrossRefGoogle Scholar
[20]Ono, H., Some problems on intermediate predicate logics, Reports on Mathematical Logic, vol. 21 (1987), pp. 55–67.Google Scholar
[21]Rybakov, V. V., Admissible rules of pretabular modal logics, Algehra i Logika, vol. 20, No. 4 (1981), pp. 440–464. (Russian)Google Scholar
[22]Segerberg, K., Propositional logics related to Heyting and Johansson, Theoria, vol. 34 (1968), pp. 26–61.CrossRefGoogle Scholar
[23]Shehtman, V. B., Rieger-Nishimura lattices, Doklady AN SSSR, vol. 241 (1978), pp. 1288–1291. (Russian)Google Scholar
[24]Shehtman, V. B., Applications of Kripke models to studies in modal and intermediate logic, Ph.D. Thesis, Moscow State University, Moscow, 1984. (Russian)Google Scholar
[25]Smorinski, C. A., Applications of Kripke models, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis (Troelstra, A. S., editor), Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, Heidelberg, and New York, 1973, pp. 324–391.CrossRefGoogle Scholar
[26]Wronski, A., Intermediate logics with the disjunction property, Reports on Mathematical Logic, vol. 1 (1973), pp. 35–51.Google Scholar