Skip to main content
×
Home
    • Aa
    • Aa

Counting the maximal intermediate constructive logics

  • Mauro Ferrari (a1) and Pierangelo Miglioli (a2)
Abstract
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”.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1] J. G. Anderson , Superconstructive propositional calculi with extra schemes containing one variable, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 18 (1972), pp. 113130.

[7] D. M. Gabbay , Semantical investigations in Heyting's intuitionistic logic, Reidel, Dordrecht, 1981.

[11] R. E. Kirk , A result on propositional logics having the disjunction properly, Notre Dame Journal of Formal Logic, vol. 23, No. 1 (1982), pp. 7174.

[13] G. Kreisel and H. Putnam , Eine Unableitbarkeitsbeweismethode für den intuit ionistischen Aussagenkalkül, Archiv für Mathematische Logik und Grundlagenforschung, vol. 3 (1957), pp. 7478.

[14] L. L. Maksimova , On maximal intermediate logics with the disjunction property, Studia Logica, vol. 45 (1986), pp. 6975.

[16] P. Miglioli , An infinite class of maximal intermediate propositional logics with the disjunction property. Archive for Mathematical Logic, vol. 31, No. 6 (1992), pp. 415432.

[18] S. Nagata , A series of successive modifications of Peirce's rule, Proceedings of the Japan Academy, Mathematical Sciences, vol. 42 (1966), pp. 859861.

[19] H. Ono , A study of intermediate predicate logics, Publications of the Research Institute for Mathematical Sciences, vol. 8 (1972), pp. 619649.

[22] K. Segerberg , Propositional logics related to Heyting and Johansson, Theoria, vol. 34 (1968), pp. 2661.

[25] C. A. Smorinski , Applications of Kripke models, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis ( A. S. Troelstra , editor), Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, Heidelberg, and New York, 1973, pp. 324391.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 1 *
Loading metrics...

Abstract views

Total abstract views: 45 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st September 2017. This data will be updated every 24 hours.