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Counting the maximal intermediate constructive logics

  • Mauro Ferrari (a1) and Pierangelo Miglioli (a2)

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”.

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[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.

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[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.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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