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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
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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”.

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

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