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Propositional quantification in the monadic fragment of intuitionistic logic

Published online by Cambridge University Press:  12 March 2014

Tomasz Połacik
Tomasz Połacik*
Affiliation:
Institute of Mathematics, University of Silesia, Katowice, Poland, E-mail: polacik@us.edu.pl
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Abstract

We study the monadic fragment of second order intuitionistic propositional logic in the language containing the standard propositional connectives and propositional quantifiers. It is proved that under the topological interpretation over any dense-in-itself metric space, the considered fragment collapses to Heyting calculus. Moreover, we prove that the topological interpretation over any dense-in-itself metric space of fragment in question coincides with the so-called Pitts' interpretation. We also prove that all the nonstandard propositional operators of the form q ↦ ∃p (q ↔ F(p)), where F is an arbitrary monadic formula of the variable p, are definable in the language of Heyting calculus under the topological interpretation of intuitionistic logic over sufficiently regular spaces.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 269 - 300
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1]Gabbay, D. M., On 2nd order intuitionistic propositional calculus with full comprehension, Archive for Mathematical Logic, vol. 16 (1974), pp. 177–186.CrossRefGoogle Scholar
[2]Ghilardi, S. and Zawadowski, M., A sheaf representation and duality for finitely presented Hey ting algebras, this Journal, vol. 60 (1995), pp. 911–939.Google Scholar
[3]Kreisel, G., Monadic operators defined by means of propositional quantification in intuitionistic logic, Reports on Mathematical Logic, vol. 12 (1981), pp. 9–15.Google Scholar
[4]Pitts, A., On an interpretation of second order quantification in the first order intuitionistic propositional logic, this Journal, vol. 57 (1992), pp. 33–52.Google Scholar
[5]Połacik, T., Pitts' quantifiers are not topological quantification, forthcoming in Notre Dame Journal of Formal Logic.Google Scholar
[6]Połacik, T., Operators defined by propositional quantification and their interpretations over Cantor space, Reports on Mathematical Logic, vol. 27 (1993), pp. 67–79.Google Scholar
[7]Tarski, A., Der Aussagenkalkül und die Topologie, Fundamenta Mathemeticae, vol. 31 (1938), pp. 103–134.CrossRefGoogle Scholar
[8]Troelstra, A. S., On a second order propositional operator in intuitionistic logic, Stadia Logica, vol. 40 (1981), pp. 113–139.CrossRefGoogle Scholar
[9]Visser, A., Bisimulations, model descriptions and propositional quantifiers, Logic Group Preprint Series 161, Department of Philosophy, Utrecht University, Utrecht, 1996.Google Scholar