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On a second order propositional operator in intuitionistic logic

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Abstract

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by

. In full topological models * is not generally definable, but over Cantor-space and the reals it can be classically shown that

; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate logic.

Over [0, 1], the operator * is (constructively and classically) undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness.

Parameters of the required kind can be obtained as so-called projections of lawless sequences.

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Troelstra, A.S. On a second order propositional operator in intuitionistic logic. Stud Logica 40, 113–139 (1981). https://doi.org/10.1007/BF01874704

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  • DOI: https://doi.org/10.1007/BF01874704

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