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On a Second Order Propositional Operator in Intuitionistic Logic

A. S. Troelstra
Studia Logica: An International Journal for Symbolic Logic
Vol. 40, No. 2 (1981), pp. 113-139
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015015
Page Count: 27
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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 *(P) ≡ ∃Q(P ↔ ⅂Q ∨ ⅂⅂Q). In full topological models * is not generally definable but over Cantor-space and the reals it can be classically shown that *(P)↔ ⅂⅂P; 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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