On the complexity of propositional quantification in intuitionistic logic

Journal of Symbolic Logic 62 (2):529-544 (1997)
We define a propositionally quantified intuitionistic logic Hπ + by a natural extension of Kripke's semantics for propositional intutionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π +, S4π +, S4.2π +, K4π +, Tπ +, Kπ + and Bπ +, studied by Fine
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DOI 10.2307/2275545
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Logics for Propositional Contingentism.Peter Fritz - 2017 - Review of Symbolic Logic 10 (2):203-236.
Expressivity of Second Order Propositional Modal Logic.ten Cate Balder - 2006 - Journal of Philosophical Logic 35 (2):209-223.
Non-Axiomatizable Second Order Intuitionistic Propositional Logic.D. Skvortsov - 1997 - Annals of Pure and Applied Logic 86 (1):33-46.

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