On second order intuitionistic propositional logic without a universal quantifier

Journal of Symbolic Logic 74 (1):157-167 (2009)
Abstract
We examine second order intuitionistic propositional logic, IPC². Let $F_\exists $ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in $F_\exists $ that is, for φ € $F_\exists $ φ is a classical tautology if and only if ¬¬φ is a tautology of IPC². We show that for each sentence φ € $F_\exists $ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As a corollary we obtain a semantic argument that the quantifier V is not definable in IPC² from ⊥, V, ^, →
Keywords second order intuitionistic propositional logic   Glivenko theorem
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