We examine second order intuitionistic propositional logic, IPC2. Let ℱ∃ be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in ℱ∃ that is, for φ∈ℱ∃, φ is a classical tautology if and only if ¬¬φ is a tautology of IPC2. We show that for each sentence φ∈ℱ∃ (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 ∀ is not definable in IPC2 from ⊥, ∨, ∧, →, ∃.
"On second order intuitionistic propositional logic without a universal quantifier." J. Symbolic Logic 74 (1) 157 - 167, March 2009. https://doi.org/10.2178/jsl/1231082306