On second order intuitionistic propositional logic without a universal quantifier

Abstract

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 ⊥, ∨, ∧, →, ∃.