Notre Dame Journal of Formal Logic 51 (4):457-473 (2010)

Abstract
We give a syntactic translation from first-order intuitionistic predicate logic into second-order intuitionistic propositional logic IPC2. The translation covers the full set of logical connectives ∧, ∨, →, ⊥, ∀, and ∃, extending our previous work, which studied the significantly simpler case of the universal-implicational fragment of predicate logic. As corollaries of our approach, we obtain simple proofs of nondefinability of ∃ from the propositional connectives and nondefinability of ∀ from ∃ in the second-order intuitionistic propositional logic. We also show that the ∀-free fragment of IPC2 is undecidable
Keywords propositional quantification   IPC2
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DOI 10.1215/00294527-2010-029
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Non-Strictly Positive Fixed Points for Classical Natural Deduction.Ralph Matthes - 2005 - Annals of Pure and Applied Logic 133 (1):205-230.

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