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.
"A Syntactic Embedding of Predicate Logic into Second-Order Propositional Logic." Notre Dame J. Formal Logic 51 (4) 457 - 473, 2010. https://doi.org/10.1215/00294527-2010-029