Zach, Richard. 2004. “Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees of Height and Arity ≤ω.” Journal of Philosophical Logic 33 (2): 155–64. https://doi.org/10.1023/B:LOGI.0000021744.10237.d0.
Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers \(\forall p\), \(\exists p\), where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most \(\omega\), the resulting logics are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel-Dummett logics with quantifiers over propositions.
Review: M. Yasuhara (Zentralblatt 1054.03011)