Decidability of Quantified Propositional Intuitionistic Logic and S4 on Trees

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p over propositions. In the context of Kripke semantics, a proposition is a subset of the worlds in a model structure which is upward closed, i.e., if h ∈ P , then h ∈ P for all h ≤ h. For propositional intuitionistic logic H, several classes of model structures are known to be complete, in particular the class of all partial orders, as well as the class of trees and some of its subclasses. Kremer (1997) has shown that the quantified propositional intuitionistic logic Hπ+ based on the class of all partial orders is recursively isomorphic to full second-order logic. He raised the question of whether the logic resulting from restriction to trees is axiomatizable—in fact, it is decidable. The main part of this note consists in establishing this fact, as well as a few related observations regarding the relationship between the formulas valid on various classes of trees. A concluding section discusses how the results transfer to a proof of decidability of modal S4 with propositional quantification on similar types of Kripke structures. (Propositionally quantified S4 on general partial orders is also known to be not axiomatizable.) Intermediate logics based on linear orders (i.e., 1-ary trees), which correspond to Gödel-Dummett logics, are also considered.