Kripke Incompleteness of Predicate Extentions of Gabbay-de Jongh’s Logic of the Finite Binary Trees

In the previous papers [4], [5], the author gave several completeness and incompleteness results on some predicate extensions with the constant domain of intermediate and modal propositional logics by means of the theory of canonical formulas (cf. [1]). However, these results are on subframe and cofinal subframe logics, and little is known for non cofinal subframe logics. In this note, we show the Kripke incompleteness of the intermediate predicate logic H∗+Tr2+K+D, where Tr2 is the axiom of Gabbayde Jongh’s logic of the finite binary trees [2] ∧2 i=0((pi ⊃ ∨ j 6=i pj) ⊃ ∨ j 6=i pj) ⊃ ∨2 i=0 pi, K is Kuroda’s axiom ∀x¬¬A(x) ⊃ ¬¬∀xA(x) and D is the axiom of constant domain ∀x(A(x) ∨B) ⊃ (∀xA(x) ∨B). Our result is the following.