A Sequence of Decidable Finitely Axiomatizable Intermediate Logics with the Disjunction Property

We are interested in extensions of the intuitionistic logic which are both decidable and have the disjunction property. Systems with the disjunction property are known, for example the Kreisel-Putnam system [1] which is I + ( 0 -* (4 v a)) ((.'-> + ) v (' ---a)) and Scott's system I + ((04') ( V -) (~ ~%~4v -0). It was shown in [3c] that the first system has the finite-model property. In this note we shall construct a sequence of intermediate logics D, with the following properties: