 An intermediate propositional logic \(\mathbf{L}\) is said to have the finite frame property if there is a class of finite frames on which the Kripke-valid formulas are exactly the theorems of \(\mathbf{L}.\) Many intermediate logics, including \(\mathbf{LC}\) and \(\mathbf{KP},\) have this property. Jankov [1968] used an infinite sequence of finite rooted Kripke frames to prove that there are continuum many intermediate logics. De Jongh, Verbrugge and Visser [2009] proved that every intermediate logic \(\mathbf{L}\) with the finite frame property is the propositional logic of \(\mathbf{HA(L)},\) that is, the class of all formulas in the language of \(\mathbf{IPC}\) all of whose arithmetical substitution instances are provable in the logical extension of \(\mathbf{HA}\) by \(\mathbf{L}.\)