Intensional logics without interative axioms

It may be that there is no way to axiomatize an intensional logic without recourse to one or more iterative axioms; then I call that logic iterative. But many familiar logics can be axiomatized by means of non-iterative axioms alone; these logics I call non-iterative. A frame, roughly speaking, is a partial interpretation for an intensional language. It provides a set 1, all subsets of which are to be regarded as eligible propositional values or truth sets for formulas. It also provides, for each intensional operator in the language, a function specifying how the propositional values of formulas compounded by means of that operator are to depend on the propositional values of their immediate constituents. A frame plus an assignment of values to propositional variables are enough to yield an assignment of propositional values to all formulas in the language. Iff every interpretation based on a certain frame assigns the propositional value 1 (truth everywhere) to a certain formula, then we say that the formula is valid in the frame. There are two different natural ways in which an intensional logic L may be said to ‘correspond’ to a class C of frames. For any class C of frames, there is a logic L that has as its theorems exactly those formulas that are valid in all frames in the class C. We say then that the class C