Kripke's model structure for D2 is a quadruple (G, K, R, N) where N is the set of normal elements, i.e., the subset of K such that for every H in N, HRH. (See [1, p. 220, p. 211].) In fact, however, this structure provides a model for E2, as the work of Lemmon shows ([2, pp. 58–62]).
To show the inadequacy of Kripke's model structure, we show that the E2 thesis □B ⊃ B, not a thesis of D2, is valid under Kripke's modelling for D2. Suppose, for a reductio argument, that □ B ⊃ B is false for some valuation ϕ in some D2 model structure. Thenyϕ(□B ⊃ B, G) = F; so by truth-functional assignments