Intermediate logics and the de Jongh property (Q627438)

Dick de Jongh proved in an unpublished paper that the propositional logic of Heyting's arithmetic, \textbf{HA}, is precisely \textbf{IPC}, that means that, for any proposition \(A(p_1,\dots, p_n)\), \(\mathbf{IPC}\vdash A(p_1,\dots, p_n)\) if and and only if \(\mathbf{HA}\vdash A(B_1,\dots, B_n)\), for all arithmetical sentences \(B_1,\dots, B_n\). Propositional intermediate logics are the logics between intuitionistic propositional logic \textbf{IPC} and classical propositional logic \textbf{CPC}. An intermediate logic is said to have the finite model property, \textbf{FMP}, if there is a class of finite Kripke frames for which it is complete. In this paper, the authors generalize de Jongh's theorem for extensions of \textbf{HA} with propositional logics that have \textbf{FMP}. \textit{C. Smoryński}, in his paper [``Applications of Kripke models'', in: A. S. Troelstra (ed.), Metamathematical investigation of intuitionistic arithmetic and analysis. Berlin: Springer. Lect. Notes Math. 344, 324--391 (1973; Zbl 0275.02025)], gave two proofs for the de Jongh theorem. The authors of this paper use Smoryński's second proof, which is more complicated but more flexible for applications.