A Second Paper "On the Interpolation Theorem for the Logic of Constant Domains"

In Gabbay [1] it is stated as an open problem whether or not Craig's Theorem holds for the logic of constant domains CD , i.e. for the extension of the intuitionistic predicate calculus, IPC , obtained by adding the schema; . Then in the later article, [2], Gabbay gives a proof of it. The proof given in [2] is via Robinson's (weak) consistency theorem and depends on relatively complicated (Kripke-) model-theoretical constructions developed in [1] (see p. 392 of [1] for a brief sketch of the method). The aim of this note is to show that the interpolation theorem for CD can also be obtained, by simple proof-theoretic methods, from §80 of Kleene's Introduction to Metamathematics [3]. GI is the classical formal system whose postulates are given on p. 442 of [3]. Let GD be the system obtained from GI by the following modifications: (1) the sequents of GD are to have at most two formulas in their succedents and (2) the intuitionistic restriction that Θ be empty is required for the succedent rules (→ ¬) and (→ ⊃). It is a simple matter to show that: , x not free in . It then follows that, using Theorem 46 of [3], if then .