Failures of the interpolation lemma in quantified modal logic

Beth's Definability Theorem, and consequently the Interpolation Lemma, fail for the version of quantified S5 that is presented in Kripke's [6]. These failures persist when the constant domain axiom-scheme ∀x□φ ≡ □∀xφ is added to S5 or, indeed, to any weaker extension of quantificational K.§1 reviews some standard material on quantificational modal logic. This is in contrast to quantified intermediate logics for, as Gabbay [6] has shown, the Interpolation Lemma holds for the logic CD with constant domains and for several of its extensions. §§2—4 establish the negative results for the systems based upon S5. §5 establishes a more general negative result and, finally, §6 considers some positive results and open problems. A basic knowledge of classical and modal quantificational logic is presupposed.Let me briefly review the relevant model theory for quantified modal logic. Further details can be found in [3] or [7].The language is obtained from the language for classical first-order logic with identity by adding a unary operator □ for necessity. The atomic formula ‘Ex’ is used as an abbreviation for ‘∃y(y = x)’ and may be read as ‘x exists’.