Interpolation property and superintuitionistic predicate logics (Q1288274)

The article deals with the interpolation problem for extensions of the first-order predicate logic. For propositional superintuitionistic logics, this problem was completely solved by \textit{L. L. Maksimova} [Algebra Logika 16, No. 6, 643-681 (1977; Zbl 0403.03047)]. The author formulates a sufficient condition that makes it possible to refute the interpolation theorem. The condition is based on an interesting explicit construction of counterexamples to the interpolation property. On the basis of this condition, it is proven that the interpolation theorem fails for a wide natural class of predicate superintuitionistic logics. Namely, for logics characterized by classes of Kripke frames such that the cardinality of all or almost all domains equals a given fixed natural \(i>2\). In particular, the interpolation property is refuted for predicate extensions of propositional logics with interpolation property. The article also contains a series of examples of superintuitionistic predicate logics with equality such that these logics possess the interpolation property whereas their pure fragments (without equality) do not possess this property.