Abstract
This chapter contains a proof of Lyndon's interpolation property (LIP) for quantified extensions of basic modal logics K, T, D, K4, and S4, and for some others, including the propositional S5 has LIP. At the same time, the quantified extension of S5, as well as other systems satisfying the Barcan formula has neither Lyndon's nor Craig's interpolation, nor Beth's property. Some examples of propositional modal logics, which have CIP but do not possess LIP are found. Craig's interpolation property is proved for a number of propositional modal logics, including the Grzegorczyk logic Grz, its extension Grz.2, and the provability logic G. A class of so-called L-conservative formulas is defined, which can be added to a propositional logic L as new axiom schemes without loss of interpolation. It is proved that the interpolation properties are preserved by transfer from predicate logics without equality to their extensions with equality.
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