INTERPOLATION, DEFINABILITY, AMALGAMATION
The definitions of interpolation and Beth's property essentially depend on the consequence relation in the logic under consideration. In modal logics, there are at least two natural logical consequence relations: provable implication and deducibility. They are equivalent in superintuitionistic logics in deduction theorem but not equivalent in normal modal logics, where only a weaker form of the deduction theorem holds. So in modal logics two forms of interpolation (CIP and IPD), and two forms (B1 and B2) of the Beth property are considered. It is shown that B1 is equivalent to CIP for all modal logics, but all other properties are not equivalent. A full diagram of interrelations of these properties as well as their algebraic equivalents is presented. In particular, IPD is equivalent to the amalgamation property (AP), CIP to the superamalgamation property (SAP), and B2 to epimorphisms surjectivity ES* of the corresponding variety of modal algebras.
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