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Failure of interpolation in combined modal logics. (English) Zbl 0968.03025
Summary: We investigate transfer of interpolation in such combinations of modal logics which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation. The same holds for combinations by taking unions, a generalization of Humberstone’s inaccessibility logic. Viewing first-order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the finite-variable fragments of first-order logic. We provide a simple condition stated only in terms of frames and bisimulations which implies failure of interpolation. Its use is exemplified in a wide range of cases.

##### MSC:
 03B45 Modal logic (including the logic of norms) 03C40 Interpolation, preservation, definability
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##### References:
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