Failure of interpolation in combined modal logics

M. Marx and C. Areces. Failure of interpolation in combined modal logics. Notre Dame Journal of Formal Logic, 39(2):253–273, 1998.

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Abstract

We investigate transfer of interpolation in such combinations of modal logic 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.

BibTeX

@Article{Marx1998,
  author =       "M. Marx and C. Areces",
  journal =      "Notre Dame Journal of Formal Logic",
  title =        "Failure of interpolation in combined modal logics",
  year =         "1998",
  number =       "2",
  pages =        "253--273",
  volume =       "39",
  abstract =     "We investigate transfer of interpolation in such
                 combinations of modal logic 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.",
}