Failure of Interpolation in Combined Modal Logics

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. In 1957, W. Craig proved the interpolation theorem for first order logic [Cra57]. Comer [Com69] showed that the property fails for all finite variable fragments except the onevariable fragment. The n-variable fragment of first order logic –for short Ln– contains all first order formulas using just n variables and containing only predica...

• Type: Journal
• Year: 1998
• Where: NDJFL
• Authors: Maarten Marx, Carlos Areces
• Keywords: Interpolation, Logic, NDJFL 1998, first Order Logic