On Definability in Multimodal Logic

Three notions of definability in multimodal logic are considered. Two are analogous to the notions of explicit definability and implicit definability introduced by Beth in the context of first-order logic. However, while by Beth’s theorem the two types of definability are equivalent for first-order logic, such an equivalence does not hold for multimodal logics. A third notion of definability, reducibility, is introduced; it is shown that in multimodal logics, explicit definability is equivalent to the combination of implicit definability and reducibility. The three notions of definability are characterized semantically using (modal) algebras. The use of algebras, rather than frames, is shown to be necessary for these characterizations. ∗We thank Samson Abramsky, Arnon Avron, Johan van Benthem, Sergiu Hart, Martin Meier, Jouku Väänänen, and an anonymous reviewer for helpful discussions. †Supported in part by NSF under grants TR-0325453 and IIS-0534064, and by AFOSR under grant FA9550-05-1-0055. ‡Support by the Israeli Science Foundation under grant 891/04 is acknowledged.