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In \cite{BoEsGoRo08--m} the authors considered finitely-valued modal
logics with Kripke style semantics where both propositions and the
accessibility relation are valued over a finite residuated
lattice. Unfortunately, the necessity operator does not satisfy in general the
normality axiom (K). In this paper we focus on the case of finite chains, and we consider a different
approach based on introducing a multimodal logic where the previous
necessity operator is replaced with a family, parametrized by truth
values different from zero, of necessity operators each one semantically
defined using the crisp accessibility relation given by the
corresponding cut of the finitely-valued original accessibility
relation. This multimodal logic is somehow more appealing than the original modal
one because axiom (K) holds for each necessity operator.
In this paper we axiomatize this multimodal logic and we
prove that, in the case the starting residuated lattice is a finite BL chain,
the modal and the multimodal languages have the same expressive power
iff this algebra is an MV chain.

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