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Refinement by interpretation replaces signature morphisms by logic interpretations as a means to translate specifications and witness refinements. The approach was recently introduced by the authors [M. A. Martins, A. Madeira, and L. S. Barbosa. Refinement via interpretation. In Proc. of 7th IEEE Int. Conf. on Software Engineering and Formal Methods, Hanoi, Vietnam, November 2009. IEEE Computer Society Press] in the context of equational specifications, in order to capture a number of relevant transformations in software design, reuse and adaptation. This paper goes a step forward and discusses the generalization of this idea to deductive systems of arbitrary dimension. This makes possible, for example, to refine sentential into equational specifications and the latter into modal ones. Moreover, the restriction to logics with finitary consequence relations is dropped which results in increased flexibility along the software development process.