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A crucial problem in abstract model checking is to find a tradeoff between constructing the "best" (the smallest) abstract model, approximating a given model, and preserving as much interesting properties over the original model as possible. In this paper, we present a method for dealing with this problem based on the definition of a new abstract satisfiability relation. This new relation allows us to analyze temporal properties with different degrees of precision, by means of a refinement process. The method subsumes the classic way of abstracting properties and the dual proposal of the authors. As a consequence, maintaining the same abstract model, we directly obtain the preservation of universal properties (as in the classic method) and the refutation of existential properties (as in the dual method). We also show the utility of this method by proving that the very important notions of completeness and precision in abstract model checking may be analyzed by using the new relation. In particular, we exploit the power of model checking to simultaneously refine both the model and the properties.