The counterpart theory is a theory of first-order logic. Lewis interprets modal claims by using a translation from quantified modal logic into the counterpart theory. However, Lewis's translation does not preserve the unsatisfiability of formulas. In this paper, an extended semantics for quantified modal logic is introduced, and the corresponding connection between models of the quantified modal logic and models of the counterpart theory is given. Based on the semantics, a faithful and full translation from quantified modal logic to the counterpart theory, which preserves the satisfiability and the unsatisfiability of formulas, is also established. Furthermore, since the counterpart theory is sound and complete, and the soundness and the completeness are preserved by the faithful and full translation, the quantified modal logic is also sound and complete.