A recent argument suggests that proper names are persistently rigid designators. Invoking the Kaplanian distinction between a world of the context of utterance and a world of the circumstance of evaluation, the argument maintains that names have to designate something only in the former, but not in the latter, implying thus that the designated objects must exist only in the former world. This paper shows that names designate something in both kinds of world and are thus obstinately rigid. This is achieved in three steps. First, the author argues that the contents of names must be available in possible worlds regardless of whether the named objects exist in them. Second, the author argues that these contents are expressed by English names in both kinds of world. Third, since Millianism suggests that names express contents by way of designating objects, the author argues that they have to designate something in both kinds of world.
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