Consider one sense in which I can imagine that Scott Soames is Mike Tyson. Suppose Soames gets into a heated argument with some two-dimensionalist at a conference, and they come to blows. Soames unleashes a barrage of punches and floors the two-dimensionalist with a devastating uppercut. As I watch this, I can imagine that Soames is Tyson. Imagining this consists in seeing his punches as having Tysonian fury and power, and perhaps imagining that he has a Tyson face tattoo. I think of Soames as having Tysonian properties. But I don't imagine that Soames is leading a double life, bulking up and fighting as Tyson, then slimming down and writing books as himself.
However, I do something different when I imagine a possible world in which Scott Soames is Mike Tyson. It's not that I imagine there being a bizarre Soames-Tyson hybrid who has a mixture of their descriptive characteristics (he knows how to box, but he can also use the diamond! sorry). This person won't succeed in being both Soames and Tyson. To imagine that Soames is Tyson, I have to imagine him living a double life. It needs to be a Morning Star - Evening Star kind of case where two seemingly different things are actually one and the same.
Maybe when I imagine that X is Y in the former case, X is taken to designate rigidly, while Y is taken as a descriptive term. In the latter case, both designate rigidly. I'm not sure what the moral of the story is beyond this, although it gives us a weird context in which proper names work as definite descriptions.
CFA May 1: Atheism and Liberal Theology
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