Essential Properties and Possible Worlds

Kit Fine famously argued for the inadequacy of possible-worlds analysis of essence. Below is a related criticism of my own. 

Call the phenomenon of a bearer having a property in all of the worlds it occupies, ‘rigid exemplification’.  What’s the relationship between essential properties and rigid exemplification? On the standard possible-worlds account, the latter constitutes an analysis of the former: the right-hand side gives a list of the constituents of the item expressed on the left-hand side:


(EPdef) F is an essential property of x =def.  x has F in all (metaphysically) possible worlds in which x exists.

A slightly weaker account of the relationship has it that while the latter isn’t an analysis of the constituents of the former, the latter and the former are yet necessarily co-extensive :

(EPiff) Necessarily, F is an essential property of x if, and only if, x has F in all (metaphysically) possible worlds in which x exists.

I worry that EPdef and EPiff are both false.  For while it’s surely necessary for a property F to be essential to a bearer x that F instantiates x in all possible worlds in which x exists, it is not also sufficient.  For it seems possible for a bearer to rigidly exemplify a non-essential property.  If one were to tell a plausible story according to which a thing has, in all metaphysically possible worlds in which it exists, a property that is not essential to it, then that would be sufficient to (defeasibly) justify this claim.   It appears that the following is just such a story.

Pinky and Winky
Suppose there are two Anselmian (or quasi-Anselmian) necessary beings, Pinky and Winky. Now although it's not constitutive of Pinky that he has a pink after-image throughout his existence, it is constitutive of Pinky that he has an aching desire to have one. And as it turns out, it's constitutive of Winky that he satisfies all the harmless desires of Pinky (because, say, it's constitutive of Winky that he's essentially loving, and since Pinky is Winky's beloved, he inevitably satisfies all his harmless desires). In this case, then, it seems that Pinky has a pink after-image in all the metaphysically possible worlds in which he exists. So by EPdef and EPiff pinkness is an essential property of Pinky.  Intuitively, however, pinkness is not an essential property of Pinky.  But if not, then EPdef and EPiff are both false.  

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