"If the requirements for being β are stricter than the requirements for being ά, then β ought to have a ‘bigger’ essence than ά…Thus, more is essential to the Shroud of Turin than to the piece of cloth [which was used as the Shroud], and the Shroud of Turin ought accordingly to have the bigger essence.” (Yablo, 1987)
We can admit that it's “necessary that the Shroud of Turin is the Shroud of Turin” (according to Steven Yablo’s paper), and that it wasn't necessary that the cloth of Turin (which was used as the Shroud) actually became the Shroud of Turin. (Therefore the Shroud has a property that the cloth didn’t have, according to Yablo, and so they aren't necessarily identical' but only “contingently identical”.) So isn’t it also necessary that the cloth of Turin was the cloth of Turin, in the same manner it's necessary that the Shroud of Turin is now the Shroud of Turin? If, on this count only, we can say that the Shroud hasn’t yet got a ‘bigger’ essence than the prior cloth.
How do we decide these essences in the first place? (So as to thereby decide which object has the ‘bigger’ essence.)
For example, it might well have been necessary that the cloth could clean things (i.e., have a functional essence); otherwise it wouldn’t have been a cloth. (Let’s take the cloth of Turin to have been a cloth created to be used as a cleaning implement.) It's not necessary, on the other hand, that the Shroud can clean things because, after all, it's now a shroud and not a cleaning cloth. Therefore it must follow that the cloth had an essence or property that the Shroud doesn't have.
Similarly, it might well have been necessary (via the sortal cleaning cloth) that the cloth wasn't black; but white instead (i.e., so that it showed up the dirt). Again, surely it's not necessary that the Shroud is white rather than black.
Yablo extracts a ‘bigger’ essence from the Shroud by treating its function as part of its essence (i.e., the function sortalised by shroud for a dead body). He disregards the cloth’s own possible functions, one of which might have been cleaning. And even if Yablo’s Turin cloth was never a cleaning cloth (but only a piece of material used for garments), it would still have had an essence/property that the Shroud doesn't have which belongs to the sortal garment material.
For example, the cloth might have been used for garments (not shrouds) and therefore it shouldn't (or couldn't) have made its wearers itch. And it might have also kept them warm too. However, a shroud, or the Shroud of Turin, needn't have these properties because the dead don't suffer from itches or cold.
So not only is Yablo’s belief that the Shroud’s essence is ‘bigger’ than the cloth’s somewhat arbitrary, it may also be the case that all deemed ontological essences are always somewhat arbitrary and also stipulated via sortals rather than discovered ontologically.
The Yablo example somewhat parallels the oft-quoted Quine example of the rational, two-legged mathematician and cyclist.
Mathematicians are, in this example, deemed to be necessarily rational: sortalised by necessarily rational being. (Does this automatically make computers capable of difficult mathematical calculations and the discovery of new proofs rational?) Cyclists, on the other hand, are deemed to be necessarily two-legged – sortalised by two-legged beings. (Although a no-legged cyclist could free-ride down hills and push the cycle up hills.) However, what if we have a mathematician who's also a cyclist – a being who falls under the two sortals: rational being and two-legged being? Quine asks:
"Is this concrete individual necessarily rational and contingently two-legged or vice versa?" (1960)
Perhaps, according to Yablo, the mathematician cyclist has a ‘bigger essence’ than a mathematician who isn’t a cyclist. (Perhaps because he has no other interests either.) This mathematical cyclist would fall under the sortals: rational being and two-legged being. But Quine thinks all this is silly. He says:
"There is no semblance of sense in rating some of his attributes as necessary and others as contingent. Some of his attributes count as important and others as unimportant, yes, some as enduring and others as fleeting; but none as necessary or contingent." (Word and Object)
The essences of the cloth of Turin and the Shroud of Turin depend on sortal specification. The cloth turned out to have a smaller essence than the Shroud simply because Yablo didn't specify it in any way; except by saying that it was the cloth of Turin and that it became the Shroud of Turn. However, Yablo does specify the Shroud (via that very sortal shroud) by saying that it shrouded the dead Christ. Again, the cloth could be specified via its material makeup. A cloth must necessarily be made up of certain materials (e.g., wool, etc.), or that it must necessarily be woven or that it mustn't retain water. Quine, therefore, had this to say on essentialism:
"An object, of itself and by whatever name or none, must be seen as having some of its traits necessarily and others contingently, despite the fact that the letter traits follow just as analytically from some ways of specifying the object as the former do from other ways of specifying it…This means adapting an invidious attitude towards certain ways of specifying x…and favouring other ways…as somehow better revealing the ‘essence’ of the object." (From a Logical Point of View, 1953, pp. 155-6)
As Gibbard (1987) might have said: The Shroud is specified via two sortals: cloth and shroud. The cloth, on the other hand, is only specified via one sortal: cloth. So, in this scheme, essences come via the sortals of objects, not the objects themselves. Indeed the Turin Shroud could come to us (or we to it) via a sortal that Yablo didn’t use.
For example, the Shroud could have been specified via the sortal objects that bear an imprint. This is a genuine sortal because there are other members of the sortal, objects that bear an imprint, other than the Shroud (e.g., white walls with their hand prints). Again, the Shroud could be specified via a sortal that the cloth certainly didn’t have: historical artefact. However, as has been said, the cloth of Turin could have come to us, or we to it, via sortals not specified by Yablo, say, cleaning cloth or garment material.