So this is a long argument setting out w "the only hope of sustaining quantified modal".
The page referred to just before your quote...
The Aristotelian notion of essence was the forerunner, no doubt, of the modern notion of intension or meaning. For Aristotle it was essential in men to be rational, accidental to be two-legged. But there is an important difference between this attitude and the doctrine of meaning. From the latter point of view it may indeed be conceded (if only for the sake of argument) that rationality is involved in the meaning of the word ‘man’ while two-leggedness is not; but two-leggedness may at the same time be viewed as involved in the meaning of ‘biped’ while rationality is not. Thus from the point of view of the doctrine of meaning it makes no sense to say of the actual individual, who is at once a man and a biped, that his rationality is essential and his two-leggedness accidental or vice versa. Things had essences. for Aristotle, but only linguistic forms lnave meanings. Meaning is what essence becomes when it is divorced from the object of reference and wedded to the word. — p.22
Quine's view of Aristotelian essentialism, it seems. So what is the "reversion to Aristotelian essentialism"? That some of the properties of a man are necessary, and some contingent.
What Smullyanis proposes (back a few paragraphs) is very close to Kripke's rigid designation. Quine would reject this on the grounds that some description must be associated with any proper name in order to "fix" it's referent. This wasn't rejected until Donnellan and Kripke's discussion of the topic, a few years later.
Quine's text is not helped by the juxtaposition of necessary and contingent, and the association of analyticity with necessity.
So let's be a bit pedantic and oppose necessity with possibility, and define these in terms of possible worlds, while also and distinctly opposing the analytic and the synthetic, such that the
analytic is understood by definition while the
synthetic is understood by checking out how things are in the world.
And as for contingency, let's leave it aside until we have a better foundation.
How's that looking?
is, then, what musty happen if modal logic is to avoid the issues with quantification that Quine raises - in this Quine is more or less correct, and the strategy Kripke adopts is pretty much the one Quine sets out - there are properties of things that are true of them in every possible world.
Whether these properties are "essential" is another question.