“Metaphysics aspires to understand reality as it is in itself, independently of the conceptual apparatus observers bring to bear on it.” (Yablo, 1987)
Allan Gibbard on Essence
The scheme that follows parallels, to some extent, the one offered by Allan Gibbard in his paper, ‘Contingent Identity’.
Conceptual essentialism is accepted by Gibbard. For example, he believes that
“variables in modal contexts shift their range of values: they range over senses…not over concrete things [like the lectern or Tony Blair].” (Gibbard, 1975)
He calls the “senses” above “individual concepts”. He also says that “proper names [e.g., ‘Tony Blair’] in modal contexts can be construed as denoting individual concepts” (1975).
Thus it's the concept of the lectern (or even the concept of Tony Blair) that has the modal quality of having set of properties E essentially, not the lectern or Blair itself or himself. It follows that concepts determine essence; or conceptual essence. However, according to Quine’s view of essentialism, necessity (or essence) applies “to the fulfilment of conditions by objects…apart from special ways of specifying them” (‘Reference and modality’, pg 151). That “special way”, presumably, would be a non-conceptual way. Gibbard himself clarifies traditional essentialism thus:
“Essentialism for a class of entities U…is the claim that for any entity e in U and any condition Ø which e fulfils, the question of whether e necessarily fulfils Ø has a definite answer apart from the way e is specified.” 
But Gibbard’s “designations” (or my concepts) determine the essences of concrete things. As Gibbard puts it:
“…it makes no sense to talk of a concrete thing as fulfilling a condition Ø in every possible world – as fulfilling Ø necessarily…apart from its designation. Essentialism, then, is false for concrete things because apart from a special designation, it is meaningless to talk of the same concrete thing in different possible worlds. It makes good sense, on the other hand, to speak of the same individual concept in different possible worlds.”
What if one denies essentialism for concepts (or other abstract entities) too, as Quine does? Here too I borrow from Gibbard and Rudolph Carnap. (Gibbard himself borrows from Carnap.) Carnap did accept analyticity in his scheme. Though his analyticity, like mine, is a question of “individual concepts”.
Conceptual essences allow the possibility of analyticity for certain statements about concrete objects. Carnap was an essentialist when it came to his “individual concepts”. These concepts, again, have essences or criteria of identity forged in terms of concepts. And because we have conceptual essences, we can “explain necessity by analyticity”. That is, in
a = b
a and b “are concepts of the same individual” (Gibbard, 1975), not variables for concrete objects.
Thus Perhaps we should write:
[Ca] = [Cb]
All bachelors are unmarried men.
According to Carnap, modal contexts were really disguised quotational contexts. (1947/1988) That is
i) Necessarily bachelors are unmarried men.
ii) “Bachelors are unmarried men” is analytic.i) is an example of de re necessity. That is, it is a statement about concrete objects: bachelors and unmarried men. ii), on the other hand, is an example of de dicto necessity. That is, it's a statement about the concepts [bachelors] and [unmarried men] and the conceptual implication articulated by the quoted sentence. (Perhaps we should say that the notion of implication here is of course the semantical one, not provability.)
It terms of essence, it's not essential that the concrete objects unmarried men are bachelors (i.e., there may be no essentiality in the world). In terms of stipulational essence, it is essential that the concept [unmarried men] implies the concept [bachelors].
Alternatively, we can used second-order modal logic to get the above points across:
a) (c) (Mc Rc)
b) (x) (Mx ٱRx)
Are we saying that a) offers us the essence of, say, mathematicians care of certain statements or concepts (i.e., “quotational analyticity”)? Or, in b), that being rational is an essential property of the concrete objects mathematicians as they are “unspecified”? That is, being a mathematician isn't essential to the variable “x”; though if the value of that variable is the class of mathematicians, then part of an x’s essence will be rationality, according to b) above.
Again, the concepts are stipulated, unlike traditional meanings (which are meant to be determinate in minds or in a platonic realm). And, as Quine said, “meaning is what essence becomes when it is divorced from the object and wedded to the word”(Quine, ‘Two Dogmas of Empiricism’).
This means that I am partly at odds with Quine in giving essences to concepts (even if not meanings). Though I agree with him in that essences don't belong to concrete objects as they are “unspecified”. The difference is, of course, that abstract meanings are, again, seen as determinate and fixed; whereas my concepts are stipulated: they aren't fixed or determinate (until stipulated) and often belong to particular conceptual schemes. According to semantic traditionalists, there is a correct and fixed answer to the question: What is the correct meaning of the word “bachelors”? Though there is no correct concept for a concrete objects outside all schemes and theories and before all acts of stipulation.
To use Saul Kripke’s words (as he used them about possible worlds): “[concepts] are given in the act of stipulation.” (Kripke, 1971)