1) Logical truths, e.g. "No unmarried man is married". These statements are true by virtue of their logical form (we could replace each appearance of 'married' with a generic 'X' and the sentence would still clearly be true).
This class is unproblematic, as we have a clear method to decide whether a statement counts as a logical truth. If a (compound) statement always comes out true, no matter what truth-values we assign to the atomic statements of the language, then it is a logical truth. (Consider "P or not-P": it is true no matter whether we assign P as 'true' or 'false'.)
2) The other, more interesting, class contains statements that can be turned into logical truths by substituting synonyms. For example, "No bachelor is married" can be turned into the logical truth above by replacing 'bachelor' with its synonym, 'unmarried man'.
So we can explain analyticity in terms of synonymy. The question Quine then raises is: can we provide an adequate account of synonymy?
An obvious strategy would be to appeal to 'definition' (as I did myself in the very first sentence of this post). But this doesn't help, because most definitions (e.g. in dictionaries) just report pre-existing synonymies, rather than stipulatively creating new ones.
Quine goes on: (pp.24-25)
Just what it means to affirm synonymy, just what the interconnections may be which are necessary and sufficient in order that two linguistic forms be properly described as synonymous, is far from clear; but, whatever these interconnections may be, ordinarily they are grounded in usage. Definitions reporting selected instances of synonymy come then as reports upon usage.
Another natural answer here would be to define X and Y as synonyms iff they can interchanged in appropriate contexts without altering the truth value (i.e. if they're interchangeable salva veritate). But what contexts must we include here? Mere extensional interchangeability is clearly insufficient - for 'featherless biped' and 'human being' are extensionally identical (true of all the same actual objects), but clearly differ in meaning. So genuine synonyms must also be intensionally identical. That is, they must satisfy something like: "Necessarily, all and only X's are Y's" (where 'necessarily' is so narrowly construed as to apply only to analytic statements). But this is just to say that "All and only X's are Y's" is analytic. So we've come full circle! We can explain either analyticity or synonymy in terms of the other, but then this other one is left unsupported.
But do they really need support? One might think these concepts are intuitive enough that we could dispense with the formal explications. But further consideration casts doubt on this. Quine points out that he has no idea whether the statement "Everything green is extended" is analytic. (The statement is certainly true, but is it true partly in virtue of how the universe is, or merely because of what the words mean?)
Further, as Everitt & Fisher point out in Modern Epistemology, scientific advances have caused us to revise statements that were once seen as being obviously analytic. For example: "Two bodies which are both falling downward cannot be moving in opposite directions" (contra spherical Earth), "For any two events A and B, either A is before B or it is not" (contra special relativity), and "If a woman gives birth to a child, she is its mother" (contra IVF).
As Everitt & Fisher explain:
What these three examples show is the way in which unforeseen scientific advances can radically change our acceptance of propositions which initially seemed immune to empirical findings. In each case, our willingness to regard the "a priori truth" differently is based on the fact that the new empirical information undermines some very general assumptions which lay behind the a priori truth. [...] Sometimes the new information shows that the old "truth" was true if taken in one way but not if taken in another, where the very idea that there are two ways in which that truth might be understood becomes intelligible only in the light of the new empirical information. (p.112, original emphasis)
Nevertheless, I'm not convinced that any of this conclusively refutes the viability of the analytic/synthetic distinction. Because it does seem plausible to me that meanings change over time, and that the three examples above really are analytic on the appropriate interpretations of 'down', 'mother', etc. (Though E&F insist that there are "no good grounds" for understanding this as a change of meaning rather than only a change of empirical beliefs, presumably the reverse is also true.) And Quine's own example simply shows that there are some borderline cases which we're not sure about. I guess that's enough for us to conclude that the distinction isn't clear-cut, at least. But I'm not sure we should want to dismiss the distinction altogether.
Having said that, I do agree with Quine (p.43) that nothing is (in principle) unrevisible:
[I]t becomes folly to seek a boundary between synthetic statements, which hold contingently on experience, and analytic statements, which hold come what may. Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. Even a statement very close to the periphery [of our 'web of beliefs'] can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of the excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy, or Einstein Newton, or Darwin Aristotle?
But given my above remarks I guess my response would be to reject the claim that analytic statements hold "come what may". For if meanings change, then what is true simply in virtue of those meanings may also change. So I think I might like to uphold a slightly muted version of the analytic/synthetic distinction, whereby even analytic statements are open to revision, since scientific advances may lead us to revise the concepts upon which those analytic statements depend. Does that sound at all plausible?
P.S. Doing things with words has also posted an essay on this topic.