Consider the following pardigmatic examples of analytic and synthetic sentences:
(1) "Dogs once existed."
(2) "Prime numbers have two distinct divisors: themselves and one."
Both of these statements feel extremely obvious to us. And, if anything we're more likely to stop asserting (2) than (1) - if some perverse person wants to count 1 as a "prime" number, that's fine with me, so (if he's insistent enough) I'll adopt his usage and hence stop saying sentence 2 (and e.g. change how I state the fundamental theorem of algebra accordingly). So - we wonder, after reading Quine, what does the further claim that (2) is analytic amount to?
Here's an idea: If someone asked me to back up my assertion of (1), I'd be surprised, but there are things I would do to support this e.g. give an example of a dog. If (bizzarely) I couldn't state any other claims in support of (1), I'd be troubled. In contrast, if asked to justify (2) I wouldn't be able to give any kind of argument for it AND I wouldn't be troubled by this, or inclined to revise. (Note: this is exactly when claims about analyticity and meaning come up in ordinary contexts - people say 'that's just what I mean by the term' when faced with skepticism about certain things.)
S is basic analytic in P's idiolect iff: either P is happy to accept S without being able to provide any further justification
S is analytic in P's idiolect iff: S is basic analytic S is derivable via some combination of premises and inferences, each of which is basic analytic.
This seems to pick out a relatively sharp class of sentences, and accord with our intuitive judgments of analyticity (at least if we assume that experience can somehow be cited as a justification [or something more sophisticated], so that direct observations don't count as analytic for the observer).
Does this refute Quine? No. For, let's think about what epistemological siginificance (this notion) analyticity has. Do we have some kind of special access to analytic truths?
Making a bunch of new sentences analytic in your idiolect is just a matter of developing the inclined to say "that's just what I mean by the word" when pressed for a justification of these sentences. And this refusal to provide extra justification doesn't somehow ensure that the sentences you assert so boldly come to express truths.
For, what bucking up your insouciance like this does, is change the facts about your use of words so that (now), if the certain of your words are meaningful at all, these sentences will express truths. Thus, it makes these sentences/inferences function as a kind of implicit definition of your terms. But, as the famous case of Tonk shows, not all implicit definitions are coherent. Also, in changing the meanings of your words in this way, you run the risk of making other non-analytic sentences that you currently accept now express falsehoods.
Thus, saying that some sentence S is analytic isn't some kind of epistemic free pass for you to accept that sentence. All it does is semantically push all your chips into the center of the table with regard to S. Whereas before you ran the risk that S would express a falsehood, now there's a better chance that S will express a truth, but if it doesn't both S and a bunch of other sentences in your language will be totally meaningless.
So, here's my current position: the analytic-synthetic distinction is real, but it doesn't give the epistemological free lunch* which the logical positivists hoped it would.
*i.e. just saying that facts about something (like math) is analytic doesn't banish mysteries about how we came to know these facts.