Quine's "Two Dogmas" is a concerted attack on the analytic/synthetic distinction. At first this might seem to be a fairly easily dispensable part of the positivist picture. But Quine shows that it is in fact crucial to the positivist view that theoretical sentences are definable in terms of observation sentences, and thus also to their defense of reductionism and foundationalism.
Quine's attack consists of two main components. There is, first of all, his discussion of various attempts to characterize the distinction precisely. (This occupies sections 2-5 of the article.) Here are a few of the twists and turns of this discussion. Which sentences are analytic? Perhaps:
(1) those which can be transformed into logical truths by substituting synonyms for synonyms.
But this simply presupposes the notion of synonymy, which is just as much in need of explanation as the notion of analyticity it is supposed to explain. So we might try:
(2) those which are true by definition (= those which can be turned into logical truths by substitution of definitions for the terms they define).
But now we need to know what constitutes a definition. If we want our definitions to accurately reflect actual usage, then we will need to make sure that the definitions only define words in other terms with which they are synonymous: but in that case we are presupposing the notion of synonymy. Gilbert Harman has used an analogy which is useful in understanding this part of Quine's argument. We may see Quine as arguing that there is no sure-fire way of distinguishing those linguistic regularities that are due to the meanings of our terms and those that are due to our beliefs. We all believe very firmly that cats are animals. But there may be no criterion by means of which to determine whether this is because we think being an animal is part of the meaning of 'cat', or whether it is just because the fact that cats are animals is a particularly obvious empirical truth. As Harman puts it, there is no distinction between our mental dictionary and our mental encyclopedia: we just have a bunch of beliefs about cats, with no sharp line between those true in virtue of meaning and those true because of the facts. (Gilbert Harman, Thought, Princeton University Press, 1973, p. 97.)
Of course, we can simply choose to use one expression as an abbreviation for a longer and more cumbersome one, as we use 'NOW' to abbreviate 'The National Organization for Women'. Even Quine might concede that it is analytic that NOW is a national organization, for instance. But (a) we very rarely use expressions explicitly and exclusively as abbreviations, so this phenomenon certainly cannot provide a general account of analyticity; (b) even in such seemingly obvious cases, it is not entirely clear that there are any analytic truths; for instance, NOW might go international, or perhaps shrink to a single state, without changing its name; again, it might decide its goals are to avoid any sort of discriminatory treatment, whether of women, of men, of racial minorities, or whatever; in such a case it would no longer be an organization specifically for women, but for purposes of recognition it might still retain its familiar name. Thus, even in cases where a certain expression begins as a mere abbreviation, it is likely to take on a life of its own.
Since (1) and (2) both turn out to presuppose that we already understand the concept of synonymy, Quine turns to that concept. What is it for two expressions to be synonymous? Here's one idea:
(3) Two expressions are synonymous if they are interchangeable salva veritate.
"salva veritate" means "saving truth," and the idea is that e1 is synonymous with e2 if and only if, in every sentence in which e1 occurs, replacing it with e2 would not change the truth value of the sentence.
Quine's response: whether this gives a sufficient condition for synonymy depends on the language for which we are trying to define synonymy. In a purely extensional language (for example, a first-order language of the type studied in introductory symbolic logic), the test will not suffice. In such a language, replacing e1 by e2 will never change the truth value of a sentence as long as e1 and e2 have the same extension (i.e. refer to the same things, if they are singular terms, or apply to the same things, if they are general terms). So to make this definition work, we'll need to specify that we have a language with intensional contexts (such as "It is analytic that _____"). What's an intensional context? A context in which substituting coextensive but nonsynonymous terms can change the truth value of the sentence. So it looks as though to make (3) work, we already need to understand the notion of synonymy! We have been led in a circle once again.
Giving up on taking synonymy as basic, Quine returns to the idea of defining analyticity directly. What about:
(4) A sentence is analytic if and only if it is true by virtue of semantic rules?
There are two cases to consider.
One is the case of an artificial language. If we design our own language, we can also specify a set of sentences that count as analytic. Quine claims that this is completely unhelpful. (We could also specify a set of sentences that we will say are "gleebish." But that doesn't mean that we have introduced a meaningful property of gleebishness.) This seems to be what Quine has in mind when he says that we could thereby define analytic-in-this-particular-language, but we still wouldn't have a useful notion of analyticity in general, one that we could apply to different languages.
A second approach in the case of artificial languages would be simply to specify that certain sentences in the language are true. (This list of truths might be regarded as axioms or postulates.) I think Quine's objection to this approach is the following. [I'm not completely certain I have this right -- I think the following is in the ballpark, but I can't promise much more than that.] Suppose that we are given a language, and that we have a way of specifying which sentences of the language are true. One way to do this would be to list a set of axioms, and say that a sentence is true if and only if it is either an axiom, or provable from the axioms. Then we could say that the axioms are analytic, while the rest of the true sentences are synthetic. The trouble with this is that there is no unique way to axiomatize a theory. We could have found a different set of axioms that would have determined the same set of true sentences. So it seems that it is arbitrary which sentences we count as axioms. (Not completely arbitrary -- some sets of axioms wouldn't do what we needed -- but arbitrary in the sense that there isn't a unique correct way to do it.)
In the case of a natural language, the problem is similar. People who understand a natural language believe that many of the sentences of that language are true. But Quine thinks there's no principled way of specifying which of the sentences they believe to be true are true by virtue of meaning, and which are empirical truths. As far as the speaker is concerned, they're all just true. We could try to construct a set of semantic rules for the language, and then say that the analytic truths are the ones that follow from the semantic rules. But (I think) Quine thinks there's no reason to think that there is a unique way to do this, so we have the same arbitrariness problem that we saw for artificial languages.
The second component of Quine's critique, found in sections 5 and 6 of "Two Dogmas," consists of a general argument against the possibility of analyticity. We may perhaps distinguish in these sections two related lines of argument.
Now, if we accept Quine's first point, then we can no longer speak of a particular sentence being confirmed or disconfirmed taken by itself. But we might still understand an analytic sentence as one immune from disconfirmation: we could say that a sentence is analytic if we would continue to regard it as true regardless of the evidence.
But Quine's response to this is that there is no special class of sentences which we will hold true come what may. We could hold any sentence to be true regardless of the evidence, if we changed our views about enough other sentences; on the other hand, we do not, or should not, regard any sentence as "immune from revision," since there may be circumstances in which the best way to revise our overall theory is to give up some sentence which had previously seemed unshakeable--even "definitions."
Quine's view, like that of the positivists, mingles semantics and epistemology. For Quine, as for the positivists and the classical empiricists, language and the world connect via experience. But, whereas for the classical empiricists simple ideas were linked to experience one by one, and whereas for the positivists observation sentences were linked to experience one by one, for Quine our entire body of theory about the world meets experience as a whole. Experiences are not linked with theoretical sentences by definitions at all; so-called "definitions" are just more bits of theory no more sacrosanct than the rest, equally open to revision if that seems the best way to improve the theory. So Quine in a sense completes the development of a strictly empiricist conception of the relation between language and the world. (At least it is arguable that Quine has made empiricism as holistic as it can get. For an attempt to out-Quine Quine, you might look at Davidson's "On the Very Idea of a Conceptual Scheme" (in his Inquiries into Truth and Interpretation, Oxford, 1984). Davidson argues that what he calls the scheme-content distinction is a "third dogma" of empiricism. Quine doesn't agree: see his "On the Very Idea of a Third Dogma," in his book Theories and Things.)