Some Background
     Empiricism is the (epistemological) doctrine that all knowledge has its ultimate source and justification in sensory experience.  Traditional empiricism faced the problem of fitting into this bedrock mathematical and logical statements which seem to be justified apriori and are furthermore necessary.  The problem is that apriori knowledge - if it existed - would give the lie to the claim that all knowledge derives from experience, and necessity is apparently not given in experience.  The Logical Positivists wanted to hang on to empiricism and not compromise it the way Hume seemed to while preserving the intuition that logic and mathematics were special.  Their solution: to make them conventional, a matter of linguistic decision.  This meant they didn’t need to acknowledge apriority as a special form of knowledge.  Since a crucial mark of convention is arbitrariness, though, taking these truths to be conventional would seem to require denying their necessity.  Still, it allowed them to account for their apparent inevitability: the conventions in question are constitutive: if you change them you change the language, you're no longer playing the same game.

     In “Truth By Convention” Quine argues that whatever the separation between mathematical/logical truths and the rest comes to, it cannot be drawn in terms of convention.  His way of adhering to unmitigated empiricism is simply to deny the apriority and necessity of logical and mathematical truths, thereby abandoning the traditional segregation.  These truths do not have a unique status.  (See Mill.) Why do they appear to?  Because they are so deeply embedded.  So he reverts to a psychological explanation (away).

     Now, the attack in "TBC" may seem a rather indirect attack on the doctrine of analyticity (see Creath in Noûs `87 who thinks it's not even an attack on analyticity).  In "Two Dogmas" and “Carnap and Logical Truth” Quine mounts a much more direct, frontal attack on the doctrine by going after the analytic/synthetic distinction itself.  There he attacks the explanatory value of separating ordinary language statements into ones that are true in virtue of meaning alone and those that are true in virtue of both meaning and fact.
    One can attack the analytic/synthetic distinction in at least three different ways:

(i) by claiming that the analytic/synthetic distinction is itself meaningless;
(ii) by claiming that it has no philosophical value (that it can't explain what it purports to explain);
(iii) by denying that there are any analytic statements - all statements are synthetic (this is Dummett's interpretation).
 

The Attack on Analyticity in "Two Dogmas"

     The first task Quine sets for himself in “Two Dogmas” is to undermine the distinction between analytic and synthetic statements.  Let's start with analytic.  Let's have some examples:

“No unmarried man is married.”  “No bachelor is married.”  “If a thing is red, then it's colored.”  “Either Nixon was impeached or Nixon wasn't impeached.”

     What about synthetic truths?

“The average rainfall in L.A. is about 12".  “Bush is the U.S. president.“ There are 20 students in this class today.”

                                                                                              analyticity

 logical truth +
    synonymy

     Quine suggests that maybe the verificationist theory of meaning might help characterize analyticity.  Remember what the LPs said about analytic statements: in terms of the Verification Critierion, such statements would be verified come what may; whatever observations we make, however the world is, an analytic statement is always verified.
     Can we use this as a definition of analyticity? It would be as good as the verificationist theory of meaning.  Quine argues that the notion of individual verification conditions can at best attach to observation sentences.  Nonobservation sentences are interconnected; only a whole body of nonobservation sentences can be verified/falsified, not individual sentences.  (This is the application of Duhem’s Thesis to our everyday language.)

Recapitulation of the Attack on the 1st Dogma

     It is agreed by everyone that all truth depends partly on the meaning of the statement to be evaluated and partly on the way the world is.  Proponents of the analytic/synthetic distinction want to delimit a class of statements whose truth depends solely on meaning, and not on the way the world is.  They would cite mathematical and logical truths as a prime example.  Ask yourselves:  How could we vary the world so as to make the statement "A horse is a horse" or "2 + 2 = 4" false?  The same can be said about so-called truths of language, like "Bachelors are unmarried men".  You might say in response:  Why doesn't the fact that we can't conceive how the world could vary so as to make these false simply show that these are necessary features of the world which cannot be changed?  How would things be different in that case?  So the claim that such truths are true in virtue of meaning alone is not at all explanatory.  Quine, of course, wouldn't subscribe to this alternative explanation.  The point is that this is an equally good - or rather, equally lame explanation.
     Another fact cited in defense of the distinction is this:  If someone denies one of these statements that are supposed to be analytic we usually take it to show that she doesn't understand it and is attaching a different meaning to the sentence.  That's supposed to show that these sentences depend for their truth only on their meaning. Quine would reject the inference here.   We can agree with Strawson and Grice<1>  that there is some distinction to be drawn here - ordinary speakers make a distinction.  But the analytic/synthetic distinction is supposed to explain something to us.  And Quine rejects the explanation.  He would say: why take the undeniability of these sentences to be a sign that they're true in virtue of meaning rather than a sign that they are synthetic sentences expressing inescapable and obvious truths about the world?  This gets us to a dead end.  There seems to be no way to answer the question one way or the other.  So Quine urges us to abandon the notion of `true in virtue of meaning alone'.  The Duhem Thesis is then brought in by Quine to press even further the point that even logical and mathematical truths could be rejected - their seeming inevitability is simply a consequence of their being closer to the interior.  So the distinction loses its alleged epistemological significance - it's a matter of degree only.
     Recall the epistemological significance accorded to the analytic/synthetic distinction by the positivists: it was supposed to solve the problem of explaining the apparently special status of logic and mathematics.  I would argue that Quine’s principla aim is to deny that the distinction can have the epistemological significance accorded to it by the positivists - that it can carry the burden of explaining the apparently special status of logic and mathematics.
     Now,  you may wonder why, in attempting to challenge the idea, Quine at least appears to actually deny that there is a distinction here.  Putnam's explanation <2> : Quine got carried away.  But maybe there's a deeper, more interesting explanation.
     It might help if we notice how Quine addresses the problem of explaining the status of logic and math.  Quine ends up denying the apparent datum: he rejects the idea that logic and math have a truly special distinct status.  In his "Reply to Hellman" (Schilpp vol. p. 207) he says something like this: If reductionism is true, then you would need to explain the special status of logic and math.  This is because, as long as empirical meaning is explained in terms of reduction to sensory terms, the statements of logic and math would end up having no empirical meaning (and so for the LPs, no meaning).  So we'd have to explain their meaningfulness in some other way.  But once you give up reductionism, you see that they're not at all special, so there's no need to explain anything - their apparent necessity can be explained by their centrality and subject-neutrality - so there's no need to invoke analyticity.   (This connects nicely the attack on the second dogma (reductionism) with the attack on the first one (analyticity), understood as an explanatory dogma.

     There are some threads of positivism in Quine's attack itself: he argues that the distinction has no experimental support and concludes that it therefore has no philosophical value or cognitive significance.  But you might think that here this is acceptable.  For, the distinction was supposed to do some philosophical work; so it seems perfectly legitimate to criticize it by showing it doesn't.

The Attack on the Second Dogma

     The second dogma Quine wants to undermine is the verificationist dogma of radical reductionism.  Recall the two strands in the original Verificationism defended by the LPs:
     A: The meaning of the sentence is given by the conditions under which we can recognize it as true or false or by the method for confiriming or disconfirming it empirically.
     B: Radical Reductionism: verification conditions are given in purely sensory terms.  Some sentences, the so-called atomic sentences, are associated directly with sense-data - "Red here now", for example.  The rest are supposed to be reduced via translation to such sentences.  (Carnap in Aufbau: "Every significant statement can be translated into a sense-datum language."
     When A. and B. are combined we get the view that meaningful sentences have associated with them two sets of verifying and falsifying sense-experiences.

     Quine first denies that most sentences have individual verification conditions which appeal to sensory experience.  Only observation sentences may be said to be verified in that way.  Nonobservation sentences typically depend for their verification on the truth of other sentences.  Only large complexes of sentences can be verified/ falsified.  Sentences are typically interconnected with one another in truth-relevant ways.  What's more, there is no simple, direct route via translation from a nonobservation sentence to an observation sentence or even a bunch of them.  The connections are tenuous and multifarious.  This gets at the second strand of verificationism, which is the second dogma of empiricism.
 The alternative picture Quine draws on the basis of these observations is this: language is an articulated network.  Experience makes contact at the periphery, where are located the observation setnences.  The interior contains nonobservation sentences.  The interior is fully articulated.  All sentences are interconnected.  We saw that Quine has rejected the view that sentences can be verified or falsified individually.  What is verifiable/falsifiable is the totality of our sentences.  Now, according to verificationism, what is meaningful must be verifible/falsifiable.  And Quine accepts that.  But this means that, for him, the unit of significance, what is meaningful is the whole of language.  Quine speaks of a person's total theory:  that is, all the sentences you hold to be true about the world.  In these terms, the unit of significance is the person's total theory.  This is Quine's Holism.<3>
     Let's look more carefully at the picture.  The farther we get from the periphery, at which direct contact with reality is supposedly made, the less observational sentences are.  The most deeply embedded sentences are mathematical statements and logical laws.
     Suppose a person is led to change the truth value of some sentence.  Such a change would lead to changes in the tv of other sentences, due to the interconnectedness.
     Sometimes, a conflict between sentences may arise.  Suppose you see a building rising in the air.  The truth of the sentence "Building x is rising in the air" would stand in conflict with the truth of the law of gravity, which is embedded pretty deeply within the network.  In such a case, a revision has to be made, to maintain consistency.  Which one should we - or would we - revise?  Well, a rule of thumb would say:  Make those revisions which would lead to the least disturbances in the network.  Typically, this would mean throwing out the recalcitrant observation sentences.  But not always.  Repeated observation of some unbelievable phenomenon - or an especially acute observation, which we cannot bring ourselves to throw out - may well lead us in the end to throw out a physical law (and all the sentences which imply it).  Considerations about reasonable procedures of revision lead Quine to two bold theses:
             (1) Any sentence can be held true come what may
             (2) No sentences is immune to revision.
Re (1): Even a wild observation can be upheld at the cost of massive revisions.
Re (2): No analytic sentences (findings in quantum mechanics have led people to abandon even the law of excluded middle).
 

1  Cf.  their famous Phil. Review 1956 paper, reprinted in Travis&Rosenberg’s anthology.
2  Cf.  “The Analytic and the Synthetic”, in Mind, Language and Reality.  Putnam takes Quine to be arguing that there are no analytic truths, and thinks he went too far, because there are obviously analytic statements (`Bachelors are unmarried men' would be a prime example), and that they are clearly different from synthetic statements.  But he thinks Quine was absolutely right to think that no philosophical ice can be cut (and no philosophical bread baked, or philosophical windows washed) using the distinction, and he thinks philosophers have been misled by using the distinction, so there is no harm in denying (albeit wrongly) that there are any analytic statements.  (In fact, it's much less harmful than taking seriously the fact that there is a difference between the analytic and the synthetic.)
3 For an earlier statement of holism, see Ayer, LT&L, p.94.