The Analyticity of Carnap Conditionals
In section 24.D of his "Replies and systematic expositions" in the Schilpp volume, Carnap argues that every theory can be split into a component "representing the factual content of the theory", and another component serving as "analytic meaning postulates [...] for the theoretical terms". In fact, he doesn't speak about every theory, but it seems that what he says is true in general.
Take everything you believe about water, and call that your water theory. Your theory presumably contains things like "water fills our lakes and rivers", "water boils at around 100 °C under normal conditions", "water consists of H2O", and so on. All that is plainly empirical. Now the factual component of your theory, according to Carnap, is its Ramsey sentence: the theory with all occurrences of "water" replaced by a variable and prefixed by an existential quantifier binding that variable. The analytic meaning postulate then is the material conditional of the Ramsey sentence as antecedent and the theory itself as consequent. Let's call that the Carnap conditional of the theory.
Is the Carnap conditional really analytic? Arguably yes. Or at any rate a priori. For how could you find out that it is false? Since a material conditional is false iff the antecedent is true and the consequent false, you'd have to discover that 1) something satisfies everything you believe about water, and yet that 2) this something is not water. But how could that be? You can only discover that something is not water by discovering that it fails to satisfy some condition necessary for being water. You can't discover that something isn't water even though it satisfies all conditions.
Carnap appears to have a different, more interesting argument for the analyticity and a priority of Carnap conditionals: he proves that among observation sentences, Carnap conditionals only logically entail logical truths. For present purposes, we can take an observation sentence (briefly, O-sentence) to be any sentence that doesn't contain the relevant 'theoretical' terms. So Carnap proves that among sentences not containing the word "water", the Carnap conditional of your water theory entails only logical truths. And this, it seems, shows that no empirical observation could possibly undermine the conditional, for there presumably are no empirical observations that can only be described using the word "water". At any rate, no observation about the stuff in our rivers and lakes, about chemistry, about causal chains of communication or the like could entail the falsity of the Carnap conditional.
Carnap's proof is simple and ingeneous. First, a lemma due to Ramsey:
Lemma: for any theory, its Ramsey sentence
logically entails exactly the same O-sentences as
Proof:
1. Suppose.
2. Then.
3. So.
4. So.
5. And then.
And now Carnap's proof:
Claim: for any theorylogically entails only logically true O-sentences.
Proof:
1. By (Lemma), the Ramsey sentence of the Carnap conditional, i.e., entails exactly the same O-sentences as the Carnap conditional itself.
2. The Ramsey sentence of the Carnap conditional is a logical truth.
3. Since logical truths only entail logical truths, the claim follows.
Great, so we don't even have to rely on the vague intuition that the Carnap sentence of your water theory is a priori, we actually have a proof!
No. To see what's wrong, take another water theory: "there is a glass of water in my kitchen". The Carnap conditional of that theory is
C) if there is a glass of something in my kitchen, then there is a glass of water in my kitchen.
And clearly that is not a priori.
Carnap's proof still goes through: Among sentences not containing "water", (C) logically entails only logical truths. But from that it doesn't follow that (C) cannot be refuted by empirical observations. It only follows that the refutation will never be logical: there is no sentence not containing "water" that logically entails the negation of (C). But there are many such sentence that a priori or inductively entail the negation of (C), like: "there is only one glass in my kitchen, and that glass contains milk". To get from this to the negation of (C), you need the "water"-involving truth "milk is not water".
The upshot is that we can't rely on Carnap's proof to have established that any old Carnap conditional is analytic or a priori. And to be fair, Carnap claims no such thing.
(Likewise, we can't rely on Ramsey's proof to have established that any old theory is a priori equivalent to its Ramsey sentence: "there is a glass of water in my kitchen" is not a priori equivalent to "there is a glass of something in my kitchen".)