The claim is that:
Now I'm not going to make this claim independently of the definition of the three concepts; any one of them could be construed as distinct from the others. So I'll begin by offering some definitions, then argue the case, and then argue the merits of the definitions.
A bit if common ground for all these concepts. What kind of thing is it that can be Analytic (etc.)? I suppose the obvious answer is propositions, however the notion of proposition is sufficiently unclear that I propose to talk about the analyticty (etc.) of statements. A statement is to be understood in this context as a triple, a language (with a well defined semantics), a sentence, and sufficient context (in which the sentence is deemed to have been asserted) to settle the meaning of the sentence.
Now a statement is Analytic if its truth can be establish by purely logical means. It is necessary if it is true in every possible world, and it is a priori if it can be known without the benefit of any empirical evidence. In the latter I want to de-emphasise the significance of the problematic word "know". Our interest here is in the kinds of evidence which are relevant to establishing the truth of the statement, not the precise conditions under which we can be said to have knowledge.
Note that the statement itself is to be considered given. In general our knowledge of the meaning of a sentence in a language is a posteriori, but this does not entail that our knowledge of the statement need be. Similarly the meta-linguistic fact that some statement is necessary will itself be contingent.
Now let us consider the claim that Hesperus is Phosphorous, argued by Kripke in his book Naming and Necessity to be necessary but a posteriori. (see also the Chicago Philosophy Project for online discussion of this book.)
Kripke's argument depends upon his claim that (even) in natural languages, names are rigid designators. This is a claim about the meaning of natural languages which seems to me not beyond doubt. However, my intent here is to argue against the existence of necessary propositions which are not also a priori. This can be done independently of the semantics of names in natural languages. However, Kripke's position on the semantics of names may well have materially contributed to his confusion about the epistemological status of the statement in question.
The argument against Kripke's position here is very brief. We accept that if names are indeed rigid designators then the statement "hesperus is phosphorous" is necessary. Contrary to Kripke however, we maintain that it must also be analytic and a priori.
If names are rigid designators, then we cannot claim to know the meaning of a statement unless we know what each name denotes. If we know the meaning of a true identity between names, then we must also know that both sides of the identity denote the same entity and hence that it is true. The truth follows from the meaning of the statement, and once the meaning of the statement is known can be known by purely logical means and without further reference to empirical evidence.
Kripke's insistence that the statement is a posteriori is based on the evident fact that we know the truth of the statement only as a result of empirical enquiry. This however, in the strange circumstance which Kripke has devised by his theory about the meaning of names, is entirely due to the resulting semantic opacity of names in natural languages. If we do not know the meaning of a name until we know what it denotes, then we may find ourselves having to go to some considerable pains to discover the meaning of a sentence such as hesperus is phosphorous. A difficulty of this kind however, does not render all statements using names of doubtful reference a posteriori.
If we rejected Kripke's position on the meaning of names, and adopted Russell's view that they are covert descriptions, we find that the epistemological and modal status of identities between names change in step. The statement becomes, perhaps more plausibly, contingent, synthetic and a posteriori.
This is not presented as an argument in favour of Russell's position. Natural languages are probably too complex, and possibly insufficiently coherent, for either generalisation about the meaning of names to be true.