Semi-public A-intensions
Some days ago, Christian and I had an interesting discussion about two-dimensionalism. While I don't agree with many of his criticisms (forthcoming in Synthese), I do agree that two-dimensionalism works best if both dimensions belong to an expression's public meaning. I think that Christian thinks that this holds only for context-dependent expressions. I think it holds almost universally. But this may be a matter of terminology: For me it is part of the meaning of 'the liquid that actually flows in rivers' that this would not denote H2O if it would turn out that XYZ flows in rivers, whereas for Christian this is a metasemantic fact. Anyway, problems for two-dimensionalism come when the first dimension doesn't belong to public meaning.
Suppose one day I'll meet Swampman and the brain in a vat. We will be able to communicate perfectly well. How could this be possible if we spoke completely different languages? Perhaps there are aspects of meaning which aren't directly relevant for understanding and communication, but this certainly can't be true for all aspects. So there is at least one sense of 'meaning' in which the words of Swampman, Vatman, and I have the same meaning. To grasp this kind of meaning, it suffices to have the right mental states, whatever these are. I would have hoped that this kind of meaning could be identified with the A-intension (or two-dimensional intension) of two-dimensional semantics. But it seems that it can't.
For A-intensions may differ from speaker to speaker. Perhaps your A-intension of 'water' picks out the liquid in the rivers, and mine the liquid in the seas. Then if Swampman is your physical duplicate, the A-intension of his 'water' is not the same as the A-intension of my 'water'. Hence A-intensions are not what his 'water' and my 'water' have in common.
(For the same reason, twodimensionalism seems to deliver only private apriority: What is a priori for me may be a posteriori for you. Armchair metaphysics then won't help public metaphysics.)
Maybe there is a way out: Add to every private A-intension of an expression E a clause stating that all competent speakers' A-intensions of E (counterfactually) necessarily corefer. So your intension of water becomes something like 'the liquid in the rivers and the stuff we all denote by 'water''. A-intensions are still private, but now they can be used to explain why Swampman, Vatman and I understand each other: We know that each of us uses 'water' to denote exactly what our private A-intension assigns to 'water'. Similarly, if I derive a priori from a physical theory that my 'water' denotes H2O, this is immediately relevant for public metaphysic, because it implies that everybody's 'water' denotes H2O.
On this account, grasping the meaning of 'water' has two parts: First, you must know that 'water' substance-rigidly denotes whatever satisfies a particular description D, and second, you must know that all speakers know that 'water' substance-rigidly denotes whatever satisfies some particular description D', which denotes the dame liquid as D.
(D of course needn't be purely qualitative. It may contain indexicals ('what flows in our rivers') and demonstratives ('that stuff'). That's the point of taking contexts (or possibilia or centered worlds) as domains of A-intensions.)
I just realized that this account resembles what Lewis says about naming the colours in 'Naming the Colours'.