Trouble with Centered Worlds
I would like to say that
If X necessarily entails all truths, and if X's A-intension coincides with its C-intension, then X a priori entails all truths.
For suppose X -> P is not a priori for some truth P. Then X -> P is an a posteriori necessity. So we need information about the actual world to know what C-intension X -> P expresses, and whether it is true. But by assumption, this information is already contained in X, and since X's C-intension coincides with its A-intension, it cannot be hidden away in X so that we'd need further information to find out that X contains that information. Hence X a priori entails X -> P; and so X -> P is itself a priori.
But A-intensions are centered while C-intensions are not, and this introduces a problem.
The problem is that any sentence whose C-intension contains just the actual world will necessitate all truths, no matter whether it contains any indexical information. If the sentence contains no indexical information, its A-intension will contain all centerings of the actual world. (Maybe the conjunction of all non-indexical truths is such a sentence.) But knowing that sentence may not suffice to infer a priori all truths, not even all non-indexical truths: if our universe contains a twin earth, we may need to know where in the world we are in order to know whether water is H2O or XYZ.
It doesn't help to demand that X must also necessitate all indexical truths: if X necessitates all non-indexical truths, it probably necessitates all indexical ones anyway. For the C-intension of "the watery stuff of my surroundings is H2O" is the same as that of "the watery stuff in WS's surroundings is H2O". So if X necessitates the latter, it automatically also necessitates the former.
Maybe I should say that X together with certain indexical truths a priori entails everything. But what are the relevant indexical truths? All indexical truths suffice to entail everything even without the help of X. Intuitively, the additional information is just what must be added to X to tell me where, when and who I am in the worlds not ruled out by X. But how should this information be given? If, for instance, I don't have a name in some world, should the additional information contain a description that uniquely picks me out (and if so, what description)? Or should I just leave it open how the information is given and say that X together with any further truth that suffices for self-location a priori entails everything?
Or maybe it would be better to put the information into X, as follows:
If X necessarily entails all truths and X's A-intension doesn't contain more than one center of any world, and if modulo centers, X's A-intension coincides with its C-intension, then X a priori entails all truths.
But that sounds terribly complicated. The original version would have been easy to apply: all we have to check is whether X contains a rigid designator; if not, then if everything supervenes on X, everything is a priori entailed by X. With the present version, we also have to check whether X's A-intension contains more than one center of any world, and how do we do that?
Or maybe instead of patching X I should substract from the entailed truths:
If X necessarily entails all truths and, modulo centers, its A-intension coincides with its C-intension, then X a priori entails all truths that contain neither indexicals nor rigid expressions.
But that sounds too weak. Maybe I should add that every truth containing rigid expressions is a priori entailed by a truth not containing rigid expressions. But I'm not entirely sure that's true if the "truths" quantified over here are simply true sentences of English -- which I would like to maintain. (One can easily construct a language with too few non-rigid expressions in which the claim is false; and I can't prove that English is not a language like that.)
So what to do?
Let's say an epistemically complete truth (for a speaker at a time) is one that a priori entails all truths (for that speaker at that time), that a metaphysically complete truth is one that necessitates all truths, and that a neutral expression is one with the same primary and secondary (A- and C-) intensions. Then I would put the point by saying that if M is a metaphysically complete neutral truth, and L is locating information (relative to M) for a speaker at a time, then M&L is epistemically complete (for that speaker at that time).
Here L is locating information relative to M for a speaker and a time if L has the form 'I am D1' and 'now is D2' where these both express truths for the speaker, and where D1 or D2 are singular terms or descriptions such that M a priori entails 'exactly one individual is D1' and 'exactly one time is D2'. It is natural to stipulate that both D1 and D2 should themselves be neutral, though there are then complexities needed to handle symmetrical worlds. Of course there will be more than one such D1 and D2, but I don't see why that poses any more problem than the fact that there will be more than one such M.
If you want to weaken the consequent instead, you could do so by saying that if M is a metaphysically complete neutral truth, then M a priori entails all neutral truths. I don't see that this is too weak, as M won't a priori entail most non-neutral truths. It's plausible that non-neutral truths are themselves entailed by neutral truths plus locating information (in a rich enough language), but that seems like a separate claim. And if you want to build in that claim, you might as well go back to the formulation above.