Unique centers
Continuing the topic of the last post, suppose I'm certain that no-one else in the history of the universe ever had (or will have) exactly the experiences that I have now. Then I can 'translate' any centered proposition into an uncentered propositions in such a way that the translation is certain to preserve truth-values. For instance, "it is raining" gets translated into "it is raining at all times and places where someone has such-and-such experiences". In this case, one might think, purely centered information can never affect my uncentered beliefs. For purely centered information only distinguishes between multiple centers within a single world; but if no world has multiple possible centers, then there is nothing to learn from such information. (This line of reasoning is related to what Mike Titelbaum says in his forthcoming paper "The Relevance of Self-Locating Beliefs", though I don't think Mike would endorse the argument I present here.)
Purely uncentered information can be understood as conditional information. A purely uncentered proposition says that if the universe is like this, then you are here; if it is like that, then you are there -- without saying how the universe actually is. In general, the purely centered fragment of any proposition E is the material conditional S(E) -> E, where S(E) is the strongest uncentered proposition entailed by E. For example, if E says that it's raining, then S(E) says that it's raining at some point somewhere, and the purely centered fragment of E says that if it is raining at some point somewhere, then it is raining (here and now).
The above suspicion could now be expressed as follows. If none of my belief worlds has multiple possible centers, then I already know all relevant conditionals; I know exactly that if this is my universe, then I am here -- at the only place in that universe where someone has such-and-such experiences. Hence there is nothing I could learn from conditionals of that form. All my ignorance is ignorance about which universe I inhabit.
Not so. The argument assumes that if I already know that A -> B, then I cannot find out that A -> ~B. But if A is false, the conditionals are both true, and I may well find out ~A in precisely this way. The purely centered information that if I'm at universe w then I'm at center c can tell me something new about the universe if I already knew that if I'm at w then I'm at a different center c'. In effect, the information tells me that I'm not at w. Even though the information is purely centered, it allows me to rule out an uncentered possibility.
Remember the thunder example from the previous post, where theory A claimed that it is thundering all the time, theory B claimed that it is thundering only once in the history of the universe, and you learned that it is thundering. The purely centered fragment of this information is that if it is thundering at some point, then it is thundering now. Suppose that is all you learn. Suppose also that each of your initial belief worlds has a unique center you deem possible. Some of these worlds are a) worlds where it thunders all the time, including at that center. Others are b) worlds where it thunders only once, including at the center. Yet others are c) worlds where it thunders once, but not at the center. (Others are worlds where it never thunders, or where it thunders twice, etc.) When you learn the conditional that if it thunders at some point, then it thunders now, you can rule out worlds of type (c), but none of type (a) or (b). Since theory A holds in all and only the (a) worlds, you thereby increase your credence in A.