If we want to model rational degrees of belief as probabilities, the objects of belief should form a Boolean algebra. Let's call the elements of this algebra propositions and its atoms (or ultrafilters) worlds. Every proposition can be represented as a set of worlds. But what are these worlds? For many applications, they can't be qualitative possibilities about the universe as a whole, since this would not allow us to model de se beliefs. A popular response is to identify the worlds with triples of a possible universe, a time and an individual. I prefer to say that they are maximally specific properties, or ways a thing might be. David Chalmers (in discussion, and in various papers, e.g. here and there) objects that these accounts are not fine-grained enough, as revealed by David Austin's "two tubes" scenario. Let's see.
First some more background. The two tubes case belongs in the category of "Frege cases" where a subject attributes one property to an object x, a possibly different property to an object y, without knowing that in fact x is the same object as y. Such cases spell trouble for the view that all there is to the relevant beliefs is captured by saying which properties the subject attributes to which objects. The trouble is especially pressing for the explanation of behaviour. Why does Pierre try to buy a ticket to London at a (French) travel agency in London? Why does Ralph reveal confidential information to Ortcutt, if he thinks he is a spy? But there is also trouble within epistemology, so to speak. If Pierre is such a good logician, why doesn't he realise that he holds contradictory opinions about London? How can we understand the fact that the ancient Babylonians received evidence confirming that Hesperus is Phosphorus, but not that Hesperus is Hesperus?
In most Frege cases, the answer is that we must somehow take into account that the subject attributes the relevant properties to the relevant objects only under a particular "guise" or "mode of presentation". Ralph attributes spyhood to Ortcutt qua man in the brown hat but not qua man on the beach. In Bayesian epistemology and decision theory, subjective probabilities should therefore be sensitive to guises. Perhaps they attach to pairs of a "singular proposition" and a guise. A simpler view, which I prefer, is to drop the singular propositions and have the probabilities attach to guises only: by the lights of Ralph's doxastic state, it is probable that the man in the brown hat is a spy, and not that the man on the beach is a spy, and that's all we need to know about the case to predict his behaviour and to understand the inferences he does and doesn't draw. (Challenge: find something Ralph does that he wouldn't do if he only had such qualitative beliefs.)
In some Frege cases, however, this response doesn't work. When Perry doesn't know that he himself is the messy shopper, or when one of Lewis's two Gods doesn't know that he is the God on the tallest mountain, their ignorance cannot be modelled as ignorance of a qualitative fact about the world. One might know all qualitative facts about the world and still not know where one is in the world. This is why we need centred qualitative propositions (or centred qualitative propositions paired with singular propositions, if we don't want to go guises-only) as objects of subjective probability. Roughly speaking, each possible universe divides into many centred worlds, one for each possible answer to `who am I?' and `when is now?'. Since a property is, roughly, something that assigns to each world and time a set of individuals, a set of word-time-individual triples can also be understood, roughly, as a property. According to Chalmers, Austin's two tubes case shows that we have to expand centred worlds to also fix the answers to other questions.
So here is the case, as presented in chapter 2 of Austin's What's the meaning of this?.
Smith is the subject of a psychological experiment. He faces an opaque screen with two small eye holes, each of which leads to a tube. Smith has learned to focus his eyes independently of each other. When he looks through the tubes, he sees a red dot (which he dubs `this' or `Harold') with his left eye and an indistinguishable red dot (which he dubs `that' or `Mauve') with his right eye. He doesn't know that he sees the same dot with both eyes, because he can't tell exactly how the tubes and his eyes are oriented.
Now the topic of Austin's book is not the modelling of subjective probability in Bayesian epistemology and decision theory. Rather, his starting point is the idea that there is a relation BELIEF which holds between a subject S and a proposition P whenever one can truly utter `S believes that Q' in a context where Q linguistically expresses the proposition P. Since in a suitable context, one can truly say `Ralph believes that Ortcutt is a spy', and the proposition linguistically expressed by `Ortcutt is a spy' is arguably not a qualitative proposition (because its truth value at counterfactual situations depends only on whether Ortcutt himself is a spy in those situations), it follows that the objects of the BELIEF relation are not always qualitative. Nothing I said above contradicts this claim, since I haven't been talking about BELIEF. So we have to be a bit careful when looking at Austin's arguments against various ways of handling the two tubes case. In particular, his main objection against the idea (attributed to Schiffer, Chisholm and Lewis) that the objects of BELIEF are qualitative properties is that this conflicts with intuitions about the non-qualitative content linguistically expressed by sentences like `that man is a spy'. Obviously, this is not an argument against the proposal that subjective probabilities are defined on an algebra of qualitative properties. This proposal is not a claim about attitude reports or about what is linguistically expressed by this or that sentence.
Our question is whether we can account for Smith's doxastic situation, for the purposes of (say) a broadly Bayesian epistemology and decision theory, in a framework where his subjective probabilities are defined over qualitative properties, or over possible individuals at possible times in (qualitative) possible worlds.
Well, can't we run the standard response to Frege cases? Smith is acquainted in two different ways with the same object, so that when he wonders, `is this = that?', he assigns middling probability to the hypothesis that the object he is acquainted with in the first way is identical to the object he is acquainted with in the second way.
Austin considers the related proposal that for Smith, `this' and `that' are synonymous to different qualitative definite descriptions in Smith's language. Obvious candidates are `the dot I see with my left eye' and `the dot I see with my right eye'. In a footnote (n.15 on p.45), Austin tries to block this move by stipulating that Smith does not identify his two visual fields as left or right, because he falsely believes (Austin should have said: suspects) that he suffers from allesthesia, a condition in which sensations from one body side appear to come from the other.
But this complication is easily circumvented, for example by moving to `the dot I see with my left eye if I don't have allesthesia, otherwise the dot I see with my right eye'. Alternatively, we might try `the dot I see with the eye that seems to me to be my left eye'. Or `the dot I baptized "this"'.
Of course, we don't really have to find a description in Fred's language, and if we do, it doesn't matter (for our context) whether Fred would regard `this' as synonymous with the description. The question is whether we can adequately model Fred's doxastic situation by saying which possible individuals at which worlds and times he can rule out, and which of the others he deems more likely than others. Austin himself appears to accept that this is possible when he discusses Stalnaker's view in chapter 5. Here he accepts that there are different "diagonal propositions" for `this is red' and `that is red'. He raises several objections to Stalnaker, but he never suggests that these diagonal propositions don't exist.
So Austin's example seems to lend itself well to the standard treatment of Frege cases. It hardly shows that we need to add new coordinates to centred worlds.
But perhaps I've been sticking too closely to Austin's own discussion. Here is Dave Chalmers's rendition of the two tubes case, from p.625 of his Frege's Puzzle paper (2011):
Fred is looking down two tubes, one attached to each eye, and has a symmetrical experience as of two red balls. Fred is objectively omniscient and knows that in fact [the] tubes are connected to one red ball and one orange ball. He also knows his own location in the world and the current time, and so knows which centred world he inhabits. When he entertains the hypothesis That is red and that is orange (using two simultaneous perceptual demonstratives), he is not in a position to determine that it is true and has rational credence 0.5 in it, but there is no verifying centred world.
Lots of changes here. Smith has been replaced by Fred. The tubes have been reoriented to point at two different objects (balls, in fact), one of which is red, the other orange. For some reason, both balls appear red to Fred. Instead of `this' and `that', Fred is "using two simultaneous perceptual demonstratives", both of which are written as `that'. Finally, Fred is omniscient about the world as a whole as well as his own present location. Nevertheless, there is supposed to be some hypothesis to which he assigns probability 0.5. If this were correct, we would indeed have an argument for more fine-grained objects of probability. But is the scenario really coherent?
Fred's stipulated omniscience immediately entails that he does not identify the two balls by different relations or descriptions, so that the object of his uncertainty could be represented as the hypothesis that the F1 is red and the F2 is orange.
Suppose for concreteness that the ball Fred sees with his left eye is the orange one. Since Fred is omniscient, he knows this. That is, he knows that the ball he sees with his left eye is orange and that the other one is red, although they both appear red. Fred also knows that he does not suffer from allesthesia. So he knows that the ball he sees in what appears to him to be his left visual field really is the left ball, which is orange. Similarly, he knows that the ball he sees in what appears to him to be his right visual field is the red ball. But then what is he uncertain about?
Let's put ourselves into Fred's shoes. The visual experience isn't too hard to imagine. It involves two distinct sensations of what appears to be a red ball at the end of a tube, one for each eye. Hold fixed this visual image, and assume you know that the ball at the end of the left tube is orange and the other one red. Now ask yourself, `is that red and that orange?'. Don't you know the answer?
Perhaps the idea is that for some reason, Fred can't tell his two sensations apart. He can't tell which one seems to be coming from his left eye. In general, he is not aware of any feature F that distinguishes the two sensations -- otherwise he would automatically know which ball is responsible for the F sensation and which for the non-F sensation. (These are straightforward centred facts.)
But then I don't see how his two uses of `that' could determinately attach to particular ones of his sensations, and thereby to particular balls. Suppose the first `that' in Fred's question `is that identical to that?' denotes the orange ball and the second the red one. Surely this is not a primitive fact. What makes it true? Normally, when you wonder `is that identical to that?', your attention shifts from one guise (or sensation) to the other between uttering or thinking the two occurrences of `that'. Not so for Fred, otherwise he would know that the sensation first attended to comes from the orange ball and the other one from the red ball.
If the two `that's are indeterminate in reference, then there is no hypothesis expressed by `that is red and that is orange'. There are several hypotheses, all of which arguably have either probability 1 or 0.
So Austin's two tubes scenario looks like a harmless Frege case. Chalmers's scenario is stipulated to block this response, but it is not at all clear that the scenario is coherent.
One more remark on the form of these arguments. If Fred's probability function is only defined for qualitative centred worlds (say), then it is easy to find all sorts of entities for which it is undefined. One could stipulate that Fred is omniscient about qualitative matters, but nevertheless assigns probability 0.5 to the singular proposition that Aristotle was fond of dogs. It would follow that we must include singular propositions as objects of probability. Similarly, one might stipulate that Fred assigns probability 0.5 to the moon, wherefore we must include the moon as an object of probability. For these proposals to pose any serious threat to the standard centred-worlds conception, it has to be shown that the extended probabilities are needed in order for subjective probabilities to do their job. Recall the case of attitudes de se. The main reason for going beyond uncentred qualitative propositions is that an agent's rational behaviour typically can't be explained if probabilities are only assigned to such uncentred propositions. I don't see any analogous motivation from the two tubes cases.