No time for the moment to think more about this (essays to mark, abstracts to read, the initial plans for the content of the Oxford Handbook of Psychiatry to review) but an undergraduate dissertation on Dennett made me look at Searle’s discussion of intentionality and the Background in his Rediscovery of Mind [Searle 1994]. Summarising the role that the Background has in helping out intentionality, he says:
1. Intentional states do not function autonomously. They do not determine conditions of satisfaction in isolation.
2. Each intentional state requires for its functioning a Network of other intentional states. Conditions of satisfaction are determined only relative to the Network.
3. Even the Network is not enough. The Network only functions relative to a set of Background capacities.
4. These capacities are not and cannot be treated as more intentional states or part of the content of any particular intentional state.
5. The same intentional content can determine different conditions of satisfaction (such as truth conditions) relative to different Backgrounds, and relative to some Backgrounds it determines none at all. [ibid: 177]
Points 1 and 5 seem most radical. Searle says a little more to unpack these:
All conscious intentionality, all thought, perception, understanding, etc. determines conditions of satisfaction only relative to a set of capacities that are not and could not be part of that very conscious state. The actual content by itself is insufficient to determine the conditions of satisfaction. [ibid: 189]
The argument for this starts with Wittgenstein’s example of a picture of a man walking up - or is it down? - hill. Searle says: ‘nothing internal to the picture... forces the interpretation we find natural’. From this example, and from examples of linguistic interpretation - of, for example, cutting various things - Searle concludes that the intentionality of such items is determined only relative to a background of both other intentional states but also of Background capacities which are themselves non-intentional.
But the surprising transition is from cases where we can identify the bearers of content (pictures, symbols, words), and where it makes sense to think of types which fall short of the determined intentional content, to intentional mental states. When he says that the same intentional content can determine different conditions of satisfaction he must surely mean something akin to: the same type of intentional content... But if so, that’s very odd. What is such a type, short of its content?
Now of course there are philosophically charged ways of articulating such types. One can, eg, split things into narrow and wide content and where the narrow sort falls short of full content. But Searle seems to be invoking Wittgenstein (not an obvious ally of the distinction of broad and narrow content) to make his claims and seems to think they follow from much less theoretical machinery.
In fact, pretty much the only argument for the transition from symbols to mental states is this:
it is useful to have a taxonomy that captures our intuition that there is a match between thought and meaning. For example, I want to capture our ordinary intuition that the man who has the belief that Sally cut the cake has a belief with ex actly the same propositional content as the literal assertion “Sally cut the cake”. 
From this he concludes that the same intentional content can determine different conditions of satisfaction. But that just seems very odd indeed.
Popping in for a G&T, Gloria suggested that what was odd was how Searle could be thinking of how things seemed from a first person perspective (how it would seem to have such an ambiguous-between-types sort of content). But I’m not sure that the problem turns on the first person. Rather, whilst the content is indeed that which can be captured in a suitable use of the words (“Sally cut the cake”, for example) it is the content of that utterance that is the same as the state. (That that utterance might have been taken by someone not in the know in some other way does not seem to me to the point. Nor thus a similar view of the mental state in question, if you forgive that phrase.) That seems to me to be what we should say whether describing the first person view or a (correct) third person ascription.
Searle, J.R. (1994) The Rediscovery of Mind, Cambridge, Mass.: MIT Press