Information, as defined by information theory, seems to ultimately reduce to the causal relations between a system and the objects it has information about. And if this was how information was necessarily defined it would pose a problem for informational functionalism, since it would make it a kind of causal functionalism. And causal functionalism isn’t a workable theory about the mind since it is an externalist theory.
The best response to this problem is not to reject information theory, but rather to realize that what it describes by the use of the word information and what informational functionalism describes by the use of the word information are not exactly the same thing. Let me illustrate the difference with an example. Consider then a piece of paper on the ground that says “some elephants are red”, which, unbeknownst to you, was generated by a random process. Does that paper convey any information to you? Information theory says no, that since there is no reliable causal channel between elephant color and the writing on the paper it cannot convey information to you. In contrast informational functionalism would say yes, that the paper conveys the information that elephants are red (assuming you can read), but that the information happens to be wrong.
So where is the difference? Well, what is meant by information in information theory is really what we would call reliable information in everyday discourse, or perhaps knowledge. For information to be reliable there must be a connection between the actual state of affairs and the information about that state of affairs. And so when there is no connection there is no information in this sense. But when we talk about information in the context of informational functionalism we certainly don’t mean reliable information. Much of the information that is part of a system may be inaccurate, distorted, or otherwise unreliable.
Obviously the kind information that informational functionalist theories refer to will have to be relativized to a system (which is why the paper only conveyed information about elephants if it could be read). So, as a starting point, let me assume then that we have a way of determining the intentional directedness of a system given all the facts about its operation (with the further assumption that we define intentionality in an internalist fashion). (I presented one theory about how this might be done in my thesis draft. Essentially the proposal is that what corresponds to intentionality in a system is an internal structure that encodes input-output correlations. For example, the internal structure that corresponds to being intentionally directed at a tree, or more specifically, the structure that is activated when a system is intending a tree, contains the inputs that the system would receive from perceiving a tree and how those inputs might change if the system took various actions. Of course intentionality may be directed at non-perceptual things as well. For example, the intentional structure that is directed at numbers has as its inputs various mathematical objects and describes how those mathematical objects act as a result of being operated on mathematically. Naturally the mathematical inputs and outputs of this structure are themselves abstracted from the many possible perceptual inputs and behavioral outputs that are the vehicles with which we deal with numbers in the world, in contrast to the structure that corresponded to being intentionally directed at a tree, which did not deal with input abstractions, but rather direct perceptual inputs.) Information then corresponds, roughly, to intentionality. To say that a system receives new information is to say that it becomes intentionally directed at something new. And the intentional habits (or abilities) that already exist in the system are the information it already contains.
For example, my mind contains information about unicorns because I have the capacity to be intentionally directed at them (well, at possible unicorns, since real unicorns don’t exist). Which means that I have a conception of unicorns in which they have certain properties (horse-like appearance, one horn); under almost all theories about intentionality this is what it means to be intentionally directed at something (conceiving of it as having certain properties). My mind also contains the information that unicorns don’t really exist. This isn’t so much a property of my being intentionally directed at unicorns, since when we think about anything we think of it as existing, but a property of by being intentionally directed at the real world, which contains the expectation that I won’t find any unicorns in it, or evidence that unicorns really exist.
So, back to the original example, the paper that states “some elephants are red” contains information, to a system that can understand the words, because when such a system reads the paper an intentional structure that is directed at elephants that have the potential to be red (or possibly at red elephants) is either created or brought to mind, and so we would say that the paper conveys that information to that system.
And so information can be defined in a way that is independent of causation, assuming that you accept a definition of intentionality that is independent of causation.