I am currently working on a small paper on Frank Jackson's From Metaphysics to Ethics, which has gotten me into the discussion of two-dimensional semantics.

According to this approach, there are two ways a term or a sentence can be said to apply or to be true at different possible worlds. The first way one can consider what some term applies to in a possible world, is by supposing that that possible world is the actual world. The second way one can consider what some term applies to in a possible world, is by treating the world as a counterfactual world. Jackson calls a term's extension, in the first sense, the term's A-extension; and in the second sense, the term's C-extension. Likewise, the intension of a term in a world considered as actual, is called the A-intension of the term; and the intension of a term in a world considered as counterfactual is called the C-intension of the term. David Chalmers has summarized (or, at least, mentioned someone else's summary) these two ways of thinking about possibilites in the following way: in the first case, one “considers a possibility as actual”, and in the second case, one “considers a possibility as counterfactual“.

To use Jackson's own example, consider the term “water” (we accept Kripke's view that terms like “water” are rigid designators). In a possible world considered as actual, the A-extension of water is whatever substance that plays the “watery role” in that world, be that H20, XYZ or whatever. In a possible world considered as counterfactual, what the term “water” denotes is H20 – its C-extension is only H20 – since “water” is a rigid designator.

My question is: Suppose w is the actual world, and that the substance that plays the watery role in w is XYZ. Then, of course, the A-extension of "water" in w is XYZ. But what about the C-extension of "water", under the supposition that w is the actual world? Do we take as a premise that w is the actual world, and that "water" in w denotes XYZ, and conclude that in every counterfactual world (that is, every other world than w), "water" denotes XYZ? I have a strong feeling I'm missing something here, but I'm not sure what it is.