First, some background: I use 'descriptive content' to refer to the kind of ('narrow') representational content or semantic value that is grounded in scenarios (conceptual possibilities, or centered possible worlds considered as actual) -- namely, the primary intension. For example, the descriptive content of a proper name, 'Bob', is given by some associated reference-fixing descriptive property, D, such that 'Bob = the D' is analytic (if D can be captured in our language). Equivalently: it is the set of possible persons who are such that he is Bob follows a priori from the hypothesis that his world is actual. [Read up on the links if this is hard to follow.]
Now for the argument:
Consider a perfectly symmetrical universe, containing two qualitative duplicates we may dub 'Bob' and 'Mirror-Bob'. They share all their objective intrinsic and extrinsic properties (for their surroundings are also qualitatively identical). So there is no difference between them -- no unique property that could fix the reference of my term 'Bob' (or 'Mirror-Bob') as referring to the one person rather than the other.* Probably the best way to make sense of this is to say that the two terms are indeterminate between the two referents, and so to assess the truth of any sentence involving the terms, we supervaluate over the various possibilities.
* = Aside: this problem could be overcome if we were inside this universe, for then we could appeal to the relative property of being the Bob in my vicinity, or whatever.
Consider, then, the following sentences:
(1) 'Bob is human' and (2) 'Bob = Bob'
And compare the first-term substitutions:
(1a) 'Mirror-Bob is human' and (2a) 'Mirror-Bob = Bob'
Both 1 and 1a are true, and indeed true in all the same scenarios, which is to say that they have the same descriptive content. (Intuitively: to say that Bob is human, and to say that Mirror-Bob is human, is not to describe two different scenarios. Both names actually range indeterminately over the same two people, and both apply equally determinately to any unique Bob-counterparts in other, non-symmetric, worlds.)
But 2 and 2a clearly differ in content, for the former is true and the latter false! For although it is indeterminate which of the two people each of these terms denotes, it is determinate that they denote numerically distinct people. (Either 'Bob' denotes the one guy and 'Mirror-Bob' the other, or vice versa. On either way of resolving the indeterminacy, the names turn out not to be co-referential. Hence, by supervaluation, it is determinate that they are non-co-referential.)
The problem for compositionality is that this fact -- that the two terms do not co-refer -- cannot be derived from the descriptive contents of each term alone. Indeed, we've seen that both names have the same primary intension (an intension which happens to be indeterminate at this world, but not at others). But, despite having the same content, they can potentially make significantly different contributions to the content of a whole sentence in which they are part, as the comparison of 2 with 2a shows.
(Can anyone suggest a neater way to make sense of this puzzle case?)