In his John Locke Lectures, Kit Fine proposes a new solution to Frege's Puzzle (see in particular lecture 2 (warning: 'RTF' format -- unless you use a perfect intrinsic duplicate of Kit Fine's computer, that means you probably have to guess all the logical symbols)).
The puzzle, according to Fine, is that there is an intuitive semantic difference between "Cicero = Cicero" and "Cicero = Tully". That is puzzling on the assumption that the semantic contribution names make to sentences is only their referent.
Fine's new solution is to restrict compositionality: the semantic value of an identity sentence is not a function of the semantic values of the singular terms it contains. The semantic value of "Cicero = Cicero", according to Fine, is the singular proposition <Cicero, identity, Cicero> together with a 'semantic connection' linking the first element of the triple with the third, whereas the value of "Cicero = Tully" is that same singular proposition without the semantic connection. Whether the semantic value of a sentence contains a semantic connection is determined somehow by the modes of presentation ideal competent speakers associate with the names in the sentence -- but these modes of presentations, or 'takes', are not part of the semantics, and only casually mentioned along the way.
Fine's general conclusion is that there is something fundamentally wrong with the way we used to do semantics: talking only about the semantic values of individual expressions, we always ignoged the irreducible semantic relations or connections between them.
I think Fine misses the real impact of Frege's puzzle. The puzzle is not about identity in particular. It is about how, in general, replacing co-referential singular terms in a sentence often results in a sentence with different cognitive significance (and not just if the sentence contains intensional operators). Consider
- The morning star is visible in the morning sky.
- The evening star is visible in the morning sky.
- molecules of water contain H atoms.
- molecules of H2O contain H atoms.
These sentences differ in their significance, and a competent speaker might regard one of each pair as true and the other as false. But this time, the difference cannot be explained by semantic connections linking certain coreferential terms.
Besides, Fine's lectures are a good example of a certain approach to the philosophy of language one might call semantic phenomenology. On this account, the aim of the project is to systematize raw intuitions about the meaning of various linguistic items. Interestingly, Fine has strong intuitions not only about the meanings of names and predicates, but also about the meanings of things like sequences of free variables. I think the whole project is a bad idea, as intuitions about meaning are just too unsystematic for it to deliver anything of interest.