Soames (in Understanding Truth) argues that the Tarskian truth predicate fails to help capture the meaning of the language in the way that the ordinary-language truth predicate may. Nevermind the details of the argument (which looks sound to me). I wonder, however, whether the Tarskian truth predicate might not continue to help capture the meaning of the language when one adds the information that it is a truth predicate. If so, then Tarski has produced a reduction of truth facts to a naturalistic truth predicate and a non-naturalistic higher order is-a-truth-predicate predicate.
I don't know if what I've said is true. But whether it is true or not, it highlights an interesting pattern one might have in a philosophical analysis. We start with a "problematic" (because semantic, non-natural, normative, vel caetera.) natural language predicate P. We find a reduction of P to an "unproblematic" predicate Q that is extensionally correct. But then to use Q to capture the facts that were captured with P, it turns out we also need the higher order fact that Q is a P-predicate which we cannot reduce (unless we want to embark on a regress). The reduction from P to Q might, nonetheless, be informative.
Here is an example of how this might work. Take the natural language binary predicate good for. Consider some non-normative reduction of this, such as the claim that x is good for y means that y desires x (nobody defends this anymore, but it'll be nicely illustrative). Let's assume this is extensionally correct (everybody working on well-being knows the counterexamples), and let's assume that "desires" is not normative (the Socrates of the Gorgias might disagree). Nonetheless, something is left out by the reduction. For while it is a trivial truth that it is an instance of benevolence to promote someone's good, it is a non-trivial truth that it is an instance of benevolence to promote someone's desire-satisfaction. So we lose information about situations by moving from is good for to is desired by. But, assuming extensional correctness, we can put this information back when we add the higher order fact that is desired by is the good-predicate.
Actually, the above is simplistic. For what is good for something or someone is dependent on the relevant kind of that something or someone. It is good for a basketball player to be very tall and thin, but may not be good for a swimmer—even if one and the same person is a basketball player and a swimmer. However, the above pattern does allow for a generalization. We have a higher-order binary predicate: ... is a (the?) good predicate for kind .... And then we have first-order "good-predicates" for different kinds. The first-order "good-predicates" might be non-normative and naturalistic. But they fail to capture the relevant information. For instance, the good predicate for the kind basketball player will "typically imply" tallness (i.e., it will entail a disjunction of tallness and some other qualities, such as even more exceptional jumping skills, which other qualities are rare). But this reductive good predicate fails to capture all the information in the ordinary predicate good for basketball players, because it fails to capture the fact that the qualities it implies, or typically implies, are good for basketball players, as opposed to swimmers, say! In fact, it might even turn out that the predicate makes no reference to basketball, simply giving a conjunction of general skills. (And that it makes reference to basketball would anyway not be enough to capture the fact that it is a predicate that it is good for a basketball player to have. After all, the predicate might also make mention of other sports, such as swimming. For it might be good for a basketball player as such to occasionally engage in other sports to develop certain skills and muscles.) So in addition to that reductive predicate, we need the higher order fact that it is a (or, the) good predicate for the kind basketball player.
I suspect that this point can be made with respect to a lot of reductive analyses. The philosophy of mind would be an interesting place to try to apply this.