Here is the final post (for the time being) on structured propositions. As promised, this is to be an account of the truth-conditions of structured propositions, presupposing a certain reasonably contentious take on the metaphysics of linguistic representation (metasemantics). It's going to be compatible with the view that structured propositions are nothing but certain n-tuples: lists of their components. (See earlier posts if you're getting concerned about other factors, e.g. the potential arbitriness in the choice of which n-tuples are to be identified with the structured proposition that Dummett is a philosopher.)
Here's a very natural way of thinking of what the relation between *sentences* and truth-conditions are, on a structured propositions picture. It's that metaphysically, the relation of "S having truth-conditions C" breaks down into two more fundamental relations: "S denoting struc prop p" and "struc prop p having truth-conditions C". The thought is something like: primarily, sentences express thoughts (=struc propositions), and thoughts themselves are the sorts of things that have intrinsic/essential representational properties. Derivatively, sentences are true or false of situations, by expressing thoughts that are true or false of those situations. As I say, it's a natural picture.
In the previous posting, I've been talking as though this direction-of-explanation was ok, and that the truth-conditions of structured propositions should have explanatory priority over the truth-conditions of sentences, so we get the neat separation into the contingent feature of linguistic representation (which struc prop a sentence latches onto) and the necessary feature (what the TCs are, given the struc prop expressed).
The way I want to think of things, something like the reverse holds. Here’s the way I think of the metaphysics of linguistic representation. In the beginning, there were patterns of assent and dissent. Assent to certain sentences is systematically associated with certain states of the world (coarse-grained propositions, if you like) perhaps by conventions of truthfulness and trust (cf. Lewis's "Language and Languages"). What it is for expressions E in a communal language to have semantic value V is for E to be paired with V under the optimally eligible semantic theory fitting with that association of sentences with coarse-grained propositions.
That's a lot to take in all at one go, but it's basically the picture of linguistic representation as fixed by considerations of charity/usage and eligibility/naturalness that lots of people at the moment seem to find appealing. The most striking feature---which it shares with other members of the "radical interpretation" approach to metasemantics---is that rather than starting from the referential properties of lexical items like names and predicates, it depicts linguistic content as fixed holistically by how well it meshes with patterns of usage. (There's lots to say here to unpack these metaphors, and work out what sort of metaphysical story of representation is being appealed to: that's something I went into quite a bit in my thesis---my take on it is that it's something close to a fictionalist proposal).
This metasemantics, I think, should be neutral between various semantic frameworks for generating the truth conditions. With a bit of tweaking, you can fit in a Davidsonian T-theoretic semantic theory into this picture (as suggested by, um... Davidson). Someone who likes interpretational semantics but isn't a fan of structured propositions might take the semantic values of names to be objects, and the semantic values of sentences to be coarse-grained propositions, and say that it's these properties that get fixed via best semantic theory of the patterns of assent and dissent (that's Lewis's take).
However, if you think that to adequately account for the complexities of natural language you need a more sophisticated, structured proposition, theory, this story also allows for it. The meaning-fixing semantic theory assign objects to names, and structured propositions to sentences, together with a clause specifying how the structured propositions are to be paired up with coarse-grained propositions. Without the second part of the story, we'd end up with an association between sentences and structured propositions, but we wouldn't make connection with the patterns of assent and dissent if these take the form of associations of sentences with *coarse grained* propositions (as on Lewis's convention-based story). So on this radical interpretation story where the targetted semantic theories take a struc prop form, we get a simultaneous fix on *both* the denotation relation between sentences and struc props, and the relation between struc props and coarse-grained truth-conditions.
Let's indulge in a bit of "big-picture" metaphor-ing. It’d be misleading to think of this overall story as the analysis of sentential truth-conditions into a prior, and independently understood, notion of the truth-conditions of structured propositions, just as it's wrong on the radical interpretation picture to think of sentential content as "analyzed in terms of" a prior, and independently understood, notion of subsentential reference. Relative to the position sketched, it’s more illuminating to think of the pairing of structured and coarse-grained propositions as playing a purely instrumental role in smoothing the theory of the representational features of language. It's language which is the “genuine” representational phenomenon in the vicinity: the truth-conditional features attributed to struc propositions are a mere byproduct.
Again speaking metaphorically, it's not that sentences get to have truth-conditions in a merely derivative sense. Rather, structured propositions have truth-conditions in a merely derivative sense: the structured proposition has truth-conditions C if it is paired with C under the optimal overall theory of linguistic representation.
For all we've said, it may turn out that the same assignment of truth-conditions to set-theoretic expressions will always be optimal, no matter which language is in play. If so, then it might be that there's a sense in which structured propositions have "absolute" truth-conditions, not relative to this or that language. But, realistically, one'd expect some indeterminacy in what struc props play the role (recall the Benacerraf point King makes, and the equally fitness of [a,F] and [F,a] to play that "that a is F" role). And it's not immediately clear why the choice to go one way for one natural language should constrain way this element is deployed in another language. So it's at least prima facie open that it's not definitely the case that the same structured propositions, with the same TCs, are used in the semantics of both French and English.
It's entirely in the spirit of the current proposal that we think of we identify [a,F] with the structured proposition that a is F only relative to a given natural language, and that this creature only has the truth-conditions it does relative to that language. This is all of a piece with the thought that the structured proposition's role is instrumental to the theory of linguistic representation, and not self-standing.
Ok. So with all this on the table, I'm going to return to read the book that prompted all this, and try to figure out whether there's a theoretical need for structured propositions with representational properties richer than those attributed by the view just sketched.
[update: interestingly, it turns out that King's book doesn't give the representational properties of propositions explanatory priority over the representational properties of sentences. His view is that the proposition that Dummett thinks is (very crudely, and suppressing details) the fact that in some actual language there is a sentence of (thus-and-such a structure) of which the first element is a word referring to Dummett and the second element is a predicate expressing thinking. So clearly semantic properties of words are going to be prior to the representational properties of propositions, since those semantic properties are components of the proposition. But more than this, from what I can make out, King's thought is that if there was a time where humans spoke a language without attitude-ascriptions and the like, then sentences would have truth-conditions, and the proposition-like facts would be "hanging around" them, but the proposition-like facts wouldn't have any representational role. Once we start making attitude ascriptions, we implicitly treat the proposition-like structure as if it had the same TCs as sentences, and (by something like a charity/eligibility story) the "propositional relation" element acquires semantic significance and the proposition-like structure gets to have truth-conditions for the first time.
That's very close to the overall package I'm sketching above. What's significant dialectically, perhaps, is that this story can explain TCs for all sorts of apparently non-semantic entities, like sets. So I'm thinking it really might be the Benacerraf point that's bearing the weight in ruling out set-theoretic entities as struc propns---as explained previously, I don't go along with *that*.]