(1) A term t is directly referential iff the semantic content of t is the designatum of t.
(2) The semantic content of a term t is X iff for every linguistic environment E, the semantic content that t contributes to E(t) is X [i.e Semantic Innocence].
(3) Thus, a term t is directly referential iff for every linguistic environment E, the semantic content that t contributes to E(t) is the designatum of t.
(4) In a bound environment (e.g. ∀xFx) a variable does not contribute its designatum (since 'Fx' must contribute a set of assignments).
(5) Thus, variables are not directly referential.
I'm interested in how people who advocate direct reference theory maneuver out of this argument. Denying either premise (1) or premise (2) are really the only options for not accepting the conclusion.
Premise (1) is an attempted definition of "direct reference". So perhaps I got the rough definition wrong. One alternative idea that would avoid the conclusion is this: a term t is "directly referential" iff the designatum of t is "directly" determined by the interpretation or assignment function, in the sense that its designatum is assigned independently of the world and time parameters. But to my ear this is a definition of rigidity de jure---it's a fact about a terms designatum across worlds/times and how it is built in by law that it be independent of worlds/times. Direct reference, however, is supposed to make a further claim about "semantic content". Kaplan defines it thusly: "When what is said in using an indexical in a context c is to be evaluated with respect to an arbitrary circumstance, the relevant object is always the referent of the indexical with respect to the context c." It's a familiar point that "direct reference" simply amounts to rigidity de jure, unless we construe it in terms of Russellian structured contents. But I'm not sure how one might want to tinker with the definition to avoid the conclusion.
In any case, I think the real action is with premise (2). And it seems that many will somehow want to deny it, and thus deny semantic innocence (e.g. see Salmon's "A theory of bondage", which develops a Fregean referential shift semantics for variable binding (although he doesn't exactly endorse the occurrence-based semantics he outlines)).
But if a term can be "directly referential" even though the value it contributes across all environments (i.e. its compositional semantic value) isn't its designatum, what is the significance of calling the term "directly referential"? For example, the direct reference theory of names is controversial precisely because of its commitments on embedded occurrences, e.g. names embedded in belief reports. (Perhaps this is where the distinction between Millianism and Direct Reference is important.) Can one accept that 'Hesperus' and 'Phosphorus' are directly referential (and thus have the same "semantic content" in some important sense) even though the semantic content of their occurrences in belief contexts are different? It's hard for me to get a grip on "semantic content", if it is detached from the semantic contribution an expression makes to the complex expressions of which it is a constituent.
I'm left wondering what "direct reference" is exactly and what important (and plausible) thesis about language it is committed to. Any ideas?