Granted, grammatical structure is linguistic; but so is lexicon. The lexicon is used in talking about the world; but so is grammatical structure. A logical truth, staying true as it does under all lexical substitutions, admittedly depends upon none of those features of the world that are reflected in lexical distinctions; but may it not depend on other features of the world, features that our language reflects in grammatical constructions rather than its lexicon? It would be pointless to protest that grammar varies from language to language, for so does lexicon. Perhaps the logical truths owe their truth to certain traits of reality which are reflected in one way by the grammar of our language, in another way by the grammar of another language, and in a third way by the combined grammar and lexicon of a third language. (ibid., my italics)
Logical truths are about the world, or are true because they “reflect” features of the world, because their grammatical structures are about the world or reflect features of the world. In what sense could a grammatical structure possibly be about the world, say anything about the world, or “reflect” features of the world? First, we should note that grammatical structures are not about the world in the same way that sentences, names, or predicates are. Grammatical structures as such aren’t true or false like (truth-apt) sentences. By all appearances, grammatical structures don’t refer to anything in the world; Tarski’s definition of truth gets along just fine without assigning semantic values to grammatical structures. Nor do they have (Fregean) senses on any theory that I know of. Nor, intuitively, are they meaningful. If a person were to speak or write down a grammatical structure – say, by speaking or writing a sequence of particles and schematic variables for grammatical categories – I can’t see why anyone would want to say that she, or her utterance or inscription, meant anything.
We might want to say that grammatical structures say something about the world in a different sense – viz., in the sense that sentences with the same non-logical constants but different grammatical structures have different truth-conditions. “(Ax)(Cat(x))” says something different from “~(Ax)(Cat(x))” because of the difference in grammatical structure between the two. We might say that a negation symbol says that the negated sentence is false, a universal quantifier over a variable says that the sentence in the scope of the quantifier is true for all values of the bound variable, and so on.* In this way, by specifying what all of the particles or logical constants say, we can state more or less precisely what an entire grammatical structure, paired with a particular sentence instantiating it, says about the world. But two observations are in order. First, it is not clear how we should construe what the grammatical structures of atomic sentences say.** Second, and more importantly, the worldliness of a grammatical structure, in this sense, is dependent on the worldliness of the non-logical constants in the sentence instantiating it. For instance, the grammatical structure of “~(Cat(Dora))” says something about the world because “(Cat(Dora))” says something about the world – it is either true or false depending on the actual features of the thing called “Dora”. The grammatical structures of uninterpreted schemata say nothing about the world, because the talk of truth, falsity, and values of variables in our sketchy specification of what the particles say presupposes an interpretation of the lexical items in the sentence. When we admit, with Quine, that a logical truth “admittedly depends upon none of those features of the world that are reflected in lexical distinctions”, then the grammatical structure of a logical truth cannot derive its worldliness from the worldliness of the “lexical distinctions” marked by the non-logical constants in a sentence instantiating the structure. Briefly, since the worldliness of grammatical structures derives from the worldliness of the terms in the sentences instantiating them, and since the putative worldliness of logical truths does not depend on the worldliness of these terms, the grammatical structures of logical truths have nothing from which to derive their putative worldliness.
There might be some way of being about the world or reflecting features of the world that I haven’t grasped yet. Perhaps we should understand Quine as saying that we might as well postulate a sui generis mode of being about the world specific to grammatical structures. I can only say in response to this that we might as well not, both for the sake of not multiplying features of the world beyond necessity and for the sake of keeping “about the world” intelligible. Lastly, someone might say that the grammatical structure of logical truths such as “it is raining or it isn’t raining” is about the world because it reflects the fact about the world that things, in general, are or aren’t the case. But this begs the question against the proponent of the linguistic theory of logical truth. What is at issue is whether this fact is about the world.
* - It is much easier to fill in the “and so on” for a formalized language than for a natural language. What does a particle like “if” say? It seems we need a worked-out semantics for conditionals to fill out a proposal like this. If you aren’t satisfied by my hand-waving here, then that probably goes to show that it is even more difficult to make Quine’s argument come out sound.
** - This probably isn’t such a big deal, since the only atomic sentence that is a logical truth is “x = x”, and Quine seems to reckon “=” a particle.