the differences between Fregian logic and contemporary logical intuitions — Leontiskos
I'm not much concerned about this, but the single most interesting point, and relevant to this thread, is that the assertion stroke disappeared. Frege thought it was necessary and later logicians universally (?) don't. I'm no historian, so I'm not quite sure how this happened. I always assumed that Frege was just wrong, and that we can do formal logic as if there were an implicit assertion stroke at the start of each line. And if we can do that, it becomes clearer that what we're really doing is manipulating symbols according to rules, as when we do mathematics, and interpretation can wait until later.
(On the other hand, natural deduction systems often use some notation for assumptions and their later discharge, I think, so that's a way of singling out formulae you are
not asserting simpliciter.)
Now for the hair-splitting.
Suppose we put the tool in our shed for storage. In doing this have we used it for an alternative purpose? No, because we are preserving the tool in order that it may be used for its singular purpose at a later date. — Leontiskos
In most cases, but also I think obviously false as a rule. You might own a tool that you have no intention of ever using, an antique gun, for instance. Men have workshops with peg-boarded walls where lots of tools are impressively arrayed and the three or four they actually use are laying on the workbench or in a drawer. I could go on.
This hair isn't worth splitting though, because most sentences asserted are not ready-to-hand like a screwdriver, but one-offs. So there's no sense talking about storage and retrieval in the first place. (Words, on the other hand, ...) Where you do see the same sentence bouncing around repeatedly is in argument and discussion, and in quotation. That means we have the option of exploring whether there are different kinds of assertion; maybe assertion in an argument is a slightly different beast from the sort of extemporaneous sharing of information we do all day.
we never handle statements independent of assertions, even when we are not asserting them. — Leontiskos
And I'm just not sure what you're reaching for here with "handle", or "independent of", for that matter. Now and then I think you're making a sort of psychological or cognitive point: Hume noted that to conceive of an object is to conceive of it as existing; you almost seem sometimes to be saying that to conceive of a statement is to conceive of it as being asserted. Which might be true, but I don't believe this is what you're saying, or what the point of saying it would be. So what kind of "handling" of statements are you talking about, and how are possible assertions implicated?
We've sort of begun talking about the assertibility of a statement as an affordance, in direct analogy to screwdrivers. But we could instead think of the way simple objects in the Tractatus are said to sort of carry with them the possible states of affairs they could enter into. Just so, a sentence in a given language has what we might think of as chemical properties: there are other sentences it will have an affinity for, and bond with readily to create a narrative or an argument; there are sentences it will repel, sentences that if they bond it will reconfigure both into new configurations with new possibilities, and so on. Philosophers tend to treat statements as having built-in "affirm" and "deny" buttons, but that's surely a somewhat impoverished view, once you consider the wealth of ways sentences relate to each other.
The hairs that remain to be split are no fun: there's
(1) the sentence;
(2) the sentence in a given context;
(3) the elaborated sentence with indexicals resolved by context;
(4) the sentence as uttered;
(5) the sentence as uttered in a specific context;
(6) the actual uttering of the sentence;
(7) all the intentions involved (which Grice admits are infinite, though it's a pretty model and gets something clearly right).
I'm sure I'm leaving some out. I'm not sure which of these we've been talking about, which Frege has, which points made depend on whether you're talking about one or the other and which don't. We may have no choice but to wade into some of this -- though I'll note again that this is the sort of crap you don't have to worry about in mathematics, where Frege's machine is both happy and indispensable.
I'd like to say something useful about this last difficulty but it will have to wait.