@Zanna Do you prefer I not say more about logic and set theory for now because you're in the midst of rereading previous messages (or otherwise prefer not to continue with it)?

if you want to say more about it, I will definitely read it and appreciate it

We don't really insist a sentence can't have a truth value unless there is only one thing it could possibly mean.

This is in regard to what you were saying there.

One might imagine a whole conversation containing multiple declarations of facts that are actually intended as metaphors for something else.

We don't usually want to consider "Smith is away" as translating symbolically to an open sentence on the grounds that it is more precisely said "Smith is now away" where "now" formalizes to a variable.

A traditional approach is to insist that, strictly speaking, no sentences are really true or false, but that instead sentences express something more abstract, propositions, that a sentence may express one proposition today and another tomorrow, that a sentence may fail to express a proposition at all, and that it is propositions that are true or false.

I am mentioning that approach because it is traditional and in some ways appealing, not to argue for it.

Another approach, which I think is better but still not quite perfect, is to say that it is not some sentences that are true or false, but specific occasions of their utterances.

I think that phrasing ("specific occasions of their utterances") is due to Quine, but I am not sure (about the specific phrasing).

In standard logic, we want the sentences we can write symbolically and that do not have any free variable (i.e., given any variable that appears in the sentence, the variable is bound to some quantifier in the sentence) to all be true or false. We want p ∨ ¬p to be true, which requires either that p be true or that ¬p be true, which requires that either p be true or p be false.

I say "in standard logic" because there are some logics that do not have the law of the excluded middle.

Suppose we have a first-order theory whose signature has one unary predicate F, no other predicates, no constants, and no function symbols, and that we have the axiom:

∃x Fx

We say that's true. But in real life, we haven't even said what that means.

Also, we can reason from it as follows: There exists at least one x for which Fx. Consider some such x. Call it a. We know Fa. (and so on)

Even considering formal systems with the same power to prove things, some of them allow a formal proof with steps exactly analogous to the above, and some don't.

As used above, the formula ∃x Fx has no free variables, because the only variable, x, is bound to a quantifier everywhere it appears. (So it is a "closed sentence.") Its subformula, Fx, has one free variable. (So it is an "open sentence.") It instantiation, Fa, has no free variables. (So it is a "closed sentence.") But there is no specific thing a refers to. It's just a name introduced literally for the sake of argument.

Furthermore, we say things like this even when we disbelieve that there is any such object.