@Zanna This is why I have avoided claiming that natural-language sentences like "She is an astronomer" are open sentences.

Both "She is an astronomer" and "Alice is an astronomer" depend on context to be understood.

Also, when we use logic, we give some some interpretation to these things. Saying "Aa" is true if and only if Alice is an astronomer is like saying "Alice is an astronomer" is true if and only if Alice is an astronomer, and both are saying something about the relationship between things we can say and facts in the world.

A sentence that can be expressed in formal logic is neither true nor false if it has any free variables, but even in the case of a sentence that can be expressed in formal logic and in which all variables (which could be all zero of its variables if it has none) are bound, its truth value sometimes depends on the meaning of one's predicates, function symbols, and constants.

At least part, and perhaps all, of this context can be said to be is supplied by what universe of discourse one is using.

When our universe of discourse is sets, the things we say are implicitly about sets.

With that said...

@Zanna Are you sure you would always know who you are talking about? And if so, would that necessarily be who in the sense of a single concrete individual?

In natural language, where we talk about pronouns and antecedents ("antecedent" has another meaning in logic: in p → q, the antecedent is p, the consequent is q, and the whole thing, p → q, is said to be a conditional), an antecedent may be a specific person or thing, like "she" in:

> Alice gave a talk yesterday. She is an astronomer.

But it need not be a specific person or thing:

> Every time we've had a guest speaker, she was an astronomer.

> No matter what calligrapher you choose, he will not be available for hire until Tuesday.

> An antecedent may be a specific person or thing, but it need not be a specific person or thing.

All three of those sentences can be translated into a language of first-order logic, though there are some subtleties that should be considered in doing it.

The resulting formulas in a language of first-order logic will contain quantifiers. Since you can always eliminate universal quantifiers in favor of existential quantifiers and vice versa, I want to be careful in saying specifically what quantifiers one must use.

But the most natural and useful translations will probably express the first two sentences using universal quantification and the third sentence using existential quantification (I would express it as the conjunction of two existentially quantified subformulas).

The translations into FOL would have no free variables and could, presumably, be taken as true or false. However, the subformulas that are quantified over would each have at least one free variable; they would not have a truth value.

For example, there is some question as to precisely what "Every time we've had a guest speaker, she was an astronomer" expresses, and also it is not obvious what degree of granularity is most useful in translating it. A very coarse and perhaps adequate way to translate it is something like ∀x (Gx → Ax) ("For all x, if we have had x as a guest speaker, x is an astronomer.")

There's something that does not seem right about this translation, because it does not talk about time and so does not account for the situation where someone was a guest speaker and only later became an astronomer (and secondarily because it does not talk about us: it folds whoever "we" refers to into the predicate "G"). It may or may not be sufficient. I am giving it as a simple example, but you should feel free to produce a better translation or to ask me to do so.

In that formula, the whole formula ∀x (Gx → Ax) has no free variables, so it (presumably) has a truth value, while the subformula Gx → Ax, which is also a well-formed formula (unlike, for example, the substring → Ax), has at least one free variable (specifically, it has exactly one free variable) and therefore does not have a truth value.

Using the terminology of open and closed sentences, ∀x (Gx → Ax) is a closed sentence and Gx → Ax is an open sentence. Instantiating Gx → Ax with a name (suppose a refers to Alice; it is not a variable) also produces a closed sentence--that is, Ga → Aa is a closed sentence.

@Zanna If you're talking about rereading, it may be that what I have said is not well enough organized that reading it multiple times provides clarity. I am not sure.

When I read that message, I had not assumed you were rereading everything we've said here about logic and set theory. I think I had the idea that you were specifically rereading the things you'd said that about.

In a sense it is good you're rereading more, in that I think the overall quality of what I have written here about logic is higher than the quality of that recent part specifically. But I don't think the organization of it necessarily makes much sense. If there are things that don't make sense to you, and you can identify what they are and say something about why they do not, then it might be better for you to mention those things. Even if there are many such things.

(And some might be things you do understand but where my terminology is unclear or were I have mistakenly said something wrong.)