Yes

@EliahKagan Sorry, I should've said that, if what I substitute for "$s$" is a closed definite description then the resulting sentence may say there is a unique such object and say something about it. More generally, it makes some claim involving those ideas. This subsequent message requires similar correction.

My goal, from this message to that message, was to clarify how it is that a formula with (arbitrarily many occurrences of) one free variable makes a claim that is true or false of each individual object.

My intention was that then, from this message to that one, it would be clear what the connection is between such formulas and notions like "everything you can say about a set $s$".

I don't think this error, regarding the precise effect of substituting with definite descriptions, prevented that core idea from coming across. Nonetheless I do sometimes wish I could manage to say a thing that is actually true. :)

@EliahKagan Like, take our universe of discourse as integers, and suppose for simplicity of presentation that we have "$\mathrm{Even}$", "$\mathrm{Odd}$", and "$\mathrm{Prime}$" predicates, as well as more common arithmetic notations.

The sentence "$y = 2 \vee y = 4$" means "$y$ is equal to $2$ or $y$ is equal to $4$" (i.e., that $y$ is $2$ or $4$), and the term "$ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Even}(x))$" means "the only even prime".

So substituting that term for "$y$" in the formula yields the sentence $$[ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Even}(x))] = 2 \vee [ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Even}(x))] = 4$$ which means, "The only even prime is $2$ or the only even prime is $4$." For that sentence to be true, there must indeed be exactly one even prime.

Now for an example of substituting a definite description where the resulting sentence does not have the effect of asserting that it succeeds at referring to an object:

The sentence "$\neg \mathrm{Even}(y) \wedge \neg \mathrm{Odd}(y)$" means "It's not the case that $y$ is even and it's not the case that $y$ is odd." It's tempting, and usually appropriate, to read such a sentence as, "$y$ is neither even nor odd." Our universe of discourse being integers, no matter what number you put for $y$ in that sentence, what you get will be false, because every integer is even or odd.

The term "$ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Odd}(x))$" means "the only odd prime." That term fails to refer to an object--there are odd primes, but there is no object that is the unique odd prime. Substituting that term for "$y$" in the formula "$\neg \mathrm{Even}(y) \wedge \neg \mathrm{Odd}(y)$" yields the sentence: $$\neg \mathrm{Even}[ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Odd}(x))] \wedge \neg \mathrm{Odd}[ɿx\, (\mathrm{Prime}(x) \wedge \mathrm{Odd}(x))]$$

But that sentence doesn't express "The only odd prime is neither even nor odd." It doesn't claim there's an odd prime and try to say something about it. Instead, it claims that it's not the case that the only odd prime is even and also not the case that the only odd prime is odd. That sentence is true.

(I mean, it doesn't claim there's a unique odd prime and try to say something about it. It also doesn't claims there's any odd prime, of course, but that's not what I meant to point out.)

@EliahKagan hahaha

my brain is just not managing to stay on through all these symbols and things :(

@EliahKagan it's true but it seems quite uninformative

ok have read those messages again and feeling slightly closer to understanding them

@Zanna Do you want to write some things in set-builder notation?