Let's say that I want to translate into first-order logic the sentence
(a) "There is a composer who is liked by anyone who likes any composer at all"
I'm having trouble translating the "anyone who likes any composer at all" part.
First of all, the sentence is quite ambiguous: does "x likes any composer at all" mean that x likes all composers or does it mean that x likes at least one composer?
Consider the following two sentences
(b) I like any music at all
(c) If you know any programming language at all, you're the perfect candidate for the job
Now, "any at all" clearly stands for "all" in (b), meaning that (b) is equivalent to "I like all kinds of music". However, "any at all" stands for "at least one" in (c), meaning that (c) is equivalent to "if you know at least one programming language, then you're the perfect candidate for the job" (not to "if you know all programming languages, you're the perfect candidate for the job")
Assuming that "any at all" in (a) stands for "all" how can I translate (a) into first-order logic? And how could I translate (a) if "any at all" stood for "at least one"?
Let's say that "any at all" stands for "all". Would it be correct to say that (a) is equivalent to
(a.1) If for every x who is a composer there is a y who likes x then there is a z such that z is a composer and y likes z
thus translating (a) into: $(\forall x)(Cx \land (\exists y)(Lyx))\to(\exists z)(Cz \land Lyz)$
"Lyz" meaning "y likes z"
"Cx" meaning "x is a composer"
"$\to$" being the material conditional
(how should (a) be translated into first-order logic in the case in which "any at all" means "at least one"?)
Yes, yes it is a bit ambiguous. However, I would favour it as : "anyone who likes some composer (don't who or how many, but not none)".
In any case, your statement of $\overline{(\forall x)(Cx ∧ \overline{(\exists y)(Lyx)})}\to\overline{(\exists z)(Cz ∧ Lyz)}$ reads as : "There is some composer who is liked by unbound variable $y$, if everyone is composer and liked by someone." Among other issues, I have overlined the scopes of the quantifiers to make it clear why the instance of $y$ is unbound in the consequent of the implication.
Completely rethink the structure.
"There is a composer who is liked by anyone who likes any composer at all"
$\exists z~(C(z)\wedge \ldots)$ "There is a composer who is/does $\ldots$"
$\forall y~(\ldots\to \ldots)$ "Anyone, who is $\ldots$, does $\ldots$"
$\forall x~(L(\ldots,x)\wedge C(x) \to \ldots)$ "Every composer, who is loved by $\ldots$, does $\ldots$"
Put it together, then perhaps put it into prenex form.