Can the "some but not all" quantifier define either the existential or universal quantifier by a first-order formula?

Question

I realize this question has already been asked, here: Quantifier 'for some but not all', but there was no formal proof of it. So, can anyone rigorously prove that the quantifier "some but not all" cannot define either the existential or universal quantifier by a first-order-logic formula, even if we restrict our attention to domains of discourse with at least two elements? Note, I do not want to add a constant symbol to our language, as Greg Nisbet did to his answer in the linked question. I do allow using the equality binary predicate, though, so basically a first-order-with-equality formula.

Answer 1

Consider any homogeneous (in the sense that its automorphism group acts transitively) structure $M$ over any first order language. If $\varphi(x)$ is any formula with one free variable, then either $M\models\forall x \varphi(x)$ or $M\models\forall x\neg\varphi(x)$. So, if $M\models\neg Yx\varphi(x)$, where $Y$ is a "some but not all" quantifier. It follows that any sentence which uses only "some but not all" quantifiers has the same truth value in all homogeneous structures (since any such sentence is a Boolean combination of sentences of the form $Yx\varphi(x)$). But it is not true that all homogeneous structures (even restricted to those with more than $1$ element) satisfy the same ordinary first-order sentences, so ordinary quantifiers cannot be expressed in terms of "some but not all" quantifiers. (For instance, there is no way to express "there are exactly two elements" using "some but not all" quantifiers, since there are homogeneous structures with two elements but also homogeneous structures with more than two elements. Or, if $P$ is a unary predicate, there is no way to express $\forall x P(x)$, since this is true in some homogeneous structures and false in others.)