TITLE: Quantifier 'for some but not all'
QUESTION [8 upvotes]: Let's consider the quantifier corresponding to the expression 'for some but not all'. Is it possible to define the universal quantifier in terms of this quantifier and sentence connectives only?

REPLY [2 votes]: Strictly as stated this is not possible without resorting to a trick of some kind.
I'll use $Y x \mathop. \varphi(x)$ as the quantifier for some but not all, following its representation in the logical hexagon.
If your domain has only a single element in it, then the value of $Y x \mathop. \varphi(x)$ is always false. However $\varphi$ might hold or might fail for the sole element of the domain.
Here's a partial solution.

Suppose the domain has cardinality at least $1$ and that we have a constant symbol $c$.
$$ \exists x \mathop. \varphi(x) \;\;\text{iff}\;\; \varphi(c) \lor (Y x \mathop. \varphi(x)) $$
As proof, suppose $\varphi$ holds for $c$, then $\varphi(c)$ holds and $\exists x \mathop. \varphi(x)$ holds.
Suppose $\varphi$ does not hold for $c$, but for holds for some $d \neq c$. In this case, $\exists x \mathop. \varphi(x)$ holds and $Y x . \varphi(x)$ holds.
Suppose $\varphi$ does not hold for any element in the domain. Then $\exists x \mathop. \varphi(x)$ fails and $\varphi(c)$ fails and $Y x \mathop. \varphi(x)$ also fails.