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?
You want to define $\forall$ and $\exists$ in term of the expression :
'for some but not all'.
1) May we assume that the translation of the expression is :
$\exists x \phi(x) \land \lnot \forall x \phi(x)$ ?
This sentence implies (if I'm right) that :
the universe is not empty
something in the universe is $\phi$
something is not $\phi$
there are at least two thing in the universe.
But the sentence is also equivalent to :
$\exists x \phi(x) \land \exists x \lnot \phi(x)$
2) If my "translation" is right and if my inferences are also right, I doubt that we can define $\exists$.
The $\exists$ quantifier implies the existence of at least one object in the universe, whlie the new "quantifier" implies the existence of at least two objects.