Quantifier 'for some but not all' 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.