How would you formalise: "There are no geniuses but Newton was a genius"?
I thought it could be: $$\neg\forall x(Px) \land Pa$$ and also $$\neg\exists x (Px) \land Pa$$These both seem to make sense and formalise the sentence, however they are not identical statements. The way it is worded is really confusing!
On
One way to formalize this would be $\bot$, since that statement is a contradiction ... Unless of course the point of the sentence is that there are no geniuses now, though we have had geniuses in the past, like Newton.
One way to formalize that would be to use a predicate $G(x,t)$ which says that $x$ is a genius at time $t$. So then:
$\neg \exists x G(x,t_{now}) \land \exists t (t<t_{now} \land G(newton,t))$
The second one looks right to me. If by $Px$ you mean "$x$ is a genius" and $a$ is Newton, then your statements translate to:
1) it's not true that every person is a genius, and it's true that Newton is a genius;
2) there's doesn't exist anyone who is a genius, and it's true that Newton is a genius