Existential Quantifiers translated into categorical statements?

Question

I've been recently trying to translate the categorical statements into the quantifiers ($\forall$ and $\exists$).

Attempts

I believe I can make the E statement as $$\nexists s:p,$$the A statement as $$\neg\nexists s:p,$$ the I statement as $$\exists s:p,$$ and the O statement as $$\neg\exists s:p.$$

Question

Are my formulas correct? If not, how might I translate the categorical statements correctly?

Answer 1

Here is the way I would do it in AEIO order: $$\forall x:S(x)\implies P(x)$$ $$\forall x:S(x)\implies \neg P(x)$$ $$\exists x:S(x)\wedge P(x)$$ $$\exists x:S(x)\wedge \neg P(x)$$