In this mapping, S and P stand for sets when on the right side of a colon.
I will write SP for S intersect P, /= for inequality, and 0 for the null
set.
All S is P (A): SP = S
Some S is P (I): SP /= 0
No S is P (E): SP = 0
Some S is not P (O): SP /= S
Then existential import is simply the assertion that S /= 0, and we can
understand I- as asserting that S, which may be null, has a non-null
intersection with P. But plainly no set P can have a non-null intersection
with 0, and so from SP /= 0 we can deduce that S /= 0. Therefore I-
is false if S = 0, and to assert anything useful we need I+.