Is existential import only about implications on subject or other nouns/objects too?

Question

Question 1

This mentions "...belief in the existence of members of the subject class". Is this necessarily true ?

Consider "Some pizza has pepperoni on it".

You get the following. Which is one is an existential import ?

1. Pizza exists
2. Non-pizza exists
3. pepperoni exists
4. Non-pepperoni exist
5. Cross on Sentence 1 through 4

Now,

 Question 2

This claims existential import is rejected per Boole. So why are texts teaching it still ? Is it because Aristotle had it but Boole rejected it so the mention it for legacy reasons ?

Answer 1

The so-called "problem of existential import" arises in the context of traditional, i.e. Aristotelian interpretation of the Square of Opposition.

According to it, an A proposition : "All swans are white", and an I proposition : "Some swan is white", are in a subalternation relation, i.e. the subaltern (I) must be True if the superaltern (A) is.

This means that the inference from A to I is - according to Aristotle - valid.

The standard translation of the categorical proposition A:

"All S are P" ("all swans are white")

is: $(\forall x)(Sx → Px)$, while the corresponding I :

"Some S is P" ("some swan is white")

is: $(\exists x) (Sx \land Px)$.

In modern terms, the conditional corresponding to A is true also when there are no swans at all, while the I is not.

This is the gist of

"the truth of the proposition requires a belief in the existence of members of the subject class".

The traditional interpretation of Syllogism assumes that the class corresponding to the term in subject position is not empty.

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There is no issue with "Some pizza has pepperoni on it"; it will be symbolized with $\exists x (Pizza(x) \land WithPep(x))$.

Obviously, the statement will be true only if there are pizzas.