EAO third figure has the form :
M-P
M-S
S-P
As an example :
No men are roses
All men are cabbages
Some cabbages are not roses
I'm new to logic and I'm wondering how above syllogism is valid.
Suppose there are $2$ men, $500$ roses, and $100$ cabbages;
then the second premise does imply there exist 2 cabbages that are not roses. No issues so far.
But what if I choose $0$ men ? The first two premises seem to not care about this ?
There is a well-known distinction between unconditionally valid categorical syllogisms and conditionally valid categorical syllogisms.
EAO-3 is an example of a conditionally valid syllogism: if you assume that every class (category) of objects is non-empty (this assumption is sometimes called the Assumption of Categorical Import), then the syllogism is valid. But if you do not make that assumption, then you can indeed get a counterexample by using an empty category, as you found.
Indeed, any syllogism where both premises are universal statements, but the conclusion is an existential statement, can at best be conditionally valid.
