Comment
Valid means true in all cases.
Thus, the schema :
Some A are B
Some B are C
Therefore : Some A are C
is not valid exactly for the reason you have stated :
it becomes true [not valid] if at least one element lies in A ∩ B ∩ C.
Correct..., but if this is not, the argument does not conclude, and this means that the argument form is not valid.
This distinction between true and valid is the core of Hurley's explanation of Rule 4 (and is the core of formal logic, since Aristotle).
Consider the fallacy :
Drawing an affirmative conclusion from a negative premise,
and consider the invalid :
All crows are birds
Some wolves are not crows
Therefore : Some wolves are birds.
This is a counter-example to the "purported" rule :
it is possible to (validly) conclude in the affermative when a premise is negative.
This means that, from the fact :
if a schema has one premise negative and the conclusion affermative, then it is not valid,
by contraposition we have :
if a schema is valid, then not (one premise negative and the conclusion affermative) i.e. either the conclusion is negative or all the premise must be afermative.
