*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.