*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.

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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.