Source: A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley.
Please observe my amelioration (and so change) of Hurley's notation.
[p 287:] Rule 4: A Negative Premise Requires a Negative Conclusion, [...]
Fallacy: Drawing an affirmative conclusion from a negative premise [...]
[p 288:] [...] The logic behind Rule 4 may be seen as follows. If MIN, MAJ, and MID once again designate the Minor, Major, and Middle Terms, an affirmative conclusion always states that the MIN class is contained either wholly or partially in the MAJ class.
The only way that such a conclusion can follow is if the MIN class is contained either wholly or partially in the MID class,
[1.] and the MID class wholly in the MAJ class.
In other words, it follows only when both premises are affirmative. But if, for example, the MIN class is contained either wholly or partially in the MID class, and the MID class is separate either wholly or partially from the MAJ class, such a conclusion will never follow. Thus, an affirmative conclusion cannot be drawn from negative premises.
I pursue only intuition; please do not answer with formal proofs or Truth Tables.
The argument above does not convince me, because why must 1 be true (ie: why must the MID be contained wholly in MAJ)? An affirmative conclusion can follow even if the MID is contained partially in MAJ.
For example, consider an III syllogism ♦which is invalid:
Some A are B. Some B are C. → Some A are C.
However, this becomes valid if at least one element lies in A ∩ B ∩ C.
♦[p 288:] As a result of the interaction of these first four rules, it turns out that no valid syllogism can have two particular premises.