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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.

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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.

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