All Radicals are covetous.
All Radicals are poor.
.'. All poor men are covetous.
The conclusion here is certainly not warranted by our premisses. For in them we spoke only of some poor men, since the predicate of an affirmative proposition is undistributed.
§ 590. Rule 5 is simply another way of stating the third axiom of mediate inference. To know that two terms disagree with the same third term gives us no ground for any inference as to whether they agree or disagree with one another, e.g.
Ruminants are not oviparous.
Sheep are not oviparous.
For ought that can be inferred from the premisses, sheep may or may not be ruminants.
§ 591. This rule may sometimes be violated in appearance, though not in reality. For instance, the following is perfectly legitimate reasoning.
No remedy for corruption is effectual that does not render it
useless.
Nothing but the ballot renders corruption useless.
.'. Nothing but the ballot is an effectual remedy for corruption.
But on looking into this we find that there are four terms—
No not-A is B.
No not-C is A.
.'. No not-C is B.
The violation of Rule 5 is here rendered possible by the additional violation of Rule 2. In order to have the middle term the same in both premisses we are obliged to make the minor affirmative, thus