1. But in the last figure, when the terms are universally joined to the middle, and both the propositions are categoric, if either of them is necessary, the conclusion also will be necessary. If, however, one of the propositions is privative, but the other categoric; when the privative is necessary, the conclusion also will be necessary. But when the categoric proposition is necessary, the conclusion will not be necessary.

2. For, in the first place, let both the propositions be necessary, and let A and B be present with every C; and let the proposition AC be necessary. Since, therefore, B is present with every C, C also will be present with a certain B, because a universal is converted into a particular proposition. Hence if A is necessarily present with every C, and C is present with a certain B, A also is necessarily present with a certain B; for B is under (i.e. is something belonging to) C. The first figure, therefore, will again be produced.

3. In a similar manner it may be demonstrated if the proposition B C is necessary; for C is converted with a certain A. Hence if B is necessarily present with every C, but C is present with a certain A, B also will be necessarily present with a certain A,

4. Again, let the proposition A C be privative, but the proposition B C affirmative; and let the privative proposition be necessary. Since, therefore, an affirmative proposition may be converted, C will be present with a certain B, but A will necessarily be present with no C, and also will necessarily not be present with a certain B; for B is under C.

5. But if the categoric proposition is necessary, the conclusion will not be necessary. For let B G be a categoric and necessary proposition; but let the proposition A C be privative and not necessary. Since, therefore, an affirmative proposition may be converted, C also will necessarily be present with a certain B; so that if A is present with no C, but C is present with a certain B, A also will not be present with a certain B; yet not from necessity. For it was demonstrated in the first figure that a privative proposition not being necessary, neither will the conclusion be necessary.

6. Farther still, this will also be evident from the terms. For let A be good; B animal; and C horse. It may, therefore, happen that good may be present with no horse; but animal is necessarily present with every horse. It is not, however, necessary that a certain animal should not be good, since it may happen that every animal is good.

No horse is good:

It is necessary that every horse should be an animal:

Therefore, some animal is not good.

Or, if this is not possible, another term must be posited, as to wake, or to sleep; for every animal is the recipient of these.

No horse wakes:

It is necessary that every horse should be an animal:

Therefore, some animal does not sleep.

No horse sleeps:

It is necessary that every horse should be an animal:

Therefore, some animal does not wake.

7. If, therefore, the terms are universally joined to the middle, it has been shown when the conclusion will be necessary.

8. But if one of the terms is universally predicated of the middle and the other partially, both, indeed, being categoric; when the universal proposition becomes necessary, the conclusion also will be necessary. The demonstration, however, is the same as before; for a partial categoric proposition may also be converted. If, therefore, it is necessary that B should be present with every C, but A is under C, it is necessary that B should be present with a certain A. For this proposition may be converted.

9. The like also will take place, if the proposition AC is necessary and universal; for B is under C.

10. But if the partial proposition is necessary, the conclusion will not be necessary.

11. For let the proposition B C be partial and necessary, and let A be present with every C, yet not from necessity. The proposition, therefore, B C being converted, the first figure will be produced: and the universal proposition is not necessary; but the partial is necessary. When, however, the propositions thus subsist, the conclusion is not necessary. Hence neither in the terms now posited will the conclusion be necessary.

Every C is A:

It is necessary that some C should be B:

Therefore, some B is A.

Every C is A:

It is necessary that some B should be C:

Therefore, some B is A.

12. Farther still, this also is evident from the terms. For let A be wakefulness; B be biped; and C be animal. It is necessary, therefore, that B should be present with a certain C, but A may happen to be present with every C, and A is not necessarily present with B. For it is not necessary that a certain biped should sleep or wake.

Every animal wakes:

It is necessary that some animal should be biped:

Therefore, some biped wakes.

13. In a similar manner also, the demonstration may be framed through the same terms, if the proposition A should be partial and necessary.

It is necessary that some animal should be a biped:

Every animal wakes:

Therefore, something that wakes is a biped.

Every animal wakes:

It is necessary that some biped should be an animal:

Therefore, some biped wakes.

14. But if one of the terms is categoric, and the other privative, when the universal proposition is privative and necessary, the conclusion also will be necessary. For if A is contingent to no C, but B is present with a certain C, it is necessary that A should not be present with a certain B.

15. But when the affirmative proposition is necessary, whether it be universal or partial, or privative partial, the conclusion will not be necessary. For we may say that other things are the same, as we have mentioned before. Let the terms, however, when the universal categoric proposition is necessary, be wakefulness, animal, man; and the middle be man.

Some man does not wake:

It is necessary that every man should be an animal:

Therefore, some animal docs not wake.

But when the partial categoric proposition is necessary, let the terms be wakefulness, animal, white. For it is necessary that animal should be present with something white: but it happens that wakefulness is not present with anything white; and it is not necessary that wakefulness should not be present with a certain animal.

Nothing white wakes:

It is necessary that something white should be an animal:

Therefore, some animal does not wake.

But when the privative partial proposition is necessary; let the terms be biped, motion, animal; and the middle be animal.

It is necessary that some animal should not be a biped:

Every animal is moved:

Therefore, something which is moved is not a biped.

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