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Aristotle, Prior Analytics. Book I, Chapter 10

1. In the second figure, however, if the privative proposition is necessary, the conclusion also will be necessary; but if the categoric proposition is necessary, the conclusion will not be necessary.

2. For, in the first place, let the primitive proposition be necessary, and let it not be possible for A to be present with any B, but let it be present with C alone. Since, therefore, a privative proposition may be converted, neither can be be present with any A.  But A is present with every C; so that B cannot be present with any C. For C is under A.

3. In a similar manner also the conclusion will be necessary if negation is added to C. For if A cannot be present with any C, neither can C be present with any A. But A is present with every B; so that neither can C be present with any B. For again, the first figure will be produced. Hence neither can B be present with C; since it is in a similar manner converted.

4. But if the categoric proposition is necessary, the conclusion will not be necessary. For let A be present with every B from necessity, and let it alone not be present with any C. The primitive proposition, therefore, being converted, the first figure will be produced. But it was shown in the first figure, that when the major privative proposition is not necessary, neither will the conclusion be necessary. Hence neither in these will the conclusion be necessary.

5. Again, if the conclusion is necessary, it will happen that C is necessarily not present with a certain A. For if B is necessarily present with no C, neither will C be necessarily present with any B. But B is necessarily present with a certain a comma if a is present from necessity with every B. Hence it is necessary that C should not be present with a certain A. Nothing, however, hinders an A of that kind from being assumed, which may be present with every C.

6. Farther still, it may also be shown from an exposition of the terms, that the conclusion is not simply necessary, but that it necessarily is, these being posited. For instance, let A be animal, B, man, C, white, and let the propositions be similarly assumed. For it will happen that animal is present with nothing white. Neither, therefore, will man be present with anything white; yet not from necessity. For it may happen that man may be white, yet not so long as animal is present with nothing white. Hence these things being admitted, the conclusion will be necessary, but will not be simply necessary.

7. The like will also take place in particular syllogisms. For when the primitive proposition is universal and necessary, the conclusion also will be necessary. But when the categoric proposition is universal and necessary, but the privative is particular and not necessary; the conclusion will not be necessary.

8. In the first place, therefore, let the privative proposition be universal and necessary, and let it not be possible for A to be present with any B, but let it be present with a certain C. Since, therefore a primitive proposition may be converted, B also cannot be present with any A. But A is present with a certain C. Hence B is necessarily not present with a certain C.

9. Again, let the categoric proposition be universal and necessary, and let the categorical (that is, affirmation) be added to be. If, therefore, A is necessarily present with every B, but is not present with a certain C; it is evident, that B is not present with a certain C; but not from necessity. For there will be the same terms in order to the demonstration, as were assumed in universal syllogisms.

10. Neither will the conclusion be necessary, if the privative assertion is necessary when assumed in a part. For the demonstration may be made through the same terms.

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