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Aristotle, Prior Analytics. Book I, Chapter 07

1. It is likewise manifest, that in all the figures, when a syllogism is not produced, both the terms being categoric or privative, and particular, nothing necessary, in short, will be inferred.

2. But if the one is categoric, and the other privative, the privative being universally assumed, a syllogism will always be produced of the less extreme with the greater.

3. For instance, if A is present with every, or with a certain B, but B is present with no C. For the propositions being converted, it is necessary that C should not be present with a certain A.

4. The like also will take place in other figures; for a syllogism will always be produced through conversion.

5. It is likewise manifest, that when an indefinite assertion is assumed for a particular attributive, it will produce the same syllogism in all the figures.

6. It is also evident, that all imperfect syllogisms are perfected through the first figure. For all of them receive their completion either demonstratively, or through the impossible; but in both ways the first figure will be produced. And if, indeed, they receive their completion demonstratively, the first figure will be produced, because thus all of them will be perfected through conversion; and conversion will produce the first figure. But if they are demonstrated through the impossible, still the first figure will be produced, because the false being posited, a syllogism will be formed in the first figure. Thus, for instance, in the last figure, if A and B are present with every C, it may be demonstrated that A is present with some B. For if a is present with no B, but B is present with every C, A will be present with no C: but it was supposed that A is present with every C. The like will also take place in other instances.

7. It is also possible to reduce all syllogisms to universal syllogisms of the first figure.

8. For it is evident, that through these the syllogisms in the second figure are perfected except that all of them are not similarly perfected: but the universal are perfected, the privative assertion being converted; And each of those that are particular, through a deduction to the impossible.

9. But particular syllogisms in the first figure, are perfected, indeed, through themselves. They may, however, be demonstrated in the second figure, by a deduction to the impossible. For instance, if A is present with every B, but B is present with a certain C, it may be shown that A will be present with a certain C. For if A is present with no C, but is present with every B: B will be present with no C; for we know this through the second figure. In a similar manner there will be a demonstration in a primitive syllogism. For if A is present with no B, but B is present with a certain C; A will not be present with a certain C. For if a is present with every C, and with no B; B will be present with no C: but this was the middle figure. Hence, since all the syllogisms in the middle figure, are reduced to universal syllogisms in the first figure; but particular syllogisms in the first figure, are reduced to syllogisms in the middle figure; It is evident, that particular syllogisms in the first figure are reduced to universal syllogisms in the first figure.

10. But the syllogism in the third figure, the terms, indeed, being universal, are immediately perfected through those syllogisms.

11. When, however, the terms are assumed in a part, they are perfected through particular syllogisms in the first figure. But these are reduced to those; so that particular syllogisms also in the third figure, are reduced to the same.

12. It is evident, therefore, that all of them may be reduced to the universal syllogisms in the first figure. 13. Hence we have shown how those syllogisms subsist which exhibit being present with, or not being present with; as well by themselves, those which are from the same figure, as with reference to each other, those which are from different figures.

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