Simple Conclusions

Chapter 11

In the last figure when the terms are related universally to the middle, and both premises are affirmative.

If one of the two is necessary, then the conclusion will be necessary.

But if one is negative, the other affirmative, whenever the negative is necessary the conclusion also will be necessary, but whenever the affirmative is necessary the conclusion will not be necessary.

First let both the premises be affirmative, and let A and B belong to all C, and let Ac be necessary.

Since then B belongs to all C, C also will belong to some B, because the universal is convertible into the particular: consequently if A belongs necessarily to all C, and C belongs to some B, it is necessary that A should belong to some B also. For B is under C. The first figure then is formed.

A similar proof will be given also if BC is necessary. For C is convertible with some A: consequently if B belongs necessarily to all C, it will belong necessarily also to some A. Again let AC be negative, BC affirmative, and let the negative premise be necessary. Since then C is convertible with some B, but A necessarily belongs to no C, A will necessarily not belong to some B either: for B is under C.

But if the affirmative is necessary, the conclusion will not be necessary.

Suppose BC is affirmative and necessary, while AC is negative and not necessary.

Since then the affirmative is convertible, C also will belong to some B necessarily: consequently if A belongs to none of the Cs, while C belongs to some of the Bs, A will not belong to some of the Bs-but not of necessity; for it has been proved, in the case of the first figure, that if the negative premise is not necessary, neither will the conclusion be necessary. Further, the point may be made clear by considering the terms. Let the term A be ‘good’, let that which B signifies be ‘animal’, let the term C be ‘horse’. It is possible then that the term good should belong to no horse, and it is necessary that the term animal should belong to every horse: but it is not necessary that some animal should not be good, since it is possible for every animal to be good. Or if that is not possible, take as the term ‘awake’ or ‘asleep’: for every animal can accept these.

If, then, the premises are universal, we have stated when the conclusion will be necessary. But if one premise is universal, the other particular, and if both are affirmative, whenever the universal is necessary the conclusion also must be necessary. The demonstration is the same as before; for the particular affirmative also is convertible. If then it is necessary that B should belong to all C, and A falls under C, it is necessary that B should belong to some A. But if B must belong to some A, then A must belong to some B: for conversion is possible. Similarly also if AC should be necessary and universal: for B falls under C. But if the particular premise is necessary, the conclusion will not be necessary. Let the premise BC be both particular and necessary, and let A belong to all C, not however necessarily. If the proposition BC is converted the first figure is formed, and the universal premise is not necessary, but the particular is necessary. But when the premises were thus, the conclusion (as we proved was not necessary: consequently it is not here either. Further, the point is clear if we look at the terms. Let A be waking, B biped, and C animal.

It is necessary that B should belong to some C, but it is possible for A to belong to C, and that A should belong to B is not necessary. For there is no necessity that some biped should be asleep or awake. Similarly and by means of the same terms proof can be made, should the proposition Ac be both particular and necessary.

But if one premise is affirmative, the other negative, whenever the universal is both negative and necessary the conclusion also will be necessary. For if it is not possible that A should belong to any C, but B belongs to some C, it is necessary that A should not belong to some B.

But whenever the affirmative proposition is necessary, whether universal or particular, or the negative is particular, the conclusion will not be necessary. The proof of this by reduction will be the same as before; but if terms are wanted, when the universal affirmative is necessary, take the terms ‘waking’-‘animal’-‘man’, ‘man’ being middle, and when the affirmative is particular and necessary, take the terms ‘waking’-‘animal’-‘white’: for it is necessary that animal should belong to some white thing, but it is possible that waking should belong to none, and it is not necessary that waking should not belong to some animal.

But when the negative proposition being particular is necessary, take the terms ‘biped’, ‘moving’, ‘animal’, ‘animal’ being middle.

Chapter 12

A simple conclusion is not reached unless both premises are simple assertions, but a necessary conclusion is possible although one only of the premises is necessary.

But in both cases, whether the syllogisms are affirmative or negative, it is necessary that one premise should be similar to the conclusion. I mean by ‘similar’, if the conclusion is a simple assertion, the premise must be simple; if the conclusion is necessary, the premise must be necessary.

Consequently this also is clear, that the conclusion will be neither necessary nor simple unless a necessary or simple premise is assumed.