Necessary Premises

Chapter 9

Sometimes, when one premise is necessary the conclusion is also necessary.

But not when either premise is necessary, but only when the major is. For example if A is taken as necessarily belonging or not belonging to B, but B is taken as simply belonging to C.

If the premises are taken in this way, A will necessarily belong or not belong to C.

For since necessarily belongs, or does not belong, to every B, and since C is one of the Bs, it is clear that for C also the positive or the negative relation to A will hold necessarily. But if the major premise is not necessary, but the minor is necessary, the conclusion will not be necessary. For if it were, it would result both through the first figure and through the third that A belongs necessarily to some B. But this is false; for B may be such that it is possible that A should belong to none of it.

Further, an example also makes it clear that the conclusion not be necessary, e.g. if A were movement, B animal, C man: man is an animal necessarily, but an animal does not move necessarily, nor does man. Similarly also if the major premise is negative; for the proof is the same.

In particular syllogisms, if the universal premise is necessary, then the conclusion will be necessary; but if the particular, the conclusion will not be necessary, whether the universal premise is negative or affirmative.

First let the universal be necessary, and let A belong to all B necessarily, but let B simply belong to some C: it is necessary then that A belongs to some C necessarily: for C falls under B, and A was assumed to belong necessarily to all B.

Similarly also if the syllogism should be negative: for the proof will be the same. But if the particular premise is necessary, the conclusion will not be necessary: for from the denial of such a conclusion nothing impossible results, just as it does not in the universal syllogisms. The same is true of negative syllogisms.

Try the terms movement, animal, white.

Chapter 10

In the second figure, if the negative premise is necessary, then the conclusion will be necessary, but if the affirmative, not necessary. First let the negative be necessary; let A be possible of no B, and simply belong to C.

Since then the negative statement is convertible, B is possible of no A.

But A belongs to all C; consequently B is possible of no C. For C falls under A. The same result would be obtained if the minor premise were negative: for if A is possible be of no C, C is possible of no A: but A belongs to all B, consequently C is possible of none of the Bs: for again we have obtained the first figure. Neither then is B possible of C: for conversion is possible without modifying the relation.

But if the affirmative premise is necessary, the conclusion will not be necessary. Let A belong to all B necessarily, but to no C simply. If then the negative premise is converted, the first figure results.

But it has been proved in the case of the first figure that if the negative major premise is not necessary the conclusion will not be necessary either. Therefore the same result will obtain here. Further, if the conclusion is necessary, it follows that C necessarily does not belong to some A.

For if B necessarily belongs to no C, C will necessarily belong to no B. But B at any rate must belong to some A, if it is true (as was assumed) that A necessarily belongs to all B.

Consequently it is necessary that C does not belong to some A. But nothing prevents such an A being taken that it is possible for C to belong to all of it. Further one might show by an exposition of terms that the conclusion is not necessary without qualification, though it is a necessary conclusion from the premises. For example let A be animal, B man, C white, and let the premises be assumed to correspond to what we had before: it is possible that animal should belong to nothing white. Man then will not belong to anything white, but not necessarily: for it is possible for man to be born white, not however so long as animal belongs to nothing white.

Consequently under these conditions the conclusion will be necessary, but it is not necessary without qualification.

Similar results will obtain also in particular syllogisms. For whenever the negative premise is both universal and necessary, then the conclusion will be necessary: but whenever the affirmative premise is universal, the negative particular, the conclusion will not be necessary.

First then let the negative premise be both universal and necessary: let it be possible for no B that A should belong to it, and let A simply belong to some C. Since the negative statement is convertible, it will be possible for no A that B should belong to it: but A belongs to some C; consequently B necessarily does not belong to some of the Cs.

Let the affirmative premise be both universal and necessary, and let the major premise be affirmative. If then A necessarily belongs to all B, but does not belong to some C, it is clear that B will not belong to some C, but not necessarily.

For the same terms can be used to demonstrate the point, which were used in the universal syllogisms. Nor again, if the negative statement is necessary but particular, will the conclusion be necessary. The point can be demonstrated by means of the same terms.