Minor Premise

If one premise is a simple proposition, the other a problematic, whenever the major premise indicates possibility all the syllogisms will be perfect and establish possibility in the sense defined.

But whenever the minor premise indicates possibility all the syllogisms will be imperfect, and those which are negative will establish not possibility according to the definition, but that the major does not necessarily belong to any, or to all, of the minor. For if this is so, we say it is possible that it should belong to none or not to all. Let A be possible for all B, and let B belong to all C. Since C falls under B, and A is possible for all B, clearly it is possible for all C also.

So a perfect syllogism results. Likewise if the premise AB is negative, and the premise BC is affirmative, the former stating possible, the latter simple attribution, a perfect syllogism results proving that A possibly belongs to no C.

Perfect syllogisms result if the minor premise states simple belonging: but that syllogisms will result if the modality of the premises is reversed, must be proved per impossibile.

At the same time it will be evident that they are imperfect: for the proof proceeds not from the premises assumed. First we must state that if B’s being follows necessarily from A’s being, B’s possibility will follow necessarily from A’s possibility. Suppose, the terms being so related, that A is possible, and B is impossible. If then that which is possible, when it is possible for it to be, might happen, and if that which is impossible, when it is impossible, could not happen, and if at the same time A is possible and B impossible, it would be possible for A to happen without B, and if to happen, then to be. For that which has happened, when it has happened, is.

But we must take the impossible and the possible not only in the sphere of becoming, but also in the spheres of truth and predictability, and the various other spheres in which we speak of the possible: for it will be alike in all.

We must understand the statement that B’s being depends on A’s being, not as meaning that if some single thing A is, B will be: for nothing follows of necessity from the being of some one thing, but from two at least, i.e. when the premises are related in the manner stated to be that of the syllogism. For if C is predicated of D, and D of F, then C is necessarily predicated of F. And if each is possible, the conclusion also is possible.

If then, for example, one should indicate the premises by A, and the conclusion by B, it would not only result that if A is necessary B is necessary, but also that if A is possible, B is possible.

Since this is proved it is evident that if a false and not impossible assumption is made, the consequence of the assumption will also be false and not impossible: e.g. if A is false, but not impossible, and if B is the consequence of A, B also will be false but not impossible.

For since it has been proved that if B’s being is the consequence of A’s being, then B’s possibility will follow from A’s possibility (and A is assumed to be possible), consequently B will be possible: for if it were impossible, the same thing would at the same time be possible and impossible.

Since we have defined these points, let A belong to all B, and B be possible for all C: it is necessary then that should be a possible attribute for all C. Suppose that it is not possible, but assume that B belongs to all C: this is false but not impossible. If then A is not possible for C but B belongs to all C, then A is not possible for all B: for a syllogism is formed in the third degree.

But it was assumed that A is a possible attribute for all B. It is necessary then that A is possible for all C.

For though the assumption we made is false and not impossible, the conclusion is impossible. It is possible also in the first figure to bring about the impossibility, by assuming that B belongs to C. For if B belongs to all C, and A is possible for all B, then A would be possible for all C.

But the assumption was made that A is not possible for all C. We must understand ’that which belongs to all’ with no limitation in respect of time, e.g. to the present or to a particular period, but simply without qualification.

For it is by the help of such premises that we make syllogisms, since if the premise is understood with reference to the present moment, there cannot be a syllogism. For nothing perhaps prevents ‘man’ belonging at a particular time to everything that is moving, i.e. if nothing else were moving: but ‘moving’ is possible for every horse; yet ‘man’ is possible for no horse.

Further let the major term be ‘animal’, the middle ‘moving’, the minor ‘man’. The premises then will be as before, but the conclusion necessary, not possible.

Man is necessarily animal. The universal must be understood simply, without limitation in respect of time.

Let the premise AB be universal and negative, and assume that A belongs to no B, but B possibly belongs to all C. These propositions being laid down, it is necessary that A possibly belongs to no C. Suppose that it cannot belong, and that B belongs to C, as above. It is necessary then that A belongs to some B: for we have a syllogism in the third figure: but this is impossible.

Thus it will be possible for A to belong to no C; for if at is supposed false, the consequence is an impossible one. This syllogism then does not establish that which is possible according to the definition, but that which does not necessarily belong to any part of the subject (for this is the contradictory of the assumption which was made: for it was supposed that A necessarily belongs to some C, but the syllogism per impossibile establishes the contradictory which is opposed to this). Further, it is clear also from an example that the conclusion will not establish possibility.

Let A be ‘raven’, B ‘intelligent’, and C ‘man’. A then belongs to no B: for no intelligent thing is a raven. But B is possible for all C: for every man may possibly be intelligent. But A necessarily belongs to no C: so the conclusion does not establish possibility. But neither is it always necessary. Let A be ‘moving’, B ‘science’, C ‘man’. A then will belong to no B; but B is possible for all C. And the conclusion will not be necessary. For it is not necessary that no man should move; rather it is not necessary that any man should move. Clearly then the conclusion establishes that one term does not necessarily belong to any instance of another term. But we must take our terms better.

If the minor premise is negative and indicates possibility, from the actual premises taken there can be no syllogism, but if the problematic premise is converted, a syllogism will be possible, as before. Let A belong to all B, and let B possibly belong to no C. If the terms are arranged thus, nothing necessarily follows: but if the proposition BC is converted and it is assumed that B is possible for all C, a syllogism results as before: for the terms are in the same relative positions.

Likewise if both the relations are negative, if the major premise states that A does not belong to B, and the minor premise indicates that B may possibly belong to no C. Through the premises actually taken nothing necessary results in any way; but if the problematic premise is converted, we shall have a syllogism. Suppose that A belongs to no B, and B may possibly belong to no C. Through these comes nothing necessary. But if B is assumed to be possible for all C (and this is true) and if the premise AB remains as before, we shall again have the same syllogism. But if it be assumed that B does not belong to any C, instead of possibly not belonging, there cannot be a syllogism anyhow, whether the premise AB is negative or affirmative. As common instances of a necessary and positive relation we may take the terms white-animal-snow: of a necessary and negative relation, white-animal-pitch. Clearly then if the terms are universal, and one of the premises is assertoric, the other problematic, whenever the minor premise is problematic a syllogism always results, only sometimes it results from the premises that are taken, sometimes it requires the conversion of one premise. We have stated when each of these happens and the reason why.

But if one of the relations is universal, the other particular, then whenever the major premise is universal and problematic, whether affirmative or negative, and the particular is affirmative and assertoric, there will be a perfect syllogism, just as when the terms are universal.

The demonstration is the same as before. But whenever the major premise is universal, but assertoric, not problematic, and the minor is particular and problematic, whether both premises are negative or affirmative, or one is negative, the other affirmative, in all cases there will be an imperfect syllogism. Only some of them will be proved per impossibile, others by the conversion of the problematic premise, as has been shown above. And a syllogism will be possible by means of conversion when the major premise is universal and assertoric, whether positive or negative, and the minor particular, negative, and problematic, e.g. if A belongs to all B or to no B, and B may possibly not belong to some C. For if the premise BC is converted in respect of possibility, a syllogism results.

But whenever the particular premise is assertoric and negative, there cannot be a syllogism.

As instances of the positive relation we may take the terms white-animal-snow; of the negative, white-animal-pitch. For the demonstration must be made through the indefinite nature of the particular premise. But if the minor premise is universal, and the major particular, whether either premise is negative or affirmative, problematic or assertoric, nohow is a syllogism possible. Nor is a syllogism possible when the premises are particular or indefinite, whether problematic or assertoric, or the one problematic, the other assertoric. The demonstration is the same as above. As instances of the necessary and positive relation we may take the terms animalwhite-man; of the necessary and negative relation, animal-white-garment. It is evident then that if the major premise is universal, a syllogism always results, but if the minor is universal nothing at all can ever be proved.