by Aristotle
[25] Clearly then if the terms are universal, and one of the propositions is simple, the other possible, whenever the proposition relating to the minor extreme is possible, a deduction always results, only sometimes it results from the propositions that are taken, sometimes it requires the conversion of one proposition. We have [30] stated when each of these happens and the reason why. But if one of the relations is universal, the other particular, then whenever the one relating to the major extreme is universal and possible, whether affirmative or negative, and the particular is affirmative and simple, there will be a perfect deduction, just as when the terms are [35] universal. The demonstration is the same as before. But whenever the one relating to the major extreme is universal, but simple rather than possible, and the other is particular and possible, whether both are negative or affirmative, or one is negative, the other affirmative, in all cases there will be an imperfect deduction. Only some of them will be proved per impossibile, others by the conversion of the possible [35b1] proposition, as has been shown above. And a deduction will be possible by means of conversion when the proposition relating to the major extreme is universal and simple, and the other particular, negative, and possible, e.g. if A belongs to every B [5] or to no B, and B may not belong to some C. For if BC is converted in respect of possibility, a deduction results. But whenever the particular is simple and negative, there cannot be a deduction. As instances of the positive relation we may take the terms white, animal, snow; of the negative, white, animal, pitch. For the [10] demonstration must be made through the indefinite nature of the particular proposition. But if the universal relates to the minor extreme, and the particular to the major, whether either is negative or affirmative, possible or simple, in no way is a deduction possible. Nor is a deduction possible when the propositions are [15] particular or indefinite, whether possible or simple, or the one possible, the other simple. The demonstration is the same as above. As instances of the necessary and positive relation we may take the terms animal, white, man; of the impossible relation, animal, white, garment. It is evident then that if the proposition relating to [20] the major extreme is universal, a deduction always results, but if the one relating to the minor is universal nothing at all can ever be proved.
16 · Whenever one proposition indicates necessity, the other possibility, there will be a deduction when the terms are related as before; and a perfect [25] deduction when the necessity relates to the minor extreme. If the terms are affirmative the conclusion will be possible, not simple, whether they are universal or not; but if one is affirmative, the other negative, when the affirmative is necessary the conclusion will be possible, not negative and simple; but when the negative is necessary the conclusion will be both possible negative, and simple negative, [30] whether the terms are universal or not. Possibility in the conclusion must be understood in the same manner as before. There cannot be a deduction to a necessary negative; for ‘not necessarily to belong’ is different from ‘necessarily not [35] to belong’.
If the terms are affirmative, clearly the conclusion which follows is not necessary. Suppose A necessarily belongs to every B, and let B be possible for every C. We shall have an imperfect deduction that A may belong to every C. That it is imperfect is clear from the proof; for it will be proved in the same manner as above. [36a1] Again, let A be possible for every B, and let B necessarily belong to every C. We shall then have a deduction that A may belong to every C, not that A does belong to [5] every C; and it is perfect, not imperfect; for it is perfected directly through the original propositions.
But if the propositions are not similar in quality, suppose first that the negative is necessary, and let A be possible for no B, but let B be possible for every C. It is necessary then that A belongs to no C. For suppose A to belong to every C or to some [10] C. Now we assumed that A is not possible for any B. Since then the negative is convertible, B is not possible for any A. But A is supposed to belong to every C or to some C. Consequently B will not be possible for any C or for every C. But it was [15] originally laid down that B is possible for every C. And it is clear that the possibility of not belonging can be deduced, since the fact of not belonging can be. Again, let the affirmative proposition be necessary, and let A possibly not belong to any B, and let B necessarily belong to every C. The deduction will be perfect, but it will [20] establish a possible negative, not a simple negative. For the proposition relating to the major was assumed in this way; and further it is not possible to prove per impossibile. For if it were supposed that A belongs to some C, and it is laid down that A possibly does not belong to any B, no impossible relation between B and C [25] follows from this. But if the negative relates to the minor extreme, when it indicates possibility a deduction is possible by conversion, as above; but when impossibility, not. Nor again when both are negative, and the one relating to the minor is not possible. The same terms as before serve both for the positive relation, white, [30] animal, snow, and for the negative relation, white, animal, pitch.
The same relation will obtain in particular deductions. Whenever the negative is necessary, the conclusion will be negative and simple: e.g. if it is not possible that [35] A should belong to any B, but B may belong to some C, it is necessary that A should not belong to some C. For if A belongs to every C, but cannot belong to any B, neither can B belong to any A. So if A belongs to every C, B can belong to no C. But it was laid down that B may belong to some C. But when the particular affirmative [36b1] in the negative deduction, i.e. BC, or the universal in the affirmative i.e. AB, is necessary, there will not be a simple conclusion. The demonstration is the same as before. But if the term relating to the minor extreme is universal, and possible, whether affirmative or negative, and the particular is necessary, there cannot be a [5] deduction. Terms where the relation is positive and necessary: animal, white, man; where it is necessary and negative: animal, white, garment. But when the universal is necessary, the particular possible, if the universal is negative we may take the [10] terms animal, white, raven to illustrate the positive relation, or animal, white, pitch to illustrate the negative; and if the universal is affirmative we may take the terms animal, white, swan to illustrate the positive relation, and animal, white, snow to illustrate the impossible relation. Nor again is a deduction possible when the propositions are indefinite, or both particular. Terms applicable in either case to illustrate the positive relation are animal, white, man; to illustrate the negative, [15] animal, white, inanimate. For the relation of animal to some white, and of white to some inanimate, is both necessary and positive and necessary and negative. Similarly if the relation is possible; so the terms may be used for all cases.
Clearly then from what has been said a deduction results or not from similar [20] relations of the terms whether we are dealing with simple or with necessary propositions, with this exception, that if the negative proposition is simple the conclusion is possible, but if the negative is necessary the conclusion is both possible and negative simple. [It is clear also that all deductions are imperfect and are [25] perfected by means of the figures above mentioned.]11
17 · In the second figure whenever both propositions are possible, no deduction is possible, whether they are affirmative or negative, universal or particular. But when one indicates belonging, the other possibility, if the affirmative indicates belonging no deduction is possible, but if the universal negative does a [30] conclusion can always be drawn. Similarly when one proposition is necessary, the other possible. Here also we must understand the term ‘possible’ in the conclusions in the same sense as before.
First we must prove that the negative possible proposition is not convertible, [35] e.g. if A may belong to no B, it does not follow that B may belong to no A. For suppose it to follow and assume that B may belong to no A. Since then possible affirmations are convertible with negations, whether they are contraries or contradictories, and since B may belong to no A, it is clear that B may belong to [37a1] every A. But this is false; for if all this can be tha
t, it does not follow that all that can be this: consequently the negative proposition is not convertible. Further, there is no reason why A may not belong to no B, while B necessarily does not belong to some [5] A; e.g. it is possible that no man should be white (for it is also possible that every man should be white), but it is not true to say that it is possible that no white thing should be a man; for many white things are necessarily not men, and the necessary (as we saw) is other than the possible.
Moreover it is not possible to prove the convertibility of these propositions by a reductio ad absurdum, i.e. by claiming that since it is false that B may belong to no [10] A, it is true that it cannot belong to no A (for the one statement is the contradictory of the other); but if this is so, it is true that B necessarily belongs to some A; and consequently A necessarily belongs to some B—but this is impossible. The argument cannot be admitted; for it does not follow that some A is necessarily B, if it is not possible that no A should be B. For the latter expression is used in two ways, [15] one if some A is necessarily B, another if some A is necessarily not B. For it is not true to say that that which necessarily does not belong to some of the As may not belong to every A, just as it is not true to say that what necessarily belongs to some A may belong to every A. If any one then should claim that because it is not possible [20] for C to belong to every D, it necessarily does not belong to some D, he would make a false assumption; for it does belong to every D, but because in some cases it belongs necessarily, therefore we say that it is not possible for it to belong to every. Hence both ‘necessarily belongs to some’ and ‘necessarily does not belong to some’ are opposed to ‘may belong to every’. Similarly also they are opposed to ‘may belong to [25] no’. It is clear then that in relation to what is possible and not possible, in the sense originally defined, we must assume, not that A necessarily belongs to some B, but that A necessarily does not belong to some B. But if this is assumed, no impossibility results; consequently there is no deduction. It is clear from what has been said that [30] the negative is not convertible.
This being proved, suppose it possible that A may belong to no B and every C. By means of conversion no deduction will result; for such a proposition, as has been said, is not convertible. Nor can a proof be obtained by a reductio; for if it is assumed that B cannot not belong to every C, no false consequence results; for A [35] may belong both to every C and to no C. In general, if there is a deduction, it is clear that its conclusion will be possible because neither of the propositions is simple; and this must be either affirmative or negative. But neither is possible. Suppose the [37b1] conclusion is affirmative: it will be proved by an example that the predicate cannot belong to the subject. Suppose the conclusion is negative: it will be proved that it is not possible but necessary. Let A be white, B man, C horse. It is possible then for A [5] to belong to all of the one and to none of the other. But it is not possible for B to belong or not to belong to C. That it is not possible for it to belong, is clear. For no horse is a man. Neither is it possible for it not to belong. For it is necessary that no horse should be a man, but the necessary we found to be different from the possible. [10] No deduction then results. A similar proof can be given if the negative is the other way about, or if both are affirmative or negative. The demonstration can be made by means of the same terms. And whenever one is universal, the other particular, or [15] both are particular or indefinite, or in whatever other way the propositions can be altered, the proof will always proceed through the same terms. Clearly then, if both the propositions are possible, no deduction results.
18 · But if one indicates belonging, the other possibility, if the affirmative [20] indicates belonging and the negative possibility no deduction will be possible, whether the terms are universal or particular. The proof is the same as above, and by means of the same terms. But when the affirmative indicates possibility, and the [25] negative belonging, we shall have a deduction. Suppose A belongs to no B, but can belong to every C. If the negative is converted, B will belong to no A. But A ex hypothesi can belong to every C: so a deduction is made, proving by means of the first figure that B may belong to no C. Similarly also if the negative relates to C. But [30] if both are negative, one indicating non-belonging, the other possibility, nothing follows necessarily from these premisses as they stand, but if the possible proposition is converted a deduction is formed to prove that B may belong to no C, [35] as before; for we shall again have the first figure. But if both are affirmative, no deduction will be possible. Terms for when the relation is positive: health, animal, man; for when it is negative: health, horse, man.
The same will hold good if the deductions are particular. Whenever the [38a1] affirmative is simple, whether universal or particular, no deduction is possible (this is proved similarly and by the same examples as above), but when the negative is, a conclusion can be drawn by means of conversion, as before. Again if both the [5] relations are negative, and the simple is universal, although no conclusion follows from the actual propositions, a deduction can be obtained by converting the possible as before. But if the negative is simple, but particular, no deduction is possible, whether the other proposition is affirmative or negative. Nor can a conclusion be [10] drawn when both are indefinite, whether affirmative or negative, or particular. The proof is the same and by the same terms.
19 · If one of the propositions indicates necessity, the other possibility, then if the negative is necessary there is a deduction not merely that it can not belong but also that it does not belong; but if the affirmative is necessary, no conclusion is [15] possible. Suppose that A necessarily belongs to no B, but may belong to every C. If the negative is converted B will belong to no A; but A ex hypothesi may belong to every C: so once more a conclusion is drawn by the first figure that B may belong to [20] no C. But at the same time it is clear that B will not belong to any C. For assume that it does; then if A cannot belong to any B, and B belongs to some C, A cannot belong to some C; but ex hypothesi it may belong to all. A similar proof can be given [25] if the negative relates to C.
Again let the affirmative be necessary, and the other possible; i.e. suppose that A may belong to no B, but necessarily belongs to every C. When the terms are arranged in this way no deduction is possible. For it turns out that B necessarily [30] does not belong to C. Let A be white, B man, C swan. White then necessarily belongs to swan, but may belong to no man; and man necessarily belongs to no swan. Clearly then we cannot draw a possible conclusion; for that which is necessary is [35] admittedly distinct from that which is possible. Nor again can we draw a necessary conclusion: for that presupposes that both propositions are necessary, or at any rate the negative one. Further it is possible also, when the terms are so arranged, that B should belong to C; for nothing prevents C falling under B, A being possible for [40] every B, and necessarily belonging to C; e.g. if C is awake, B animal, A motion. For motion necessarily belongs to what is awake, and is possible for every animal; and [38b1] everything that is awake is animal. Clearly then the conclusion cannot be negative and simple, if the relation must be positive when the terms are related as above. Nor can the opposite affirmations be established: consequently no deduction is possible. A similar proof is possible if the affirmative is the other way about. [5]
But if the propositions are similar in quality, when they are negative a deduction can always be formed by converting the possible as before. Suppose A necessarily does not belong to B, and possibly may not belong to C: if the [10] propositions are converted B belongs to no A, and A may possibly belong to every C; thus we have the first figure. Similarly if the negative relates to C. But if they are affirmative there cannot be a deduction. Clearly the conclusion cannot be a negative simple or a negative necessary proposition because no negative has been [15] laid down either in the simple or in the necessary mode. Nor can the conclusion be a possible negative proposition. For if the terms are so related, B necessarily will not belong to C; e.g. suppose that A is white, B swan, C man. Nor can the opposi
te [20] affirmations be established, since we have shown that B necessarily does not belong to C. A deduction then is not possible at all.
Similar relations will obtain in particular deductions. For whenever the [25] negative is universal and necessary, a deduction will always be possible to prove both that it may and that it does not (the proof proceeds by conversion); but when the affirmative is universal and necessary, no conclusion can be drawn. This can be proved in the same way as for universal deductions, and by the same terms. Nor is a conclusion possible when both are affirmative: this also may be proved as above. But [30] when both are negative, and the one which signifies non-belonging is universal and necessary, though nothing follows necessarily from the premisses as they are stated, [35] a conclusion can be drawn as above if the possible proposition is converted. But if both are indefinite or particular, no deduction can be formed. The same proof will serve, and the same terms.