by Aristotle
It is clear then from what has been said that if the terms are related to one [28a1] another in the way stated, a deduction results of necessity; and if there is a deduction, the terms must be so related. But it is evident also that all the deductions in this figure are imperfect; for all are made perfect by certain supplementary [5] assumptions, which either are contained in the terms of necessity or are assumed as hypotheses, i.e. when we prove per impossibile. And it is evident that an affirmative deduction is not attained by means of this figure, but all are negative, whether universal or particular.
6 · But if one term belongs to all, and another to none, of a third, or if both [10] belong to all, or to none, of it, I call such a figure the third; by middle term in it I mean that of which both are predicated, by extremes I mean the predicates, by the major extreme that which is further from the middle, by the minor that which is nearer to it. The middle term stands outside the extremes, and is last in position. A [15] deduction cannot be perfect in this figure either, but it may be potential whether the terms are related universally or not to the middle term.
If they are universal, whenever both P and R belong to every S, it follows that P will necessarily belong to some R. For, since the affirmative is convertible, S will belong to some R: consequently since P belongs to every S, and S to some R, P must [20] belong to some R; for a deduction in the first figure is produced. It is possible to demonstrate this both per impossibile and by exposition. For if both P and R belong to every S, should one of the Ss, e.g. N, be taken, both P and R will belong to this, [25] and thus P will belong to some R.
If R belongs to every S, and P to no S, there will be a deduction that P will necessarily not belong to some R. This may be demonstrated in the same way as before by converting the proposition RS. It might be proved also per impossibile, as [30] in the former cases. But if R belongs to no S, P to every S, there will be no deduction. Terms for the positive relation are animal, horse, man; for the negative relation animal, inanimate, man.
Nor can there be a deduction when both terms are asserted of no S. Terms for the positive relation are animal, horse, inanimate; for the negative relation man, [35] horse, inanimate—inanimate being the middle term.
It is clear then in this figure also when a deduction will be possible and when not, if the terms are related universally. For whenever both the terms are affirmative, there will be a deduction that one extreme belongs to some of the other; [28b1] but when they are negative, no deduction will be possible. But when one is negative, the other affirmative, if the major is negative, the minor affirmative, there will be a deduction that the one extreme does not belong to some of the other; but if the relation is reversed, no deduction will be possible.
[5] If one term is related universally to the middle, the other in part only, when both are affirmative there must be a deduction, no matter which is universal. For if R belongs to every S, P to some S, P must belong to some R. For since the [10] affirmative is convertible, S will belong to some P; consequently since R belongs to every S, and S to some P, R must also belong to some P; therefore P must belong to some R. Again if R belongs to some S, and P to every S, P must belong to some R. This may be demonstrated in the same way as the preceding. And it is possible to [15] demonstrate it also per impossibile and by exposition, as in the former cases.
But if one term is affirmative, the other negative, and if the affirmative is universal, a deduction will be possible whenever the minor term is affirmative. For if R belongs to every S, but P does not belong to some S, it is necessary that P does not belong to some R. For if P belongs to every R, and R belongs to every S, then P [20] will belong to every S; but we assumed that it did not. Proof is possible also without reduction, if one of the Ss be taken to which P does not belong.
But whenever the major is affirmative, no deduction will be possible, e.g. if P belongs to every S, and R does not belong to some S. Terms for the universal affirmative relation are animate, man, animal. For the universal negative relation it [25] is not possible to get terms, if R belongs to some S, and does not belong to some S. For if P belongs to every S, and R to some S, then P will belong to some R; but we assumed that it belongs to no R. We must put the matter as before. Since its not belonging to some is indefinite, it is true to say of that which belongs to none that it [30] does not belong to some. But if R belongs to no S, no deduction is possible, as has been shown. Clearly then no deduction will be possible here.
But if the negative term is universal, whenever the major is negative and the minor affirmative there will be a deduction. For if P belongs to no S, and R belongs to some S, P will not belong to some R; for we shall have the first figure again, if the [35] proposition RS is converted.
But when the minor is negative, there will be no deduction. Terms for the positive relation are animal, man, wild; for the negative relation, animal, science, wild—the middle in both being the term wild.
Nor is a deduction possible when both are stated in the negative, but one is universal, the other particular. When the minor is related universally to the middle, [29a1] take the terms animal, science, wild; animal, man, wild. When the major is related universally to the middle, take as terms for a negative relation raven, snow, white. For a positive relation terms cannot be found, if R belongs to some S, and does not [5] belong to some S. For if P belongs to every R, and R to some S, then P belongs to some S; but we assumed that it belongs to no S. Our point, then, must be proved from the indefinite nature of the particular statement.
Nor is a deduction possible at all, if each of the extremes belongs to some of the middle, or does not belong, or one belongs and the other does not, or one belongs to some, the other not to all, or if they are indefinite. Common terms for all are animal, man, white; animal, inanimate, white. [10]
It is clear then in this figure also when a deduction will be possible, and when not; and that if the terms are as stated, a deduction results of necessity, and if there is a deduction, the terms must be so related. It is clear also that all the deductions in this figure are imperfect (for all are made perfect by certain supplementary [15] assumptions), and that it will not be possible to deduce a universal conclusion by means of this figure, whether negative or affirmative.
7 · It is evident also that in all the figures, whenever a deduction does not result, if both the terms are affirmative or negative nothing necessary follows at all, [20] but if one is affirmative, the other negative, and if the negative is assumed universally, a deduction always results relating the minor to the major term, e.g. if A belongs to every or some B, and B belongs to no C; for if the propositions are converted it is necessary that C does not belong to some A. Similarly also in the [25] other figures; a deduction always results by means of conversion. It is evident also that the substitution of an indefinite for a particular affirmative will effect the same deduction in all the figures.
It is clear too that all the imperfect deductions are made perfect by means of [30] the first figure. For all are brought to a conclusion either probatively or per impossibile, in both ways the first figure is formed: if they are made perfect probatively, because (as we saw) all are brought to a conclusion by means of conversion, and conversion produces the first figure; if they are proved per [35] impossibile, because on the assumption of the false statement the deduction comes about by means of the first figure, e.g. in the last figure, if A and B belong to every C, it follows that A belongs to some B; for if A belonged to no B, and B belongs to every C, A would belong to no C; but (as we stated) it belongs to every C. Similarly also with the rest.
It is possible also to reduce all deductions to the universal deductions in the [29b1] first figure. Those in the second figure are clearly made perfect by these, though not all in the same way; the universal ones are made perfect by converting the negative premiss, each of the particular by reductio ad impossibile. In the first figure [5] particular deductions are indeed made perfect by themselves, but it is possible also to prove them by means of the seco
nd figure, reducing them ad impossibile, e.g. if A belongs to every B, and B to some C, it follows that A belongs to some C. For if it belonged to no C, and belongs to every B, then B will belong to C: this we know by [10] means of the second figure. Similarly also demonstration will be possible in the case of the negative. For if A belongs to no B, and B belongs to some C, A will not belong to some C; for if it belonged to every C, and belongs to B, then B will belong to no C; and this (as we saw) is the middle figure. Consequently, since all deductions in the [15] middle figure can be reduced to universal deductions in the first figure, and since particular deductions in the first figure can be reduced to deductions in the middle figure, it is clear that particular deductions can be reduced to universal deductions in the first figure. Deductions in the third figure, if the terms are universal, are [20] directly made perfect by means of those deductions; but, when one of the propositions is particular, by means of the particular deductions in the first figure and these (we have seen) may be reduced to the universal deductions in the first figure; consequently also the particular deductions in the third figure may be so [25] reduced. It is clear then that all may be reduced to the universal deductions in the first figure.
We have stated then how deductions which prove that something belongs or does not belong to something else are constituted, both how those of the same figure are constituted in themselves, and how those of different figures are related to one another.
8 · Since there is a difference according as something belongs, necessarily [30] belongs, or may belong (for many things belong, but not necessarily, others neither necessarily nor indeed at all, but it is possible for them to belong), it is clear that there will be different deductions for each of these, and deductions with differently related terms, one concluding from what is necessary, another from what is, a third [35] from what is possible.
In the case of what is necessary, things are pretty much the same as in the case of what belongs; for when the terms are put in the same way, then, whether something belongs or necessarily belongs (or does not belong), a deduction will or will not result alike in both cases, the only difference being the addition of the [30a1] expression ‘necessarily’ to the terms. For the negative is convertible alike in both cases, and we should give the same account of the expressions ‘to be in something as in a whole’ and ‘to be predicated of every’. Thus in the other cases, the conclusion [5] will be proved to be necessary by means of conversion, in the same manner as in the case of simple predication. But in the middle figure when the universal is affirmative, and the particular negative, and again in the third figure when the universal is affirmative and the particular negative, the demonstration will not take the same form, but it is necessary by the exposition of a part of the subject, to which [10] in each case the predicate does not belong, to make the deduction in reference to this: with terms so chosen the conclusion will be necessary. But if the relation is necessary in respect of the part exposed, it must hold of some of that term in which this part is included; for the part exposed is just some of that. And each of the resulting deductions is in the appropriate figure.
[15] 9 · It happens sometimes also that when one proposition is necessary the deduction is necessary, not however when either is necessary, but only when the one related to the major is, e.g. if A is taken as necessarily belonging or not belonging to B, but B is taken as simply belonging to C; for if the propositions are taken in this [20] way, A will necessarily belong or not belong to C. For since A 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 AB is not necessary, but BC is necessary, the conclusion will not be necessary. For if it were, it [25] 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 will not be necessary, e.g. if A were movement, B animal, C man; man is [30] an animal necessarily, but an animal does not move necessarily, nor does man. Similarly also if AB is negative; for the proof is the same.
In particular deductions, if the universal is necessary, then the conclusion will be necessary; but if the particular, the conclusion will not be necessary, whether the [35] universal proposition is negative or affirmative. First let the universal be necessary, and let A belong to every 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 every B. Similarly also if the deduction should be [30b1] negative; for the proof will be the same. But if the particular 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 deductions. The same is true of [5] negatives too. Try the terms movement, animal, white.
10 · In the second figure, if the negative proposition 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 [10] the negative is convertible, B is possible of no A. But A belongs to every C; consequently B is possible of no C For C falls under A. The same result would be obtained if the negative refers to C; for if A is possible of no C, C is possible of no A; [15] but A belongs to every B, consequently C is possible of no B; for again we have obtained the first figure. Neither then is B possible of C; for conversion is possible as before.
But if the affirmative proposition is necessary, the conclusion will not be necessary. Let A belong to every B necessarily, but to no C simply. If then the [20] negative is converted, the first figure results. But it has been proved in the case of the first figure that if the negative related to the major 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 [25] 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 every 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 [30] 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 necessary given the premisses. For example let A be animal, B man, C white, and let the propositions be assumed in the same way as before: it is possible that animal should belong to [35] nothing white. Man then will not belong to anything white, but not necessarily; for it is possible for a man to become white, not however so long as animal belongs to nothing white. Consequently given these premisses the conclusion will be necessary, but it is not necessary without qualification.
Similar results will obtain also in particular deductions. For whenever the [31a1] negative proposition is both universal and necessary, then the conclusion will be necessary; but whenever the affirmative is universal and the negative particular, the [5] conclusion will not be necessary. First then let the negative be both universal and necessary: let it be possible for no B that A should belong to it, and let A belong to some C. Since the negative 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 [10] some C. Again let the affirmative be both universal and necessary, and let the affirmative refer to B. If then A necessarily belongs to every 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 [15] deductions. Nor again, if the negative is necessary but particular, will the conclusion be necessary. The point can be demonstrated by means of the same terms.
11 · In the last figure when the terms are related universally to the middle, [20] and both propositions are affirmative, if o
ne 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 [25] propositions be affirmative, and let A and B belong to every C, and let AC be necessary. Since then B belongs to every C, C also will belong to some B, because the universal is convertible into the particular; consequently if A belongs necessarily to every C, and C belongs to some B, it is necessary that A should belong to some B [30] 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 every C, it will belong necessarily also to some A.
Again let AC be negative, BC affirmative, and let the negative be necessary. [35] 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. For suppose BC is affirmative and [40] 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 no [31b1] C while C belongs to some B, A will not belong to some B—but not of necessity; for it has been proved, in the case of the first figure, that if the negative proposition is not necessary, neither will the conclusion be necessary. Further, the point may be [5] made clear by considering the terms. Let A be good, B animal, C horse. It is possible then that good should belong to no horse, and it is necessary that 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 [10] term awake or asleep; for every animal can accept these.