It is possible to prove these results also by reduction ad impossibile. (15)
It is clear then that a syllogism is formed when the terms are so related, but not a perfect syllogism; for necessity is not perfectly established merely from the original premisses; others also are needed.
But if M is predicated of every N and O, there cannot be a syllogism. Terms to illustrate a positive relation between the extremes are substance, animal, man; a negative relation, substance, animal, (20) number—substance being the middle term.
Nor is a syllogism possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man: a negative relation, line, animal, stone.
It is clear then that if a syllogism is formed when the terms are universally related, the terms must be related as we stated at the outset:33 for if they are otherwise related no necessary consequence follows. (25)
If the middle term is related universally to one of the extremes, a particular negative syllogism must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement: by ‘an opposite manner’ I mean, if the universal statement is negative, the particular is affirmative: if the universal is affirmative, (30) the particular is negative. For if M belongs to no N, but to some O, it is necessary that N does not belong to some O.34 For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure. (35) Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O:35 for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O. And if M belongs to all N but not to all O, we shall conclude that N does not belong to all O: the proof is the same as the above. [27b] But if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, (5) substance, raven; animal, white, raven. Nor will there be a conclusion when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit: a negative relation, animal, substance, science.
If then the universal statement is opposed to the particular, (10) we have stated when a syllogism will be possible and when not: but if the premisses are similar in form, I mean both negative or both affirmative, a syllogism will not be possible anyhow. First let them be negative, and let the major premiss be universal, e. g. let M belong to no N, (15) and not to some O. It is possible then for N to belong either to all O or to no O. Terms to illustrate the negative relation are black, snow, animal. But it is not possible to find terms of which the extremes are related positively and universally, if M belongs to some O, and does not belong to some O. For if N belonged to all O, but M to no N, then M would belong to no O: but we assumed that it belongs to some O. (20) In this way then it is not admissible to take terms: our point must be proved from the indefinite nature of the particular statement. For since it is true that M does not belong to some O, even if it belongs to no O, and since if it belongs to no O a syllogism is (as we have seen36 not possible, clearly it will not be possible now either.
Again let the premisses be affirmative, and let the major premiss as before be universal, e. g. let M belong to all N and to some O. (25) It is possible then for N to belong to all O or to no O. Terms to illustrate the negative relation are white, swan, stone. But it is not possible to take terms to illustrate the universal affirmative relation, for the reason already stated:37 the point must be proved from the indefinite nature of the particular statement. But if the minor premiss is universal, (30) and M belongs to no O, and not to some N, it is possible for N to belong either to all O or to no O. Terms for the positive relation are white, animal, raven: for the negative relation, white, stone, raven. If the premisses are affirmative, terms for the negative relation are white, animal, snow; for the positive relation, white, animal, swan. Evidently then, whenever the premisses are similar in form, (35) and one is universal, the other particular, a syllogism cannot be formed anyhow. Nor is one possible if the middle term belongs to some of each of the extremes, or does not belong to some of either, or belongs to some of the one, not to some of the other, or belongs to neither universally, or is related to them indefinitely. Common terms for all the above are white, animal, man: white, animal, inanimate.
[28a] It is clear then from what has been said that if the terms are related to one another in the way stated, a syllogism results of necessity; and if there is a syllogism, the terms must be so related. But it is evident also that all the syllogisms in this figure are imperfect: for all are made perfect by certain supplementary statements, (5) 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 conclusion 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, (10) or if both 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 the predicates 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. (15) A syllogism cannot be perfect in this figure either, but it may be valid whether the terms are related universally or not to the middle term.
If they are universal, whenever both P and R belong to all S, it follows that P will necessarily belong to some R.38 For, since the affirmative statement is convertible, S will belong to some R: consequently since P belongs to all S, and S to some R, P must belong to some R: for a syllogism in the first figure is produced. (20) It is possible to demonstrate this also per impossibile and by exposition. For if both P and R belong to all S, should one of the Ss, e. g. N, be taken, both P and R will belong to this, and thus P will belong to some R. (25)
If R belongs to all S, and P to no S, there will be a syllogism to prove that P will necessarily not belong to some R.39 This may be demonstrated in the same way as before by converting the premiss RS.40 It might be proved also per impossibile, as in the former cases. (30) But if R belongs to no S, P to all S, there will be no syllogism. Terms for the positive relation are animal, horse, man: for the negative relation animal, inanimate, man.
Nor can there be a syllogism when both terms are asserted of no S. Terms for the positive relation are animal, horse, inanimate; for the negative relation man, horse, inanimate—inanimate being the middle term. (35)
It is clear then in this figure also when a syllogism will be possible and when not, if the terms are related universally. For whenever both the terms are affirmative, there will be a syllogism to prove that one extreme belongs to some of the other; but when they are negative, no syllogism will be possible. [28b] But when one is negative, the other affirmative, if the major is negative, the minor affirmative, there will be a syllogism to prove that the one extreme does not belong to some of the other: but if the relation is reversed, no syllogism will be possible.
If one term is related universally to the middle, (5) the other in part only, when both are affirmative there must be a syllogism, no matter which of the premisses is universal. For if R belongs to all S, P to some S, P must belong to some R.41 For since the affirmative statement is convertible S will belong to some P: consequently since R belongs to all S, (10) 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 all S, P must belong to some R.42 This may be demonstrated in the same way as the preceding. And it is possible to demonstrate it also per impossibile and by exposition, (15) as in the former cases. But if one term is affirmative, the other negative, and if the affirmative is universal, a syllogism will be possible whenever the minor term is
affirmative. For if R belongs to all S, but P does not belong to some S, it is necessary that P does not belong to some R.43 For if P belongs to all R, and R belongs to all S, then P will belong to all S: but we assumed that it did not. (20) Proof is possible also without reduction ad impossibile, if one of the Ss be taken to which P does not belong.
But whenever the major is affirmative, no syllogism will be possible, e. g. if P belongs to all 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 is not possible to get terms, (25) if R belongs to some S, and does not belong to some S. For if P belongs to all 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.44 Since the expression ‘it does not belong to some’ is indefinite, it may be used truly of that also which belongs to none. But if R belongs to no S, (30) no syllogism is possible, as has been shown.45 Clearly then no syllogism will be possible here.
But if the negative term is universal, whenever the major is negative and the minor affirmative there will be a syllogism. For if P belongs to no S, and R belongs to some S, P will not belong to some R:46 for we shall have the first figure again, (35) if the premiss RS is converted.
But when the minor is negative, there will be no syllogism. 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 syllogism possible when both are stated in the negative, but one is universal, the other particular. When the minor is related universally to the middle, take the terms animal, science, wild; animal, man, wild. [29a] 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 belong to some S. For if P belongs to all R, and R to some S, (5) 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 syllogism possible anyhow, if each of the extremes belongs to some of the middle, or does not belong, or one belongs and the other does not to some of the middle, or one belongs to some of the middle, the other not to all, or if the premisses are indefinite. Common terms for all are animal, man, white: animal, inanimate, (10) white.
It is clear then in this figure also when a syllogism will be possible, and when not; and that if the terms are as stated, a syllogism results of necessity, and if there is a syllogism, the terms must be so related. It is clear also that all the syllogisms in this figure are imperfect (for all are made perfect by certain supplementary assumptions), (15) and that it will not be possible to reach a universal conclusion by means of this figure, whether negative or affirmative.
7 It is evident also that in all the figures, whenever a proper syllogism 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 stated universally, a syllogism always results relating the minor to the major term, e. g. if A belongs to all or some B, and B belongs to no C: for if the premisses are converted it is necessary that C does not belong to some A.47 Similarly also in the other figures: a syllogism always results by means of conversion. (25) It is evident also that the substitution of an indefinite for a particular affirmative will effect the same syllogism in all the figures.
It is clear too that all the imperfect syllogisms are made perfect by means of the first figure. (30) For all are brought to a conclusion either ostensively or per impossibile. In both ways the first figure is formed: if they are made perfect ostensively, because (as we saw) all are brought to a conclusion by means of conversion, (35) and conversion produces the first figure: if they are proved per impossibile, because on the assumption of the false statement the syllogism comes about by means of the first figure, e. g. in the last figure, if A and B belong to all C, it follows that A belongs to some B: for if A belonged to no B, and B belongs to all C, A would belong to no C: but (as we stated) it belongs to all C. Similarly also with the rest.
[29b] It is possible also to reduce all syllogisms to the universal syllogisms in the first figure. Those in the second figure are clearly made perfect by these, though not all in the same way; the universal syllogisms are made perfect by converting the negative premiss, (5) each of the particular syllogisms by reduction ad impossibile. In the first figure particular syllogisms are indeed made perfect by themselves, but it is possible also to prove them by means of the second figure, reducing them ad impossibile, e. g. if A belongs to all B, and B to some C, it follows that A belongs to some C. For if it belonged to no C, and belongs to all B, then B will belong to no C: this we know by means of the second figure. (10) 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 all C, and belongs to no B, then B will belong to no C: and this (as we saw) is the middle figure. (15) Consequently, since all syllogisms in the middle figure can be reduced to universal syllogisms in the first figure, and since particular syllogisms in the first figure can be reduced to syllogisms in the middle figure, it is clear that particular syllogisms48 can be reduced to universal syllogisms in the first figure. Syllogisms in the third figure, if the terms are universal, (20) are directly made perfect by means of those syllogisms;49 but, when one of the premisses is particular, by means of the particular syllogisms in the first figure: and these (we have seen) may be reduced to the universal syllogisms in the first figure: consequently also the particular syllogisms in the third figure may be so reduced. It is clear then that all syllogisms may be reduced to the universal syllogisms in the first figure. (25)
We have stated then how syllogisms which prove that something belongs or does not belong to something else are constituted, both how syllogisms of the same figure are constituted in themselves, and how syllogisms of different figures are related to one another.…
13 Perhaps enough has been said about the proof of necessity, (15) how it comes about and how it differs from the proof of a simple statement. [32a] We proceed to discuss that which is possible, when and how and by what means it can be proved. I use the terms ‘to be possible’ and ‘the possible’ of that which is not necessary but, being assumed, results in nothing impossible. (20) We say indeed ambiguously of the necessary that it is possible. But that my definition of the possible is correct is clear from the phrases by which we deny or on the contrary affirm possibility. For the expressions ‘it is not possible to belong’, ‘it is impossible to belong’, and ‘it is necessary not to belong’ are either identical or follow from one another; consequently their opposites also, ‘it is possible to belong’, ‘it is not impossible to belong’, (25) and ‘it is not necessary not to belong’, will either be identical or follow from one another. For of everything the affirmation or the denial holds good. That which is possible then will be not necessary and that which is not necessary will be possible. It results that all premisses in the mode of possibility are convertible into one another. (30) I mean not that the affirmative are convertible into the negative, but that those which are affirmative in form admit of conversion by opposition, e. g. ‘it is possible to belong’ may be converted into ‘it is possible not to belong’, and ‘it is possible for A to belong to all B’ into ‘it is possible for A to belong to no B’ or ‘not to all B’, and ‘it is possible for A to belong to some B’ into ‘it is possible for A not to belong to some B’. (35) And similarly the other propositions in this mode can be converted. For since that which is possible is not necessary, and that which is not necessary may possibly not belong, it is clear that if it is possible that A should belong to B, it is possible also that it should not belong to B: and if it is possible that it should belong to all, it is also possible that it should not belong to all. The same holds good in the case of particular affirmations: (40) for the proof
is identical. [32b] And such premisses are affirmative and not negative: for ‘to be possible’ is in the same rank as ‘to be’, as was said above.50
Having made these distinctions we next point out that the expression ‘to be possible’ is used in two ways. In one it means to happen generally and fall short of necessity, (5) e. g. man’s turning grey or growing or decaying, or generally what naturally belongs to a thing (for this has not its necessity unbroken, since man’s existence is not continuous for ever, although if a man does exist, it comes about either necessarily or generally). In another sense the expression means the indefinite, (10) which can be both thus and not thus, e. g. an animal’s walking or an earthquake’s taking place while it is walking, or generally what happens by chance: for none of these inclines by nature in the one way more than in the opposite.
That which is possible in each of its two senses is convertible into its opposite, (15) not however in the same way: but what is natural is convertible because it does not necessarily belong (for in this sense it is possible that a man should not grow grey) and what is indefinite is convertible because it inclines this way no more than that. Science and demonstrative syllogism are not concerned with things which are indefinite, because the middle term is uncertain; but they are concerned with things that are natural, (20) and as a rule arguments and inquiries are made about things which are possible in this sense. Syllogisms indeed can be made about the former, but it is unusual at any rate to inquire about them.
The Basic Works of Aristotle (Modern Library Classics) Page 12