Prior Analytics · Book I

Aristotle · a new plain-English translation from the original language

1| Prior Analytics, Book 1. First we must say what our inquiry is about and what it concerns: it is about demonstration and demonstrative knowledge. Then we must define what a premise is, what a term is, and what a syllogism is, and what kind of syllogism is complete and what kind incomplete; and after that, what it is for one thing to be in another as in a whole, or not to be, and what we mean by saying something holds of every case, or of none. A premise, then, is a statement affirming or denying something of something; and this is either universal, particular, or indefinite. By universal I mean that something holds of every case or of none; by particular, that it holds of some case, or does not hold of some, or does not hold of every case; by indefinite, that it holds or does not hold without any specification of universal or particular, as in "the same knowledge covers contraries" or "pleasure is not a good." A demonstrative premise differs from a dialectical one in this: the demonstrative premise is the taking of one side of a contradiction (for the one who demonstrates does not ask a question but takes something as given), whereas the dialectical premise is a question about a contradiction. But this will make no difference to whether a syllogism results from either: for both the one who demonstrates and the one who asks a question reason by taking something as holding, or not holding, of something.

2| So, without qualification, a syllogistic premise will be an affirmation or denial of something about something in the manner stated; it is demonstrative if it is true and has been taken through the primary hypotheses; and it is dialectical, for the one asking, a question about a contradiction, but for the one reasoning from it, the taking of what appears to be the case and is generally accepted, as has been said in the Topics. What a premise is, then, and how the syllogistic, the demonstrative, and the dialectical premise differ, will be stated with precision later; for our present purpose let what has been said suffice for now. I call a term that into which a premise is resolved -- that is, both the predicate and that of which it is predicated -- with "is" or "is not" added. A syllogism is an argument in which, certain things being posited, something other than what was posited follows of necessity from their being so. By "from their being so" I mean that it results because of them, and by "resulting because of them" I mean that no further term from outside is needed for the necessity to come about. I call a syllogism complete if it needs nothing else besides what has been taken in order for the necessity to be apparent; incomplete, if it needs one or more further things which are necessary given the underlying terms but have not been taken through premises. For one thing to be in another as in a whole, and for one thing to be predicated of every case of another, are the same. We say that something is predicated of every case when nothing can be found belonging to the subject of which the other term will not be said; and "of no case" is understood likewise.

3| Since every premise states either that something holds, or that it holds of necessity, or that it may possibly hold, and since of these some are affirmative and some negative according to each kind of statement, and again, among the affirmative and negative premises some are universal, some particular, and some indefinite -- it is necessary that a universal negative premise about what holds converts with respect to its terms: for instance, if no pleasure is good, then no good will be pleasure either. A universal affirmative premise must also convert, but not universally -- only in part: for instance, if every pleasure is good, then some good is pleasure. Among particular premises, the affirmative must convert in part as well (for if some pleasure is good, then some good will be pleasure); but the negative need not convert (for it is not the case that, if man does not belong to some animal, animal does not belong to some man). First, then, let the premise A-B be a universal negative. If, then, A holds of no B, then B will hold of no A either; for if it held of some, say of C, it would not be true that A holds of no B, since C would be one of the B's. But if A holds of every B, then B will hold of some A; for if it held of none, then A would hold of no B either -- but it was assumed to hold of every B. The same holds if the premise is particular: for if A holds of some B, then B must hold of some A; for if it held of none, A would hold of no B either. But if A does not hold of some

4| B, it is not necessary that B does not hold of some A -- for instance, let B be animal and A man: man does not hold of every animal, but animal holds of every man.

5| The same will hold in the case of necessary premises. For the universal negative converts universally, while of the affirmative premises each converts only in part. For if it is necessary that A holds of no B, it is also necessary that B holds of no A; for if it could hold of some, then A too could hold of some B. And if A holds of necessity of every B or of some B, then B must hold of necessity of some A; for if it were not necessary, then A would not hold of necessity of some B either. But the particular negative does not convert, for the same reason we gave before. Now with regard to possible premises, since "possible" is said in many ways (for we call both the necessary and the not-necessary and the potential "possible"), in the case of affirmative premises the conversion will work the same way throughout. For if A may hold of every B or of some B, then B may hold of some A; for if it could hold of none, then A could hold of no B either -- this has been shown already. But with negative premises it is not the same: rather, whenever "possible" is said in the sense of holding of necessity, or of not necessarily not holding, the conversion works the same way -- for instance if someone says that man may possibly not be a horse, or that white may possibly hold of no garment (for of these, the one of necessity does not hold, while the other need not hold, and the premise converts in the same way:

6| for if it is possible that horse holds of no man, then it is also possible that man holds of no horse; and if it is possible that white holds of no garment, then it is also possible that garment holds of nothing white -- for if it held of necessity of some, then white would hold of some garment of necessity, since this has been shown already), and likewise for the particular negative. But whenever "possible" is said in the sense of what holds for the most part and by nature -- the sense in which we define the possible -- the negative conversions will not work the same way; rather, the universal negative premise does not convert, while the particular negative does convert. This will become clear when we speak about the possible. For now, let this much be evident to us in addition to what has been said: that "it is possible to hold of nothing" or "of not every case" has the grammatical form of an affirmative statement (for "is possible" is arranged like "is," and "is," whatever it is added to as predicate, always and in every case makes an affirmation -- for instance "is not-good" or "is not-white" or, without qualification, "is not-this"; this too will be shown in what follows) -- and with respect to conversion these will behave in the same way as the others do.

7| Now that these points have been distinguished, let us next say through what means, when, and how every syllogism comes about; afterward we must speak about demonstration. We must speak about syllogism before demonstration, because syllogism is the more general thing: demonstration is a kind of syllogism, but not every syllogism is a demonstration. Whenever three terms are related to one another in such a way that the last is contained in the whole of the middle, and the middle is contained in the whole of the first (or excluded from it), it is necessary that there be a perfect syllogism of the extremes. I call "middle" that term which is itself contained in another and contains another in itself, and which also becomes middle in position; and I call "extremes" both the term that is in another and the term in which another is. For if A is predicated of every B, and B of every C, then A must be predicated of every C: we have already said earlier what we mean by "of every." Similarly too, if A belongs to no B, and B belongs to every C, then A will belong to no C. But if the first follows every one of the middle, and the middle belongs to none of the last, there will be no syllogism of the extremes; for nothing necessary results from these being so, since the first can belong to every one or to none of the last, so that neither the particular nor the universal conclusion becomes necessary; and if nothing is necessary through these premises, there will be no syllogism. Terms for belonging to every: animal-man-horse; for belonging to none: animal-man-stone. Nor again is there a syllogism when neither the first belongs to the middle nor the middle to any of the last. Nor in this case either will there be a syllogism. Terms for belonging: knowledge-line-medicine;

8| for not belonging: knowledge-line-unit. When the terms are universal, then, it is clear in this figure when there will be a syllogism and when there will not, and that when there is a syllogism the terms must be related as we have said, and that if they are so related, there will be a syllogism. But if one of the terms is universal and the other particular in relation to the other term, then whenever the universal is set at the major extreme, whether affirmative or privative, and the particular is set at the minor extreme, affirmative, it is necessary that there be a perfect syllogism; but whenever the universal is set at the minor extreme, or the terms are related in any other way, it is impossible. I call "major extreme" that in which the middle is, and "minor extreme" that which falls under the middle. Let A belong to every B, and B to some C. Then, if to be predicated of every is what was said at the start, A must belong to some C. And if A belongs to no B, and B belongs to some C, A must fail to belong to some C; for we have defined what we mean by "of none."

9| so that there will be a perfect syllogism. Similarly too, if the premise BC were indefinite, so long as it is affirmative: for the same syllogism results whether it is taken as indefinite or as particular. But if the universal is set at the minor extreme, whether affirmative or privative, there will be no syllogism, whether the indefinite or particular premise is affirmative or negative-for instance if A belongs, or does not belong, to some B, and B belongs to every C. Terms for belonging: good-state-practical wisdom; for not belonging: good-state-ignorance. Again, if B belongs to no C, and A belongs, or does not belong, or does not belong to every, some B, there will be no syllogism in this case either. Terms: white-horse-swan, white-horse-raven. The same terms serve also if B is indefinite. Nor when the term next to the major extreme is universal, whether affirmative or privative, and the term next to the minor extreme is privative and particular, will there be a syllogism-for instance if A belongs to every B, and B does not belong to some C, or does not belong

10| to every C. For whatever thing the middle fails to belong to, the first will follow that thing either universally or not at all. Suppose the terms are animal-man-white; then, among the things of which "white" is not predicated, let swan and snow be taken: then "animal" is predicated of the one, of all of it, and of the other, of none of it, so that there is no syllogism. Again, let A belong to no B, and let B fail to belong to some C; and let the terms be soulless-man-white; then take, among the things of which "white" is not predicated, man, swan, and snow: for "soulless" is predicated of all of one and of none of the other. Further, since "B does not belong to some C" is indefinite, and it is true both when B belongs to none of C and when it does not belong to every C-that it does not belong to some-and since, when such terms are taken that B belongs to none of C, no syllogism results (for this has been said already), it is clear that no syllogism will result from the terms being related in this way, for it would have resulted in these cases too. It will be shown in the same way also if the universal is set as privative. Nor when both intervals are particular, whether both affirmative, or both privative, or one affirmative and the other privative, or one indefinite and the other definite, or both indefinite, will there be a syllogism in any way.

11| Terms common to all these cases: animal-white-horse, animal-white-stone. It is clear, then, from what has been said, that if there is to be a syllogism in this figure in the particular mode, the terms must be related as we have stated; for if they are related otherwise, no syllogism results in any way. It is clear too that all the syllogisms in this figure are perfect (for they are all completed by means of what was assumed at the start), and that all the problems are demonstrated through this figure: belonging to every, belonging to none, belonging to some, and not belonging to some. I call this sort of figure the first. But whenever the same term belongs to one thing wholly and to another not at all, or to both wholly, or to neither, I call this sort of figure the second; and in it I call the middle the term that is predicated of both, the extremes the terms of which this is said, the major extreme the one lying next to the middle, and the minor extreme the one further from the middle. The middle is placed outside the extremes, and is first in position.

12| A perfect syllogism, then, will never result in this figure, but it will be possible whether the terms are universal or not universal. If the terms are universal, there will be a syllogism whenever the middle belongs to one of them wholly and to the other not at all, whichever of the two the privative premise concerns; but in no other way. For let M be predicated of no N, and of every X. Since the privative premise converts, N will belong to no M; but M was assumed to belong to every X; so N will belong to no X-for this has been shown already. Again, if M belongs to every N and to no X, neither will X belong to any N (for if M belongs to no X, neither does X belong to any M; but M belonged to every N; therefore X will belong to no N-for again the first figure has resulted); and since the privative premise converts, N too will belong to no X, so that the same syllogism results. These things can also be shown by leading to the impossible. It is clear, then, that a syllogism results when the terms are so related, but it is not perfect;

13| For the necessary conclusion is brought about not only from the original premises but also from others. But if M is predicated of every N and of every X, there will be no syllogism. Terms for belonging: substance—animal—man; for not belonging: substance—animal—number; the middle term is substance. Nor will there be a syllogism when M is predicated of neither N nor X at all. Terms for belonging: line—animal—man; for not belonging: line—animal—stone. It is clear, then, that if there is a syllogism when the terms are universal, the terms must be related as we stated at the outset; for if they are related otherwise, the necessary conclusion does not result. But if the middle term is universal in relation to one of the extremes—when it becomes universal in relation to the major term, either affirmatively or privatively, while in relation to the minor term it is particular and opposite to the universal premise (I call it opposite in this sense: if the universal premise is privative, the particular is affirmative; if the universal premise is affirmative, the particular is privative)—then a particular privative syllogism must result. For if M belongs to no N but belongs to some X, then N must not belong to some X. For since the privative premise converts, N will belong to no M;

14| but M was assumed to belong to some X; hence N will not belong to some X. For a syllogism results through the first figure. Again, if M belongs to every N but does not belong to some X, N must not belong to some X. For if it belonged to every X, and M is also predicated of every N, then M would have to belong to every X; but it was assumed not to belong to some X. And if M belongs to every N but not to every X, there will be a syllogism that N does not belong to every X; the proof is the same. But if M is predicated of every X and not of every N, there will be no syllogism. Terms: animal—substance—raven; animal—white—raven. Nor will there be a syllogism when M is predicated of no X and of some N. Terms for belonging: animal—substance—unit; for not belonging: animal—substance—knowledge. When, then, the universal premise is opposite to the particular one, we have stated when there will be a syllogism and when there will not. But when the premises are of the same form—both privative or both affirmative—there will never be a syllogism. Let them first be privative, and let the universal premise be placed in relation to the major extreme, so that M belongs to no N but does not belong to some X;

15| then N can belong to every X or to no X. Terms for not belonging: black—snow—animal; but terms for belonging cannot be found, if M belongs to some X and does not belong to some other X. For if N belonged to every X, and M to no N, then M would belong to no X; but it was assumed to belong to some X. So in this way terms cannot be found, but the point must be shown from an indefinite premise. For since it is true that M does not belong to some X even when it belongs to no X, and since when M belongs to no X there was no syllogism, it is clear that there will be none now either. Again, let the premises be affirmative, and let the universal premise be placed in the same way, so that M belongs to every N but to some X. Then N can belong to every X or to no X. Terms for belonging to none: white—swan—stone; but terms for belonging to every one cannot be found, for the same reason as before, and the point must be shown from an indefinite premise. But if the universal premise is in relation to the minor extreme, and M belongs to no X but does not belong to some N, then N can belong

16| to every X or to no X. Terms for belonging: white—animal—raven; for not belonging: white—stone—raven. But if the premises are affirmative, terms for not belonging: white—animal—snow; for belonging: white—animal—swan. It is clear, then, that when the premises are of the same form, one universal and the other particular, no syllogism ever results. Nor does one result if each term belongs or does not belong to some part, or belongs to one and not to the other, or to neither wholly, or is stated indefinitely. Terms common to all these cases: white—animal—man; white—animal—inanimate. It is clear, then, from what has been said, that if the terms stand to one another as has been stated, a syllogism results of necessity; and if there is a syllogism, the terms must stand in this way. It is also clear that all the syllogisms in this figure are imperfect (for all of them are completed by the assumption of something further, which either belongs to the terms of necessity or is laid down as a hypothesis, as when we prove through the impossible), and that no affirmative syllogism results through this figure, but all are privative, both the universal and the particular ones.

17| But if one term belongs to every instance of the same subject and another to none of it, or both belong to every instance or to none, I call a figure of this kind the third; and I call the middle term in it the one of which both terms are predicated, and I call the extremes the terms predicated of it; the major extreme is the one farther from the middle, the minor the one nearer. The middle term is placed outside the extremes, last in position. No perfect syllogism results in this figure either, but one will be possible whether the terms are universal or not universal in relation to the middle term. When they are universal, then, whenever both P and R belong to every S, it will follow of necessity that P belongs to some R. For since the affirmative premise converts, S will belong to some R; so that, since P belongs to every S and S belongs to some R, P must belong to some R; for a syllogism results through the first figure. The proof can also be made through the impossible and by exposition. For if both belong to every S, then if some one of the S's is taken, say N, both P and R will belong to it, so that P will belong

18| to some R. And if R belongs to every S but P belongs to no S, there will be a syllogism that P will not belong to some R of necessity; for the manner of proof is the same when the premise R-S is converted. It could also be shown through the impossible, just as in the previous cases. But if R belongs to no S and P belongs to every S, there will be no syllogism. Terms for belonging: animal—horse—man; for not belonging: animal—inanimate—man. Nor will there be a syllogism when both are said of no S. Terms for belonging: animal—horse—inanimate; for not belonging: man—horse—inanimate; the middle term is inanimate. It is clear, then, in this figure too, when there will be a syllogism and when there will not, if the terms are universal. For when both terms are affirmative, there will be a syllogism that the extreme belongs to some instance of the other extreme; when they are privative, there will not be one. But when one is privative and the other affirmative, if the major term becomes privative and the other affirmative, there will be a syllogism that the extreme does not belong to some instance of the other extreme; but if it is the other way around, there will not be one. But if one premise is universal in relation to the middle term and the other particular, then, if both are affirmative, a syllogism must result, whichever of the two terms is universal.

19| For if R belongs to every S, and Π belongs to some S, then Π must belong to some R. For since the affirmative converts, S will belong to some Π; so, since R belongs to every S and S belongs to some Π, R will also belong to some Π; therefore Π belongs to some R. Again, if R belongs to some S and Π belongs to every S, then Π must belong to some R; for the manner of proof is the same. It can also be demonstrated through the impossible and by exposition, as in the earlier cases. If one premise is affirmative and the other privative, with the affirmative universal, then whenever the minor is affirmative, there will be a syllogism. For if P belongs to every S, and Π does not belong to some S, then Π must not belong to some R. For if it belonged to every R, then, since R belongs to every S, P would belong to every S as well;

20| but it did not so belong. This can also be shown without reduction to the impossible, if something is taken among the S's to which Π does not belong. But when the major is affirmative, there will be no syllogism, for example if P belongs to every S but R does not belong to some S. Terms for belonging to every: animate—man—animal. But for belonging to none it is not possible to get terms, if P belongs to some S and not to some S; for if Π belongs to every R, and R belongs to some S, then Π will belong to some S as well; but it was assumed to belong to none. Instead we must proceed as in the earlier cases; for since 'not belonging to some' is indefinite, it is also true to say of what belongs to none that it does not belong to some; but when it belongs to none, there was no syllogism. So it is clear there will be no syllogism. If the privative premise is universal among the terms, then whenever the major is privative and the minor affirmative, there will be a syllogism. For if P belongs to no S, and R belongs to some S, then Π will not belong to some R; for again the first figure will result once the premise R–S is converted.

21| But when the minor is privative, there will be no syllogism. Terms for belonging: animal—man—wild. For not belonging: animal—knowledge—wild; the middle term in both is 'wild.' Nor when both premises are taken as privative, one universal and the other particular. Terms, when the minor is universal in relation to the middle: animal—knowledge—wild, animal—man—wild; when the major is universal, terms for not belonging: raven—snow—white. For belonging, it is not possible to get terms, if R belongs to some S and not to some S; for if Π belongs to every R, and R belongs to some S, then Π would belong to some S; but it was assumed to belong to none. Instead it must be shown from the indefinite premise. Nor again if each premise belongs to some of the middle term, or does not belong, or one belongs and the other does not, or one belongs to some and the other not to every, or both are taken indefinitely — in no case will there be a syllogism. Terms common to all these: animal—man—white, animal—inanimate—white. It is clear, then, in this figure too, when there will be a syllogism and when not, and that when the terms stand as described a syllogism necessarily results, and if there is a syllogism, the terms must stand so. It is also clear that all the syllogisms in this figure are incomplete (for they are all completed by the assumption of something additional), and that it is not possible to reach a universal conclusion through this figure, neither privative nor affirmative.

22| It is also clear that in all the figures, whenever no syllogism results, if both terms are affirmative or both privative, nothing necessary follows at all; but if one is affirmative and the other privative, with the privative taken universally, there always results a syllogism of the minor extreme in relation to the major, for example if A belongs to every B or to some B, and B belongs to no C; for once the premises are converted, C must not belong to some A. The same holds for the other figures too; for a syllogism always results through conversion. It is also clear that substituting the indefinite premise for the particular affirmative will produce the same syllogism in all the figures. It is also clear that all the incomplete syllogisms are completed through the first figure. For they are all brought to a conclusion either directly or through the impossible, and in both ways the first figure results: when completed directly, because they were all brought to a conclusion through conversion, and conversion produced the first figure; and when demonstrated through the impossible, because once the false premise is posited, the syllogism comes about through the first figure — for example in the last figure, if A and B both belong to every C, it can be shown that A belongs to some B; for if it belonged to none, and B belongs to every C, then A would belong to no C; but it belonged to every C. The same holds in the other cases too. It is also possible to reduce all syllogisms to the universal syllogisms in the first figure. Those in the second figure are clearly completed through those, though not all in the same way, but the universal ones through conversion of the universal privative premise, and each of the particular ones through reduction to the impossible. The particular syllogisms in the first figure are completed

23| through themselves as well, but it is also possible to demonstrate them through the second figure by reduction to the impossible, for example if A belongs to every B, and B belongs to some C, to show that A belongs to some C; for if it belonged to none, and A belongs to every B, then B would belong to no C — for this we know through the second figure. The demonstration will proceed similarly for the privative case too. For if A belongs to no B, and B belongs to some C, then A will not belong to some C; for if it belonged to every C, and A belongs to no B, then B would belong to no C — and this was the middle figure. So then, since the syllogisms in the middle figure are all reduced to the universal syllogisms in the first figure, and the particular syllogisms in the first figure are reduced to those in the middle figure, it is clear that the particular syllogisms too will be reduced to the universal syllogisms in the first figure. Those in the third figure, when the terms are universal, are completed immediately through those syllogisms; but when they are taken as particular, through the particular syllogisms in the first figure; and these were reduced to those, so that the particular syllogisms in the third figure are reduced to them as well. It is clear, then, that all will be reduced to the universal syllogisms in the first figure. We have now stated how matters stand for the syllogisms that show belonging or not belonging — both those within the same figure taken by themselves, and those from different figures compared with one another.

24| Since belonging, belonging of necessity, and possibly belonging are different (for many things belong but not of necessity, and other things neither belong of necessity nor belong at all, but can possibly belong), it is clear that the syllogism for each of these will also be different, and that the terms will not be arranged in the same way — one syllogism will be from necessary premises, another from premises of belonging, another from premises of possibility. Now for necessary premises the case is much the same as for premises of belonging: for when the terms are laid down in the same way, there will or will not be a syllogism alike for belonging and for necessarily belonging or not belonging, except that it will differ by the addition, to the terms, of necessarily belonging or not belonging. For the negative converts in the same way, and we will render "being in the whole" and "said of all" in the same way. In the other cases, then, the necessity of the conclusion will be shown in the same manner, through conversion, as in the case of belonging. But in the middle figure, when the universal premise is affirmative and the particular one negative, and again in the third figure, when the universal is predicative and the particular negative, the demonstration will not proceed in the same way; instead one must set out that thing to which each term fails to belong, and construct the syllogism about it — for the syllogism will then be necessary about these; and if it is necessary of the thing set out, it is necessary also of some part of that original term, since the thing set out is precisely some part of it. Each of these two syllogisms comes about in its own proper figure.

25| It also happens sometimes that, though only one of the premises is necessary, the conclusion still comes out necessary — but not from just any premise being necessary, only from the one connected to the major term. For example, if A has been taken as belonging or not belonging to B of necessity, while B merely belongs to C (without necessity), then, the premises being taken this way, A will of necessity belong or not belong to C. For since A of necessity belongs or does not belong to every B, and C is some part of the Bs, clearly one or the other of these will hold of C as well, of necessity. But if A-B is not necessary, while B-C is necessary, the conclusion will not be necessary. For if it were, it would follow that something belongs to some B of necessity, both through the first figure and through the third — and that is false, since B may be of such a kind that it is possible for it to belong to nothing. Further, this is clear from terms as well: let A be motion, B be animal, and C be man. A man is of necessity an animal, but an animal is not of necessity in motion, nor is a man. The same holds if A-B is negative, for the demonstration is the same. In the case of particular syllogisms, if the universal premise is necessary, the conclusion too will be necessary; but if the particular premise is necessary, the conclusion will not be necessary, whether the universal premise is negative or affirmative.

26| Let the universal premise be necessary first: let A belong to every B of necessity, and let B belong only to some C (without necessity). Then A must belong to some C of necessity, since C falls under B, and A belonged to every B of necessity. The same holds if the syllogism is negative, for the demonstration will be the same. But if the particular premise is necessary, the conclusion will not be necessary (for nothing impossible results), just as in the universal syllogisms. The same holds again for the negative cases. Terms: motion — animal — white.

27| In the second figure, if the negative premise is necessary, the conclusion too will be necessary; but if the affirmative premise is necessary, the conclusion will not be necessary. Let the negative premise be necessary first: let A be possible to belong to no B, but let it merely belong to every C (without necessity). Then, since the negative converts, it is also not possible for B to belong to any A; but A belongs to every C, so it is not possible for B to belong to any C, since C falls under A. Likewise if the negative premise is placed with C: if it is not possible for A to belong to any C, then it is not possible for C to belong to any A either; but A belongs to every B, so it is not possible for C to belong to any B, for we again get the first figure. Therefore B does not belong to C either, since it converts in the same way. But if the affirmative premise is necessary, the conclusion will not be necessary. Let A belong to every B of necessity, but let it merely belong to no C (without necessity). Converting the negative premise, we get the first figure; and it has been shown in the first figure that when the negative premise attached to the minor term is not necessary, the conclusion is not necessary either — so it will not be necessary of necessity in these cases either. Further, if the conclusion were necessary, it would follow that C fails to belong to some A of necessity. For if B belongs to no C of necessity, then C too will belong to no B of necessity. But B must belong to some A, at least, since A belonged to every B of necessity. So C must fail to belong to some A of necessity. But nothing prevents A being taken as something

28| to which C can possibly belong, in every case. Further, by setting out terms one can show that the conclusion is not necessary without qualification, but only necessary given these particular terms. For example, let A be animal, B be man, and C be white, and let the premises be taken in the same way: it is possible for animal to belong to no white thing. Man, then, will belong to no white thing, but not of necessity, since it is possible for a man to become white, only not so long as animal belongs to no white thing. So given these particular terms the conclusion will be necessary, but not necessary without qualification. The same holds for particular syllogisms too. For when the negative premise is both universal and necessary, the conclusion will also be necessary; but when the affirmative premise is universal and the negative premise particular, the conclusion will not be necessary. Let the negative premise be universal and necessary first: let it be possible for A to belong to no B, and let A belong to some C. Then, since the negative converts, it is not possible for B to belong to any A either; but A belongs to some C, so B will of necessity fail to belong to some C. Again, let the affirmative premise be universal and necessary, and let the affirmative be the one attached to B. If A belongs to every B of necessity, but does not belong to some C, then it is clear that B will not belong to some C — but not of necessity, since the same terms will serve for the demonstration as in the universal syllogisms. Nor, again, if the negative premise, taken as particular, is necessary, will the conclusion be necessary; for the demonstration proceeds through the same terms.

29| In the last figure, when the terms are universal with respect to the middle term and both premises are affirmative, then if either one is necessary, the conclusion too will be necessary. But if one is negative and the other affirmative, then when the negative one is necessary, the conclusion will also be necessary, but when the affirmative one is necessary, it will not be necessary. Let both premises be affirmative first, and let A and B both belong to every C, with

30| and let A-C be necessary. Since B belongs to every C, C will also belong to some B, because the universal converts with the particular; so that if it belongs to every C of necessity, and C belongs to some B, then A must also belong to some B of necessity — for B falls under C. This, then, gives the first figure. The same will be shown in the same way if B-C is necessary: for C converts with some A, so that if B belongs to every C of necessity, it will also belong to some A of necessity. Again, let C be privative and B-C affirmative, and let the privative be necessary. Since C converts with some B, and A belongs to no C of necessity, A will not belong to some B of necessity either — for B falls under C. But if the affirmative is necessary, the conclusion will not be necessary. For let B-C be affirmative and necessary, and let C be privative and not necessary. Since the affirmative converts, C will also belong to some B of necessity, so that if A belongs to no C and C belongs to some B, A will not belong to some B —

31| but not of necessity. For it has been shown in the first figure that when the privative premise is not necessary, the conclusion will not be necessary either. This is also clear through terms. Let A be good, B be animal, and C be horse. It is possible for good to belong to no horse, while it is necessary that animal belong to every horse; but it is not necessary that some animal not be good, since it is possible for every animal to be good — or, if that is not possible, let being awake or being asleep serve as the term, since every animal is receptive of these. If, then, the terms are universal with respect to the middle term, it has been stated when the conclusion will be necessary. But if one term is universal and the other particular, and both premises are affirmative, then whenever the universal premise is necessary, the conclusion will also be necessary. The demonstration is the same as before, for the particular affirmative also converts. If, then, it is necessary that B belong to every

32| C, and this falls under C, it is necessary that B belong to some A. And if B belongs to something, A too must belong to some B of necessity, for it converts. Likewise also if A-C should be necessary and universal, for B falls under C. But if the particular premise is necessary, the conclusion will not be necessary. For let B-C be particular and necessary, and let A belong to every C, though not of necessity. Then, once B-C is converted, the first figure results, and the universal premise is not necessary while the particular premise is necessary. But when the premises stand this way, the conclusion was not necessary, and so it is not necessary in this case either. This is also clear from terms. Let A be being-awake, B be two-footed, and C be animal. Then B must belong to some C of necessity, while A can possibly belong to C, and A need not belong to B of necessity — for it is not necessary that something two-footed be either asleep or awake.

33| It will be shown in the same way, and through the same terms, if A-C should be particular and necessary. But if one of the terms is affirmative and the other privative: when the universal premise is privative and necessary, the conclusion too will be necessary. For if A can belong to no C, while B belongs to some C, then A must not belong to some B of necessity. But when the affirmative premise is posited as necessary, whether universal or particular, or when the privative premise is particular, the conclusion will not be necessary. The rest is the same as what we said before; and for terms — when the universal affirmative is necessary: being-awake, animal, man, with man as middle; when the particular affirmative is necessary: being-awake, animal, white — for it is necessary that animal belong to some white thing, but being-awake can belong to none, and it is not necessary that being-awake fail to belong to some animal. And when the particular privative is necessary: two-footed, moving, animal, with animal as middle. It is clear, then, that there is no syllogism concerning simple belonging unless both premises are premises of simple belonging, but there is one concerning necessity even if only one premise is necessary. In

34| both cases — whether the syllogisms are affirmative or privative — one of the two premises must be similar to the conclusion. By 'similar' I mean: if the conclusion is of simple belonging, the premise must be of simple belonging; if it is necessary, the premise must be necessary. So this too is clear: the conclusion cannot be either necessary or a matter of simple belonging unless a necessary premise, or a premise of simple belonging, has been taken. Concerning the necessary, then — how it comes about, and what difference

35| it has in relation to simple belonging, has been stated at sufficient length. Concerning the contingent, let us next say when, how, and through what sort of premises there will be a syllogism. I call 'contingent' — and 'the contingent' — that which, not being necessary, but posited as belonging, will involve nothing impossible on that account; for we also apply the word 'contingent' to the necessary, but homonymously. It turns out that all premises expressed in terms of contingency convert into one another. I do not mean the affirmative converting with the negative, but those which have an affirmative form in respect of their opposition — for instance, 'contingently belongs' converts with 'contingently does not belong,' and 'contingently belongs to all' with 'contingently belongs to none' and 'not to all,' and 'to some' with 'not to some.' The same holds for the others as well. For since the contingent is not necessary, and what is not necessary admits of not belonging, it is clear that if it is contingent for A to belong to B, it is also contingent for it not to belong; and if it is contingent for it to belong to all, it is also contingent for it not to belong to all. The same holds for particular affirmations too, for the demonstration is the same. Such premises are affirmative, not privative; for 'being contingent' is ranked together with 'being,' in the same way as was said before.

36| Now that these points have been distinguished, let us say again that 'being contingent' is spoken of in two ways: in one way, of what happens for the most part and falls short of necessity — for instance, a man going gray, or growing, or declining, or in general whatever belongs to a thing by nature (for this does not have necessity in an unbroken way, because a man does not always exist, yet when a man does exist, it holds either of necessity or for the most part); in the other way, of what is indeterminate, which is capable of being so or not being so — for instance, an animal walking, or an earthquake occurring while it is walking, or in general whatever comes about by chance; for such a thing is no more disposed by nature to happen this way than the opposite way. Each kind of the contingent converts with its opposite premises, but not in the same manner: the natural kind converts with 'not belonging of necessity' (for in this way it is contingent that a man not go gray), while the indeterminate kind converts with 'no more this way than that.' Scientific knowledge and demonstrative syllogism do not concern the indeterminate, because the middle term is disorderly, but they do concern the natural kind, and arguments and inquiries are, for the most part, concerned with things contingent in this way —

37| But regarding those cases, a syllogism can come about, though it is not usually the object of inquiry. These points will be settled more fully later on; for now let us say when, how, and what sort of syllogism will result from premises of possibility. Since it is possible to understand "this is possible of that" in two ways—either of that to which the thing actually belongs, or of that to which it is possible for it to belong (for "A is possible of that of which B holds" signifies one or the other of these: either that of which B is said, or that of which it is possible for B to be said, and it makes no difference whether we say "A is possible of that of which B holds" or "A may possibly belong to everything that B may belong to")—it is clear that "A is possible of all B" could be said in two ways. Let us first say, then, what sort of syllogism will result, and of what kind, if B is possible of that of which C holds, and A is possible of that of which B holds; for in this way both premises are taken in respect of possibility, whereas when A is possible of that of which B actually holds, one premise is assertoric and the other is a premise of possibility. So we must begin from the premises that share the same form, as in the other cases too.

38| When, then, A is possible of all B and B is possible of all C, there will be a perfect syllogism that A is possible of all C. This is clear from the definition: for this is what we meant by "possible of belonging to all." Similarly too, if A is possible of no B, and B is possible of all C, then A is possible of no C; for A's not being possible of anything that B is possible of—this was what we meant by leaving out none of what falls under B's possibility. But when A is possible of all B, and B is possible of no C, no syllogism results from the premises as taken; but when the B–C premise is converted according to possibility, the same syllogism results as before. For since it is possible for B to belong to no C, it is also possible for it to belong to all C; this was stated earlier. So if B is possible of all C, and A is possible of all B, again the same syllogism results. And similarly too if the negation is attached to both premises together with possibility. I mean, for example, if A is possible of no B and B is possible of no C:

39| through the premises as taken, no syllogism results, but when they are converted, again the same one results as before. It is clear, then, that when the negation is attached to the minor term, or to both premises, either no syllogism results, or one results but is not perfect; for the necessary conclusion is reached only by conversion. But if one of the premises is taken as universal and the other as particular, when the universal one is placed toward the major term, the syllogism will be perfect. For if A is possible of all B, and B is possible of some C, then A is possible of some C. This is clear from the definition of possibility. Again, if A is possible of no B, and B is possible of belonging to some C, then A must be possible of not belonging to some C; the proof is the same. But if the particular premise is taken as negative, and the universal one as affirmative, with the same arrangement (that is, A is possible of all B, and B is possible of not belonging to some C), then through

40| the premises as taken, no evident syllogism results; but when the particular premise is converted and it is instead taken that B is possible of belonging to some C, the same conclusion results as before, just as in the cases from the start. But if the premise toward the major term is taken as particular, and the one toward the minor as universal, then whether both are taken as affirmative, or as negative, or as differing in form, or both indefinite or particular, there will be no syllogism at all; for nothing prevents B from extending more widely than A and not being predicated of an equal range. Let C be taken as that of which B extends beyond A: for it is possible for A to belong to it neither in every case, nor in none, nor in some, nor not in some, given that premises of possibility convert and B is possible of more things than it actually belongs to. This is also clear from terms: for with the premises so arranged, it is possible for the first term to belong to the last in no case, and also necessary for it to belong to it in every case. Terms common to all these cases: for necessary belonging, animal–white–man; for impossibility, animal–white–garment. It is clear, then, that when the terms stand in this way, no syllogism results.

41| For every syllogism is one either of actual belonging, or of necessity, or of possibility. That it is not one of actual belonging or of necessity is clear: for the affirmative conclusion is refuted by the negative example, and the negative by the affirmative one. It remains, then, that it should be one of possibility; but this is impossible, for it has been shown that, with the terms so arranged, the first term must belong to the last in every case, and also may possibly belong to it in no case. So there could be no syllogism of possibility; for the necessary was not the possible. It is clear that, when the terms are universal in premises of possibility, a syllogism always results in the first figure, whether the premises are affirmative or negative—perfect if they are affirmative, imperfect if they are negative. But possibility must be taken not in the sense of necessity, but according to the distinction we have stated. Sometimes this distinction goes unnoticed. If one of the premises is taken as assertoric and the other as a premise of possibility, then when the premise toward the major term signifies possibility,

42| all the syllogisms will be perfect, and of possibility according to the distinction we stated; but when it is the premise toward the minor term that signifies possibility, all the syllogisms will be imperfect, and the negative ones among them will be of possibility not in the sense of the distinction stated, but of belonging to none, or not to all, by necessity; for if it belongs to none, or not to all, by necessity, we say that it is also possible for it to belong to none and not to all. For let A be possible of all B, and let B be assumed to belong actually to all C. Since, then, C falls under B, and A is possible of all B, it is clear that A is possible of all C as well. A perfect syllogism results. Similarly, if the A–B premise is negative and the B–C premise is affirmative, one taken as a premise of possibility and the other as assertoric, a perfect syllogism will result that A is possible of belonging to no C. It is clear, then, that when the assertoric premise is placed toward the minor term, perfect syllogisms result; but that syllogisms will result when the arrangement is the opposite must be shown by reduction to the impossible. At the same time it will also be clear that they are imperfect, since the proof does not proceed directly from the premises as taken. But first it must be said that if, when A is the case, B must be the case, then, when A is possible, B will also be possible by necessity.

43| For suppose things stand thus: let A stand for what is possible, and B for what is impossible. If, then, the possible, when it is possible for it to be, might come to be, while the impossible, when it is impossible, could not come to be, and if the possible A and the impossible B held at the same time, then it would be possible for A to come to be without B, and if it could come to be, it could also be; for what has come to be, is, at the time when it has come to be. We must take the impossible and the possible not only with respect to coming to be, but also with respect to being truly said and to actually holding, and in however many other senses the possible is spoken of; for the argument will hold alike in all of them. Further, when we say that if A is, B is, we must not take this to mean that B will be if some single thing, A, is the case; for nothing follows of necessity from a single thing's being the case, but only from at least two, namely when the premises stand as was stated in the case of the syllogism. For if the

44| C holds of D, and D holds of Z, then C holds of Z of necessity; and if each premise is possible, the conclusion too is possible. So if one were to set A as standing for the premises and B for the conclusion, it would follow not only that, when a necessary thing is the case, B is also necessarily the case, but also that when a possible thing is the case, B is possible. Now that this has been shown, it is clear that if a false but not impossible thing is assumed, then what results because of that assumption will likewise be false and not impossible. For example, if the assumption is false but not impossible, and B is the case whenever A is the case, then B too will be false but not impossible. For since it has been shown that if B is the case whenever A is the case, then B will also be possible whenever A is possible, and A is assumed to be possible, B too will be possible; for if it were impossible, the same thing would be at once possible and impossible. Now that these points have been settled, let A hold of every B, and let B be possible for every C. Then A must be possible for every C. For suppose it is not possible, and let B be posited as actually holding of every C;

45| this is false, but not impossible. So if A is not possible for every C, while B holds actually of every C, then it results that A does not hold of every B; for a syllogism comes about through the third figure. But it was assumed that A holds of every B. A must therefore be possible for every C; for when a false but not impossible thing is assumed, what results is impossible. We must take 'holding of every term' without defining it by reference to time — as 'now' or 'at this time' — but without qualification; for it is through premises of this sort that we construct our syllogisms, since if the premise is taken with reference to 'now' there will be no syllogism. For nothing, perhaps, prevents 'man' from holding at some time of everything that is moving,

46| if, say, nothing else were moving; and 'moving' is possible for every horse; but 'man' is possible for no horse. Again, let the first term be 'animal,' the middle 'moving,' and the last 'man.' The premises, then, will hold in the same way, but the conclusion is necessary, not possible; for man is of necessity an animal. It is clear, then, that the universal premise must be taken without qualification, and not by defining it with reference to time. Again, let A-B be a universal negative premise, and let A be taken as holding of no B, while B is possible for every C. Given these, A must be possible for no C. For suppose it is not possible, and let B be posited as actually holding of C, as before. Then A must hold of some B; for a syllogism comes about through the third figure; but this is impossible. So A would be possible for no C; for when a false thing is assumed, what results is impossible. This syllogism, then, is not of the possible in the defined sense, but of what holds of no term of necessity (for this is the contradictory of the hypothesis that was made; for it was assumed that A holds of some C of necessity, and the syllogism through the impossible proves the opposite assertion).

47| Further, it is also clear from the terms that the conclusion will not be possible. Let A be 'raven,' B 'thinking,' and C 'man.' A holds of no B; for nothing that thinks is a raven. And B is possible for every C; for thinking is possible for every man. But A holds of necessity of no C; the conclusion, then, is not possible. But neither is it always necessary. Let A be 'moving,' B 'knowledge,' and C 'man.' A will hold of no B, B is possible for every C, and the conclusion will not be necessary; for it is not necessary that no man moves, though it is not necessary that some man moves either. It is clear, then, that the conclusion is of the kind 'holds of no term of necessity.' The terms, however, must be chosen better. If the negative premise is posited with reference to the minor extreme, signifying possibility, then from the premises themselves as taken

48| there will be no syllogism, but if the premise concerning possibility is converted, there will be one, just as in the earlier cases. For let A hold of every B, and let B be possible for no C. With the terms standing thus, nothing will be necessary; but if B-C is converted and B is taken as possible for every C, a syllogism results, just as before; for the terms stand in a like position. And in the same way, when both intervals are negative — if A holds of no B, and B is possible for no C — from the premises themselves as taken nothing necessary results at all, but if the premise concerning possibility is converted, there will be a syllogism. For let A be taken as holding of no B, and let B be possible for no C. Through these, then, nothing is necessary; but if B is taken as possible for every C — which is also true — and the A-B premise stands as before, again the same syllogism will result. But if it is posited that B does not hold of every C, and that it is not possible for it not to hold, there will be no syllogism at all, whether the A-B premise is negative or affirmative.

49| Common terms: for necessary holding, 'white—animal—snow'; for not being possible, 'white—animal—pitch.' It is clear, then, that when the terms are universal, and one of the premises is taken as actual and the other as of possibility, then whenever the premise concerning the minor extreme is taken as possible, a syllogism always results — sometimes directly from the premises as taken, sometimes only when the premise is converted. When each of these occurs, and for what reason, we have stated. If one of the intervals is taken as universal and the other as particular: when the premise concerning the minor extreme is posited as universal and possible — whether negative or affirmative — and the particular one as affirmative and actual, there will be a perfect syllogism, just as when the terms are universal. The proof is the same as before. But when the premise concerning the major extreme is universal, actual and not of possibility, and the other is particular and of possibility, whether both are posited as negative or as affirmative, or

50| one negative and the other affirmative, in every case there will be an imperfect syllogism — except that some will be proved through the impossible, and others also through the conversion of the possibility-premise, just as in the earlier cases. There will be a syllogism through conversion whenever the universal premise, posited with reference to the major extreme, signifies actual holding, and the particular premise, being negative, is taken as concerning possibility — for instance, if A holds, or does not hold, of every B, and it is possible for B not to hold of some C; for when B-C is converted with respect to possibility, a syllogism results. But when the particular premise is taken as signifying actual not-holding, there will be no syllogism. Terms for holding: 'white—animal—snow'; for not holding: 'white—animal—pitch'; for the proof must be taken through the indefinite premise. But if the universal premise is posited with reference to the minor extreme, and the particular one with reference to the major — whether negative or affirmative, and whether either one is of possibility or actual — there will be no syllogism at all. Nor, when both premises are posited as particular or indefinite — whether taken as concerning possibility, or as actual, or with these interchanged — will there be a syllogism in this case either. The proof is the same as in the earlier cases. Common terms: for necessary holding, 'animal—white—man'; for not being possible, 'animal—white—cloak.' It is clear, then, that when the premise concerning the major extreme is posited as universal, a syllogism always results, but when it is the premise concerning the minor extreme, none ever results in any case.

51| When one of the premises signifies belonging by necessity and the other signifies possibility, the syllogism will hold in the same way as when the terms are related in the same manner, and it will be perfect when the necessary premise is placed at the minor extreme. The conclusion, if the terms are affirmative, will be of possibility and not of belonging, whether they are taken universally or not universally. But if one is affirmative and the other negative, then whenever the affirmative is the necessary one, the conclusion will be of possibility and not of not-belonging; but whenever the negative one is necessary, it will be both of the possibility of not belonging and of not belonging

52| outright, whether the terms are taken universally or not universally. And possibility in the conclusion must be taken in the same way as in the earlier cases. But there will be no syllogism concluding to not-belonging-by-necessity; for 'not belonging by necessity' — that is, not necessarily belonging — is one thing, and necessarily not belonging is another. That the conclusion does not come out necessary when the terms are affirmative is clear. For let A belong to all B by necessity, and let it be possible for B to belong to all C. Then there will be an imperfect syllogism that A may possibly belong to all C. That it is imperfect is clear from the proof, for it will be shown in the same way as in the earlier cases. Again, let it be possible for A to belong to all B, and let B belong to all C by necessity. Then there will be a syllogism that A may possibly belong to all C, but not that it belongs — and it will be perfect, not imperfect, for it is completed straightaway through the original premises. If the premises are not of the same quality, let the negative be the necessary one first, and let it be possible for A to belong to no B, and let it be possible for B to belong to all C.

53| Then A must belong to no C. For suppose it belongs either to all or to some C; but it was assumed that A is possible for no B. Since, then, the negative converts, B too is possible for no A; but A was supposed to belong to C either wholly or partly, so that B could possibly belong to no C, or not to all of it; but it was assumed from the start that B is possible for all C. It is clear too that a syllogism concluding to the possibility of not belonging comes about as well, if indeed one concluding to not belonging does. Again, let the affirmative premise be the necessary one, and let it be possible for A to belong to no B, and let B belong to all C by necessity. Then the syllogism will be perfect, but not concluding to not belonging — rather to the possibility of not belonging; for the premise taken from the major extreme was taken in that way, and it is not possible to lead the argument to the impossible. For if A were supposed to belong to some C, while it is also assumed that A is possible for

54| no B, nothing impossible results from these. But if the negative is placed at the minor extreme, then when it signifies possibility, there will be a syllogism by conversion, just as in the previous cases; but when it signifies non-possibility, there will not be. Nor when both premises are negative and the one at the minor is not one of possibility. The terms are the same: for belonging, white–animal–snow; for not belonging, white–animal–pitch. It will hold the same way for the particular syllogisms too. For whenever the negative premise is the necessary one, the conclusion too will be of not belonging. For example, if it is possible for A to belong to no B, and possible for B to belong to some C, then A must not belong to some C. For if A belongs to all C, while A is possible for no B, then B too is possible for no A; so that if A belongs to all C, B is possible for no C; but it was assumed that B is possible for some C. But whenever the particular affirmative is the necessary premise — the one in the negative syllogism, such as BC — or the universal one in the affirmative syllogism, such as AB, there will be no syllogism concluding to belonging.

55| The proof is the same as in the earlier cases. But if the universal premise, whether affirmative or negative, is placed at the minor extreme as one of possibility, and the particular is necessary [at the minor extreme], there will be no syllogism (the terms for necessary belonging are animal–white–man, and for impossibility of belonging, animal–white–garment). But when the universal is the necessary premise and the particular is one of possibility: if the universal is negative, the terms for belonging are animal–white–raven, and for not belonging, animal–white–pitch; if it is affirmative, the terms for belonging are animal–white–swan, and for impossibility, animal–white–snow. Nor again, when the premises are taken indefinite, or both particular, will there be a syllogism in this way either. The common terms are: for belonging, animal–white–man; for not belonging, animal–white–inanimate. For animal belongs to some white thing, and white to some inanimate thing, both necessarily and — in another case — impossibly; and likewise for possibility, so that these terms are useful for every case. It is clear, then, from what has been said, that with the terms related in the same way, a syllogism does or does not come about alike in the case of belonging and in the case of necessity — except that when the negative premise is taken as one of belonging, the syllogism was of possibility, whereas when it is taken as one of necessity, the syllogism was both of possibility and of not belonging.

56| In the second figure, when both premises are taken as premises of possibility, there will be no syllogism at all, whether they are taken affirmative or negative, universal or particular. But when one premise signifies belonging and the other possibility, then, if the affirmative signifies belonging, there will never be a syllogism, but if the universal negative signifies belonging, there always will be. It holds the same way when one of the premises is taken as necessary and the other as possible. And in these cases too the possibility in the conclusions must be understood as in the previous cases. First, then, it must be shown that the negative premise of possibility does not convert — for instance, if A is possible for no B, it is not necessary that B too be possible for no A. For suppose it does, and let B be possible for no A. Now since, in the case of possibility, affirmations convert with negations — both the contrary ones and the opposite ones — and B is possible for no A, it is clear that B would also be possible for all A. But this is false; for it does not follow that, because this belongs possibly to all of that, this belongs to that necessarily. So the negative does not convert. Further, nothing prevents A from being possible for no B, while B necessarily does not belong to some A — for instance, it is possible for white to belong to no man (since it is also possible for it to belong), but it is not true to say that man is possible for no white thing; for to many white things man necessarily does not belong, and the necessary

57| was not the same as the possible. But neither will it be shown to convert by way of the impossible — suppose someone were to claim: since it is false that B is possible for no A, it is true that it is not possible for none (for the two are affirmation and negation of each other), and if this is true, it is true that B necessarily belongs to some A; so A too would belong to some B; but this is impossible. For it does not follow that, if it is not possible for B to belong to no A, B necessarily belongs to some A. For 'it is not possible to belong to none' is said in two ways: one, if it necessarily belongs to some; the other, if it necessarily does not belong to some. For of what necessarily does not belong to some A, it is not true to say that it is possible for it not to belong to all, just as, of what necessarily belongs to some, it is not true to say that it is possible for it to belong to all. If, then, someone were to claim that, since it is not possible for C to belong to all D, C necessarily does not belong to some D, he would be assuming something false; for it does belong to all D, but because it belongs to some of them by necessity, we say for this reason that it is not possible for it to belong to all. So both 'necessarily belonging to some' and 'necessarily not belonging to some' are opposed to 'possibly belonging to all.'

58| The same holds for the possible-of-belonging-to-none premise as well. It is clear, then, that with respect to the possible and the not-possible as we distinguished at the start, what we must take is not that something necessarily belongs to some subject, but that something necessarily does not belong to some subject. When this is taken, nothing impossible results, so no syllogism is produced. It is evident, then, from what has been said, that the privative premise does not convert. Now that this has been shown, and as possibly belonging to all C. Then no syllogism will result through conversion, since it has been stated that a premise of this kind does not convert. Nor will one result through the impossible: for if we posit that B is ⟨not⟩ possible ⟨not⟩ to belong to all C, nothing false results, since it is possible for the same thing to belong both to all C and to none of it. And in general, if there is a syllogism, it is clear that it would concern the possible, since neither premise has been taken as an assertion of belonging,

59| and such a conclusion would be either affirmative or privative; but neither is possible. For if the affirmative is posited, it will be shown by means of terms that belonging is not possible; and if the privative, that the conclusion is not possible but necessary. Let A be white, B man, and C horse. Then A, white, can belong to all of one and to none of the other. But B cannot possibly either belong or not belong to C. That it cannot belong is clear, since no horse is a man. But neither is it possible for it not to belong: for it is necessary that no horse be a man, and the necessary was not the same as the possible. No syllogism results, then. The same will be shown if the privative premise is posited the other way round, and if both premises are taken as affirmative or both as privative — for the demonstration will proceed through the same terms — and likewise when one premise is universal and the other particular, or both particular, or indefinite, or however else the premises may be varied: the demonstration will always proceed through the same terms. It is evident, then, that when both premises are posited as possible, no syllogism results at all.

60| If one premise signifies belonging and the other possibility, then when the affirmative is posited as one of belonging and the privative as one of possibility, no syllogism will ever result, whether the terms are taken universally or in part (the demonstration is the same and proceeds through the same terms). But when the affirmative is one of possibility and the privative one of belonging, there will be a syllogism. For let A be taken as belonging to no B, and as possibly belonging to all C. Then, the privative premise being converted, B will belong to no A; and A was possible for all C; so a syllogism results, through the first figure, that B is possible for no C. The same holds if the privative is posited with respect to C instead. But if both premises are privative, one signifying not-belonging and the other possibility, nothing necessary results from the premises as taken; but when the premise concerning possibility is converted, a syllogism results that B is possible for no C, just as in the cases before —

61| for again it will be the first figure. But if both premises are posited as affirmative, no syllogism results. The terms for belonging are health—animal—man, and for not-belonging, health—horse—man. The same will hold for the particular syllogisms as well. For when the affirmative is one of belonging, whether taken universally or in part, no syllogism results (this too is shown in the same way and through the same terms as before); but when it is the privative that is one of belonging, there will be a syllogism through conversion, just as before. Again, if both intervals are taken as privative, with the not-belonging one universal, nothing necessary results from the premises themselves; but when the possibility-premise is converted, a syllogism results, just as before. But if the privative is one of belonging and is taken in part, no syllogism results, whether the other premise is affirmative or privative. Nor does one result when both premises are taken indefinite — whether affirmative or negative — or particular. The demonstration is the same and proceeds through the same terms.

62| If one of the premises signifies necessity and the other possibility, then when the privative is the necessary one, there will be a syllogism, concluding not only that the term is possible not to belong but that it does not belong; but when the affirmative is the necessary one, there will be no syllogism. For let it be posited that A necessarily belongs to no B, and possibly belongs to all C. Then, the privative premise being converted, B will belong to no A; and A was possible for all C; so again, through the first figure, a syllogism results that B is possible for no C. And at the same time it is clear that B will also belong to no C. For suppose it does belong: then, if A is possible for no B, and B belongs to some C, A is not possible for some C; but it was assumed possible for all C. The same will be shown if the privative is posited with respect to C instead. Again, let the affirmative premise be the necessary one and the other possible, and let A be possible

63| for no B, and let it belong to all C of necessity. With the terms so arranged, no syllogism results at all. For it would follow that B necessarily does not belong to C. Let A be white, B man, and C swan. White necessarily belongs to swan, and possibly belongs to no man, and man necessarily belongs to no swan. That there is no syllogism concluding possibility is clear, since the necessary was not the same as the possible. But neither is there one concluding necessity, since the necessary conclusion resulted either from both premises being necessary or from the privative one being necessary. Moreover, with these terms posited it is also possible for B to belong to C: nothing prevents C from falling under B, while A is possible for all B and necessarily belongs to C — for instance, if C were the waking, B animal, and A, the term posited, motion. For motion necessarily belongs to the waking, and is possible for every animal; and everything waking is an animal. It is clear, then, that neither is there a syllogism concluding not-belonging, given that, with the terms so arranged, belonging is necessary.

64| Nor, then, is there one concluding the opposite affirmations, so that no syllogism results at all. The same will be shown if the affirmative is posited the other way round. But if the premises are alike in form, then when both are privative, a syllogism always results, by converting the premise concerning possibility, just as before. For let A be taken as necessarily not belonging to B, and as possibly not belonging to C. Then, the premises being converted, B belongs to no A, and A is possible for all C; so the first figure results. And likewise if the privative is posited with respect to C instead. But if both premises are posited as affirmative, no syllogism results. That there is none concluding not-belonging or necessarily-not-belonging is clear, since no privative premise has been taken, either as one of belonging or of necessary belonging. But neither is there one concluding possible not-belonging: for with the terms so arranged, B will necessarily not belong to C.

65| Nor is there a syllogism concluding the opposite affirmatives either, since it has been shown that B does not belong to C of necessity. So no syllogism results at all. The same will hold for the particular syllogisms too: for when the privative premise is universal and necessary, there will always be a syllogism both of possibility and of not belonging (the demonstration is by conversion), but when the affirmative premise is universal and necessary, never — for it will be shown in the same way as with the universal syllogisms, and through the same terms. Nor when both premises are taken as affirmative: for the demonstration of this case is the same as before. But when both are privative, and the one signifying not belonging is universal and necessary, the necessary conclusion will not follow from the premises as taken, but when the premise concerning possibility is converted there will be a syllogism, just as in the previous cases. But if both are taken as indefinite or particular, there will be no syllogism. The demonstration is the same, and through the same terms. It is clear, then, from what has been said, that when the universal privative premise is posited as necessary, a syllogism always results — not only concluding possible not-belonging but also actual not-belonging — but when the affirmative premise is necessary, never. And that a syllogism does or does not result in the same way whether the premises are of necessity or of belonging. And it is clear too that all these syllogisms are imperfect, and that they are completed by means of the figures already discussed.

66| In the last figure, a syllogism will result both when both premises signify possibility and when only one does. Now when the premises signify possibility, the conclusion too will be possible; and likewise when one signifies possibility and the other belonging. But when the other is posited as necessary: if it is affirmative, the conclusion will be neither necessary nor actual; but if it is privative, there will be a syllogism concluding not-belonging, just as in the previous cases — and here too the possible in the conclusions must be understood in the same way.

67| Let the premises, then, first be possible, and let both A and B be possible of belonging to every C. Since, then, the particular affirmative converts, and B is possible of every C, C will also be possible of some B. So if A is possible of every C, and C is possible of some B, then A must also be possible of some B — for the first figure results. And if A is possible of belonging to no C, while B is possible of every C, then A must be possible of not belonging to some B — for again the first figure results, by conversion. But if both premises are posited as privative, no necessity follows from the premises as taken, but when the premises are converted a syllogism will result, just as in the previous cases. For if A and B are possible of not belonging to C, then, when 'possible of not belonging' is exchanged for 'possible of belonging,' the first figure will again result, by conversion. But if one of the terms is universal and the other particular, then, the terms standing in the same relation as in the case of belonging, there will be a syllogism, and there will not be one.

68| For let A be possible of belonging to every C, and B possible of belonging to some C. Here too the first figure will result once the particular premise is converted: for if A belongs to every C possibly, and C is possible of some B, A will be possible of some B. And likewise if the universal premise is posited with B-C. Similarly too if the A-C premise is privative and the B-C premise affirmative — for again the first figure will result, by conversion. But if both are posited as privative, one universal and the other particular, no syllogism will result from the premises as taken, but there will be one once they are converted, just as in the previous cases. But when both are taken as indefinite or particular, there will be no syllogism — for it is necessary that the same thing belong to every B and to none. Terms for belonging: animal — man — white; for not belonging: horse — man — white, with white as the middle term. If one of the premises signifies belonging and the other signifies possibility, the conclusion will be that something is possible,

69| and not that it belongs, and there will be a syllogism when the terms stand in the same relation as before. For let them first be affirmative, and let A belong to every C, and let B be possible of belonging to every C. Then, once B-C is converted, the first figure will result, and the conclusion will be that A is possible of belonging to some B — for whenever, in the first figure, one of the premises signified possibility in this way, the conclusion too was possible. Similarly too if C belongs and A-C signifies possibility; and if A is privative and B-C affirmative — whichever one belongs — in both cases the conclusion will be possible: for again the first figure results, and it has been shown that when one of its premises signifies possibility, the conclusion too will be possible. But if the privative premise is posited with the minor term, or if both are taken as privative, no syllogism will result from the premises as they stand, but there will be one once they are converted, just as in the previous cases. And if one of the premises is universal and the other particular — both being affirmative, or the universal one privative and the particular one affirmative — the manner of the syllogisms will be the same,

70| for they are all completed through the first figure. So it is clear that the syllogism will conclude possibility and not belonging. But if the affirmative premise is universal and the privative one particular, the demonstration will be by way of the impossible. For let B belong to every C, and let A be possible of not belonging to some C; then A must be possible of not belonging to some B. For if A belongs to every B of necessity, and B is posited as belonging to every C, then A will belong to every C of necessity — for this has been shown before. But it was assumed that A is possible of not belonging to some C. When both premises are taken as indefinite or particular, there will be no syllogism. The demonstration is the same as before, and through the same terms.

71| If one of the premises is necessary and the other possible, then, when the terms are affirmative, there will always be a syllogism concluding possibility; but when one is affirmative and the other privative, if the affirmative one is necessary, the conclusion will be that something is possible not to belong, while if the privative one is necessary, the conclusion will be both that something is possible not to belong and that it does not belong. But there will be no syllogism concluding necessary not-belonging, just as in the other figures. Let the terms, then, first be affirmative, and let A belong to every C of necessity, and let B be possible of belonging to every C. Since, then, A belongs to every C of necessity, and C is possible of some B, A too will be possible, and not actual, of belonging to some B — for that is how it turned out in the first figure. Likewise it will be shown too if B-C is posited as necessary and A-C as possible. Again, let one be affirmative and the other privative, the affirmative one being necessary: let A be possible of belonging to no C, and let B belong to every C of necessity. Then again the first figure results —

72| For the negative premise too signifies contingency; so it is clear that the conclusion will be contingent. For whenever the premises stood this way in the first figure, the conclusion was also contingent. But if the negative premise is necessary, the conclusion will be both that it is contingent for something not to belong, and that it does not belong. For let A be posited as not belonging to C of necessity, and B as belonging to every C contingently. Then when the affirmative premise BC is converted, it will be the first figure, and the negative premise will be necessary. And when the premises stood this way, it resulted that A both is contingently not belonging to some C and does not belong to it, so that A must also not belong to some B of necessity. And when the negative is posited with the minor extreme, if it is contingent, there will be a syllogism once the premise is converted, just as in the earlier cases; but if it is necessary, there will not be one — for it is possible for it to belong to all, and also possible for it to belong to none. Terms: for belonging to all — sleep, sleeping horse, man; for belonging to none — sleep, waking horse, man.

73| The situation will be similar too if one of the terms is universal and the other particular in relation to the middle. For if both are affirmative, there will be a syllogism of contingency and not of belonging; and likewise when one is taken as negative and the other as affirmative, with the affirmative necessary. But when the negative is necessary, the conclusion too will be one of not belonging — for the manner of proof will be the same whether the terms are universal or not universal. For the syllogisms must be completed through the first figure, so that, just as in those cases, the same result must follow in these as well. And when the negative, taken as universal, is posited with the minor extreme, if it is contingent there will be a syllogism by conversion, but if it is necessary there will not be. This will be shown in the same way as in the universal cases, and through the same terms. It is clear, then, in this figure too, when and how there will be a syllogism, and when it will be one of contingency and when one of belonging. And it is clear also that all these are incomplete, and that they are completed through the first figure.

74| That the syllogisms in these figures are completed by means of the universal syllogisms in the first figure, and are reduced to them, is clear from what has been said. But that every syllogism whatsoever stands this way will now become clear, once it is shown that every syllogism comes about through one of these figures. Every demonstration and every syllogism must show either that something belongs or that it does not belong, and this either universally or in part, and further either ostensively or from a hypothesis. And the argument through the impossible is a part of that from a hypothesis. Let us then speak first about the ostensive syllogisms, for once these have been shown, it will be clear also for those leading to the impossible and, in general, for those from a hypothesis. If, then, one needed to conclude by syllogism that A belongs to B, or

75| does not belong to it, one must take something of something. Now if A is taken directly of B, what was to be proved from the start will have been taken as a premise. But if A is taken of C, and C of nothing, and nothing else of C, and C of nothing else, there will be no syllogism — for from taking one thing of one thing nothing follows of necessity. So a further premise must also be assumed. If, then, something else is taken of C, or something else is taken of A, or something else of C, nothing prevents there being a syllogism, but it will not concern B through the premises taken. Nor, when C belongs to something else, and that to something else again, and this to yet another, if the chain does not connect to B — not even then will there be a syllogism concerning B. For we said in general that there will never be a syllogism of one thing about another unless some middle term is taken which stands in some predicative relation to each of the two. For a syllogism without qualification is made out of premises; a syllogism concerning this thing is made out of premises concerning this thing; and a syllogism of this thing in relation to that thing is made through premises of this thing in relation to that thing. But it is impossible to take a premise concerning B while neither predicating nor denying anything of it, or again, in relation to A and B, to take nothing common to both but predicate or deny of each certain things peculiar to it alone.

76| So one must take some middle term common to both, which will connect the predications, if indeed there is to be a syllogism of this thing in relation to that thing. If, then, it is necessary to take something common to both, and this can happen in three ways — either by predicating A of C and C of B, or C of both, or both of C — and these are the figures already described, it is clear that every syllogism must come about through one of these figures. The same account holds even if the connection to B is made through more terms, for the figure will be the same even in the case of several. That the ostensive syllogisms are concluded through the figures already described is clear; and that those leading to the impossible are as well will become clear from the following. For all who conclude through the impossible infer the falsehood by syllogism, and prove the original point from a hypothesis, whenever something impossible results once the contradictory has been posited — for example that the diagonal is incommensurable, because it follows that

77| the odd numbers become equal to the even, once it is posited as commensurable. Now it is concluded by syllogism that the odd numbers become equal to the even, but that the diagonal is incommensurable is proved from the hypothesis, since a falsehood results because of the contradictory. For this is what it was to reason through the impossible: to show something impossible by means of the original hypothesis. So then, since in arguments led to the impossible there comes about an ostensive syllogism of the falsehood, while the original point is proved from a hypothesis, and we said earlier that the ostensive syllogisms are concluded through these figures, it is clear that the syllogisms through the impossible will also be through these figures. And likewise all the other syllogisms from a hypothesis — for in all of them the syllogism comes about with respect to the point substituted for it, while the original point is concluded through an agreement or some other hypothesis. And if this is true, then every demonstration and every syllogism must come about through the three figures already described. And once this has been shown, it is clear that every syllogism is completed through the first figure and is reduced to the universal syllogisms within it.

78| Further, in every syllogism some of the terms must be affirmative, and the universal must be present. For without the universal there will either be no syllogism, or it will not bear on the point at issue, or it will beg the original question. For let it be posited that musical pleasure is excellent. Now if one should claim that pleasure is excellent without adding 'every,' there will be no syllogism; and if one claims that some pleasure is excellent, then if it is a different pleasure, it has nothing to do with the point at issue, and if it is this very one, it takes as a premise the very thing to be proved from the start. This becomes clearer in geometrical diagrams — for example, the proof that in an isosceles triangle the angles at the base are equal. Let lines AB be drawn to the center. If, then, one takes angle AC as equal to angle BD without claiming outright that the angles of the semicircles are equal, and again takes C equal to D without adding the whole angle of the segment, and further claims that, since the whole angles are equal and equal amounts have been subtracted from them, the remaining angles E and F are equal, one will be begging the original question,

79| unless one takes it as agreed that if equals are subtracted from equals, equals remain. It is clear, then, that the universal premise must hold in every case, and that a universal conclusion is proved from terms that are all universal, while a particular conclusion is proved either way — so that if the conclusion is universal, the terms must of necessity be universal, but if the terms are universal, it is possible for the conclusion not to be universal. It is clear too that in every syllogism either both premises or one of them must become similar to the conclusion. I mean not only in being affirmative or negative, but also in being necessary, assertoric, or possible. And one must examine the other categories as well. It is also clear, in general, when there will be a syllogism and when there will not, and when it is possible and when it is complete, and that if there is a syllogism, its terms must stand in one of the ways stated.

80| It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion comes about through different terms in addition — for instance one conclusion reached both through B and through C, D, or through A, B and through A, C, D; for nothing prevents there being several middle terms for the same terms. But if this is so, the syllogisms are not one but several. Or again, when each of A, B is itself taken through a syllogism (for instance A through D, E, and again B through H), or one by induction and the other by syllogism. But even so the syllogisms are several; for there are several conclusions, namely A and B and C. If, then, there is not several but one, in this way it is possible for the same conclusion to come about through more terms, but not in the way that it comes about through A, B — that is impossible. For let E have been concluded from A, B, C, D. Then necessarily some one of them has been taken in relation to another, one as whole, the other as part; for this has been shown earlier, that if there is a syllogism, some of the terms must be related in this way. Let it, then, be related in this way to B. There is, then, some conclusion from them. This conclusion is either E, or one of C, D, or something else besides these. And if it is E, the syllogism would be from A, B alone. But as for C, D, if they stand in such a way that

81| one is as whole and the other as part, there will be some conclusion from them too, and it will be either E, or one of A, B, or something else besides these. And if it is E or one of A, B, then either there will be more syllogisms, or it turns out — as was possible — that the same thing is concluded through more terms; but if it is something else besides these, there will be more syllogisms, and unconnected with one another. But if C does not stand to D in such a way as to produce a syllogism, they will have been taken in vain, unless for the sake of induction or concealment or some other such purpose. But if from A, B there comes about, not E, but some other conclusion, and from C, D either one of these or something else besides these, then the syllogisms become more numerous, and are not of the thing proposed; for it was proposed that the syllogism be of E. And if no conclusion at all comes about from C, D, then it turns out both that they have been taken in vain and that the syllogism is not of the original point. So it is clear that every demonstration and every syllogism will proceed through three terms only.

82| This being clear, it is evident too that it proceeds from two premises and no more (for three terms make two premises), unless something further is taken in addition, as was said at the outset, for the completion of the syllogisms. It is clear, then, that in whatever argument the premises through which the governing conclusion comes about are not even in number (for some of the conclusions higher up must of necessity be premises), that argument either has not been syllogized, or has asked more questions than were necessary for its thesis. Now if syllogisms are taken with respect to their governing premises, every syllogism will consist of an even number of premises and an odd number of terms — for the terms exceed the premises by one; and the conclusions will be half the premises. But when the conclusion is reached through prosyllogisms, or through several continuous middle terms — as when B is concluded through C, D — the number of terms will likewise exceed the premises by one (for the interpolated term is placed either outside or in the middle; and either way it turns out that the intervals between terms are one fewer), while the premises are equal to the intervals; not, however,

83| will it always be that the premises are even and the terms odd, but the two alternate — when the premises are even, the terms are odd, and when the terms are even, the premises are odd; for along with a term one premise is added, whichever end the term is added at, so that since one set was even and the other odd, the two must necessarily alternate as the same addition keeps being made. But the conclusions will no longer keep the same ratio either to the terms or to the premises; for when one term is added, conclusions will be added one fewer than the terms already present — for it produces no conclusion with the last term only, but does with all the others; for instance, if D is added to B, C, two conclusions are immediately added along with it, one in relation to A and one in relation to B. Similarly with the rest. And the same holds if the term is interpolated into the middle; for it will fail to produce a syllogism with only one term. So the conclusions will be far more numerous than both the terms and the premises.

84| Since we have in hand what the syllogisms are concerned with, and how, in each figure, and in how many ways, a conclusion is shown, it is clear to us also which problem is hard and which is easy to attempt; for what is concluded in more figures and through more moods is easier, while what is concluded in fewer figures and through fewer moods is harder to attempt. Now the universal affirmative is shown through the first figure only, and through this in only one way; the negative is shown through both the first and the middle figure, through the first in only one way, and through the middle in two ways; the particular affirmative is shown through the first figure and through the last, in one way through the first, and in three ways through the last. The particular negative is shown in all the figures, but in the first in only one way, while in the middle and the last it is shown in two ways and in three ways respectively. It is clear, then, that the universal affirmative is the hardest to establish but the easiest to refute. And in general, for one refuting, universals are easier than particulars; for whether it fails to hold of any or fails to hold of some, the thesis is refuted; and of these, that it fails to hold of some is shown in all the figures, while that it holds of none is shown in two.

85| And the same holds for the negatives; for whether the thing holds of all or holds of some, the original thesis is refuted; and this was shown in two figures. But among the particular propositions, refutation is possible in only one way, by showing that the thing holds of all, or of none. For one establishing, the particulars are easier; for they are shown in more figures and through more ways. And in general one must not fail to notice that it is possible to refute the one kind through the other — universals through particulars and these through universals — but it is not possible to establish universals through particulars, though it is possible to establish particulars through universals. At the same time it is clear also that refuting is easier than establishing. How, then, every syllogism comes about, and through how many terms and premises, and how these stand in relation to one another, and further which problem is shown in each figure, and which in more figures and which in fewer, is clear from what has been said.

86| We must now say how we ourselves will always be well supplied with syllogisms for whatever is proposed, and by what route we will grasp the principles concerning each subject. For presumably it is not enough to study merely how syllogisms come to be — one must also have the ability to produce them. Of all things that are, some are of such a kind that they are never truly predicated universally of anything else (for instance Cleon and Callias, and whatever is a particular and perceptible thing), while other things are predicated of these (for each of these is both a man and an animal); other things are themselves predicated of others, but nothing prior is predicated of them; and other things both are predicated of others and have others predicated of them, for instance man is predicated of Callias, and animal of man. That some things are naturally said of nothing else is clear: nearly every perceptible thing is such that it is predicated of nothing, except accidentally — for we sometimes say that white thing is Socrates, or that the one approaching is Callias. That the ascent also comes to a stop at some point, going upward, we will discuss again; for now let this be taken as established. Of these lowest things, then, it is not possible to demonstrate some other term predicated of them, except by opinion, but they are predicated of other things; nor are particulars predicated of other things, but other things are predicated of them. The intermediate things, clearly, admit of both — for they too will be said of

87| others, and others will be said of them — and it is chiefly about these that arguments and inquiries are concerned. One must grasp the premises concerning each subject in this way: first positing the subject itself, along with its definitions and whatever is peculiar to the thing, then after that whatever follows the thing, and again whatever the thing follows, and whatever cannot belong to it. Those things which cannot possibly belong to it need not be taken separately, because the privative converts. And among the things that follow, one must distinguish those predicated in the what-it-is, those predicated as peculiar attributes, and those predicated as accidents, and among these, which hold by opinion and which in truth; for the more of these one has ready to hand, the sooner one will arrive at a conclusion, and the more true ones one has, the more one will demonstrate. One should select not what follows some particular case but what follows the thing as a whole — not, say, what follows some particular man, but what follows every man; for the syllogism proceeds through universal premises. When the premise is indefinite, it is unclear whether it is universal, but when it is determinate, this is evident. Likewise one must select, among the things the subject follows, those it follows as wholes, for the reason already stated. But what follows must not be taken to follow as a whole — I mean, for instance, that every animal follows man, or that every science follows music — but only that it follows without qualification, in the way we in fact propose it,

88| for the other way is both useless and impossible — for instance, that every man is every animal, or that justice is every good thing. Rather, 'to every' is said of that to which the thing follows. Whenever the subject for which one must take the consequents is contained under something more universal, one should not select, in its case, what follows or does not follow the universal (for that has already been taken in the case of the universal — whatever follows animal also follows man, and likewise for what does not belong to it); rather, one should take what is peculiar to each subject in particular. For there are certain things peculiar to a species beyond its genus, since the several species must have certain peculiar attributes of their own. Nor, again, should one select for the universal the things which the contained species follows — for instance, for animal, the things man follows; for if animal follows man, it must follow all those things too, but these belong more properly to the selection concerning man. One must also take what follows for the most part, and what it follows for the most part; for the syllogism concerning problems that hold for the most part is likewise drawn from premises that hold for the most part, either all of them or some — since the conclusion of any syllogism resembles its principles. Further, one should not select what follows everything, since no syllogism will come from such premises; the reason for this will become clear in what follows.

89| Those who wish to establish something of a whole subject must look, in the case of what is to be established, to the subjects of which it happens to be said, and, in the case of what it must be predicated of, to what follows that subject; for if any of these coincide, the one term must belong to the other. But if one wants to show not that it belongs to every case but to some, one must look to what each of the terms follows; for if any of these coincide, it must belong to some. When it must belong to none, then for the subject to which it must not belong, one must look to what follows it, and for the term that must not belong, to what cannot be present in it; or conversely, for the subject to which it must not belong, to what cannot be present in it, and for the term that must not belong, to what follows it. For if any of these coincide, whichever way, it cannot belong at all, the one to the other; for sometimes the resulting syllogism is in the first figure, sometimes in the middle figure. But if it must not belong to some, then for the subject to which it must not belong, one must look to what it follows, and for the term that must not belong, to what cannot belong to it; for if any of these coincide, it must fail to belong to some. Perhaps each of these points will be clearer put this way. Let the consequents of A be represented by B, the things A follows by C, and the things that cannot belong to A by D. Again, let the things belonging to E be represented by F, the things E follows by G, and the things that cannot belong to E by H. If, then, something among the C's is the same as something among the F's, A must belong to every E; for F belongs to every E, and A belongs to every C, so A belongs to every E. But if C and G are the same, A must belong to some E; for

90| A belongs to C, and E follows G in every case. But if F and D are the same, A will belong to no E, by a prosyllogism; for since the privative converts, and F is the same as D, A will belong to no F, and F belongs to every E. Again, if B and H are the same, A will belong to no E; for B belongs to every A, but will belong to none of E — since it was the same as H, and H belonged to none of the E's. But if D and G are the same, A will not belong to some E; for A will not belong to G, since it does not belong to D either, and G falls under E, so A will not belong to some of the E's. But if B is the same as G, the syllogism will be converted: E will belong to every A — for B belongs to A, and E belongs to B, since it was the same as G — but A need not belong to every E, only to some, because the universal predication converts to the particular.

91| It is clear, then, that for each problem one must look to the aforementioned classes belonging to each term; for it is through these that all the syllogisms are produced. And among the consequents, and among the things each term follows, one must look especially to the first and most universal ones — for instance, for E, to KF rather than to F alone, and for A, to KC rather than to C alone. For if A belongs to F, it belongs to both F and E; but if it does not follow this, it may still follow F. Likewise one must examine the things it follows: if it follows the first and most universal terms, it also follows the things under them; but if it does not follow these, it may still follow the things under them. It is also clear that the inquiry proceeds through three terms and two premises, and that all the syllogisms come about through the figures already discussed. For it is shown that A belongs to every E when something among the C's and F's is taken as the same; this will be the middle term, and A and E the extremes, so the first figure results. And it is shown to belong to some E when C and G are taken as the same.

92| This is the last figure; for a middle term is what produces the result. The conclusion holds of none, when Δ and Ζ are the same term. And the same is true of the first figure and the middle figure as well: of the first, because A belongs to none of Ζ (if the privative premise converts), and Ζ belongs to all of Ε; of the middle, because Δ belongs to no A and to all Ε.

93| The conclusion holds of some but not of all, when Δ and Η are the same. This too is the last figure: for Δ will belong to none of Η, and Ε will belong to all of Η. It is clear, then, that all syllogisms come about through the figures already discussed, and that we should not select everything that follows universally, since no syllogism results from such things. For it was never possible to establish a conclusion outright from things that follow a term, and it is not possible to disprove anything by means of what follows both terms universally either; for the middle term must belong to the one and not belong to the other. It is also clear that the other lines of inquiry based on such selections are useless for producing a syllogism — for instance, whether the things that follow each of the two terms are the same, or whether the things A follows, and the things that cannot belong to Ε, are the same, or again whatever cannot belong to either — for no syllogism comes about through any of these. For if the things that follow are the same, say Β and Ζ, the result is the middle figure, with both premises affirmative; but if the things A follows, and the things that cannot belong to Ε, are the same — say Γ and Θ — the result is the first figure, with a negative premise attached to the minor term.

94| But if the things that cannot belong to either term are the same — say Δ and Θ — both premises are negative, whether in the first figure or in the middle figure, and in that case there is never a syllogism at all. It is also plain that in this kind of inquiry we must take terms that are the same, not terms that are different or contrary, first because the whole point of the inspection is to find the middle term, and the middle term must be taken as the same, not as different. Further, in all the cases where a syllogism does happen to result from taking contraries, or things that cannot belong to the same thing, it will reduce to the ways already stated — for instance, if Β and Ζ are contraries, or cannot belong to the same thing. Once these are taken, there will indeed be a syllogism that A belongs to none of the term in question, but not from these premises themselves — rather through the method already described; for Β will belong to all of A and to none of Ε, so that Β must be the same as some Θ. It is clear, then, that no syllogism comes about from these inspections by themselves, but it is necessary — if Β and Ζ are contraries — that Β be the same as some of the Θs, and that the syllogism come about through that. So it turns out that those who inquire in this way end up looking, in addition, for another route to the necessary conclusion, simply because the identity of Β and the Θs escapes their notice.

95| Syllogisms that lead to the impossible stand in the same relation to demonstrative syllogisms as these others do; for they too come about through the things that follow a term and the things a term follows. And the inspection is the same in both cases: whatever is proved demonstratively can also be proved through the impossible, using the same terms, and whatever is proved through the impossible can also be proved demonstratively — for example, that A belongs to none of Ε. Suppose it is laid down that A belongs to some of Ε; then, since Β belongs to all A, and A belongs to some Ε, Β will belong to some of Ε; but it was assumed to belong to none. Again, take the case that A belongs to some Ε: if A belonged to none of Ε, and Ε belonged to all Η, then A would belong to none of Η; but it belonged to all of it. The same holds in the other problems as well; for the proof through the impossible will always and in every case proceed from the things that follow a term and the things each term follows. And for each problem the inquiry is the same whether one wants to draw the syllogism demonstratively or to lead it to the impossible; for both demonstrations proceed from the same terms — for example, if it has been shown that A belongs to none of Ε by the fact that it follows that Β also belongs to some Ε, which is impossible; if it is taken

96| that Β belongs to none of Ε but to all of the other term, it is clear that A will belong to none of Ε. Again, if it has been proved demonstratively that A belongs to none of Ε, then on the supposition that it belongs to some, it will be shown through the impossible that it belongs to none. Likewise in the other cases too: in every one of them it is necessary to take some term in common, other than the ones laid down, in relation to which the syllogism of the falsehood will be constructed, so that once this premise is converted, and the other premise kept the same, the syllogism becomes demonstrative and proceeds through the same terms. For the demonstrative syllogism differs from the one leading to the impossible in that in the demonstrative both premises are laid down as true, while in the one leading to the impossible one of them is laid down as false. These points will become clearer through what follows, when we discuss the impossible; for now let this much be plain to us — that one must look to the same things whether one wants to reason demonstratively or to lead the argument to the impossible. As for the other syllogisms from a hypothesis — those, for instance, that proceed by substitution or by quality — the inquiry will concern the terms laid down, not the ones from which we started but the ones substituted for them, though the manner of inspection is the same.

97| We must also examine and distinguish how many kinds of syllogism from a hypothesis there are. Each of the problems, then, is proved in this way; but some of them can also be syllogized in another way — for instance, universal problems by means of a particular inspection carried out from a hypothesis. For if Γ and Δ were the same, and it were assumed that Ε belongs only to the Ηs, then Α would belong to all of Ε; and again, if Δ and Η were the same, and Α were predicated only of the Ηs, it would follow that A belongs to none of Ε. It is clear, then, that this too is a way one must inspect. The same method holds for necessary and possible premises as well; for the inquiry is the same, and the syllogism will proceed through the same terms whether it concerns the possible or the actual, in corresponding order. In the case of the possible, one must also take the things that do not actually belong but are capable of belonging; for it has been shown that the syllogism of possibility comes about through these as well. The same holds likewise for the other categories. It is clear, then, from what has been said, not only that it is possible for all syllogisms to come about by this route, but also that it is impossible for them to come about by any other. For every syllogism has been shown to come about through one of the figures already discussed, and these cannot be constructed by any means other than the things that follow a term and the things each term follows; for it is from these that the premises and the choice of the middle term come, so that a syllogism cannot come about by any other means either.

98| The method, then, is the same in every case — in philosophy, and in any craft or branch of learning whatever. One must survey the things that belong to each of the two terms and the things each belongs to, be well supplied with as many of these as possible, and examine them by means of the three terms: refuting in one way, establishing in another — establishing what actually belongs, when truth is at stake, from what has been set out according to truth, but for dialectical syllogisms from premises based on reputable opinion. The starting points of syllogisms have been stated in general terms: how they stand, and how one must go hunting for them, so that we do not look to everything that is said, and do not, when establishing and refuting, look to the same things, nor establish something of all or of some while refuting it of all or of some in the same breath, but instead narrow the search to fewer, definite things, and select, for each subject among the things that are, its own — for instance, concerning the good, or concerning knowledge. Most of the starting points, however, are peculiar to each subject. This is why it falls to experience to hand over the starting points that concern each subject — I mean, for example, that astronomical experience hands over the starting points of astronomical science (for once the observed phenomena had been adequately grasped, the astronomical demonstrations were discovered in that way), and the same holds for any other craft or science whatever. So that if the things that belong to each subject have been grasped, it is then our task to bring the demonstrations readily to light. For if nothing that truly belongs to the facts has been left out of the survey, we shall be able,

99| Concerning everything, then: where there is a demonstration, one must find it and demonstrate it; where demonstration is not natural to the thing, one must make this evident. In general, then, the way in which one must select premises has pretty much been stated; we have gone through it precisely in the treatise on dialectic. That division by genera is a small part of the method we have described is easy to see. For division is, as it were, a weak syllogism: what it needs to prove, it begs, and it always draws its conclusion from something further up. This very point had escaped the notice of everyone who used the method, and they tried to persuade people that it was possible to get a demonstration of substance and of what a thing is. So they understood neither what conclusion it is possible to reach by dividing, nor that it was possible in the way we have stated. Now in demonstrations, whenever something must be concluded to belong, the middle term, through which the syllogism comes about, must always be lesser than, and not universal in relation to, the first of the extremes; but division wants the opposite, for it takes the universal as its middle term. Let animal stand for A, mortal for B, and immortal for C, and let man, whose account we must grasp, stand for D. Now the divider takes every animal to be either mortal or immortal; and this means: whatever is A is entirely either B or C. Again, in dividing, he always posits man to be an animal, so that he takes A to belong to D.

100| The syllogism, then, is that D will be entirely either B or C, so that it is necessary that man be either mortal or immortal; but that he is a mortal animal is not necessary — it is simply assumed; and this was exactly what needed to be proved by syllogism. And again, positing A as mortal animal, B as footed, and C as footless, and D as man, he takes, in the same way, A to be either in B or in C (for every mortal animal is either footed or footless), and A to belong to D (for he has taken man to be a mortal animal); so that it is necessary that man be a footed or footless animal, but that he is footed is not necessary — it is merely taken; and this again was what needed to be shown.

101| And in this way, for those who always proceed by division, it turns out that they take the universal as their middle term, while what had to be demonstrated, together with the differentiae, they take as the extreme. But in the end, that this is man, or whatever it is that is being sought, they state nothing clear enough to make it necessary; for they carry out the whole rest of the route without even supposing that the available resources exist. It is clear that by this method one cannot refute anything, nor draw a conclusion about an accident or a property, nor about a genus, nor in cases where it is unknown whether something is thus or so — for instance, whether the diagonal is incommensurable or commensurable. For if one takes that every length is either commensurable or incommensurable, and the diagonal is a length, it has been concluded by syllogism that the diagonal is either incommensurable or commensurable. But if he is going to take it as incommensurable, he will be taking exactly what needed to be proved by syllogism. So it is not possible to demonstrate it this way: this is the route, but it cannot be done by way of it. Let incommensurable-or-commensurable stand for A, length for B, diagonal for C. It is clear, then, that this method of inquiry is not suited to every investigation, and that even in the cases where it seems most fitting, it is not useful there either. From what materials demonstrations come to be, and how, and what one must look to in the case of each problem, is clear

102| from what has been said. How we shall reduce syllogisms to the figures described earlier must be discussed next, for this is still left over from our inquiry. For if we were to study the generation of syllogisms, and had the capacity to discover them, and moreover could analyze those already produced into the figures described earlier, our original purpose would be complete. At the same time it will also happen that what was said earlier is confirmed and made clearer as being so, through what will now be said; for everything true must be in every way in agreement with itself. First, then, one must try to pick out the two premises of the syllogism (for it is easier to divide into larger parts than into smaller ones, and the composite parts are fewer than the elements they are made of),

103| then examine which one is universal and which is particular, and, if both have not been stated, supply the missing one oneself. For sometimes people propose the universal premise but do not state the one contained within it, whether writing or asking questions; or they propose these premises but leave out the ones through which these are brought to a conclusion, and ask other, pointless questions instead. One must therefore examine whether something superfluous has been taken and whether something necessary has been left out, and add the one and remove the other, until one arrives at the two premises; for without these it is not possible to reduce arguments that have been questioned out in this way. In some cases it is easy to see what is missing, but in others it escapes notice, and people seem to reach a conclusion by syllogism because something necessary follows from what has been laid down — for instance, if it should be assumed that when a non-substance is destroyed a substance is not destroyed, but that when the things a thing is composed of are destroyed, the thing composed of them is destroyed too. For once these are laid down, it is necessary that a part of a substance be a substance; yet this has not been proved by syllogism through what has been assumed — rather, premises are missing. Again, if, when man exists, it is necessary that animal exists, and when animal exists, that substance exists, then when man exists it is necessary that substance exists; but this has not yet been proved by syllogism, for the premises are not arranged as we have said. We are deceived in such cases because something necessary follows from what is laid down, and a syllogism too is something necessary. But the necessary extends more widely than the syllogism: every syllogism is something necessary, but not everything necessary is a syllogism. So it is not the case that, whenever something follows once certain things have been laid down, one should try to reduce it straightaway; rather, one must first grasp the two premises, then divide them in this way into their terms, and set down as the middle term the one that is stated in both premises — for the middle term must occur in both premises, in every figure. If, then, the middle term both predicates and is predicated of something, or itself predicates while something else is denied of it, this will be the first figure;

104| but if it both predicates and is denied of something, the middle term will belong to the second figure; and if other things are predicated of it, or one thing is denied of it and another predicated, this will be the last figure. For this is how the middle term stood in each figure. And likewise even if the premises are not universal, for the determination of the middle term is the same. It is clear, then, that in an argument in which the same term is not stated more than once, no syllogism results, for no middle term has been taken. Since we have determined which kind of problem is concluded in each figure, and in which figure the universal conclusion falls and in which the particular, it is clear that one must not look to every figure, but to the one proper to each problem. As for those problems that are concluded in more than one figure, we shall recognize the figure by the position of the middle term.

105| So it often happens that we are deceived about syllogisms because of the necessary, as has been said before, but sometimes because of a similarity in how the terms are set out; and this must not escape our notice. For example, if A is said of B and B is said of C: for it would seem, the terms standing this way, that there is a syllogism, but neither anything necessary nor a syllogism results. Let A stand for always existing, B for "conceivable Aristomenes," and C for Aristomenes. Now it is true that A holds of B — for conceivable Aristomenes always exists. But also B holds of C — for Aristomenes is conceivable Aristomenes. But A does not hold of C — for Aristomenes is perishable. For no syllogism resulted with the terms standing this way; rather the premise A-B had to be taken universally. But this is false — to claim that every conceivable Aristomenes always exists, when Aristomenes is perishable. Again, let C stand for Miccalus, B for "musical Miccalus," and A for "perishing tomorrow." Now it is true to predicate B of C — for Miccalus is musical Miccalus. But also A holds of B — for musical Miccalus would perish tomorrow. But A of C is false. And this is the same case as before; for it is not universally true that musical Miccalus perishes tomorrow. And unless this is taken, there is no syllogism.

106| So this deception arises over a small point: we concede as though it made no difference to say that this holds of that, or that

107| this holds of every that — this is what we grant. But often falsehood will result from not setting out the terms of the premise correctly — for example, if A were health, B disease, and C man. For it is true to say that A may possibly belong to no B (for health holds of no disease), and again that B holds of every C (for every man is receptive of disease). It would then seem to follow that health may possibly belong to no man. The cause of this is that the terms are not correctly set out in their wording, since if the terms naming the states are substituted, there will be no syllogism — for example, if in place of "health" we put "being healthy," and in place of "disease" "being diseased." For it is not true to say that being healthy cannot hold of a person who is diseased. And unless this is taken, there is no syllogism except one about possibility; and this is not impossible — for it is possible for health to hold of no man. Again, in the middle figure the falsehood will arise in the same way: for it is possible for health to hold of no disease and of every man, so that it is possible for disease to hold of no man. And in the third figure the falsehood arises with regard to possibility, for it is possible for health and disease, knowledge and ignorance, and in general contraries, to hold of the same thing, though it is impossible for them to hold of one another. And this conflicts with what was said before; for when more than one thing could hold of the same subject, they could also hold of one another. It is clear, then, that in all these cases the deception arises from how the terms are set out; for when the terms derived from the states — 'being healthy', 'being diseased' — are substituted, no falsehood results. It is plain, then, that in premises of this kind, the term naming the state must always be substituted for the state and set down as the term.

108| We must not always insist on setting out the terms by a single word; for often there will be expressions for which no name is laid down, and this is why it is difficult to reduce syllogisms of this sort to the figures. And sometimes deception will also arise from this very search — for example, the claim that there is a syllogism of things without a middle. Let A be "two right angles," B "triangle," and C "isosceles." Now A holds of C because of B, but it no longer holds of B because of anything else — for the triangle has two right angles in its own right — so that there will be no middle term for B, though it is demonstrable. For it is clear that the middle must not always be taken to be a "this," some single term, but sometimes an expression — which is exactly what happens in the case just mentioned.

109| "The first holds of the middle" and "the middle holds of the extreme" must not be taken to mean that they will always be predicated of each other, or that the first is related to the middle in the same way as the middle is to the last term. And the same holds for "does not hold." Rather, in as many ways as "being" is said, and it is true to say this very thing, in that many ways we must think "holding" too is meant. For example, take the claim that there is a single knowledge of contraries. Let A be "there is a single knowledge," and let B be "contrary to one another." Now A holds of B, not in the sense that the contraries are a single knowledge, but in the sense that it is true to say of them that there is a single knowledge of them. Now it happens sometimes that the first is said of the middle, but the middle is not said of the third — for example, if wisdom is knowledge, and wisdom is of the good, the conclusion is that there is knowledge of the good; the good, then, is not knowledge, but wisdom is knowledge. And sometimes the middle is said of the third, but the first is not said of the middle — for example, if there is knowledge of everything that is qualified or is a contrary, and the good is both a contrary and something qualified, the conclusion is that there is knowledge of the good, but the good is not knowledge, nor is it "qualified," nor "contrary" — rather, the good is these things. And sometimes neither is the first said of the middle nor the middle of the third, while the first is sometimes said of the third and sometimes not. For example, if wherever there is knowledge there is a genus of that thing, and there is knowledge of the good, the conclusion is that there is a genus of the good; but here nothing is predicated of anything. But if wherever there is knowledge, that thing is a genus, and there is knowledge of the good, the conclusion

110| is that the good is a genus; here the first is predicated of the extreme, but the terms are not said of one another. The same manner of understanding must be applied to "not holding." For "this does not hold of that" does not always signify "this is not that"; sometimes it signifies "this is not of that" or "this is not to that." For example, that there is no motion of motion nor coming-to-be of coming-to-be, but there is of pleasure — therefore pleasure is not a coming-to-be. Or again, that there is a sign of laughter, but there is no sign of a sign, so that laughter is not a sign. And likewise in the other cases where the problem is rejected because the genus itself is spoken of in some particular way in relation to it. Again, that the right moment is not a needful time: for God has a right moment, but there is no needful time for God, since nothing is beneficial to God. For the terms to be set down must be "right moment," "needful time," and "God," while the premise must be taken according to the grammatical case of the name. For, put simply, we say this about all cases: the terms must always be set down according to the nominative forms of the names — for example, "man" or "good" or "contraries," not "of man" or "of good" or "of contraries" — but the premises must be taken according to the case appropriate to each: either "to this," as with "the equal"; or "of this," as with "the double"; or "this," as with "the one striking" or "the one seeing"; or "this one," as in "man is an animal" — or however else the name happens to fall in the premise.

111| Belonging to a given thing, and being truly said of a given thing, must be understood in as many ways as the categories have been divided, and these either in some respect or without qualification, and further either simple or combined; and likewise for not belonging. These points must be examined and distinguished more thoroughly. What is reduplicated in the premises must be placed at the first extreme term, not at the middle term. I mean, for example, if a syllogism should arise that there is knowledge of justice as being good, then 'as being good,' or 'good,' must be placed at the first term. For let A be knowledge as being good, B be good, and C be justice. It is true, then, to predicate A of B: for there is knowledge of the good, as being good. But B also holds of C: for justice is precisely something good. This, then, is how the

112| analysis comes out. But if 'as being good' were placed at B, it would not work: for A would be true of B, but B would not be true of C — for to predicate 'good, as being good' of justice is false and unintelligible. The same holds if it were shown that the healthy is knowable as being good, or that a goat-stag is, as being not-existent, or that man is perishable as being perceptible: in every case where something extra is predicated, the reduplication must be placed at the extreme term. The setting of the terms is not the same when something is proved without qualification and when it is proved as being a this, or in some respect, or in some way — I mean, for example, when it is shown that the good is knowable, as against when it is shown that it is knowable as being good. If it has been shown to be knowable without qualification, being must be set as the middle term; but if it has been shown as being good, then a certain being must be set as the middle term. For let A be knowledge as being a certain being, B be a certain being, and C be the good. It is true, then, to predicate A of B: for there was knowledge of the certain being, as being a certain being. But B also holds of C: for what C stands for is a certain being. So A will hold of C as well: there will therefore be knowledge of the good as being good; for 'a certain being' was a sign of its own proper substance. But if being had been set as the middle term, and at the extreme term being had been stated without qualification and not as a certain being, there would not be a syllogism that there is knowledge of the good as being good, but only that it is, as being — for example, with A as knowledge as being, B as being, and C as the good. It is clear, then, that in syllogisms bearing on some particular respect the terms must be taken in this way.

113| One must also substitute terms that mean the same thing — names for names, phrases for phrases, and a name for a phrase — and always take the name in place of the phrase wherever one can, since the setting-out of the terms is easier that way. For example, if it makes no difference to say that the object of supposition is not a genus of the object of opinion, or to say that the object of opinion is not precisely a kind of object of supposition (since what is signified is the same), then in place of the phrase given, 'object of supposition' and 'object of opinion' should be set down as the terms. Since it is not the same thing for pleasure to be good and for pleasure to be the good, the terms must not be set down in the same way: if the syllogism concludes that pleasure is the good, then 'the good' must be used as term; but if it concludes that pleasure is good, then 'good' must be used. And so likewise in the other cases.

114| It is not the same thing, either in fact or in what is said, to state that A belongs to everything to which B belongs, and to state that A belongs to everything of which B is truly said to hold of all; for nothing prevents B from belonging to C without belonging to all of it. For example, let B be beautiful and C be white. If, then, beautiful belongs to some white thing, it is true to say that beautiful belongs to the white — but perhaps not to all of it. Now if A belongs to B, but not to everything of which B is truly said, then, whether B belongs to all of C or only belongs to it at all, it need not follow that A belongs — not merely that it fails to belong to all of C, but that it need not belong to it at all. But if A belongs to everything of which B is truly said, it will follow that A is said of everything of which B is said of all. If, however, A is said only of that of which B is said of all, nothing prevents B from belonging to C while A does not belong to all of it, or does not belong to it at all. With three terms, then, it is clear that for A to be said of everything of which B is said, means this: of everything of which B is said, A too is said of all of it.

115| And if B is said of everything, A too holds in this way; but if B is not said of everything, A need not be said of everything either. One should not think that anything absurd results from setting out a particular case; for we make no use of its being some particular this. It is like the geometer who says that this line is a foot long and straight and without breadth, when it is not — but he does not use it in his reasoning as though he were drawing his conclusion from those features. For, in general, whenever there is nothing standing to another as whole to part, and nothing further standing to it as part to whole, the one giving the proof proves nothing from premises of that kind, so that no syllogism results. We use setting-out the way we use perception, addressing ourselves to the learner; it is not that demonstration is impossible without these, in the way it is impossible without the premises from which the syllogism is drawn. Let it not escape us that within one and the same syllogism not all the conclusions come through a single figure, but one through this figure and another through a different one. It is clear, then, that the analyses too must be carried out accordingly. And since not every problem is proved in every figure, but each kind is assigned to a particular one, it is clear from the conclusion in which figure one must look for it.

116| As for arguments directed against a definition, whenever they happen to be arguing against just one of the elements contained in the definition, that element being argued against should be set down as the term, and not the whole formula; for then one will be less likely to be thrown into confusion by its length. For example, if someone showed that water is a drinkable liquid, then 'drinkable' and 'water' should be set down as the terms.

117| Further, one should not try to reduce arguments from a hypothesis; for it is not possible to reduce them from the premises as laid down. For they have not been proved by means of a syllogism, but are all agreed to by convention. For example, suppose someone lays down the hypothesis that, if there is not a single capacity for contraries, there is not a single knowledge of them either, and then argues that not every capacity is a capacity for contraries — for instance, the capacity for the healthy and for the diseased, since then the same thing would be at once healthy and diseased. That there is not a single capacity for all contraries has, then, been shown; but that there is not a single knowledge of them has not been shown. Yet one is compelled to grant it — but not on the basis of a syllogism, only on the basis of the hypothesis. This conclusion, then, cannot be reduced; but that there is not a single capacity can be — for this, presumably, was an actual syllogism, while the other was a hypothesis. The same holds too for arguments concluded through the impossible: these cannot be analyzed as a whole either, but the reduction to the impossible can be analyzed, since that part is shown by a syllogism, while the other part cannot, since it is concluded from a hypothesis. They differ from the cases mentioned before in that, in those cases, the other party must have agreed in advance if he is going to accept the conclusion — for instance, that if a single capacity for contraries were shown, the knowledge of them would also be the same. But here people concede the point even without having agreed to it beforehand, because the falsehood is evident — for example, when the diagonal is assumed to be commensurable, it follows that odd numbers turn out equal to even ones. There are also many other arguments concluded from a hypothesis, which need to be examined and clearly marked out. What, then, are the

118| The differences among these, and the number of ways something can come about from a hypothesis, we will discuss later; for now let this much be clear to us, that such syllogisms cannot be resolved into the figures. And we have stated the reason why. As for the problems that are proved in more than one figure, if one has been syllogized in one of them, it is possible to reduce the syllogism to the other — for instance, the negative syllogism in the first figure to the second, and the one in the middle figure to the first — though not all of them, only some. This will become clear in what follows. For if A belongs to no B, and B belongs to every C, then A belongs to no C. This, then, is the first figure; but if the negative premise is converted, it will be the second figure: for B belongs to no A, and to every C. Similarly too if the syllogism is not universal but particular, for instance if A belongs to no B and B belongs to some C: when the negative premise is converted, it will be the middle figure. Of the syllogisms in the second figure, the universal ones will be reduced to the first, but of the particular ones only one will be. For let A belong to no B, and to every C. When the negative premise is converted, then, it will be the first figure: for B belongs to no A, and A will belong to every C.

119| But if the affirmative premise is with B, and the negative is with the first term, then C must be posited as the first term: for this belongs to no A, and A belongs to every B, so that C belongs to no B. Nor, then, does B belong to no C: for the negative premise converts. But if the syllogism is particular, then when the negative premise is with the major extreme, it will be reduced to the first figure — for instance if A belongs to no B, and to some C: when the negative premise is converted, it will be the first figure, for B will belong to no A, and A to some C. But when the affirmative premise is the one that is with the major extreme, it will not be resolved — for instance if A belongs to every B, but not to every C: for B does not admit of conversion, and even if it did, there would be no syllogism from it. Again, the syllogisms in the third figure will not all be resolved into the first, but those in the first are all resolved into the third. For let A belong to every B, and B to some C. Since, then, the particular affirmative converts,

120| C will belong to some B; and A belonged to every B, so that the third figure comes about. And if the syllogism is negative, likewise: for the particular affirmative converts, so that A will belong to no B, and C will belong to some B. Of the syllogisms in the last figure, only one does not resolve into the first — when the negative premise is not posited as universal — but all the others do resolve. For let A and B both be predicated of every C. Then C will convert with each of them in the particular: it belongs, then, to some B. So the first figure will result, if A belongs to every C and C to some B. And if A belongs to every C and B to some C, the argument is the same, for C converts with B. But if B belongs to every C and A to some C, then B must be posited as the first term: for B belongs to every C, and C to some A, so that B belongs to some A. And since the particular converts, A too will belong to some B.

121| And if the syllogism is negative, with the terms universal, it must be taken in the same way. For let B belong to every C, and A to none. Then C will belong to some B, and A to no C, so that C will be the middle term. Likewise too if the negative is universal and the affirmative particular: for A belongs to no C, and C will belong to some B. But if the negative premise is taken as particular, there will be no resolution — for instance if B belongs to every C, but does not belong to some C: for when B C is converted, both premises will be particular. It is clear also that, for resolving the figures into one another, it is the premise with the minor extreme that must be converted in both figures: for it is by changing this premise that the transition came about. Of the syllogisms in the middle figure, one is resolved into the third and the other is not. For when the universal premise is negative, it is resolved. For if A belongs to no B, and to some C, both premises convert alike with respect to A, so that B belongs to no A, and C to some A;

122| A, then, is the middle term. But when A belongs to every B, and does not belong to some C, there will be no resolution: for neither of the premises is universal as a result of conversion. And the syllogisms from the third figure will also be resolved into the middle figure when the negative premise is universal — for instance if A belongs to no C, and B to some or to every C. For C will belong to no A, and to some B. But if the negative premise is particular, there will be no resolution: for the particular negative does not admit of conversion. It is clear, then, that the same syllogisms do not resolve into these figures that also did not resolve into the first, and that when the syllogisms are led back into the first figure, these alone are completed through the impossible. How, then, one must reduce syllogisms, and that the figures resolve into one another, is clear from what has been said.

123| It makes some difference, in establishing or refuting a claim, whether one supposes that not being this and being not-this signify the same thing or something different — for instance, not being white as against being not-white. For they do not signify the same thing, nor is being not-white the negation of being white, but rather not being white is. The account of this is as follows. It stands the same way with can walk in relation to can not-walk as with is white in relation to is not-white, and knows the good in relation to knows the not-good. For knows the good and is one who knows the good differ in no way, nor does can walk differ from is one who is able to walk; so the same holds for their opposites — cannot walk and is not one who is able to walk. If, then, is not one who is able to walk signifies the same as is able to not-walk or is able not to walk, then these will belong to the same subject at the same time (for the same person is able both to walk and not to walk, and is knowledgeable both of the good and of the not-good) — but an affirmation and its opposite negation do not belong to the same subject at the same time. Just as, then, not knowing the good and knowing the not-good are not the same, so too being not-good and not being good are not the same. For when one pair of proportional terms differs, so does the other pair. Nor are being unequal and not being equal the same: for the one has something underlying it, namely the thing that is unequal, and

124| this is the unequal, while to another neither belongs. That is why not everything is either equal or unequal, but everything is either equal or not-equal. Further, "it is a not-white piece of wood" and "it is not a white piece of wood" do not hold together. For if it is a not-white piece of wood, it will be wood; but what is not a white piece of wood need not be wood at all. So it is clear that "it is not-good" is not the negation of "it is good." If, then, of any one thing either the affirmation or the negation is true, then, if this is not the negation, clearly it would be some kind of affirmation. But every affirmation has a negation; so this one too has one, and that is "it is not not-good." These stand in the following order relative to one another. Let A be "being good," B "not being good," C "being not-good" (under B), and D "not being not-good" (under A). Then of everything either A or B will hold, and never both of the same thing; and either C or D, and never both of the same thing. And whatever has C must have B in every case (for if it is true to say that something is not-white, it is also true that it is not white

125| — for it is impossible to be white and to be not-white at the same time, or to be a not-white piece of wood and to be a white piece of wood, so that if the affirmation does not hold, the negation will); but C does not always follow B (for what is not wood at all will not be a not-white piece of wood either). Conversely, then, whatever has A has D in every case (for it has either C or D; and since it is not possible to be not-white and white at the same time, D will hold — for of what is white it is true to say that it is not not-white); but A does not hold of everything that has D (for of what is not wood at all it is not true to say A, namely that it is a white piece of wood, so that D is true but A is not true, namely that it is a white piece of wood). And it is clear that A and C never hold of the same thing, while B and D can hold of the same thing. Privations stand to their predicates in the same way, on this same scheme. Let "equal" be A, "not-equal" be B, "unequal" be C, "not-unequal" be D.

126| And in the case of many things too, of which the same thing holds of some but not of others, the negation would likewise be true — that not all are white, or that each is not white; but that each is not-white, or that all are not-white, is false. Likewise the negation of "every animal is white" is not "every animal is not-white" (for both are false), but "not every animal is white." Since it is clear that "it is not-white" and "it is not white" signify different things, and the one is an affirmation, the other a negation, it is evident that the way of demonstrating each is not the same — for instance, that whatever is an animal is not white, or may not be white, is one thing, and that it is true to say of it "not-white" is another; for this is what it is to be not-white. But that it is true to say "it is white" or "it is not-white" is shown in the same way; for both are demonstrated constructively through the first figure. For "true" is ranked in the same way as "is." For the negation of "it is true to say white" is not "it is true to say not-white," but "it is not true to say white." So if it will be true to say that whatever is a man is either musical or not-musical, then of whatever is an animal one must take it as being either musical or being not-musical, and this has been demonstrated. But that whatever is a man is not musical is shown destructively, by the three ways already stated.

127| In general, whenever A and B are so related that they cannot belong to the same thing at once, but of necessity one or the other belongs to everything, and again C and D likewise, and A follows C without converting with it, then D will follow B as well, and will not convert with it either; and A and D can belong to the same thing, but B and C cannot. First, then, that D follows B is clear from this: since one or the other of C and D necessarily belongs to everything, and whatever has B cannot have C, because C brings A along with it, and A and B cannot belong to the same thing, it is clear that D will follow. Again, since C does not convert with A, and either C or D belongs to everything, it is possible for A and D to belong to the same thing. But B and C cannot, because A follows along with C; for something impossible would result. It is clear, then, that B does not convert with D either, since D and A can belong to the same thing at once. But it sometimes happens, even with terms arranged in this way,

128| that one is deceived through not taking correctly the opposites of which one or the other must belong to everything; for instance, if A and B cannot belong to the same thing at once, but of necessity whatever does not have the one has the other, and again C and D likewise, and A follows everything that has C, it will result that whatever has D necessarily has B, which is false. For let the negation of A and B be taken as Z, and again of C and D as Θ. Then of necessity everything has either A or Z; for it has either the affirmation or the negation. And again either C or Θ; for these are the affirmation and the negation. And whatever has C has A lying under it in every case. So whatever has Z has Θ in every case. Again, since of Z and B one or the other belongs to everything, and likewise of Θ and D, and Θ follows Z, then B will follow D as well; for this we know, from what was shown. If, then, A follows C, then B follows D too. But this is false; for in cases so arranged the relation of following was the reverse. For presumably it is not necessary that everything have either A or Z, nor either Z or B. For Z is not the negation of A. For the negation of "good" is "not-good"; but "not-good" is not the same as "neither good nor not-good." And likewise with C and D; for the two negations taken are different.

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