1 · First we must state the subject of the enquiry and what it is about: the [24a10] subject is demonstration, and it is about demonstrative understanding.1 Next we must determine what a proposition2 is, what a term is, and what a deduction3 is (and what sort of deduction is perfect and what imperfect); and after that, what it is for one thing to be or not be in another as a whole, and what we mean by being predicated of every or of no. [15]
A proposition, then, is a statement affirming or denying something of something; and this is either universal or particular or indefinite. By universal I mean a statement that something belongs to all or none of something; by particular that it belongs to some or not to some or not to all; by indefinite that it does or does not belong, without any mark of being universal or particular, e.g. ‘contraries are [20] subjects of the same science’, or ‘pleasure is not good’. A demonstrative proposition differs from a dialectical one, because a demonstrative proposition is the assumption of one of two contradictory statements (the demonstrator does not ask for his premiss, but lays it down), whereas a dialectical proposition choice between two contradictories. But this will make no difference to the production of a deduction in [25] either case; for both the demonstrator and the dialectician argue deductively after assuming that something does or does not belong to something. Therefore a deductive proposition without qualification will be an affirmation or denial of something concerning something in the way we have described; it will be demonstrative, if it is true and assumed on the basis of the first principles of its science; it will be dialectical if it asks for a choice between two contradictories (if one is [24b10] enquiring) or if it assumes what is apparent and reputable, as we said in the Topics4 (if one is deducing). Thus as to what a proposition is and how deductive, demonstrative and dialectical propositions differ, we have now said enough for our present purposes—we shall discuss the matter with precision later on.5 [15]
I call a term that into which the proposition is resolved, i.e. both the predicate and that of which it is predicated, ‘is’ or ‘is not’ being added.
A deduction is a discourse in which, certain things being stated, something [20] other than what is stated follows of necessity from their being so. I mean by the last phrase that it follows because of them, and by this, that no further term is required from without in order to make the consequence necessary.
I call perfect a deduction which needs nothing other than what has been stated to make the necessity evident; a deduction is imperfect if it needs either one or more [25] things, which are indeed the necessary consequences of the terms set down, but have not been assumed in the propositions.
That one term should be in another as in a whole is the same as for the other to be predicated of all of the first. And we say that one term is predicated of all of another, whenever nothing can be found of which the other term cannot be asserted; ‘to be predicated of none’ must be understood in the same way.
[25a1] 2 · Every proposition states that something either belongs or must belong or may belong; of these some are affirmative, others negative, in respect of each of the three modes; again some affirmative and negative propositions are universal, others [5] particular, others indefinite. It is necessary then that in universal attribution the terms of the negative proposition should be convertible, e.g. if no pleasure is good, then no good will be pleasure; the terms of the affirmative must be convertible, not however universally, but in part, e.g. if every pleasure is good, some good must be [10] pleasure; the particular affirmative must convert in part (for if some pleasure is good, then some good will be pleasure); but the particular negative need not convert, for if some animal is not man, it does not follow that some man is not animal.
First then take a universal negative with the terms A and B. Now if A belongs [15] to no B, B will not belong to any A; for if it does belong to some B (say to C), it will not be true that A belongs to no B—for C is one of the Bs. And if A belongs to every B, then B will belong to some A; for if it belongs to none, then A will belong to no [20] B—but it was laid down that it belongs to every B. Similarly if the proposition is particular: if A belongs to some B, it is necessary for B to belong to some A; for if it belongs to none, A will belong to no B. But if A does not belong to some B, it is not necessary that B should not belong to some A: e.g., if B is animal and A man; for [25] man does not belong to every animal, but animal belongs to every man.
3 · The same manner of conversion will hold good also in respect of necessary propositions. The universal negative converts universally; each of the affirmatives [30] converts into a particular. If it is necessary that A belongs to no B, it is necessary also that B belongs to no A. For if it is possible that it belongs to some A, it would be possible also that A belongs to some B. If A belongs to all or some B of necessity, it is necessary also that B belongs to some A; for if there were no necessity, neither would A belong to some B of necessity. But the particular negative does not convert, [35] for the same reason which we have already stated.
In respect of possible propositions, since possibility is used in several ways (for we say that what is necessary and what is not necessary and what is potential is possible), affirmative statements will all convert in a similar manner. For if it is possible that A belongs to all or some B, it will be possible that B belongs to some A. [25b1] For if it could belong to none, then A could belong to no B. This has been already proved. But in negative statements the case is different. Whatever is said to be possible, either because it necessarily belongs or because it does not necessarily not belong, admits of conversion like other negative statements, e.g. if one should say, it [5] is possible that the man is not a horse, or that no garment is white. For in the former case the one necessarily does not belong to the other; in the latter there is no necessity that it should: and the proposition converts like other negative statements. For if it is possible for no man to be a horse, it is also admissible for no horse to be a man; and if it is admissible for no garment to be white, it is also admissible for [10] nothing white to be a garment. For if some white thing must be a garment, then some garment will necessarily be white. This has been already proved. The particular negative is similar. But if anything is said to be possible because it is the general rule and natural (and it is in this way we define the possible), the negative [15] propositions can no longer be converted in the same way: the universal negative does not convert, and the particular does. This will be plain when we speak about the possible.6 At present we may take this much as clear in addition to what has been said: the statements that it is possible that A belongs to no B or does not belong to [20] some B are affirmative in form; for the expression ‘is possible’ ranks along with ‘is’, and ‘is’ makes an affirmative always and in every case, whatever the terms to which it is added in predication, e.g. ‘it is not-good’ or ‘it is not-white’ or in a word ‘it is not-this’. But this also will be proved in the sequel.7 In conversion these will behave like the other affirmative propositions. [25]
4 · After these distinctions we now state by what means, when, and how every deduction is produced; subsequently we must speak of demonstration. Deduction should be discussed before demonstration, because deduction is the more general: a demonstration is a sort of deduction, but not every deduction is a [30] demonstration.
Whenever three terms are so related to one another that the last is in the middle as in a whole, and the middle is either in, or not in, the first as in a whole, the extremes must be related by a perfect deduction. I call that term middle which both [35] is itself in another and contains another in itself: in position also this comes in the middle. By extremes I mean both that term which is itself in another and that in which another is contained. If A is predicated of every B, and B of every C, A must [26a1] be predicated of every C: we have already explained what we mean by ‘predicated of every’. Similarly also, if A is predicated of no B, and B of every C, it is necessary that A will belong to no C.
But if the first term belongs to all the middle, but the middle to none of the last term, there will be no deduction in respect of the extremes; for nothing necessary [5] follows from the terms being so related; for it is possible that the first should belong either to all or to none of the last, so that neither a particular nor a universal conclusion is necessary. But if there is no necessary consequence, there cannot be a deduction by means of these propositions. As an example of a universal affirmative relation between the extremes we may take the terms animal, man, horse; of a universal negative relation, the terms animal, man, stone. Nor again can a [10] deduction be formed when neither the first term belongs to any of the middle, nor the middle to any of the last. As an example of a positive relation between the extremes take the terms science, line, medicine: of a negative relation science, line, unit.
If then the terms are universally related, it is clear in this figure when a deduction will be possible and when not, and that if a deduction is possible the terms [15] must be related as described, and if they are so related there will be a deduction.
But if one term is related universally, the other in part only, to its subject, there must be a perfect deduction whenever universality is posited with reference to the major term either affirmatively or negatively, and particularity with reference to [20] the minor term affirmatively; but whenever the universality is posited in relation to the minor term, or the terms are related in any other way, a deduction is impossible. I call that term the major in which the middle is contained and that term the minor which comes under the middle. Let A belong to every B and B to some C. Then if ‘predicated of every’ means what was said above, it is necessary that A belongs to [25] some C. And if A belongs to no B and B to some C, it is necessary that A does not belong to some C. (The meaning of ‘predicated of none’ has also been defined.) So there will be a perfect deduction. This holds good also if deduction BC should be indefinite, provided that it is affirmative; for we shall have the same deduction [30] whether it is indefinite or particular.
But if the universality is posited with respect to the minor term either affirmatively or negatively, a deduction will not be possible, whether the other is affirmative or negative, indefinite or particular: e.g. if A belongs or does not belong to some B, and B belongs to every C. As an example of a positive relation between [35] the extremes take the terms good, state, wisdom; of a negative relation, good, state, ignorance. Again if B belongs to no C, and A belongs or does not belong to some B (or does not belong to every B), there cannot be a deduction. Take the terms white, horse, swan; white, horse, raven. The same terms may be taken also if BA is indefinite.
[26b1] Nor when the proposition relating to the major extreme is universal, whether affirmative or negative, and that to the minor is negative and particular, can there be a deduction: e.g. if A belongs to every B, and B does not belong to some C or not [5] to every C. For the first term may be predicable both of all and of none of the term to some of which the middle does not belong. Suppose the terms are animal, man, white: next take some of the white things of which man is not predicated—swan and snow: animal is predicated of all of the one, but of none of the other. Consequently there cannot be a deduction. Again let A belong to no B, but let B not belong to some [10] C. Take the terms inanimate, man, white: then take some white things of which man is not predicated—swan and snow: inanimate is predicated of all of the one, of none of the other.
Further since it is indefinite to say that B does not belong to some C, and it is [15] true that it does not belong to some C both if it belongs to none and if it does not belong to every, and since if terms are assumed such that it belongs to none, no deduction follows (this has already been stated), it is clear that this arrangement of terms will not afford a deduction: otherwise one would have been possible in the other case too. A similar proof may also be given if the universal proposition is negative. [20]
Nor can there in any way be a deduction if both the relations are particular, either positively or negatively, or the one positively and the other negatively, or one indefinite and the other definite, or both indefinite. Terms common to all the above are animal, white, horse; animal, white, stone. [25]
It is clear then from what has been said that if there is a deduction in this figure with a particular conclusion, the terms must be related as we have stated: if they are related otherwise, no deduction is possible at all. It is evident also that all the deductions in this figure are perfect (for they are all completed by means of [30] what was originally assumed) and that all conclusions are proved by this figure, viz. universal and particular, affirmative and negative. Such a figure I call the first.
5 · Whenever the same thing belongs to all of one subject, and to none of another, or to all of each subject or to none of either, I call such a figure the second; [35] by middle term in it I mean that which is predicated by both subjects, by extremes the terms of which this is said, by major extreme that which lies near the middle, by minor that which is further away from the middle. The middle term stands outside the extremes, and is first in position. A deduction cannot ever be perfect in this [27a1] figure, but it may be potential whether the terms are related universally or not.
If then the terms are related universally a deduction will be possible, whenever the middle belongs to all of one subject and to none of another (it does not matter which has the negative relation), but in no other way. Let M be predicated of no N, [5] but of every O. Since, then, the negative is convertible, N will belong to no M; but M was assumed to belong to every O: consequently N will belong to no O. This has already been proved. Again if M belongs to every N, but to no O then O will belong to no N. For if M belongs to no O, O belongs to no M; but M (as was said) belongs to [10] every N: O then will belong to no N; for the first figure has again been formed. But since the negative is convertible, N will belong to no O. Thus it will be the same deduction.
It is possible to prove these results also by reductio ad impossibile. [15]
It is clear then that a deduction is formed when the terms are so related, but not a perfect one; for the necessity is not perfectly established merely from the original assumptions; others also are needed.
But if M is predicated of every N and O, there will not be a deduction. Terms to illustrate a positive relation between the extremes are substance, animal, man; a [20] negative relation, substance, animal, number—substance being the middle term.
Nor is a deduction possible when M is predicated neither of any N nor of any O. Terms to illustrate a positive relation are line, animal, man; a negative relation, line, animal, stone.
It is clear then that if a deduction is formed when the terms are universally related, the terms must be related as we stated at the outset; for if they are [25] otherwise related no necessary consequence follows.
If the middle term is related universally to one of the extremes, a particular negative deduction must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement (by ‘an opposite manner’ I mean, [30] if the universal statement is negative, the particular is affirmative: if the universal is affirmative, the particular is negative). For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative is convertible, N [35] will belong to no M; but M was admitted to belong to some O: therefore N will not belong to some O; for a deduction is found by means of the first figure. Again if M belongs to every N, but not to some O, it is necessary that N does not belong to some O; for if N belongs to every O, and M is predicated also of every N, M must belong [27b1] to every O; but we assumed that M does not belong to some O. And if M belongs to every N but not to every O, we shall conclude that N does not belong to every O: the proof is the same as the above. But if M is predicated of every O, but not of every N, [5] there will be no deduction. Take the terms animal, substance, raven; animal, white raven. Nor will there be a deduction when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit; a negative relation, animal, substance, science.
If then the universal statement is opposed to the particular, we have stated [10] when a deduction will be possible and when not; but if the premisses are similar in form, I mean both negative or both affirmative, a deduction will not be possible at all. First let them be negative, and let the universality apply to the major term, i.e. [15] let M belong to no N, and not to some O. It is possible then for N to belong either to every O or to no O. Terms to illustrate the negative relation are black, snow, animal. But it is not possible to find terms of which the extremes are related positively and universally, if M belongs to some O, and does not belong to some O. For if N belonged to every O, but M to no N, then M would belong to no O; but we assumed [20] that it belongs to some O. In this way then it is not admissible to take terms: our point must be proved from the indefinite nature of the particular statement. For since it is true that M does not belong to some O, even if it belongs to no O, and since if it belongs to no O a deduction is (as we have seen) not possible, clearly it will not be possible now either.
Again let the propositions be affirmative, and let the universality apply as [25] before, i.e. let M belong to every N and to some O. It is possible then for N to belong to every O or to no O. Terms to illustrate the negative relation are white, swan, stone. But it is not possible to take terms to illustrate the universal affirmative relation, for the reason already stated: the point must be proved from the indefinite nature of the particular statement. And if the universality applies to the minor extreme, and M belongs to no O, and not to some N, it is possible for N to belong [30] either to every O or to no O. Terms for the positive relation are white, animal, raven; for the negative relation, white, stone, raven. If the propositions are positive, terms for the negative relation are white, animal, snow; for the positive relation, white, animal, swan. Evidently then, whenever the propositions are similar in form, and one is universal, the other particular, a deduction cannot be formed at all. Nor is one [35] possible if the middle term belongs to some of each of the extremes, or does not belong to some of either, or belongs to some of the one, not to some of the other, or belongs to neither universally, or is related to them indefinitely. Common terms for all the above are white, animal, man; white, animal, inanimate.
It is clear then from what has been said that if the terms are related to one [28a1] another in the way stated, a deduction results of necessity; and if there is a deduction, the terms must be so related. But it is evident also that all the deductions in this figure are imperfect; for all are made perfect by certain supplementary [5] assumptions, which either are contained in the terms of necessity or are assumed as hypotheses, i.e. when we prove per impossibile. And it is evident that an affirmative deduction is not attained by means of this figure, but all are negative, whether universal or particular.
6 · But if one term belongs to all, and another to none, of a third, or if both [10] belong to all, or to none, of it, I call such a figure the third; by middle term in it I mean that of which both are predicated, by extremes I mean the predicates, by the major extreme that which is further from the middle, by the minor that which is nearer to it. The middle term stands outside the extremes, and is last in position. A [15] deduction cannot be perfect in this figure either, but it may be potential whether the terms are related universally or not to the middle term.
If they are universal, whenever both P and R belong to every S, it follows that P will necessarily belong to some R. For, since the affirmative is convertible, S will belong to some R: consequently since P belongs to every S, and S to some R, P must [20] belong to some R; for a deduction in the first figure is produced. It is possible to demonstrate this both per impossibile and by exposition. For if both P and R belong to every S, should one of the Ss, e.g. N, be taken, both P and R will belong to this, [25] and thus P will belong to some R.
If R belongs to every S, and P to no S, there will be a deduction that P will necessarily not belong to some R. This may be demonstrated in the same way as before by converting the proposition RS. It might be proved also per impossibile, as [30] in the former cases. But if R belongs to no S, P to every S, there will be no deduction. Terms for the positive relation are animal, horse, man; for the negative relation animal, inanimate, man.
Nor can there be a deduction when both terms are asserted of no S. Terms for the positive relation are animal, horse, inanimate; for the negative relation man, [35] horse, inanimate—inanimate being the middle term.
It is clear then in this figure also when a deduction will be possible and when not, if the terms are related universally. For whenever both the terms are affirmative, there will be a deduction that one extreme belongs to some of the other; [28b1] but when they are negative, no deduction will be possible. But when one is negative, the other affirmative, if the major is negative, the minor affirmative, there will be a deduction that the one extreme does not belong to some of the other; but if the relation is reversed, no deduction will be possible.
[5] If one term is related universally to the middle, the other in part only, when both are affirmative there must be a deduction, no matter which is universal. For if R belongs to every S, P to some S, P must belong to some R. For since the [10] affirmative is convertible, S will belong to some P; consequently since R belongs to every S, and S to some P, R must also belong to some P; therefore P must belong to some R. Again if R belongs to some S, and P to every S, P must belong to some R. This may be demonstrated in the same way as the preceding. And it is possible to [15] demonstrate it also per impossibile and by exposition, as in the former cases.
But if one term is affirmative, the other negative, and if the affirmative is universal, a deduction will be possible whenever the minor term is affirmative. For if R belongs to every S, but P does not belong to some S, it is necessary that P does not belong to some R. For if P belongs to every R, and R belongs to every S, then P [20] will belong to every S; but we assumed that it did not. Proof is possible also without reduction, if one of the Ss be taken to which P does not belong.
But whenever the major is affirmative, no deduction will be possible, e.g. if P belongs to every S, and R does not belong to some S. Terms for the universal affirmative relation are animate, man, animal. For the universal negative relation it [25] is not possible to get terms, if R belongs to some S, and does not belong to some S. For if P belongs to every S, and R to some S, then P will belong to some R; but we assumed that it belongs to no R. We must put the matter as before. Since its not belonging to some is indefinite, it is true to say of that which belongs to none that it [30] does not belong to some. But if R belongs to no S, no deduction is possible, as has been shown. Clearly then no deduction will be possible here.
But if the negative term is universal, whenever the major is negative and the minor affirmative there will be a deduction. For if P belongs to no S, and R belongs to some S, P will not belong to some R; for we shall have the first figure again, if the [35] proposition RS is converted.
But when the minor is negative, there will be no deduction. Terms for the positive relation are animal, man, wild; for the negative relation, animal, science, wild—the middle in both being the term wild.
Nor is a deduction possible when both are stated in the negative, but one is universal, the other particular. When the minor is related universally to the middle, [29a1] take the terms animal, science, wild; animal, man, wild. When the major is related universally to the middle, take as terms for a negative relation raven, snow, white. For a positive relation terms cannot be found, if R belongs to some S, and does not [5] belong to some S. For if P belongs to every R, and R to some S, then P belongs to some S; but we assumed that it belongs to no S. Our point, then, must be proved from the indefinite nature of the particular statement.
Nor is a deduction possible at all, if each of the extremes belongs to some of the middle, or does not belong, or one belongs and the other does not, or one belongs to some, the other not to all, or if they are indefinite. Common terms for all are animal, man, white; animal, inanimate, white. [10]
It is clear then in this figure also when a deduction will be possible, and when not; and that if the terms are as stated, a deduction results of necessity, and if there is a deduction, the terms must be so related. It is clear also that all the deductions in this figure are imperfect (for all are made perfect by certain supplementary [15] assumptions), and that it will not be possible to deduce a universal conclusion by means of this figure, whether negative or affirmative.
7 · It is evident also that in all the figures, whenever a deduction does not result, if both the terms are affirmative or negative nothing necessary follows at all, [20] but if one is affirmative, the other negative, and if the negative is assumed universally, a deduction always results relating the minor to the major term, e.g. if A belongs to every or some B, and B belongs to no C; for if the propositions are converted it is necessary that C does not belong to some A. Similarly also in the [25] other figures; a deduction always results by means of conversion. It is evident also that the substitution of an indefinite for a particular affirmative will effect the same deduction in all the figures.
It is clear too that all the imperfect deductions are made perfect by means of [30] the first figure. For all are brought to a conclusion either probatively or per impossibile, in both ways the first figure is formed: if they are made perfect probatively, because (as we saw) all are brought to a conclusion by means of conversion, and conversion produces the first figure; if they are proved per [35] impossibile, because on the assumption of the false statement the deduction comes about by means of the first figure, e.g. in the last figure, if A and B belong to every C, it follows that A belongs to some B; for if A belonged to no B, and B belongs to every C, A would belong to no C; but (as we stated) it belongs to every C. Similarly also with the rest.
It is possible also to reduce all deductions to the universal deductions in the [29b1] first figure. Those in the second figure are clearly made perfect by these, though not all in the same way; the universal ones are made perfect by converting the negative premiss, each of the particular by reductio ad impossibile. In the first figure [5] particular deductions are indeed made perfect by themselves, but it is possible also to prove them by means of the second figure, reducing them ad impossibile, e.g. if A belongs to every B, and B to some C, it follows that A belongs to some C. For if it belonged to no C, and belongs to every B, then B will belong to C: this we know by [10] means of the second figure. Similarly also demonstration will be possible in the case of the negative. For if A belongs to no B, and B belongs to some C, A will not belong to some C; for if it belonged to every C, and belongs to B, then B will belong to no C; and this (as we saw) is the middle figure. Consequently, since all deductions in the [15] middle figure can be reduced to universal deductions in the first figure, and since particular deductions in the first figure can be reduced to deductions in the middle figure, it is clear that particular deductions can be reduced to universal deductions in the first figure. Deductions in the third figure, if the terms are universal, are [20] directly made perfect by means of those deductions; but, when one of the propositions is particular, by means of the particular deductions in the first figure and these (we have seen) may be reduced to the universal deductions in the first figure; consequently also the particular deductions in the third figure may be so [25] reduced. It is clear then that all may be reduced to the universal deductions in the first figure.
We have stated then how deductions which prove that something belongs or does not belong to something else are constituted, both how those of the same figure are constituted in themselves, and how those of different figures are related to one another.
8 · Since there is a difference according as something belongs, necessarily [30] belongs, or may belong (for many things belong, but not necessarily, others neither necessarily nor indeed at all, but it is possible for them to belong), it is clear that there will be different deductions for each of these, and deductions with differently related terms, one concluding from what is necessary, another from what is, a third [35] from what is possible.
In the case of what is necessary, things are pretty much the same as in the case of what belongs; for when the terms are put in the same way, then, whether something belongs or necessarily belongs (or does not belong), a deduction will or will not result alike in both cases, the only difference being the addition of the [30a1] expression ‘necessarily’ to the terms. For the negative is convertible alike in both cases, and we should give the same account of the expressions ‘to be in something as in a whole’ and ‘to be predicated of every’. Thus in the other cases, the conclusion [5] will be proved to be necessary by means of conversion, in the same manner as in the case of simple predication. But in the middle figure when the universal is affirmative, and the particular negative, and again in the third figure when the universal is affirmative and the particular negative, the demonstration will not take the same form, but it is necessary by the exposition of a part of the subject, to which [10] in each case the predicate does not belong, to make the deduction in reference to this: with terms so chosen the conclusion will be necessary. But if the relation is necessary in respect of the part exposed, it must hold of some of that term in which this part is included; for the part exposed is just some of that. And each of the resulting deductions is in the appropriate figure.
[15] 9 · It happens sometimes also that when one proposition is necessary the deduction is necessary, not however when either is necessary, but only when the one related to the major is, e.g. if A is taken as necessarily belonging or not belonging to B, but B is taken as simply belonging to C; for if the propositions are taken in this [20] way, A will necessarily belong or not belong to C. For since A necessarily belongs, or does not belong, to every B, and since C is one of the Bs, it is clear that for C also the positive or the negative relation to A will hold necessarily. But if AB is not necessary, but BC is necessary, the conclusion will not be necessary. For if it were, it [25] would result both through the first figure and through the third that A belongs necessarily to some B. But this is false; for B may be such that it is possible that A should belong to none of it. Further, an example also makes it clear that the conclusion will not be necessary, e.g. if A were movement, B animal, C man; man is [30] an animal necessarily, but an animal does not move necessarily, nor does man. Similarly also if AB is negative; for the proof is the same.
In particular deductions, if the universal is necessary, then the conclusion will be necessary; but if the particular, the conclusion will not be necessary, whether the [35] universal proposition is negative or affirmative. First let the universal be necessary, and let A belong to every B necessarily, but let B simply belong to some C: it is necessary then that A belongs to some C necessarily; for C falls under B, and A was assumed to belong necessarily to every B. Similarly also if the deduction should be [30b1] negative; for the proof will be the same. But if the particular is necessary, the conclusion will not be necessary; for from the denial of such a conclusion nothing impossible results, just as it does not in the universal deductions. The same is true of [5] negatives too. Try the terms movement, animal, white.
10 · In the second figure, if the negative proposition is necessary, then the conclusion will be necessary, but if the affirmative, not necessary. First let the negative be necessary; let A be possible of no B, and simply belong to C. Since then [10] the negative is convertible, B is possible of no A. But A belongs to every C; consequently B is possible of no C For C falls under A. The same result would be obtained if the negative refers to C; for if A is possible of no C, C is possible of no A; [15] but A belongs to every B, consequently C is possible of no B; for again we have obtained the first figure. Neither then is B possible of C; for conversion is possible as before.
But if the affirmative proposition is necessary, the conclusion will not be necessary. Let A belong to every B necessarily, but to no C simply. If then the [20] negative is converted, the first figure results. But it has been proved in the case of the first figure that if the negative related to the major is not necessary the conclusion will not be necessary either. Therefore the same result will obtain here. Further, if the conclusion is necessary, it follows that C necessarily does not belong [25] to some A. For if B necessarily belongs to no C, C will necessarily belong to no B. But B at any rate must belong to some A, if it is true (as was assumed) that A necessarily belongs to every B. Consequently it is necessary that C does not belong to some A. But nothing prevents such an A being taken that it is possible for C to [30] belong to all of it. Further one might show by an exposition of terms that the conclusion is not necessary without qualification, though it is necessary given the premisses. For example let A be animal, B man, C white, and let the propositions be assumed in the same way as before: it is possible that animal should belong to [35] nothing white. Man then will not belong to anything white, but not necessarily; for it is possible for a man to become white, not however so long as animal belongs to nothing white. Consequently given these premisses the conclusion will be necessary, but it is not necessary without qualification.
Similar results will obtain also in particular deductions. For whenever the [31a1] negative proposition is both universal and necessary, then the conclusion will be necessary; but whenever the affirmative is universal and the negative particular, the [5] conclusion will not be necessary. First then let the negative be both universal and necessary: let it be possible for no B that A should belong to it, and let A belong to some C. Since the negative is convertible, it will be possible for no A that B should belong to it; but A belongs to some C; consequently B necessarily does not belong to [10] some C. Again let the affirmative be both universal and necessary, and let the affirmative refer to B. If then A necessarily belongs to every B, but does not belong to some C, it is clear that B will not belong to some C, but not necessarily. For the same terms can be used to demonstrate the point, which were used in the universal [15] deductions. Nor again, if the negative is necessary but particular, will the conclusion be necessary. The point can be demonstrated by means of the same terms.
11 · In the last figure when the terms are related universally to the middle, [20] and both propositions are affirmative, if one of the two is necessary, then the conclusion will be necessary. But if one is negative, the other affirmative, whenever the negative is necessary the conclusion also will be necessary, but whenever the affirmative is necessary the conclusion will not be necessary. First let both the [25] propositions be affirmative, and let A and B belong to every C, and let AC be necessary. Since then B belongs to every C, C also will belong to some B, because the universal is convertible into the particular; consequently if A belongs necessarily to every C, and C belongs to some B, it is necessary that A should belong to some B [30] also. For B is under C. The first figure then is formed. A similar proof will be given also if BC is necessary. For C is convertible with some A; consequently if B belongs necessarily to every C, it will belong necessarily also to some A.
Again let AC be negative, BC affirmative, and let the negative be necessary. [35] Since then C is convertible with some B, but A necessarily belongs to no C, A will necessarily not belong to some B either; for B is under C. But if the affirmative is necessary, the conclusion will not be necessary. For suppose BC is affirmative and [40] necessary, while AC is negative and not necessary. Since then the affirmative is convertible, C also will belong to some B necessarily; consequently if A belongs to no [31b1] C while C belongs to some B, A will not belong to some B—but not of necessity; for it has been proved, in the case of the first figure, that if the negative proposition is not necessary, neither will the conclusion be necessary. Further, the point may be [5] made clear by considering the terms. Let A be good, B animal, C horse. It is possible then that good should belong to no horse, and it is necessary that animal should belong to every horse; but it is not necessary that some animal should not be good, since it is possible for every animal to be good. Or if that is not possible, take as the [10] term awake or asleep; for every animal can accept these.
If, then, the terms are universal in relation to the middle, we have stated when the conclusion will be necessary. But if one is universal, the other particular, and if both are affirmative, whenever the universal is necessary the conclusion also must [15] be necessary. The demonstration is the same as before; for the particular affirmative also is convertible. If then it is necessary that B should belong to every C, and A falls under C, it is necessary that B should belong to some A. But if B must belong to some A, then A must belong to some B; for conversion is possible. Similarly also if AC should be necessary and universal; for B falls under C. But if the particular is [20] necessary, the conclusion will not be necessary. Let BC be both particular and necessary, and let A belong to every C, not however necessarily. If BC is converted the first figure is formed, and the universal proposition is not necessary, but the particular is necessary. But when the propositions were thus, the conclusion (as we [25] proved) was not necessary; consequently it is not here either. Further, the point is clear if we look at the terms. Let A be waking, B biped, and C animal. It is necessary that B should belong to some C, but it is possible for A to belong to C, and that A should belong to B is not necessary. For there is no necessity that some biped should [30] be asleep or awake. Similarly and by means of the same terms proof can be made, should AC be both particular and necessary.
But if one of the terms is affirmative, the other negative, whenever the universal is both negative and necessary the conclusion also will be necessary. For if it is not possible that A should belong to any C, but B belongs to some C, it is [35] necessary that A should not belong to some B. But whenever the affirmative is necessary, whether universal or particular, or the negative is particular, the conclusion will not be necessary. The rest of the proof of this will be the same as before; but if terms are wanted, when the affirmative is universal and necessary, take the terms waking, animal, man, man being middle, and when the affirmative is [32a1] particular and necessary, take the terms waking, animal, white; for it is necessary that animal should belong to some white thing, but it is possible that waking should belong to none, and it is not necessary that waking should not belong to some animal. But when the negative is particular and necessary, take the terms biped, moving, animal, animal being middle. [5]
12 · It is clear then that a deduction that something belongs is not reached unless both propositions state that something belongs, but a necessary conclusion is possible even if one only of the propositions is necessary. But in both cases, whether the deductions are affirmative or negative, it is necessary that one proposition [10] should be similar to the conclusion. I mean by ‘similar’, if the conclusion states that something belongs, the proposition must too; if the conclusion is necessary, the proposition must be necessary. Consequently this also is clear, that the conclusion will be neither necessary nor simple unless a necessary or simple proposition is assumed.
13 · Perhaps enough has been said about necessity, how it comes about and [15] how it differs from belonging. We proceed to discuss that which is possible, when and how and by what means it can be proved. I use the terms ‘to be possible’ and ‘the possible’ of that which is not necessary but, being assumed, results in nothing impossible. We say indeed, homonymously, of the necessary that it is possible. [But [20] that my definition of the possible is correct is clear from the contradictory negations and affirmations. For the expressions ‘it is not possible to belong’, ‘it is impossible to belong’, and ‘it is necessary not to belong’ are either identical or follow from one [25] another; consequently their opposites also, ‘it is possible to belong’, ‘it is not impossible to belong’, and ‘it is not necessary not to belong’, will either be identical or follow from one another. For of everything the affirmation or the denial holds good. That which is possible then will be not necessary and that which is not [30] necessary will be possible.]8 It results that all propositions in the mode of possibility are convertible into one another. I mean not that the affirmative are convertible into the negative, but that those which are affirmative in form admit of conversion by opposition, e.g. ‘it is possible to belong’ may be converted into ‘it is possible not to belong’, and ‘it is possible to belong to every’ into ‘it is possible to belong to no’ or ‘not to every’, and ‘it is possible to belong to some’ into ‘it is possible not to belong to [35] some’. And similarly for the others. For since that which is possible is not necessary, and that which is not necessary may possibly not belong, it is clear that if it is possible that A should belong to B, it is possible also that it should not belong to B; and if it is possible that it should belong to every, it is also possible that it should not belong to every. The same holds good in the case of particular affirmations; for the [32b1] proof is identical. And such propositions are affirmative and not negative; for ‘to be possible’ is in the same rank as ‘to be’, as was said above.
Having made these distinctions we next point out that ‘to be possible’ is used in [5] two ways. In one it means to happen for the most part and fall short of necessity, e.g. a man’s turning grey or growing or decaying, or generally what naturally belongs to a thing (for this has not its necessity unbroken, since a man does not exist forever, [10] although if a man does exist, it comes about either necessarily or for the most part). In another way it means the indefinite, which can be both thus and not thus, e.g. an animal’s walking or an earthquake’s taking place while it is walking, or generally what happens by chance; for none of these inclines by nature in the one way more than in the opposite.
That which is possible in each of its two ways is convertible into its opposite, [15] not however in the same way: what is natural is convertible because it does not necessarily belong (for in this sense it is possible that a man should not grow grey) and what is indefinite is convertible because it inclines this way no more than that. Science and demonstrative deductions are not concerned with things which are indefinite, because the middle term is uncertain; but they are concerned with things [20] that are natural, and as a rule arguments and inquiries are made about things which are possible in this sense. Deductions indeed can be made about the former, but it is unusual at any rate to inquire about them.
These matters will be treated more definitely in the sequel;9 our business at present is to state when and how and what deductions can be made from possible [25] propositions. The expression ‘it is possible for this to belong to that’ may be taken in two ways: either ‘to which that belongs’ or ‘to which it may belong’; for ‘A may be said of that of which B’ means one or other of these—either ‘of which B is said’ or ‘of [30] which it may be said’; and there is no difference between ‘A may be said of that of which B’ and ‘A may belong to every B’ It is clear then that the expression ‘A may possibly belong to every B’ might be used in two ways. First then we must state the nature and characteristics of the deduction which arises if B is possible of the subject of C, and A is possible of the subject of B. For thus both propositions are assumed in the mode of possibility; but whenever A is possible of the subject of B, [35] one proposition is simple, the other possible. Consequently we must start from propositions which are similar in form, as in the other cases.
14 · Whenever A may belong to every B, and B to every C, there will be a perfect deduction that A may belong to every C. This is clear from the definition; for it was in this way that we explained ‘to be possible to belong to every’. Similarly if it [33a1] is possible for A to belong to no B, and for B to belong to every C, then it is possible for A to belong to no C. For the statement that it is possible for A not to belong to that of which B may be true means (as we saw) that none of those things which can fall under B is left out of account. But whenever A may belong to every B, and B [5] may belong to no C, then indeed no deduction results from the propositions assumed; but if BC is converted after the manner of possibility, the same deduction results as before. For since it is possible that B should belong to no C, it is possible also that it should belong to every C. This has been stated above. Consequently if B [10] is possible for every C, and A is possible for every B, the same deduction again results. Similarly if in both propositions the negative is joined with ‘it is possible’: e.g. if A may belong to no B, and B to no C. No deduction results from the assumed [15] propositions, but if they are converted we shall have the same deduction as before. It is clear then that if the negation relates either to the minor extreme or to both the propositions, either no deduction results, or if one does it is not perfect. For the necessity results from the conversion. [20]
But if one of the propositions is universal, the other particular, when one relating to the major extreme is universal there will be a deduction. For if A is possible for every B, and B for some C, then A is possible for some C. This is clear from the definition of being possible. Again if A may belong to no B, and B may [25] belong to some C, it is necessary that A may not belong to some of the Cs. The proof is the same as above. But if the particular proposition is negative, and the universal is affirmative, and they are in the same position as before, e.g. A is possible for every B, B may not belong to some C, then an evident deduction does not result from the [30] assumed propositions; but if the particular is converted and it is laid down that B may belong to some C, we shall have the same conclusion as before, as in the cases given at the beginning.
But if the proposition relating to the major extreme is particular, the minor [35] universal, whether both are affirmative, or negative, or different in quality, or if both are indefinite or particular, in no way will a deduction be possible. For nothing prevents B from reaching beyond A, so that as predicates they cover unequal areas. Let C be that by which B extends beyond A. To C it is not possible that A should belong—either to all or to none or to some or not to some, since propositions in the [33b1] mode of possibility are convertible and it is possible for B to belong to more things than A. Further, this is obvious if we take terms; for if the propositions are as [5] assumed, the first term is both possible for none of the last and must belong to all of it. Take as terms common to all the cases under consideration animal, white, man, where the first belongs necessarily to the last; animal, white, garment, where it is not possible that the first should belong to the last. It is clear then that if the terms are related in this manner, no deduction results. For every deduction proves that [10] something belongs either simply or necessarily or possibly. It is clear that there is no proof of the first or of the second. For the affirmative is destroyed by the negative, and the negative by the affirmative. There remains the proof of possibility. But this is impossible. For it has been proved that if the terms are related in this manner it is [15] both necessary that the first should belong to all the last and not possible that it should belong to any. Consequently there cannot be a deduction to prove the possibility; for the necessary (as we stated) is not possible.
It is clear that if the terms are universal in possible propositions a deduction [20] always results in the first figure, whether they are affirmative or negative, but that a perfect deduction results in the first case, an imperfect in the second. But possibility must be understood according to the definition laid down, not as covering necessity. This is sometimes forgotten.
[25] 15 · If one proposition is simple, the other possible, whenever the one related to the major extreme indicates possibility all the deductions will be perfect and establish possibility in the sense defined; but whenever the one related to the minor indicates possibility all the deductions will be imperfect, and those which are [30] negative will establish not possibility according to the definition, but that something does not necessarily belong to any, or to every. For if something does not necessarily belong to any or to every, we say it is possible that it should belong to none or not to every. Let A be possible for every B, and let B belong to every C. Since C falls under [35] B, and A is possible for every B, clearly it is possible for every C also. So a perfect deduction results. Likewise if the proposition AB is negative, and BC is affirmative, the former stating possible, the latter simple attribution, a perfect deduction results proving that A possibly belongs to no C.
[34a1] It is clear that perfect deductions result if the proposition related to the minor term states simple belonging; but that deductions will result in the opposite case, must be proved per impossibile. At the same time it will be evident that they are imperfect; for the proof proceeds not from the propositions assumed. First we must [5] state that if B’s being follows necessarily from A’s being, B’s possibility will follow necessarily from A’s possibility. For suppose, the terms being so related, that A is possible, and B is impossible. If then that which is possible, when it is possible for it to be, might happen, and if that which is impossible, when it is impossible, could not [10] happen, and if at the same time A is possible and B impossible, it would be possible for A to happen without B, and if to happen, then to be. For that which has happened, when it has happened, is. But we must take the impossible and the possible not only in the sphere of becoming, but also in the spheres of truth and [15] belonging, and the various other spheres in which we speak of the possible; for it will be alike in all. Further we must understand the statement that B’s being follows from A’s being, not as meaning that if some single thing A is, B will be; for nothing follows of necessity from the being of some one thing, but from two at least, i.e. when the propositions are related in the manner stated to be that of a deduction. For if C is predicated of D, and D of F, then C is necessarily predicated of F. And if each [20] is possible, the conclusion also is possible. If then, for example, one should indicate the propositions by A, and the conclusion by B, it would not only result that if A is necessary, B is necessary, but also that if A is possible, B is possible.
Since this is proved it is evident that if a false and not impossible assumption is [25] made, the consequence of the assumption will also be false and not impossible: e.g. if A is false, but not impossible, and if B follows from A, B also will be false but not impossible. For since it has been proved that if B’s being follows from A’s being, [30] then B’s possibility will follow from A’s possibility, and A is assumed to be possible, consequently B will be possible; for if it were impossible, the same thing would at the same time be possible and impossible.
Since we have clarified these points, let A belong to every B, and B be possible for every C: it is necessary then that A should possibly belong to every C. Suppose [35] that it is not possible, but assume that B belongs to every C: this is false but not impossible. If then A is not possible for every C but B belongs to every C, then A is not possible for every B; for a deduction is formed in the third figure. But it was [40] assumed that A possibly belonged to every B. It is necessary then that A is possible for every C. For though the assumption we made is false and not impossible, the [34b1] conclusion is impossible. [It is possible also in the first figure to bring about the impossibility, by assuming that B belongs to C. For if B belongs to every C, and A is possible for every B, then A would be possible for every C. But the assumption was [5] made that A is not possible for every C.]10
We must understand ‘that which belongs to every’ with no limitation in respect of time, e.g. to the present or to a particular period, but without qualification. For it is by the help of such propositions that we make deductions, since if the proposition is understood with reference to the present moment, there cannot be a deduction. [10] For nothing perhaps prevents man belonging at a particular time to everything that is moving, i.e. if nothing else were moving; but moving is possible for every horse; yet man is possible for no horse. Further let the first term be animal, the middle moving, the last man. The propositions then will be as before, but the conclusion [15] necessary, not possible. For man is necessarily animal. It is clear then that the universal must be understood without qualification, and not limited in respect of time.
Again let the proposition AB be universal and negative, and assume that A belongs to no B, but B possibly belongs to every C. These being laid down, it is [20] necessary that A possibly belongs to no C. Suppose that it cannot belong, and that B belongs to C, as above. It is necessary then that A belongs to some B; for we have a deduction in the third figure; but this is impossible. Thus it will be possible for A to [25] belong to no C; for if that is supposed false, the consequence is impossible. This deduction then does not establish possibility according to the definition, but that it belongs necessarily to none (for this is the contradictory of the assumption which [30] was made; for it was supposed that A necessarily belongs to some C, but a deduction per impossibile establishes the contradictory assertion). Further, it is clear also from an example that the conclusion will not establish possibility. Let A be raven, B intelligent, and C man. A then belongs to no B; for no intelligent thing is a raven. [35] But B is possible for every C; for every man may be intelligent. But A necessarily belongs to no C; so the conclusion does not establish possibility. But neither is it always necessary. Let A be moving, B science, C man. A then will belong to no B [40] but B is possible for every C. And the conclusion will not be necessary. For it is not necessary that no man should move; indeed it is not necessary that some man should [35a1] move. Clearly then the conclusion establishes that it belongs necessarily to none. But we must take our terms better.
If the negative relates to the minor extreme and indicates possibility, from the [5] actual propositions taken there can be no deduction, but if the possible proposition is converted, a deduction will be possible, as before. Let A belong to every B, and let B possibly belong to no C. If the terms are arranged thus, nothing necessarily follows; [10] but if BC is converted and it is assumed that B is possible for every C, a deduction results as before; for the terms are in the same relative positions. Likewise if both the relations are negative, if AB indicates that it does not belong, and BC that it possibly belongs to none. Through the propositions actually taken nothing necessary [15] results in any way; but if the possible proposition is converted, we shall have a deduction. Suppose that A belongs to no B, and B may belong to no C. Through these comes nothing necessary. But if B is assumed to be possible for every C (and this is true) and if the proposition AB remains as before, we shall again have the [20] same deduction. But if it be assumed that B does not belong to every C, instead of possibly not belonging, there cannot be a deduction at all, whether the proposition AB is negative or affirmative. As common instances of a necessary and positive relation we may take the terms white, animal, snow; of an impossible relation, white, animal, pitch.
[25] Clearly then if the terms are universal, and one of the propositions is simple, the other possible, whenever the proposition relating to the minor extreme is possible, a deduction always results, only sometimes it results from the propositions that are taken, sometimes it requires the conversion of one proposition. We have [30] stated when each of these happens and the reason why. But if one of the relations is universal, the other particular, then whenever the one relating to the major extreme is universal and possible, whether affirmative or negative, and the particular is affirmative and simple, there will be a perfect deduction, just as when the terms are [35] universal. The demonstration is the same as before. But whenever the one relating to the major extreme is universal, but simple rather than possible, and the other is particular and possible, whether both are negative or affirmative, or one is negative, the other affirmative, in all cases there will be an imperfect deduction. Only some of them will be proved per impossibile, others by the conversion of the possible [35b1] proposition, as has been shown above. And a deduction will be possible by means of conversion when the proposition relating to the major extreme is universal and simple, and the other particular, negative, and possible, e.g. if A belongs to every B [5] or to no B, and B may not belong to some C. For if BC is converted in respect of possibility, a deduction results. But whenever the particular is simple and negative, there cannot be a deduction. As instances of the positive relation we may take the terms white, animal, snow; of the negative, white, animal, pitch. For the [10] demonstration must be made through the indefinite nature of the particular proposition. But if the universal relates to the minor extreme, and the particular to the major, whether either is negative or affirmative, possible or simple, in no way is a deduction possible. Nor is a deduction possible when the propositions are [15] particular or indefinite, whether possible or simple, or the one possible, the other simple. The demonstration is the same as above. As instances of the necessary and positive relation we may take the terms animal, white, man; of the impossible relation, animal, white, garment. It is evident then that if the proposition relating to [20] the major extreme is universal, a deduction always results, but if the one relating to the minor is universal nothing at all can ever be proved.
16 · Whenever one proposition indicates necessity, the other possibility, there will be a deduction when the terms are related as before; and a perfect [25] deduction when the necessity relates to the minor extreme. If the terms are affirmative the conclusion will be possible, not simple, whether they are universal or not; but if one is affirmative, the other negative, when the affirmative is necessary the conclusion will be possible, not negative and simple; but when the negative is necessary the conclusion will be both possible negative, and simple negative, [30] whether the terms are universal or not. Possibility in the conclusion must be understood in the same manner as before. There cannot be a deduction to a necessary negative; for ‘not necessarily to belong’ is different from ‘necessarily not [35] to belong’.
If the terms are affirmative, clearly the conclusion which follows is not necessary. Suppose A necessarily belongs to every B, and let B be possible for every C. We shall have an imperfect deduction that A may belong to every C. That it is imperfect is clear from the proof; for it will be proved in the same manner as above. [36a1] Again, let A be possible for every B, and let B necessarily belong to every C. We shall then have a deduction that A may belong to every C, not that A does belong to [5] every C; and it is perfect, not imperfect; for it is perfected directly through the original propositions.
But if the propositions are not similar in quality, suppose first that the negative is necessary, and let A be possible for no B, but let B be possible for every C. It is necessary then that A belongs to no C. For suppose A to belong to every C or to some [10] C. Now we assumed that A is not possible for any B. Since then the negative is convertible, B is not possible for any A. But A is supposed to belong to every C or to some C. Consequently B will not be possible for any C or for every C. But it was [15] originally laid down that B is possible for every C. And it is clear that the possibility of not belonging can be deduced, since the fact of not belonging can be. Again, let the affirmative proposition be necessary, and let A possibly not belong to any B, and let B necessarily belong to every C. The deduction will be perfect, but it will [20] establish a possible negative, not a simple negative. For the proposition relating to the major was assumed in this way; and further it is not possible to prove per impossibile. For if it were supposed that A belongs to some C, and it is laid down that A possibly does not belong to any B, no impossible relation between B and C [25] follows from this. But if the negative relates to the minor extreme, when it indicates possibility a deduction is possible by conversion, as above; but when impossibility, not. Nor again when both are negative, and the one relating to the minor is not possible. The same terms as before serve both for the positive relation, white, [30] animal, snow, and for the negative relation, white, animal, pitch.
The same relation will obtain in particular deductions. Whenever the negative is necessary, the conclusion will be negative and simple: e.g. if it is not possible that [35] A should belong to any B, but B may belong to some C, it is necessary that A should not belong to some C. For if A belongs to every C, but cannot belong to any B, neither can B belong to any A. So if A belongs to every C, B can belong to no C. But it was laid down that B may belong to some C. But when the particular affirmative [36b1] in the negative deduction, i.e. BC, or the universal in the affirmative i.e. AB, is necessary, there will not be a simple conclusion. The demonstration is the same as before. But if the term relating to the minor extreme is universal, and possible, whether affirmative or negative, and the particular is necessary, there cannot be a [5] deduction. Terms where the relation is positive and necessary: animal, white, man; where it is necessary and negative: animal, white, garment. But when the universal is necessary, the particular possible, if the universal is negative we may take the [10] terms animal, white, raven to illustrate the positive relation, or animal, white, pitch to illustrate the negative; and if the universal is affirmative we may take the terms animal, white, swan to illustrate the positive relation, and animal, white, snow to illustrate the impossible relation. Nor again is a deduction possible when the propositions are indefinite, or both particular. Terms applicable in either case to illustrate the positive relation are animal, white, man; to illustrate the negative, [15] animal, white, inanimate. For the relation of animal to some white, and of white to some inanimate, is both necessary and positive and necessary and negative. Similarly if the relation is possible; so the terms may be used for all cases.
Clearly then from what has been said a deduction results or not from similar [20] relations of the terms whether we are dealing with simple or with necessary propositions, with this exception, that if the negative proposition is simple the conclusion is possible, but if the negative is necessary the conclusion is both possible and negative simple. [It is clear also that all deductions are imperfect and are [25] perfected by means of the figures above mentioned.]11
17 · In the second figure whenever both propositions are possible, no deduction is possible, whether they are affirmative or negative, universal or particular. But when one indicates belonging, the other possibility, if the affirmative indicates belonging no deduction is possible, but if the universal negative does a [30] conclusion can always be drawn. Similarly when one proposition is necessary, the other possible. Here also we must understand the term ‘possible’ in the conclusions in the same sense as before.
First we must prove that the negative possible proposition is not convertible, [35] e.g. if A may belong to no B, it does not follow that B may belong to no A. For suppose it to follow and assume that B may belong to no A. Since then possible affirmations are convertible with negations, whether they are contraries or contradictories, and since B may belong to no A, it is clear that B may belong to [37a1] every A. But this is false; for if all this can be that, it does not follow that all that can be this: consequently the negative proposition is not convertible. Further, there is no reason why A may not belong to no B, while B necessarily does not belong to some [5] A; e.g. it is possible that no man should be white (for it is also possible that every man should be white), but it is not true to say that it is possible that no white thing should be a man; for many white things are necessarily not men, and the necessary (as we saw) is other than the possible.
Moreover it is not possible to prove the convertibility of these propositions by a reductio ad absurdum, i.e. by claiming that since it is false that B may belong to no [10] A, it is true that it cannot belong to no A (for the one statement is the contradictory of the other); but if this is so, it is true that B necessarily belongs to some A; and consequently A necessarily belongs to some B—but this is impossible. The argument cannot be admitted; for it does not follow that some A is necessarily B, if it is not possible that no A should be B. For the latter expression is used in two ways, [15] one if some A is necessarily B, another if some A is necessarily not B. For it is not true to say that that which necessarily does not belong to some of the As may not belong to every A, just as it is not true to say that what necessarily belongs to some A may belong to every A. If any one then should claim that because it is not possible [20] for C to belong to every D, it necessarily does not belong to some D, he would make a false assumption; for it does belong to every D, but because in some cases it belongs necessarily, therefore we say that it is not possible for it to belong to every. Hence both ‘necessarily belongs to some’ and ‘necessarily does not belong to some’ are opposed to ‘may belong to every’. Similarly also they are opposed to ‘may belong to [25] no’. It is clear then that in relation to what is possible and not possible, in the sense originally defined, we must assume, not that A necessarily belongs to some B, but that A necessarily does not belong to some B. But if this is assumed, no impossibility results; consequently there is no deduction. It is clear from what has been said that [30] the negative is not convertible.
This being proved, suppose it possible that A may belong to no B and every C. By means of conversion no deduction will result; for such a proposition, as has been said, is not convertible. Nor can a proof be obtained by a reductio; for if it is assumed that B cannot not belong to every C, no false consequence results; for A [35] may belong both to every C and to no C. In general, if there is a deduction, it is clear that its conclusion will be possible because neither of the propositions is simple; and this must be either affirmative or negative. But neither is possible. Suppose the [37b1] conclusion is affirmative: it will be proved by an example that the predicate cannot belong to the subject. Suppose the conclusion is negative: it will be proved that it is not possible but necessary. Let A be white, B man, C horse. It is possible then for A [5] to belong to all of the one and to none of the other. But it is not possible for B to belong or not to belong to C. That it is not possible for it to belong, is clear. For no horse is a man. Neither is it possible for it not to belong. For it is necessary that no horse should be a man, but the necessary we found to be different from the possible. [10] No deduction then results. A similar proof can be given if the negative is the other way about, or if both are affirmative or negative. The demonstration can be made by means of the same terms. And whenever one is universal, the other particular, or [15] both are particular or indefinite, or in whatever other way the propositions can be altered, the proof will always proceed through the same terms. Clearly then, if both the propositions are possible, no deduction results.
18 · But if one indicates belonging, the other possibility, if the affirmative [20] indicates belonging and the negative possibility no deduction will be possible, whether the terms are universal or particular. The proof is the same as above, and by means of the same terms. But when the affirmative indicates possibility, and the [25] negative belonging, we shall have a deduction. Suppose A belongs to no B, but can belong to every C. If the negative is converted, B will belong to no A. But A ex hypothesi can belong to every C: so a deduction is made, proving by means of the first figure that B may belong to no C. Similarly also if the negative relates to C. But [30] if both are negative, one indicating non-belonging, the other possibility, nothing follows necessarily from these premisses as they stand, but if the possible proposition is converted a deduction is formed to prove that B may belong to no C, [35] as before; for we shall again have the first figure. But if both are affirmative, no deduction will be possible. Terms for when the relation is positive: health, animal, man; for when it is negative: health, horse, man.
The same will hold good if the deductions are particular. Whenever the [38a1] affirmative is simple, whether universal or particular, no deduction is possible (this is proved similarly and by the same examples as above), but when the negative is, a conclusion can be drawn by means of conversion, as before. Again if both the [5] relations are negative, and the simple is universal, although no conclusion follows from the actual propositions, a deduction can be obtained by converting the possible as before. But if the negative is simple, but particular, no deduction is possible, whether the other proposition is affirmative or negative. Nor can a conclusion be [10] drawn when both are indefinite, whether affirmative or negative, or particular. The proof is the same and by the same terms.
19 · If one of the propositions indicates necessity, the other possibility, then if the negative is necessary there is a deduction not merely that it can not belong but also that it does not belong; but if the affirmative is necessary, no conclusion is [15] possible. Suppose that A necessarily belongs to no B, but may belong to every C. If the negative is converted B will belong to no A; but A ex hypothesi may belong to every C: so once more a conclusion is drawn by the first figure that B may belong to [20] no C. But at the same time it is clear that B will not belong to any C. For assume that it does; then if A cannot belong to any B, and B belongs to some C, A cannot belong to some C; but ex hypothesi it may belong to all. A similar proof can be given [25] if the negative relates to C.
Again let the affirmative be necessary, and the other possible; i.e. suppose that A may belong to no B, but necessarily belongs to every C. When the terms are arranged in this way no deduction is possible. For it turns out that B necessarily [30] does not belong to C. Let A be white, B man, C swan. White then necessarily belongs to swan, but may belong to no man; and man necessarily belongs to no swan. Clearly then we cannot draw a possible conclusion; for that which is necessary is [35] admittedly distinct from that which is possible. Nor again can we draw a necessary conclusion: for that presupposes that both propositions are necessary, or at any rate the negative one. Further it is possible also, when the terms are so arranged, that B should belong to C; for nothing prevents C falling under B, A being possible for [40] every B, and necessarily belonging to C; e.g. if C is awake, B animal, A motion. For motion necessarily belongs to what is awake, and is possible for every animal; and [38b1] everything that is awake is animal. Clearly then the conclusion cannot be negative and simple, if the relation must be positive when the terms are related as above. Nor can the opposite affirmations be established: consequently no deduction is possible. A similar proof is possible if the affirmative is the other way about. [5]
But if the propositions are similar in quality, when they are negative a deduction can always be formed by converting the possible as before. Suppose A necessarily does not belong to B, and possibly may not belong to C: if the [10] propositions are converted B belongs to no A, and A may possibly belong to every C; thus we have the first figure. Similarly if the negative relates to C. But if they are affirmative there cannot be a deduction. Clearly the conclusion cannot be a negative simple or a negative necessary proposition because no negative has been [15] laid down either in the simple or in the necessary mode. Nor can the conclusion be a possible negative proposition. For if the terms are so related, B necessarily will not belong to C; e.g. suppose that A is white, B swan, C man. Nor can the opposite [20] affirmations be established, since we have shown that B necessarily does not belong to C. A deduction then is not possible at all.
Similar relations will obtain in particular deductions. For whenever the [25] negative is universal and necessary, a deduction will always be possible to prove both that it may and that it does not (the proof proceeds by conversion); but when the affirmative is universal and necessary, no conclusion can be drawn. This can be proved in the same way as for universal deductions, and by the same terms. Nor is a conclusion possible when both are affirmative: this also may be proved as above. But [30] when both are negative, and the one which signifies non-belonging is universal and necessary, though nothing follows necessarily from the premisses as they are stated, [35] a conclusion can be drawn as above if the possible proposition is converted. But if both are indefinite or particular, no deduction can be formed. The same proof will serve, and the same terms.
It is clear then from what has been said that if the universal and negative proposition is necessary, a deduction is always possible, proving not merely that it [40] can not belong but also that it does not; but if the affirmative is necessary no conclusion can be drawn. It is clear too that a deduction is possible or not under the [39a1] same conditions whether simple or necessary. And it is clear that all the deductions are imperfect, and are completed by means of the figures mentioned.
[5] 20 · In the last figure a deduction is possible whether both or only one of the propositions is possible. When the propositions indicate possibility the conclusion will be possible; and also when one indicates possibility, the other belonging. But when the other is necessary, if it is affirmative the conclusion will be neither [10] necessary nor simple; but if it is negative there will be a deduction that it does not belong, as above. In these also we must understand the expression ‘possible’ in the conclusion in the same way as before.
[15] First let them be possible and suppose that both A and B may belong to every C. Since then the affirmative is convertible into a particular, and B may belong to every C, it follows that C may belong to some B. So, if A is possible for every C, and C is possible for some B, then A must be possible for some B. For we have got the [20] first figure. And if A may belong to no C, but B may belong to every C, it follows that A may not belong to some B; for we shall have the first figure again by conversion. But if both should be negative no necessary consequence will follow [25] from them as they are stated, but if the propositions are converted there will be a deduction as before. For if A and B may not belong to C, if ‘may belong’ is substituted we shall again have the first figure by means of conversion. But if one of the terms is universal, the other particular, a deduction will be possible, or not, [30] under the same arrangement of the terms as in the case of simple propositions. Suppose that A may belong to every C, and B to some C. We shall have the first figure again if the particular proposition is converted. For if A is possible for every [35] C, and C for some B, then A is possible for some B. Similarly if BC is universal. Likewise also if AC is negative, and BC affirmative; for we shall again have the first figure by conversion. But if both should be negative—the one universal and the [39b1] other particular—although no conclusion will follow from them as they are put, it will follow if they are converted, as above. But when both are indefinite or particular, no deduction can be formed; for A must belong both to every B and to no B. To illustrate the affirmative relation take the terms animal, man, white; to [5] illustrate the negative, take the terms horse, man, white, white being the middle term.
21 · If one of the propositions indicates belonging, the other possibility, the conclusion will be that it is possible, not that it belongs; and a deduction will be [10] possible under the same arrangement of the terms as before. First let them be affirmative: suppose that A belongs to every C, and B may belong to every C. If BC is converted, we shall have the first figure, and the conclusion that A may belong to some B. For when one of the propositions in the first figure indicates possibility, the [15] conclusion also (as we saw) is possible. Similarly if BC indicates belonging, AC possibility; or if AC is negative, BC affirmative, no matter which of the two is simple; in both cases the conclusion will be possible; for the first figure is obtained once more, and it has been proved that if one proposition indicates possibility in that [20] figure the conclusion also will be possible. But if the negative relates to the minor extreme, or if both are negative, no conclusion can be drawn from them as they stand, but if they are converted a deduction is obtained as before. [25]
If one of the propositions is universal, the other particular, then when both are affirmative, or when the universal is negative, the particular affirmative, we shall have the same sort of deductions; for all are completed by means of the first figure. So it is clear that the deduction will be not that it belongs but that it is possible. But [30] if the affirmative is universal, the negative particular, the proof will proceed by a reductio ad impossibile. Suppose that B belongs to every C, and A may not belong to some C: it follows that A may not belong to some B. For if A necessarily belongs [35] to every B, and B (as has been assumed) belongs to every C, A will necessarily belong to every C; for this has been proved before. But it was assumed that A may not belong to some C.
Whenever both are indefinite or particular, no deduction will be possible. The [40a1] demonstration is the same as before, and proceeds by means of the same terms.
22 · If one of the propositions is necessary, the other possible, when the [5] terms are affirmative a possible conclusion can always be drawn; when one is affirmative, the other negative, if the affirmative is necessary a possible negative can be inferred; but if the negative is necessary both a possible and a simple negative conclusion are possible. But a necessary negative conclusion will not be [10] possible, any more than in the other figures.
Suppose first that the terms are affirmative, i.e. that A necessarily belongs to every C, and B may belong to every C. Since then A must belong to every C, and C may belong to some B, it follows that A may (not does) belong to some B; for so it [15] resulted in the first figure. A similar proof may be given if BC is necessary, and AC is possible. Again suppose one is affirmative, the other negative, the affirmative being necessary, i.e. suppose A may belong to no C, but B necessarily belongs to [20] every C. We shall have the first figure once more; and—since the negative proposition indicates possibility—it is clear that the conclusion will be possible; for when the propositions stand thus in the first figure, the conclusion (as we found) is possible. But if the negative proposition is necessary, the conclusion will be not only [25] that A may not belong to some B but also that it does not belong to some B. For suppose that A necessarily does not belong to C, but B may belong to every C. If the affirmative BC is converted, we shall have the first figure, and the negative proposition is necessary. But when the propositions stood thus, it resulted that A [30] might not belong to some C, and that it did not belong to some C; consequently here it follows that A does not belong to some B. But when the negative relates to the minor extreme, if it is possible we shall have a deduction by altering the proposition, [35] as before; but if it is necessary no deduction can be formed. For A both necessarily belongs to every B, and cannot belong to any B. To illustrate the former take the terms sleep, sleeping horse, man; to illustrate the latter take the terms sleep, waking horse, man.
Similar results will obtain if one of the terms is related universally to the [40b1] middle, the other in part. If both are affirmative, the conclusion will be possible, not simple; and also when one is negative, the other affirmative, the latter being necessary. But when the negative is necessary, the conclusion also will be a simple [5] negative; for the same kind of proof can be given whether the terms are universal or not. For the deductions must be made perfect by means of the first figure, so that a result which follows in the first figure follows also in the third. But when the negative is universal and relates to the minor extreme, if it is possible a deduction [10] can be formed by means of conversion; but if it is necessary a deduction is not possible. The proof will follow the same course as for the universal deductions; and the same terms may be used.
It is clear then in this figure also when and how a deduction can be formed, and when the conclusion is possible, and when it is simple. It is evident also that all [15] deductions in this figure are imperfect, and that they are made perfect by means of the first figure.
23 · It is clear from what has been said that the deductions in these figures are made perfect by means of the universal deductions in the first figure and are [20] reduced to them. That every deduction without qualification can be so treated, will be clear presently, when it has been proved that every deduction is formed through one or other of these figures.
It is necessary that every demonstration and every deduction should prove either that something belongs or that it does not, and this either universally or in [25] part, and further either probatively or hypothetically. One sort of hypothetical proof is the reductio ad impossibile. Let us speak first of probative deductions; for after it has been proved in their case, the truth of our contention will be clear with regard to those which are proved per impossibile, and in general hypothetically.
[30] If then one wants to deduce that A belongs or does not belong to B, one must assume something of something. If now A should be assumed of B, the proposition originally in question will have been assumed. But if A should be assumed of C, but C should not be assumed of anything, nor anything of it, nor anything else of A, no [35] deduction will be possible. For nothing necessarily follows from the assumption of some one thing concerning some one thing. Thus we must take another proposition as well. If then A be assumed of something else, or something else of A, or something different of C, nothing prevents a deduction being formed, but it will not be in relation to B through the propositions taken. Nor when C belongs to something [41a1] else, and that to something else and so on, no connexion however being made with B, will a deduction be possible in relation to B. For in general we stated that no deduction can establish the attribution of one thing to another, unless some middle term is taken, which is somehow related to each by way of predication. For a deduction in general is made out of propositions, and a deduction referring to this [5] out of propositions with the same reference, and a deduction relating this to that proceeds through propositions which relate this to that. But it is impossible to take a proposition in reference to B, if we neither affirm nor deny anything of it; or again to take a proposition relating A to B, if we take nothing common, but affirm or deny peculiar attributes of each. So we must take a middle term relating to both, which [10] will connect the predications, if we are to have a deduction relating this to that. If then we must take something common in relation to both, and this is possible in three ways (either by predicating A of C, and C of B, or C of both, or both of C), and [15] these are the figures of which we have spoken, it is clear that every deduction must be made in one or other of these figures. The argument is the same if several middle terms should be necessary to establish the relation to B; for the figure will be the same whether there is one middle term or many. [20]
It is clear then that probative deductions are effected by means of the aforesaid figures; the following considerations will show that reductiones ad impossibile also are effected in the same way. For all who effect an argument per impossibile deduce what is false, and prove the original conclusion hypothetically when something impossible results from the assumption of its contradictory; e.g. that the [25] diagonal of the square is incommensurate with the side, because odd numbers are equal to evens if it is supposed to be commensurate. One deduces that odd numbers come out equal to evens, and one proves hypothetically the incommensurability of the diagonal, since a falsehood results from its contradictory. For this we found to [30] be deducing per impossibile, viz. proving something impossible by means of an hypothesis conceded at the beginning. Consequently, since the falsehood is established in reductions ad impossibile by a probative deduction, and the original conclusion is proved hypothetically, and we have already stated that probative [35] deductions are effected by means of these figures, it is evident that deductions per impossibile also will be made through these figures. Likewise all the other hypothetical deductions; for in every case the deduction leads up to the substituted proposition; but the original thesis is reached by means of a concession or some other hypothesis. But if this is true, every demonstration and every deduction must [41b1] be formed by means of the three figures mentioned above. But when this has been shown it is clear that every deduction is perfected by means of the first figure and is reducible to the universal deductions in this figure. [5]
24 · Further in every deduction one of the terms must be affirmative, and universality must be present: unless one of the premisses is universal either a deduction will not be possible, or it will not refer to the subject proposed, or the original position will be begged. Suppose we have to prove that pleasure in music is [10] good. If one should claim that pleasure is good without adding ‘every’, no deduction will be possible; if one should claim that some pleasure is good, then if it is different from pleasure in music, it is not relevant to the subject proposed; if it is this very pleasure, one is assuming that which was originally proposed. This is more obvious in geometrical proofs, e.g. that the angles at the base of an isosceles triangle are [15] equal. Suppose the lines A and B have been drawn to the centre. If then one should assume that the angle AC is equal to the angle BD, without claiming generally that angles of semicircles are equal; and again if one should assume that the angle C is equal to the angle D, without the additional assumption that every angle of a segment is equal to every other angle of the same segment; and further if one should assume that when equal angles are taken from the whole angles, which [20] are themselves equal, the remainders E and F are equal, he will beg the original position, unless he also assumes that when equals are taken from equals the remainders are equal.
It is clear then that in every deduction there must be a universal, and that a universal is proved only when all the terms are universal, while a particular is [25] proved in both cases; consequently if the conclusion is universal, the terms also must be universal, but if the terms are universal it is possible that the conclusion may not be universal. And it is clear also that in every deduction either both or one of the propositions must be like the conclusion. I mean not only in being affirmative or [30] negative, but also in being necessary, simple, or possible. We must consider also the other forms of predication.
It is clear also when a deduction in general can be made and when it cannot; and when a potential, when a perfect deduction can be formed; and that if a deduction is formed the terms must be arranged in one of the ways that have been [35] mentioned.
25 · It is clear too that every demonstration will proceed through three terms and no more, unless the same conclusion is established by different pairs of propositions; e.g. E may be established through A and B, and through C and D, or through A and B, or A and C and D. For nothing prevents there being several middles for the same terms. But in that case there is not one but several deductions. [42a1] Or again when each of A and B is obtained by deduction, e.g. A by means of D and E, and again B by means of F and G. Or one may be obtained by deduction, the other by induction. But thus also the deductions are many; for the conclusions are [5] many, e.g. A and B and C.
But if this can be called one deduction, not many, the same conclusion may be reached by several terms in this way, but it cannot be reached as C is established by means of A and B. Suppose that E is inferred from A, B, C, and D. It is necessary [10] then that of these one should be related to another as whole to part; for it has already been proved that if a deduction is formed some of its terms must be related in this way. Suppose then that A stands in this relation to B. Some conclusion then follows from them. It must either be E or one or other of C and D, or something other than these.
[15] If it is E the deduction will have A and B for its sole premisses. But if C and D are so related that one is whole, the other part, some conclusion will follow from them also; and it must be either E, or one or other of A and B, or something other than these. And if it is E, or A or B, either the deductions will be more than one, or the same thing happens to be inferred by means of several terms in the sense which we saw to be possible. But if the conclusion is other than E or A or B, the deductions [20] will be many, and unconnected with one another. But if C is not related to D as to make a deduction, the propositions will have been assumed to no purpose, unless for the sake of induction or of obscuring the argument or something of the sort.
But if from A and B there follows not E but some other conclusion, and if from [25] C and D either A or B follows or something else, then there are several deductions, and they do not establish the conclusion proposed; for we assumed that the deduction proved E. And if no conclusion follows from C and D, it turns out that these propositions have been assumed to no purpose, and the deduction does not prove the original proposition.
So it is clear that every demonstration and every deduction will proceed [30] through three terms only.
This being evident, it is clear that a conclusion follows from two propositions and not from more than two for the three terms make two propositions unless a new proposition is assumed, as was said at the beginning, to perfect the deductions. It is [35] clear therefore that in whatever deductive argument the propositions through which the main conclusion follows (for some of the preceding conclusions must be propositions) are not even in number, this argument either has not been deduced or it has assumed more than was necessary to establish its thesis.
If then deductions are taken with respect to their main propositions, every [42b1] deduction will consist of an even number of propositions and an odd number of terms (for the terms exceed the propositions by one), and the conclusions will be half the number of the propositions. But whenever a conclusion is reached by means of preliminary deductions or by means of several continuous middle terms, e.g. AB [5] by means of C and D, the number of the terms will similarly exceed that of the propositions by one (for the extra term must either be added outside or inserted; but in either case it follows that the relations of predication are one fewer than the terms related, and the propositions will be equal in number to the relations of predication). [10] The propositions however will not always be even, the terms odd; but they will alternate—when the propositions are even, the terms must be odd; when the terms are even, the propositions must be odd; for along with one term one proposition is added, if a term is added from any quarter. Consequently since the propositions were (as we saw) even, and the terms odd, we must make them alternately even and [15] odd at each addition. But the conclusions will not follow the same arrangement either in respect to the terms or to the propositions. For if one term is added, conclusions will be added less by one than the pre-existing terms; for the conclusion is drawn not in relation to the single term last added, but in relation to all the rest, [20] e.g. if to ABC the term D is added, two conclusions are thereby added, one in relation to A, the other in relation to B. Similarly with any further additions. And similarly too if the term is inserted in the middle; for a deduction will not be effected in relation to one term only. Consequently the conclusions will be much more [25] numerous than the terms or the propositions.
26 · Since we understand the subjects with which deductions are concerned, what sort of conclusion is established in each figure, and in how many ways this is done, it is evident to us both what sort of problem is difficult and what sort is easy to [30] prove. For that which is concluded in many figures and through many moods is easier; that which is concluded in few figures and through few moods is more difficult to attempt. The universal affirmative is proved by means of the first figure only and by this in only one way; the negative is proved both through the first figure [35] and through the second, through the first in one way, through the second in two. The particular affirmative is proved through the first and through the last figure, in one way through the first, in three through the last. The particular negative is proved in all the figures, but once in the first, in two ways in the second, in three in [43a1] the third. It is clear then that the universal affirmative is most difficult to establish and most easy to overthrow. In general, universals are easier game for the destroyer than particulars; for whether the predicate belongs to none or not to some, they are [5] destroyed; and the particular negative is proved in all the figures, the universal negative in two. Similarly with negatives: the original statement is destroyed, whether the predicate belongs to all or to some; and this we found possible in two figures. But particular statements can be refuted in one way only—by proving that the predicate belongs either to all or to none. But particular statements are easier to [10] establish; for proof is possible in more figures and through more moods. And in general we must not forget that it is possible to refute statements by means of one another, I mean, universal statements by means of particular, and particular statements by means of universal; but it is not possible to establish universal statements by means of particular, though it is possible to establish particular statements by means of universal. At the same time it is evident that it is easier to [15] refute than to establish.
The manner in which every deduction is produced, the number of the terms and propositions through which it proceeds, the relation of the propositions to one another, the character of the problem proved in each figure, and the number of the figures appropriate to each problem, all these matters are clear from what has been said.
[20] 27 · We must now state how we may ourselves always have a supply of deductions in reference to the problem proposed and by what road we may reach the principles relative to the problem; for no doubt we ought not only to investigate the construction of deductions, but also to have the power of making them.
[25] Of all the things which exist some are such that they cannot be predicated of anything else truly and universally, e.g. Cleon and Callias, i.e. the individual and sensible, but other things may be predicated of them (for each of these is both man and animal); and some things are themselves predicated of others, but nothing prior [30] is predicated of them; and some are predicated of others, and yet others of them, e.g. man of Callias and animal of man. It is clear then that some things are naturally not said of anything; for as a rule each sensible thing is such that it cannot be predicated [35] of anything, save incidentally—for we sometimes say that that white object is Socrates, or that that which approaches is Callias. We shall explain in another place12 that there is an upward limit also to the process of predicating; for the present we must assume this. Of these it is not possible to demonstrate another predicate, save as a matter of opinion, but these may be predicated of other things. Neither can individuals be predicated of other things, though other things can be [40] predicated of them. Whatever lies between these limits can be spoken of in both ways: they may be said of others, and others said of them. And as a rule arguments and inquiries are concerned with these things.
We must select the propositions suitable to each problem in this manner: first [43b1] we must lay down the subject and the definitions and the properties of the thing; next we must lay down those attributes which follow the thing, and again those which the thing follows, and those which cannot belong to it. (Those to which it [5] cannot belong need not be selected, because the negative is convertible.) Of the attributes which follow we must distinguish those which fall within the definition, those which are predicated as properties, and those which are predicated as accidents, and of the latter those which apparently and those which really belong. The larger the supply a man has of these, the more quickly will he reach a [10] conclusion; and in proportion as he apprehends those which are truer, the more cogently will he demonstrate.
But he must select not those which follow some of the thing but those which follow the thing as a whole, e.g. not what follows some man but what follows every man; for deduction proceeds through universal propositions. If it is indefinite, it is uncertain whether the proposition is universal, but if it is definite, the matter is [15] clear. Similarly one must select those attributes which the subject follows as wholes, for the reason given. But that which follows one must not suppose to follow as a whole, e.g. that every animal follows man or every science music, but only that it follows, without qualification, as indeed we state it in a proposition—for the other statement is useless and impossible, e.g. that every man is every animal or justice is [20] every good. But that which something follows receives the mark ‘every’. Whenever the subject, for which we must obtain the attributes that follow, is contained by something else, what follows or does not follow the universal must not be selected in dealing with the subordinate term (for these attributes have been taken in dealing [25] with the superior term; for what follows animal also follows man, and what does not belong to animal does not belong to man); but we must choose those attributes which are peculiar to each subject. For some things are peculiar to the species as distinct from the genus; for there must be attributes peculiar to the different species. Nor in the case of the universal should we select those things which the contained term follows, e.g. taking for animal what man follows. It is necessary [30] indeed, if animal follows man, that it should follow all these also. But these belong more properly to the choice of what concerns man. One must take also what follows a thing—and what it follows—for the most part; for in the case of problems about what holds for the most part, deductions depend on propositions, either all or some, [35] which hold for the most part (for the conclusion of each deduction is similar to its principles). Again, we should not select things which follow everything; for no deduction can be made from them (the reason why this is so will be made clear in what follows).
[40] 28 · If men wish to establish something about some whole, they must look to the subjects of that which is being established (the subjects of which it happens to be asserted), and the attributes which follow that of which it is to be predicated. For [44a1] if any of these subjects is the same as any of these attributes, the one must belong to the other. But if the purpose is to establish not a universal but a particular proposition, they must look for the terms which each follows; for if any of these are identical, the attribute must belong to some of the subject. Whenever the one term has to belong to none of the other, one must look to the consequents of the subject, [5] and to those attributes which cannot be present in the predicate in question; or conversely to the attributes which cannot be present in the subject, and to the consequents of the predicate. If any members of these groups are identical, one of the terms in question cannot belong to any of the other. For sometimes a deduction in the first figure results, sometimes a deduction in the second. But if the object is to establish a particular negative proposition, we must find antecedents of the subject [10] in question and attributes which cannot belong to the predicate in question. If any members of these two groups are identical, it follows that one of the terms in question does not belong to some of the other.
Perhaps each of these statements will become clearer in the following way. Suppose the consequents of A are designated by B, the antecedents of A by C, [15] attributes which cannot belong to A by D. Suppose again that the attributes of E are designated by F, the antecedents of E by G, and attributes which cannot belong to E by H. If then one of the Cs should be identical with one of the Fs, A must belong to every E; for F belongs to every E, and A to every C: consequently A belongs to every [20] E. If C and G are identical, A must belong to some E; for A follows C, and E follows every G. If F and D are identical, A will belong to none of the Es by a preliminary deduction; for since the negative is convertible, and F is identical with D, A will [25] belong to none of the Fs, but F belongs to every E. Again, if B and H are identical, A will belong to none of the Es; for B will belong to every A, but to no E; for it was assumed to be identical with H, and H belonged to none of the Es. If D and G are [30] identical, A will not belong to some of the Es; for it will not belong to G, because it does not belong to D; but G falls under E; consequently A will not belong to some of the Es. If B is identical with G, there will be a converted deduction; for E will belong to every A, since B belongs to A and E to B (for B was found to be identical with G); but that A should belong to every E is not necessary, but it must belong to some E [35] because it is possible to convert the universal statement into a particular.
It is clear then that in every problem we must look to the aforesaid relations of the subject and predicate; for all deductions proceed through these. But if we are seeking consequents and antecedents we must look especially for those which are primary and universal, e.g. in reference to E we must look to KF rather than to F [44b1] alone, and in reference to A we must look to KC rather than to C alone. For if A belongs to KF, it belongs both to F and to E; but if it does not follow KF, it may yet follow F. Similarly we must consider the antecedents of A itself; for if a term follows the primary antecedents, it will follow those also which are subordinate, but if it does not follow the former, it may yet follow the latter. [5]
It is clear too that the inquiry proceeds through the three terms and the two propositions, and that all the deductions proceed through the aforesaid figures. For it is proved that A belongs to every E, whenever an identical term is found among the Cs and Fs. This will be the middle term; A and E will be the extremes. So the [10] first figure is formed. And A will belong to some E, whenever C and G are apprehended to be the same. This is the last figure; for G becomes the middle term. And A will belong to no E, when D and F are identical. Thus we have both the first figure and the middle figure; the first, because A belongs to no F, since the negative is convertible, and F belongs to every E; the middle figure because D belongs to no [15] A, and to every E. And A will not belong to some E, whenever D and G are identical. This is the last figure; for A will belong to no G, and E will belong to every G. Clearly then all the deductions proceed through the aforesaid figures, and we must [20] not select consequents of everything, because no deduction is produced from them. For (as we saw) it is not possible at all to establish a proposition from consequents, and it is not possible to refute by means of a consequent of everything; for the middle term must belong to the one, and not belong to the other.
It is clear too that other methods of inquiry by selection are useless to produce [25] a deduction, e.g. if the consequents of the terms in question are identical, or if the antecedents of A are identical with those attributes which cannot belong to E, or if those attributes are identical which cannot belong to either term; for no deduction is produced by means of these. For if the consequents are identical, e.g. B and F, we [30] have the middle figure with both propositions affirmative; if the antecedents of A are identical with attributes which cannot belong to E, e.g. C with H, we have the first figure with its proposition relating to the minor extreme negative. If attributes which cannot belong to either term are identical, e.g. C and H, both propositions are [35] negative, either in the first or in the middle figure. But no deduction is possible in these ways.
It is evident too that we must find out which terms in this inquiry are identical, not which are different or contrary, first because the object of our investigation is the middle term, and the middle term must be not diverse but identical. Secondly, [45a1] wherever it happens that a deduction results from taking contraries or terms which cannot belong to the same thing, all arguments can be reduced to the aforesaid moods, e.g. if B and F are contraries or cannot belong to the same thing. For if these are taken, a deduction will be formed to prove that A belongs to none of the Es, not [5] however from the assumptions made but in the aforesaid mood. For B will belong to every A and to no E. Consequently B must be identical with one of the Hs. [Again, if B and G cannot belong to the same thing, it follows that A will not belong to some of [10] the Es; for then too we shall have the middle figure; for B will belong to every A and to no E. Consequently B must be identical with some of the Hs. For the fact that B and G cannot belong to the same thing differs in no way from the fact that B is [15] identical with some of the Hs; for that includes everything which cannot belong to E.]13
It is clear then that from these inquiries taken by themselves no deduction results; but if B and F are contraries B must be identical with one of the Hs, and the [20] deduction results through these terms. It turns out then that those who inquire in this manner are looking gratuitously for some other way than the necessary way because they have failed to observe the identity of the Bs with the Hs.
29 · Deductions which lead to impossible conclusions are similar to probative deductions; they also are formed by means of the consequents and antecedents [25] of the terms in question. In both cases the same inquiry is involved. For what is proved probatively may also be deduced per impossibile by means of the same terms; and what is proved per impossibile may also be proved probatively, e.g. that A belongs to no E. For suppose A to belong to some E: then since B belongs to every [30] A and A to some E, B will belong to some of the Es; but it was assumed that it belongs to none. Again we may prove that A belongs to some E; for A belonged to no E, and E belongs to every G, A will belong to none of the Gs; but it was assumed to belong to all. Similarly with the other problems. The proof per impossibile will [35] always and in all cases be from the consequents and antecedents of the terms in question. Whatever the problem, the same inquiry is necessary whether one wishes to use a probative deduction or a reduction to impossibility. For both the demonstrations start from the same terms; e.g. suppose it has been proved that A [40] belongs to no E, because it turns out that otherwise B belongs to some E and this is impossible—if now it is assumed that B belongs to no E and to every A, it is clear [45b1] that A will belong to no E. Again if it has been deduced probatively that A belongs to no E, assume that A belongs to some E and it will be proved per impossibile to belong to no E. Similarly with the rest. In all cases it is necessary to find some [5] common term other than the subjects of inquiry, to which the deduction establishing the false conclusion may relate, so that if this proposition is converted, and the other remains as it is, the deduction will be probative by means of the same terms. For the probative deduction differs from the reductio ad impossibile in this: [10] in the probative both propositions are laid down in accordance with the truth, in the reductio ad impossibile one is assumed falsely.
These points will be made clearer by the sequel,14 when we discuss reduction to impossibility: at present this much must be clear, that we must look to the same [15] terms whether we wish to use a probative deduction or a reduction to impossibility. In the other hypothetical deductions (I mean those which proceed by substitution or by positing a certain quality), the inquiry will be directed to the terms of the problem to be proved—not the terms of the original problem, but the substitutes; and the method of the inquiry will be the same as before. But we must consider and [20] determine in how many ways hypothetical deductions are possible.
Each of the problems then can be proved in the manner described; but it is possible to deduce some of them in another way, e.g. universal problems by the inquiry which leads up to a particular conclusion, with the addition of an hypothesis. For if the Cs and the Gs should be identical, but E should be assumed to belong to the Gs only, then A would belong to every E; and again if the Ds and the [25] Gs should be identical, but E should be predicated of the Gs only, it follows that A will belong to none of the Es. Clearly then we must consider the matter in this way also. The method is the same whether the relation is necessary or possible. For the inquiry will be the same, and the deduction will proceed through terms arranged in [30] the same order whether a possible or a simple proposition is proved. We must find in the case of possible relations, as well as terms that belong, terms which can belong though they actually do not; for we have proved that a deduction which establishes a possible relation proceeds through these terms as well. Similarly also with the other modes of predication. [35]
It is clear then from what has been said not only that all deductions can be formed in this way, but also that they cannot be formed in any other. For every deduction has been proved to be formed through one of the aforementioned figures, and these cannot be composed through other terms than the consequents and [40] antecedents of the terms in question; for from these we obtain the propositions and find the middle term. Consequently a deduction cannot be formed by means of [46a1] other terms.
30 · The method is the same in all cases, in philosophy and in any art or study. We must look for the attributes and the subjects of both our terms, and we [5] must supply ourselves with as many of these as possible, and consider them by means of the three terms, refuting statements in one way, establishing them in another, in the pursuit of truth starting from an arrangement of the terms in accordance with truth, while if we look for dialectical deductions we must start from plausible propositions. The principles of deductions have been stated in [10] general terms, both how they are characterized and how we must hunt for them, so as not to look to everything that is said about the terms of the problem or to the same points whether we are establishing or refuting, or again whether we are establishing of all or of some, and whether we are refuting of all or some; we must look to fewer [15] points and they must be definite. We have also stated how we must select with reference to each thing that is, e.g. about good or knowledge. But in each science the principles which are peculiar are the most numerous. Consequently it is the business of experience to give the principles which belong to each subject. I mean for example that astronomical experience supplies the principles of astronomical science; for once the phenomena were adequately apprehended, the demonstrations [20] of astronomy were discovered. Similarly with any other art or science. Consequently, if the attributes of the thing are apprehended, our business will then be to exhibit readily the demonstrations. For if none of the true attributes of things had been omitted in the survey, we should be able to discover the proof and demonstrate [25] everything which admitted of proof, and to make that clear, whose nature does not admit of proof.
Thus we have explained fairly well in general terms how we must select propositions: we have discussed the matter precisely in the treatise concerning [30] dialectic.15
31 · It is easy to see that division by genera is a small part of the method we have described; for division is, so to speak, a weak deduction; for what it ought to prove, it begs, and it always deduces something more general than the attribute in [35] question. First, this very point had escaped all those who used the method of division; and they attempted to persuade men that it was possible to make a demonstration of substance and essence. Consequently they did not understand what it is possible to deduce by division, nor did they understand that it was possible to deduce in the manner we have described. In demonstrations, when there is a need to deduce that something belongs, the middle term through which the deduction is [46b1] formed must always be inferior to and not comprehend the first of the extremes. But division has a contrary intention; for it takes the universal as middle. Let animal be the term signified by A, mortal by B, and immortal by C, and let man, whose [5] definition is to be got, be signified by D. The man who divides assumes that every animal is either mortal or immortal: i.e. whatever is A is all either B or C. Again, always dividing, he lays it down that man is an animal, so he assumes A of D as belonging to it. Now the deduction is that every D is either B or C, consequently [10] man must be either mortal or immortal, but it is not necessary that man should be a mortal animal—this is begged: and this is what ought to have been deduced. And again, taking A as mortal animal, B as footed, C as footless, and D as man, he [15] assumes in the same way that A inheres either in B or in C (for every mortal animal is either footed or footless), and he assumes A of D (for he assumed man to be a mortal animal); consequently it is necessary that man should be either a footed or a footless animal; but it is not necessary that man should be footed—this he assumes: and it is just this again which he ought to have proved. Always dividing then in this [20] way it turns out that they assume as middle the universal term, and as extremes that which ought to have been the subject of proof and the differentiae. In conclusion, they do not make it clear, and show it to be necessary, that this is man or whatever the subject of inquiry may be; for they pursue the other method altogether, never even [25] suspecting the presence of the rich supply of evidence which might be used.
It is clear that it is neither possible to refute by this method, nor to deduce about an accident or property of a thing, nor about its genus, nor in cases in which it is unknown whether it is thus or thus, e.g. whether the diagonal is incommensurate [30] or commensurate. For if he assumes that every length is either commensurate or incommensurate, and the diagonal is a length, he has deduced that the diagonal is either incommensurate or commensurate. But if he should assume that it is incommensurate, he will have assumed what he ought to have proved. He cannot then prove it; for this is his method, but proof is not possible by this method. (Let A [35] stand for incommensurate or commensurate, B for length, C for diagonal). It is clear then that this method of investigation is not suitable for every inquiry, nor is it useful in those cases in which it is thought to be most suitable.
32 · From what has been said it is clear from what elements demonstrations are formed and in what manner, and to what points we must look in each problem. Our next business is to state how we can reduce deductions to the aforementioned figures; for this part of the inquiry still remains. If we should investigate the [47a1] production of deductions and had the power of discovering them, and further if we could resolve the deductions produced into the aforementioned figures, our original project would be brought to a conclusion. It will happen at the same time that what [5] has been already said will be confirmed and its truth made clearer by what we are about to say. For everything that is true must in every respect agree with itself.
First then we must attempt to select the two propositions of the deduction (for [10] it is easier to divide into large parts than into small, and the composite parts are larger than the elements out of which they are made); next we must inquire which are universal and which particular, and if both have not been stated, we must ourselves assume the one which is missing. For sometimes men put forward the universal, but do not posit the proposition which is contained in it, either in writing [15] or in discussion: or men put these forward, but omit those through which they are inferred, and invite the concession of others to no purpose. We must inquire then whether anything unnecessary has been assumed, or anything necessary has been omitted, and we must posit the one and take away the other, until we have reached the two propositions; for unless we have these, we cannot reduce arguments put [20] forward in the way described. In some arguments it is easy to see what is wanting, but some escape us, and appear to be deductions, because something necessary results from what has been laid down, e.g. if the assumptions were made that substance is not annihilated by the annihilation of what is not substance, and that if [25] the elements out of which a thing is made are annihilated, then that which is made out of them is destroyed: these propositions being laid down, it is necessary that any part of substance is substance; this has not however been deduced from the assumptions, but propositions are wanting. Again if it is necessary that animal should exist, if man does, and that substance should exist, if animal does, it is necessary that substance should exist if man does; but as yet the conclusion has not [30] been deduced; for the propositions are not in the shape we described.
We are deceived in such cases because something necessary results from what is assumed, since deduction also is necessary. But that which is necessary is wider than deduction; for every deduction is necessary, but not everything which is necessary is a deduction. Consequently, though something results when certain [35] propositions are assumed, we must not try to reduce it directly, but must first take the two propositions, then divide them into their terms. We must take that term as middle which is stated in both the propositions; for it is necessary that the middle should be found in both in all the figures.
If then the middle term is a predicate and a subject of predication, or if it is a [47b1] predicate, and something else is denied of it, we shall have the first figure; if it both is a predicate and is denied of something, the middle figure; if other things are predicated of it, or one is denied, the other predicated, the last figure. For it was thus that we found the middle term placed in each figure. It is placed similarly too if [5] the propositions are not universal; for the middle term is determined in the same way. Clearly then, if the same term is not said more than once in the course of an argument, a deduction cannot be made; for a middle term has not been taken. Since [10] we know what sort of problem is established in each figure, and in which the universal and in what sort the particular is established, clearly we must not look for all the figures, but for that which is appropriate to the problem in hand. If it is established in more figures than one, we shall recognize the figure by the position of the middle term.
[15] 33 · Men are frequently deceived about deduction because the inference is necessary, as has been said above; sometimes they are deceived by the similarity in the positing of the terms; and this ought not to escape our notice. E.g. if A is said of B, and B of C: it would seem that a deduction is possible since the terms stand thus; [20] but nothing necessary results, nor does a deduction. Let A represent being eternal, B Aristomenes as an object of thought, C Aristomenes. It is true then that A belongs to B. For Aristomenes as an object of thought is eternal. But B also belongs to C; for [25] Aristomenes is Aristomenes as an object of thought. But A does not belong to C; for Aristomenes is perishable. For no deduction was made although the terms stood thus: that required that the proposition AB should be stated universally. But this is false, that every Aristomenes who is an object of thought is eternal, since [30] Aristomenes is perishable. Again let C stand of Miccalus, B for musical Miccalus, A for perishing to-morrow. It is true to predicate B for C; for Miccalus is musical Miccalus. Also A can be predicted of B; for musical Miccalus might perish to-morrow. But to say A of C is false at any rate. This argument then is identical [35] with the former; for it is not true universally that musical Miccalus perishes to-morrow; but unless this is assumed, no deduction (as we have shown) is possible.
This deception then arises through ignoring a small distinction. For we accept the conclusion as though it made no difference whether we said ‘This belongs to that’ or ‘That belongs to all of that’.
[48a1] 34 · Men will frequently fall into error through not setting out the terms of the proposition well, e.g. suppose A to be health, B disease, C man. It is true to say that A cannot belong to any B (for health belongs to no disease) and again that B [5] belongs to every C (for every man is capable of disease). It would seem to follow that health cannot belong to any man. The reason for this is that the terms are not set out well in expression, since if the things which are in the conditions are [10] substituted, no deduction can be made, e.g. if healthy is substituted for health and diseased for disease. For it is not true to say that being healthy cannot belong to one who is diseased. But unless this is assumed no conclusion results, save in respect of possibility; but such a conclusion is not impossible; for it is possible that health [15] should belong to no man. Again the falsity may occur in a similar way in the middle figure: it is not possible that health should belong to any disease, but it is possible that health should belong to every man, consequently it is not possible that disease should belong to any man. In the third figure the falsity results in reference to possibility. For health and disease, and knowledge and ignorance, and in general contraries, may belong to the same thing, but cannot belong to one another. This is [20] not in agreement with what was said before; for we stated that when several things could belong to the same thing, they could belong to one another.
It is evident then that in all these cases the error arises from the setting out of the terms; for if the things that are in the conditions are substituted, no falsity [25] arises. It is clear then that in such propositions what possesses the condition ought always to be substituted for the condition and taken as the term.
35 · We must not always seek to set out the terms in a single word; for we shall often have phrases to which no single name is equivalent. Hence it is difficult [30] to reduce deductions with such terms. Sometimes too error will result from such a search, e.g. the belief that deduction can establish something immediate. Let A stand for two right angles, B for triangle, C for isosceles triangle. A then belongs to C because of B; but A belongs to B not in virtue of anything else (for the triangle in [35] virtue of its own nature contains two right angles); consequently there will be no middle term for AB, although it is demonstrable. For it is clear that the middle must not always be assumed to be an individual thing, but sometimes a phrase, as happens in the case mentioned.
36 · That the first term belongs to the middle, and the middle to the [40] extreme, must not be understood in the sense that they can always be predicated of one another or that the first term will be predicated of the middle in the same way as [48b1] the middle is predicated of the last term. The same holds if the premisses are negative. But we must suppose that ‘to belong’ has as many meanings as the ways in which ‘to be’ and ‘it is true to say this is that’ are used. Take for example the statement that there is a single science of contraries. Let A stand for there being a [5] single science, and B for things which are contrary to one another. Then A belongs to B, not in the sense that contraries are a single science, but in the sense that it is true to say of the contraries that there is a single science of them.
It happens sometimes that the first term is said of the middle, but the middle is [10] not said of the third term, e.g. 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, though wisdom is knowledge. Sometimes the middle term is said of the [15] third, but the first is not said of the middle, e.g. if there is a science of everything that has a quality, or is a contrary, and the good both is a contrary and has a quality, the conclusion is that there is a science of the good—but the good is not a science, nor is that which has a quality or is a contrary, though the good is both of these. Sometimes neither the first term is said of the middle, nor the middle of the third, [20] while the first is sometimes said of the third, and sometimes not; e.g. if there is a genus of that of which there is a science, and there is a science of the good, we conclude that there is a genus of the good. But nothing is predicated of anything. And if that of which there is a science is a genus, and there is a science of the good, [25] we conclude that the good is a genus. The first term then is predicated of the extreme, but the terms are not said of one another.