by Aristotle
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 belo
ng 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. The
se 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.