by Aristotle
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.