Since one thing has one contrary, it might be asked in what sense unity is opposed to plurality, and the equal to the great and to the small. For if we always use the word “whether” in an antithesis—e.g., “whether it is white or black,” or “whether it is white or not” (but we do not ask “whether it is a man or white,” unless we are proceeding upon some assumption, and asking, for instance, whether it was Cleon who came or Socrates. This is not a necessary disjunction in any class of things, but is derived from the use in the case of opposites—for it is only opposites that cannot be true at the same time—and we have this same use here in the question “which of the two came?” 1056afor if both alternatives were possible, the question would be absurd; but even so the question falls into an antithesis: that of “one” or “many”—i.e., “whether both came, or one”)— if, then, the question “whether” is always concerned with opposites, and we can ask “whether it is greater or smaller, or equal,” what is the nature of the antithesis between “equal” and “greater or smaller”? It is contrary neither to one only, nor to both: for (a) it is no more contrary to the greater than to the smaller; (b) “equal” is contrary to “unequal,” and thus it will be contrary to more than one thing; (c) if “unequal” means the same as both “greater” and “smaller” at the same time, “equal” must still be opposed to them both: This difficulty supports the theory¹ that “the unequal” is a duality. But the result is that one thing is contrary to two; which is impossible. Further, it is apparent that “equal” is intermediate between “great” and “small,” but it is not apparent that any contrariety is intermediate, nor can it be, by definition; for it could not be complete if it were the intermediate of something, but rather it always has something intermediate between itself and the other extreme. It remains, then, that it is opposed either as negation or as privation. Now it cannot be so opposed to one of the two, for it is no more opposed to the great than to the small. Therefore it is a privative negation of both. For this reason we say “whether” with reference to both, and not to one of the two—e.g., “whether it is greater or equal,” or “whether it is equal or smaller”; there are always three alternatives. But it is not a necessary privation; for not everything is equal which is not greater or smaller, but only things which would naturally have these attributes. The equal, then, is that which is neither great nor small, but would naturally be either great or small; and it is opposed to both as a privative negation, and therefore is intermediate between them. And that which is neither good nor bad is opposed to both, but it has no name (for each of these terms has several meanings, and there is no one material which is receptive of both); that which is neither white nor black is better entitled to a name, although even this has no single name, but the colors of which this negation is privatively predicated are to a certain extent limited; for it must be either grey or buff or something similar. Therefore those persons are wrong in their criticism who imagine that all terms are used analogously, so that that which is neither a shoe nor a hand will be intermediate between “shoe” and “hand,” because that which is neither good nor bad is intermediate between good and bad—as though there must be an intermediate in all cases; but this does not necessarily follow. For the one is a joint negation of opposites where there is an intermediate and a natural interval; 1056bbut in the other case there is no question of difference, since the joint negation applies to things which are in different genera, and therefore the substrate is not one.²

A similar question might be raised about “one” and “many.” For if “many” is absolutely opposed to “one,” certain impossibilities result. (1) One will be few; for “many” is also opposed to “few.” (2) Two will be many; since “twofold” is “manifold,” and “twofold” is derived from two. Therefore one will be few; for in what relation can two be many if not in relation to one, which must therefore be few? for there can be nothing less. (3) If “much” and “little” are in plurality what “long” and “short” are in length, and if whatever is “much” is also “many,” and “many” is “much” (unless indeed there is a difference in the case of a plastic continuum³), “few” will be a plurality. Therefore one will be a plurality, if it is few; and this necessarily follows if two is many. Presumably, however, although “many” in a sense means “much,” there is a distinction; e.g., water is called “much” but not “many.” To all things, however, which are divisible the term “many” is applicable: in one sense, if there is a plurality which involves excess either absolutely or relatively (and similarly “few” is a plurality involving defect); and in another in the sense of number, in which case it is opposed to “one” only. For we say “one or many” just as if we were to say “one and ones,” or “white thing and white things,” or were to compare the things measured with the measure. Multiples, too, are spoken of in this way; for every number is “many,” because it consists of “ones,” and because every number is measurable by one; and also as being the opposite of one, and not of few. In this sense even two is many; but as a plurality involving excess either relatively or absolutely it is not many, but the first plurality. Two is, however, absolutely few; because it is the first plurality involving defect (hence Anaxagoras⁴ was not right in leaving the subject by saying “all things were together, infinite both in multitude and in smallness”; instead of “in smallness” he should have said “in fewness,”⁵ for things cannot be infinite in fewness), since fewness is constituted not by one, as some hold, but by two. In the sphere of numbers “one” is opposed to many as the measure to the measurable, i.e., as relative terms are opposed which are not of their own nature relative. We have distinguished elsewhere⁶ that things are called relative in two senses—either as being contraries, or as knowledge is related to the knowable, A being related to B because B is described in relation to A. 1057a There is no reason why one should not be fewer than something, e.g. two; for if it is fewer it is not therefore few. Plurality is, as it were, a genus of number, since number is a plurality measurable by one. And in a sense one and number are opposed; not, however, as being contrary, but as we have said some relative terms to be; for it is qua measure and measurable that they are opposed. (Hence not everything which is one is a number—e.g., a thing which is indivisible.) But although the relation between knowledge and the knowable is said to be similar to this, it turns out not to be similar. For it would seem that knowledge is a measure, and the knowable that which is measurable by it; but it happens that whereas all knowledge is knowable, the knowable is not always knowledge, because in a way knowledge is measured by the knowable.⁷ Plurality is contrary neither to the few (whose real contrary is the many, as an excessive plurality to an exceeded plurality) nor in all senses to one; but they are contrary in one sense (as has been said) as being the one divisible and the other indivisible; and in another as being relative (just as knowledge is relative to the knowable) if plurality is a number and one is the measure.

Since there can be, and in some cases is, an intermediate between contraries, intermediates must be composed of contraries; for all intermediates are in the same genus as the things between which they are intermediate. By intermediates we mean those things into which that which changes must first change. E.g., if we change from the highest string to the lowest by the smallest gradations we shall first come to the intermediate notes; and in the case of colors if we change from white to black we shall come to red and grey before we come to black; and similarly in other cases. But change from one genus into another is impossible except accidentally; e.g., from color to shape. Therefore intermediates must be in the same genus as one another and as the things between which they are intermediate. But all intermediates are between certain opposites, for it is only from these per se that change is possible. Hence there can be no intermediate between things which are not opposites; for then there would be change also between things which are not opposites. Of things which are opposites, contradiction has no intermediate term (for contradiction means this: an antithesis one term of which must apply to any given thing, and which contains no intermediate term); of the remaining types of opposites some are relative, others privative, and others contrary. Those relative opposites which are not contrary have no intermediate. The reason for this is that they are not in the same genus— 1057bfor what is intermediate between knowledge and the knowable?—but between great and small there is an intermediate. Now since intermediates are in the same genus, as has been shown, and are between contraries, they must be composed of those contraries. For the contraries must either belong to a genus or not. And if there is a genus in such a way that it is something prior to the contraries, then the differentiae which constitute the contrary species (for species consist of genus and differentiae) will be contraries in a prior sense. E.g., if white and black are contraries, and the one is a penetrative⁸ and the other a compressive color, these differentiae, “penetrative” and “compressive,” are prior, and so are opposed to each other in a prior sense. But it is the species which have contrary differentiae that are more truly contraries; the other, i.e. intermediate, species will consist of genus and differentiae. E.g., all colors which are intermediate between white and black should be described by their genus (i.e. color) and by certain differentiae. But these differentiae will not be the primary contraries; otherwise every thing will be either white or black. Therefore they will be different from the primary contraries. Therefore they will be intermediate between them, and the primary differentiae will be “the penetrative” and “the compressive.” Thus we must first investigate the contraries which are not contained in a genus, and discover of what their intermediates are composed. For things which are in the same genus must either be composed of differentiae which are not compounded with the genus, or be incomposite. Contraries are not compounded with one another, and are therefore first principles; but intermediates are either all incomposite or none of them. Now from the contraries something is generated in such a way that change will reach it before reaching the contraries themselves (for there must be something which is less in degree than one contrary and greater than the other). Therefore this also will be intermediate between the contraries. Hence all the other intermediates must be composite; for that which is greater in degree than one contrary and less than the other is in some sense a compound of the contraries of which it is said to be greater in degree than one and less than the other. And since there is nothing else homogeneous which is prior to the contraries, all intermediates must be composed of contraries. Therefore all the lower terms, both contraries and intermediates, must be composed of the primary contraries. Thus it is clear that intermediates are all in the same genus, and are between contraries, and are all composed of contraries.