by Aristotle
2 · With regard to the substance and nature of the one we must ask in which [10] of two ways it exists. This is the very question that we reviewed in our discussion of problems, viz. what the one is and how we must conceive of it, whether we must take the one itself as being a substance (as both the Pythagoreans say in earlier and Plato in later times), or there is, rather, an underlying nature and it is to be explained more intelligibly and more in the manner of the natural philosophers, of whom one [15] says the one is love, another says it is air, and another the indefinite.
If then no universal can be a substance, as has been said in our discussion of substance and being, and if being itself cannot be a substance in the sense of a one apart from the many (for it is common to the many), but is only a predicate, clearly [20] the one also cannot be a substance; for being and one are the most universal of all predicates. Therefore, on the one hand, classes are not certain entities and substances separable from other things; and on the other hand the one cannot be a class, for the same reasons for which being and substance cannot be classes.
Further, this must hold good in all categories alike. Now ‘being’ and ‘unity’ [25] have an equal number of meanings; so that since in the sphere of qualities the one is something definite—some entity—and similarly in the sphere of quantities, clearly we must also ask in general what unity is, as we must ask what being is, since it is not enough to say that its nature is just to be unity or being. But in colours the one is [30] a colour, e.g. white—the other colours are observed to be produced out of this and black, and black is the privation of white, as darkness of light. Therefore if all existent things were colours, existent things would have been a number, indeed, but of what? Clearly of colours; and the ‘one’ would have been a particular ‘one’, e.g. [35] white. And similarly if all existent things were tunes, they would have been a number, but a number of quarter-tones, and their substance would not have been number; and the one would have been something whose substance was not the one [1054a1] but the quarter-tone. And similarly if all existent things had been articulate sounds, they would have been a number of letters, and the one would have been a vowel. And if all existent things were rectilineal figures, they would have been a number of figures, and the one would have been the triangle. And the same argument applies to all other classes. Since, therefore, while there are numbers and a one both in [5] affections and in qualities and in quantities and in movement, in all cases the number is a number of particular things and the one is one something, and its substance is not to be one, the same must be true of substances; for it is true of all cases alike. That the one, then, in every class is a definite thing, and in no case is its [10] nature just this—viz. unity, is evident; but as in colours the one itself which we must seek is one colour, so too in substance the one itself is one substance.
And that in a sense unity means the same as being is clear from the fact that it follows the categories in as many ways, and is not comprised within any category, e.g. neither in substance nor in quality, but is related to them just as being is; and [15] from the fact that in ‘one man’ nothing more is predicated than in ‘man’, just as being is nothing apart from substance or quality or quantity; and to be one is just to be a particular thing.
3 · The one and the many are opposed in several ways, of which one is the [20] opposition of the one and plurality as indivisible and divisible; for that which is either divided or divisible is called a plurality, and that which is indivisible or not divided is called one. Now since opposition is of four kinds, and one of these two terms is privative in meaning, they must be contraries, and neither contradictory [25] nor correlative. And the one gets its meaning and explanation from its contrary, the indivisible from the divisible, because plurality and the divisible is more perceptible than the indivisible, so that in formula plurality is prior to the indivisible, because of the conditions of perception. To the one belong, as we indicated graphically in our [30] distinction of the contraries, the same and the like and the equal, and to plurality belong the other and the unlike and the unequal.
‘The same’ has several meanings: we sometimes mean ‘the same numerically’; again, we call a thing the same if it is one both in formula and in number, e.g. you are one with yourself both in form and in matter; and again, if the formula of its [1054b1] primary substance is one, e.g. equal straight lines are the same, and so are equal and equal-angled quadrilaterals—there are many such, but in these equality constitutes unity.
Things are like if, not being absolutely the same, nor without difference in their compound substance, they are the same in form, e.g. the larger square is like [5] the smaller, and unequal straight lines are like; they are like, but not absolutely the same. Other things are like, if, having the same form, and being things in which difference of degree is possible, they have no difference of degree. Other things, if they have a quality that is in form one and the same—e.g. whiteness—in a greater [10] or less degree, are called like because their form is one. Other things are called like if the qualities they have in common are more numerous than those in which they differ—either the qualities in general or the prominent qualities, e.g. tin is like silver, qua white, and gold is like fire, qua yellow and red.
Evidently, then, ‘other’ and ‘unlike’ also have several meanings. And the other [15] in one sense is the opposite of the same (so that everything is either the same as or other than everything else). In another sense things are other unless both their matter and their formula are one (so that you are other than your neighbour). The other in the third sense is exemplified in the objects of mathematics. ‘Other’ or ‘the same’ can for this reason be predicated of everything with regard to everything else,—but only if the things are one and existent, for the other is not the [20] contradictory of the same; which is why it is not predicated of non-existent things (while ‘not the same’ is so predicated). It is predicated of all existing things; for if a thing is both existent and one, it is naturally either one or not one. The other, then, and the same are thus opposed.
But difference is not the same as otherness. For the other and that which it is other than need not be other in some definite respect (for everything that exists is [25] either other or the same), but that which is different from anything is different in some respect, so that there must be something identical whereby they differ. And this identical thing is genus or species; for all things that differ differ either in genus or in species, in genus if the things have not their matter in common and are not generated out of each other (i.e. if they belong to different figures of predication), [30] and in species if they have the same genus (the genus is that same thing which both the different things are said to be in respect of their substance). And contraries are different, and contrariety is a kind of difference. That we are right in this supposition is shown by induction. For they are all seen to be different; they are not merely other, but some are other in genus, and others are in the same line of [1055a1] predication, and therefore in the same genus, and the same in genus. We have distinguished elsewhere what sort of things are the same or other in genus.
4 · Since things which differ may differ from one another more or less, there [5] is also a greatest difference, and this I call contrariety. That contrariety is the greatest difference is made clear by induction. For things which differ in genus have no way to one another, but are too far distant and are not comparable; and for things that differ in species the extremes from which generation takes place are the contraries; and the distance between extremes—and therefore that between the contraries—is the greatest.
[10] But that which is greatest in each class is complete. For that is greatest which cannot be exceeded, and that is complete beyond which nothing can be found. For the complete difference marks the end (just as the other things which are called complete are so called because they have attained an end), and beyond the end there is nothing; for in everything it is the extreme and includes all else, and therefore [15] there is nothing b
eyond the end, and the complete needs nothing further. From this, then, it is clear that contrariety is complete difference; and as contraries are so called in several senses, their modes of completeness will answer to the various modes of contrariety which attach to them.
This being so, evidently one thing cannot have more than one contrary, for [20] neither can there be anything more extreme than the extreme, nor can there be more than two extremes for the one interval. And in general if contrariety is a difference, and if a difference must be between two things, then the complete difference must be so too.
And the other definitions are also necessarily true of contraries. For in each case the complete difference is the greatest difference. We cannot get anything beyond it, whether the things differ in genus or in species; for it has been shown that [25] there is no difference between anything and the things outside its genus; and among these things the complete difference is the greatest. And the things in the same genus which differ most are contraries; for the complete difference is the greatest difference among these. And the things in the same receptive material which differ most are contrary; for the matter is the same for contraries. And of the things which [30] are dealt with by the same faculty the most different are contrary; for one science deals with one class of things, and in these the complete difference is the greatest.
The primary contrariety is that between state and privation—not every privation, however (for ‘privation’ has several meanings), but that which is complete. And the other contraries must be called so with reference to these, some [35] because they possess these, others because they produce or tend to produce them, others because they are acquisitions or losses of these or of other contraries. Now if the kinds of opposition are contradiction and privation and contrariety and relation, and of these the first is contradiction, and contradiction admits of no intermediate, [1055b1] while contraries admit of one, clearly contradiction and contrariety are not the same. But privation is a kind of contradiction; for what suffers privation, either in general or in some determinate way, is either that which is quite incapable of having some attribute or that which, being of such a nature as to have it, has it not; here we [5] have already a variety of meanings, which have been distinguished elsewhere. Privation, therefore, is a contradiction or incapacity which is determinate or taken along with the receptive material. This is the reason why, while contradiction does not admit of an intermediate, privation sometimes does; for everything is equal or not equal, but not everything is equal or unequal, or if it is, it is only within the [10] sphere of that which is receptive of equality. If, then, the changes which happen to the matter start from the contraries, and proceed either from the form and the possession of the form or from a privation of the form or shape, clearly all contrariety is a privation. (But perhaps not all privation is contrariety, the reason [15] being that that which suffers privation may suffer it in several ways.) For the extremes from which the changes proceed are contraries.
And this is obvious also by induction. For every contrariety involves, as one of its terms, a privation. But not all cases are alike; inequality is the privation of equality and unlikeness of likeness, and vice is the privation of excellence. But the [20] cases differ as has been said; in one case we mean simply that the thing suffers privation, in another case that it does so at a certain time or in a certain part (e.g. at a certain age or in the proper part), or throughout. This is why in some cases there is something in between (there are men who are neither good nor bad), and in others there is not (a number must be either odd or even). Further, some contraries have their subject defined, others have not.—Therefore it is evident that one of the [25] contraries is always privative; but it is enough if this is true of the first—i.e., the generic—contraries, e.g. the one and the many; for the others can be referred to these.
[30] 5 · Since one thing has one contrary, we might raise the question how the one is opposed to the many and the equal to the great and the small.—For if we use the word ‘whether’ only in an opposition, asking e.g. whether it is white or black, and whether it is white or not white (we do not ask whether it is a man or white, unless [35] we are proceeding on a prior assumption and asking e.g. whether it was Cleon or Socrates that came. But this is not necessary in any class of things. Yet even this is an extension from the case of opposites; for opposites alone cannot be present together; and we assume this incompatibility here in asking which of the two came; [1056a1] for if they might both have come, the question would have been absurd. But if they might, even so this falls just as much into an opposition—that of the one and the many, i.e. we ask whether both came or only one):—if, then, the question ‘whether’ is always concerned with opposites, and we can ask whether it is greater or less or [5] equal, what is the opposition between the greater and the less, and the equal? The equal is not contrary either to one alone or to both; for why should it be contrary to the greater rather than to the less? Further, the equal is contrary to the unequal. Therefore it will be contrary to more things than one. But if the unequal means the same as both the greater and the less together, the equal will be opposite to both [10] (and the difficulty supports those who say the unequal is a ‘two’), but it follows that one thing has two contraries, which is impossible. Again, the equal is evidently intermediate between the great and the small, but no contrary is either observed to be intermediate, nor, from its definition, can be so; for it would not be a perfect contrary if it were intermediate between any two things, but rather it always has something intermediate between itself and something else.
[15] It remains, then, that it is opposed either as negation or as privation. It cannot be opposite to one of the two; for why to the great rather than to the small? It is then the privative negation of both. Therefore also ‘whether’ is said with reference to both—not to one of the two (e.g. we ask whether it is greater or equal, or whether it [20] is equal or less); there are always three cases. But it is not a necessary privation; for not everything which is not greater or less is equal, but only the things which are of such a nature as to have these attributes. The equal, then, is that which is neither great nor small and is naturally fitted to be either great or small; and it is opposed to both as a privative negation (and therefore is also intermediate). And that which is [25] neither good nor bad is opposed to both, but has no name (for each of these has several meanings and the receptive material is not one); but that which is neither white nor black has more claim to a name. Yet even this has not one name, though the colours of which this negation is privately predicated are in a way limited; for [30] they must be either grey or yellow or something else of the kind. Therefore it is an incorrect criticism that is passed by those who think that all such phrases are used in the same way, so that that which is neither a shoe nor a hand would be intermediate between a shoe and a hand, since that which is neither good nor bad is intermediate between the good and the bad,—as if there must be an intermediate in all cases. This result does not necessarily follow. For the combined denial of opposites applies [35] when there is an intermediate and a certain natural interval; but in the other case there is no difference; for the things, the denials of which are combined, belong to [1056b1] different classes, so that the substratum is not one.
6 · We might raise similar questions about the one and the many. For if the many are absolutely opposed to the one, certain impossible results follow. One will [5] then be few; for the many are opposed also to the few. Further, two will be many, since the double is multiple, and double derives from two; therefore one will be few; for what is that in comparison with which two are called many, except one and that which is few? For there is nothing fewer. Further, if a lot and few are in plurality [10] what the long and the short are in length, and whatever is a lot is also many, and the many are a lot (unless, indeed, there is a difference in the case of an easily-bounded continuum), the few will be a plurality. Therefore one is a plurality, if it is few; and this must be so, if two are many. But perhaps, while
the many are in a sense said to [15] be a lot, it is with a difference, e.g. there is a lot of water, not many waters.
But ‘many’ is applied to the things that are divisible; in one sense it means a plurality which is excessive either absolutely or relatively (while ‘few’ is similarly a plurality which is deficient), and in another sense it means number, in which sense alone it is opposed to the one. For we say ‘one or many’, just as if one were to say [20] ‘one and ones’ or ‘white thing and white things’, or to compare the things that have been measured with the measure. It is in this sense also that multiples are so called. For each number is said to be many because it consists of ones and because each number is measurable by one; and it is many as that which is opposed to one, not to the few. In this sense, then, even two is many—not however in the sense of a [25] plurality which is excessive either relatively or absolutely; it is the first plurality. But without qualification two is few; for it is the first plurality which is deficient. (For this reason Anaxagoras was not right in leaving the subject with the statement that all things were together, boundless both in multitude and in smallness—where by ‘and in smallness’ he meant ‘and in fewness’; for they could not have been [30] boundless in fewness.) For it is not one, as some say, but two, that make a few.
The one is opposed then to the many in numbers as measure to thing measurable; and these are opposed as relatives which are not from their very nature relative. We have distinguished elsewhere the two senses in which relatives are so [35] called—some as contraries, others as knowledge to thing known, a term being called relative because another is relative to it. There is nothing to prevent one from being [1057a1] fewer than something, e.g. than two; for if it is fewer, it is not therefore few. Plurality is as it were the class to which number belongs; for number is plurality measurable by one. And one and number are in a sense opposed, not as contrary, but as we have said some relative terms are opposed; for inasmuch as one is measure and [5] the other measurable, they are opposed. This is why not everything that is one is a number, i.e. if the thing is indivisible it is not a number. But though knowledge is similarly spoken of as related to the knowable, the relation does not work out similarly, for while knowledge might be thought to be the measure, and the [10] knowable the thing measured, the fact is that all knowledge is knowable, but not all that is knowable is knowledge, because in a sense knowledge is measured by the knowable.—Plurality is contrary neither to the few (the many being contrary to this as excessive plurality to plurality exceeded), nor to the one in every sense; but in one [15] sense they are contrary, as has been said, because the former is divisible and the latter indivisible, while in another sense they are relative (as knowledge is to the knowable), if plurality is number and the one is measure.