[71a1] 1 · All teaching and all intellectual learning come about from already existing knowledge. This is evident if we consider it in every case; for the mathematical sciences are acquired in this fashion, and so is each of the other arts. [5] And similarly too with arguments—both deductive and inductive arguments proceed in this way; for both produce their teaching through what we are already aware of, the former getting their premisses as from men who grasp them, the latter proving the universal through the particular’s being clear. (And rhetorical arguments [10] too persuade in the same way; for they do so either through examples, which is induction, or through enthymemes, which is deduction.)
It is necessary to be already aware of things in two ways: of some things it is necessary to believe already that they are, of some one must grasp what the thing said is, and of others both—e.g. of the fact that everything is either affirmed or [15] denied truly, one must believe that it is; of the triangle, that it signifies this; and of the unit both (both what it signifies and that it is). For each of these is not equally clear to us.
But you can become familiar by being familiar earlier with some things but getting knowledge of the others at the very same time—i.e. of whatever happens to be under the universal of which you have knowledge. For that every triangle has [20] angles equal to two right angles was already known; but that there is a triangle in the semicircle here became familiar at the same time as the induction. (For in some cases learning occurs in this way, and the last term does not become familiar through the middle—in cases dealing with what are in fact particulars and not said of any underlying subject.)
[25] Before the induction, or before getting a deduction, you should perhaps be said to understand in a way—but in another way not. For if you did not know if it is simpliciter, how did you know that it has two right angles simpliciter? But it is clear that you understand it in this sense—that you understand it universally—but you do not understand it simpliciter. (Otherwise the puzzle in the Meno1 will result; for [30] you will learn either nothing or what you know.)
For one should not argue in the way in which some people attempt to solve it: Do you or don’t you know of every pair that it is even? And when you said Yes, they brought forward some pair of which you did not think that it was, nor therefore that it was even. For they solve it by denying that people know of every pair that it is even, but only of anything of which they know that it is a pair.—Yet they know it of [71b]1 that which they have the demonstration about and which they got their premisses about; and they got them not about everything of which they know that it is a triangle or that it is a number, but of every number and triangle simpliciter. For no proposition of such a type is assumed (that what you know to be a number… or what you know to be rectilineal . . . ), but they are assumed as holding of every [5] case.
But nothing, I think, prevents one from in a sense understanding and in a sense being ignorant of what one is learning; for what is absurd is not that you should know in some sense what you are learning, but that you should know it in this sense, i.e. in the way and sense in which you are learning it.
2 · We think we understand a thing simpliciter (and not in the sophistic fashion accidentally) whenever we think we are aware both that the explanation [10] because of which the object is is its explanation, and that it is not possible for this to be otherwise. It is clear, then, that to understand is something of this sort; for both those who do not understand and those who do understand—the former think they are themselves in such a state, and those who do understand actually are. Hence [15] that of which there is understanding simpliciter cannot be otherwise.
Now whether there is also another type of understanding we shall say later; but we say now that we do know through demonstration. By demonstration I mean a scientific deduction; and by scientific I mean one in virtue of which, by having it, we understand something.
If, then, understanding is as we posited, it is necessary for demonstrative [20] understanding in particular to depend on things which are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusion (for in this way the principles will also be appropriate to what is being proved). For there will be deduction even without these conditions, but there will not be demonstration; for it will not produce understanding.
Now they must be true because one cannot understand what is not the [25] case—e.g. that the diagonal is commensurate. And they must depend on what is primitive and non-demonstrable because otherwise you will not understand if you do not have a demonstration of them; for to understand that of which there is a demonstration non-accidentally is to have a demonstration. They must be both explanatory and more familiar and prior—explanatory because we only understand [30] when we know the explanation; and prior, if they are explanatory, and we are already aware of them not only in the sense of grasping them but also of knowing that they are.
Things are prior and more familiar in two ways; for it is not the same to be prior by nature and prior in relation to us, nor to be more familiar and more familiar [72a1] to us. I call prior and more familiar in relation to us what is nearer to perception, prior and more familiar simpliciter what is further away. What is most universal is [5] furthest away, and the particulars are nearest; and these are opposite to each other.
Depending on things that are primitive is depending on appropriate principles; for I call the same thing primitive and a principle. A principle of a demonstration is an immediate proposition, and an immediate proposition is one to which there is no other prior. A proposition is the one part of a contradiction,2 one thing said of one; it [10] is dialectical if it assumes indifferently either part, demonstrative if it determinately assumes the one that is true.3 [A statement is either part of a contradiction.]4 A contradiction is an opposition of which of itself excludes any intermediate; and the part of a contradiction saying something of something is an affirmation, the one saying something from something is a denial.
[15] An immediate deductive principle I call a posit if one cannot prove it but it is not necessary for anyone who is to learn anything to grasp it; and one which it is necessary for anyone who is going to learn anything whatever to grasp, I call an axiom (for there are some such things); for we are accustomed to use this name especially of such things. A posit which assumes either of the parts of a [20] contradiction—i.e., I mean, that something is or that something is not—I call a supposition; one without this, a definition. For a definition is a posit (for the arithmetician posits that a unit is what is quantitatively indivisible) but not a supposition (for what a unit is and that a unit is are not the same).
[25] Since one should both be convinced of and know the object by having a deduction of the sort we call a demonstration, and since this is the case when these things on which the deduction depends are the case, it is necessary not only to be already aware of the primitives (either all or some of them) but actually to be better aware of them. For a thing always belongs better to that thing because of which it [30] belongs—e.g. that because of which we love is better loved. Hence if we know and are convinced because of the primitives, we both know and are convinced of them better, since it is because of them that we know and are convinced of what is posterior.
It is not possible to be better convinced than one is of what one knows, of what one in fact neither knows nor is more happily disposed toward than if one in fact knew. But this will result if someone who is convinced because of a demonstration is [35] not already aware of the primitives, for it is necessary to be better convinced of the principles (either all or some of them) than of the conclusion.
Anyone who is going to have understanding through demonstration must not only be familiar with the principles and better convinced of them than of what is [72b1] being proved, but also there must be no other thing more convincing to him or more familiar among the opposites of the principles on which a deduction of the contrary error may depend—if anyone who understands simpliciter must be unpersuadable.
3 · Now some think that because one must understand the primitives there is [5] no understanding at all; others that there is, but that there are demonstrations of everything. Neither of these views is either true or necessary.
For the one party, supposing that one cannot understand in another way,5 claim that we are led back ad infinitum on the grounds that we would not understand what is posterior because of what is prior if there are no primitives; and they argue correctly, for it is impossible to go through infinitely many things. And if [10] it comes to a stop and there are principles, they say that these are unknowable since there is no demonstration of them, which alone they say is understanding; but if one cannot know the primitives, neither can what depends on them be understood simpliciter or properly, but only on the supposition that they are the case.
The other party agrees about understanding; for it, they say, occurs only [15] through demonstration. But they argue that nothing prevents there being demonstration of everything; for it is possible for the demonstration to come about in a circle and reciprocally.
But we say that neither is all understanding demonstrative, but in the case of the immediates it is non-demonstrable—and that this is necessary is evident; for if it [20] is necessary to understand the things which are prior and on which the demonstration depends, and it comes to a stop at some time, it is necessary for these immediates to be non-demonstrable. So as to that we argue thus; and we also say that there is not only understanding but also some principle of understanding by which we become familiar with the definitions.
And that it is impossible to demonstrate simpliciter in a circle is clear, if [25] demonstration must depend on what is prior and more familiar; for it is impossible for the same things at the same time to be prior and posterior to the same things—unless one is so in another way (i.e. one in relation to us, the other simpliciter), which induction makes familiar. But if so, knowing simpliciter will not [30] have been properly defined, but will be twofold. Or is the other demonstration not demonstration simpliciter in that it comes from about what is more familiar to us?
There results for those who say that demonstration is circular not only what has just been described, but also that they say nothing other than that this is the case if this is the case—and it is easy to prove everything in this way. It is clear that [35] this results if we posit three terms. (For it makes no difference to say that it bends back through many terms or through few, or through few or two.) For whenever if A is the case, of necessity B is, and if this then C, then if A is the case C will be the case. Thus given that if A is the case it is necessary that B is, and if this is that A is (for that is what being circular is)—let A be C: so to say that if B is the case A is, is [73a1] to say that C is, and this implies that if A is the case C is. But C is the same as A. Hence it results that those who assert that demonstration is circular say nothing but that if A is the case A is the case. And it is easy to prove everything in this way. [5]
Moreover, not even this is possible except in the case of things which follow one another, as properties do. Now if a single thing is laid down, it has been proved6 that it is never necessary that anything else should be the case (by a single thing I mean [10] that neither if one term nor if one posit is posited . . .), but two posits are the first and fewest from which it is possible, if at all, actually to deduce something. Now if A follows B and C, and these follow one another and A, in this way it is possible to prove all the postulates reciprocally in the first figure, as was proved in the account [15] of deduction.7 (And it was also proved that in the other figures either no deduction comes about or none about what was assumed.) But one cannot in any way prove circularly things which are not counterpredicated; hence, since there are few such things in demonstrations, it is evident that it is both empty and impossible to say that demonstration is reciprocal and that because of this there can be demonstration [20] of everything.
4 · Since it is impossible for that of which there is understanding simpliciter to be otherwise, what is understandable in virtue of demonstrative understanding will be necessary (it is demonstrative if we have it by having a demonstration). Demonstration, therefore, is deduction from what is necessary. We must therefore [25] grasp on what things and what sort of things demonstrations depend. And first let us define what we mean by holding of every case and what by in itself and what by universally.
Now I say that something holds of every case if it does not hold in some cases and not others, nor at some times and not at others; e.g. if animal holds of every [30] man, then if it is true to call this a man, it is true to call him an animal too; and if he is now the one, he is the other too; and the same goes if there is a point in every line. Evidence: when asked if something holds of every case, we bring our objections in this way—either if in some cases it does not hold or if at some time it does not.
One thing belongs to another in itself both if it belongs to it in what it is—e.g. [35] line to triangle and point to line (for their substance depends on these and they belong in the account which says what they are)—and also if the things it belongs to themselves belong in the account which makes clear what it is—e.g. straight belongs to line and so does curved, and odd and even to number, and prime and [73b1] composite, and equilateral and oblong; and for all these there belongs in the account which says what they are in the one case line, and in the others number. And similarly in other cases too it is such things that I say belong to something in itself; [5] and what belongs in neither way I call accidental, e.g. musical or white to animal.
Again, what is not said of some other underlying subject—as what is walking is something different walking (and white),8 while a substance, and whatever signifies some ‘this,’ is just what it is without being something else. Thus things which are not said of an underlying subject I call things in themselves, and those which are said of an underlying subject I call accidentals.
[10] Again, in another way what belongs to something because of itself belongs to it in itself, and what does not belong because of itself is accidental—e.g. if it lightened when he was walking, that was accidental; for it was not because of his walking that it lightened, but that, we say, was accidental. But if because of itself, then in itself—e.g. if something died while being sacrificed, it died in the sacrifice since it [15] died because of being sacrificed, and it was not accidental that it died while being sacrificed.
Whatever, therefore, in the case of what is understandable simpliciter, is said to belong to things in themselves in the sense of inhering in the predicates or of being inhered in, holds both because of themselves and from necessity. For it is not possible for them not to belong, either simpliciter or as regards the opposites—e.g. straight or crooked to line, and odd or even to number. For the contrary is either a [20] privation or a contradiction in the same genus—e.g. even is what is not odd among numbers, in so far as it follows. Hence if it is necessary to affirm or deny, it is necessary too for what belongs in itself to belong.
Now let holding of every case and in itself be defined in this fashion; I call [25] universal whatever belongs to something both of every case and in itself and as such. It is evident, therefore, that whatever is universal belongs from necessity to its objects. (To belong in itself and as such are the same thing—e.g. point and straight belong to line in itself (for they belong to it as line), and two right angles belong to [30] triangle as triangle (for the triangle is in itself equal to two right angles).)
Something holds universally whenever it is proved of a chance case and primitively; e.g. having two right angles neither holds universally of figure (yet one may prove of a figure that it has two right angles—but not of a chance figure, nor [35] does one use a chance figure in proving it; for the quadrangle is a figure but it does not have angles equal to two right angles)—and a chance isosceles does have angles equal to two right angles, but not primitively—the triangle is prior. If, then, a chance case is proved primitively to have two right angles or whatever else, it belongs universally to this primitively, and of this the demonstration holds [74a1] universally in itself; but of the others it holds in some fashion not in itself, nor does it hold of the isosceles universally, but with a wider extension.
5 · It must not escape our notice that it often happens that we make mistakes and that what is being proved does not belong primitively and universally in the way [5] in which it seems to be being proved universally and primitively. We make this error when either we cannot grasp anything higher apart from the particular, or we can but it is nameless for objects different in sort, or that of which it is proved is in fact a whole which is a part of something else. (For the demonstration will hold for the [10] parts and it will hold of every case, but nevertheless the demonstration will not hold of this primitively and universally—I say a demonstration is of this primitively and as such when it is of it primitively and universally.)
Now if someone were to prove that right angles do not meet, the demonstration would seem to hold of this because of its holding of all right angles. But that is not [15] so, if it comes about not because they are equal in this way but in so far as they are equal in any way at all.
And if there were no triangles other than the isosceles, having two right angles would seem to belong to it as isosceles.
And it might seem that proportion alternates for things as numbers and as lines and as solids and as times—as once it used to be proved separately, though it is [20] possible for it to be proved of all cases by a single demonstration. But because all these things—numbers, lengths, times, solids—do not constitute a single named item and differ in sort from one another, it used to be taken separately. But now it is proved universally; for it did not belong to things as lines or as numbers, but as this which they suppose to belong universally.
[25] For this reason, even if you prove of each triangle either by one or by different demonstrations that each has two right angles—separately of the equilateral and the scalene and the isosceles—you do not yet know of the triangle that it has two right angles, except in the sophistic fashion, nor do you know it of triangle universally,9 not even if there is no other triangle apart from these. For you do not [30] know it of the triangle as triangle, nor even of every triangle (except in respect of number; but not of every one in respect of sort, even if there is none of which you do not know it.)
So when do you not know universally, and when do you know simpliciter? Well, clearly you would know simpliciter if it were the same thing to be a triangle and to be equilateral (either for each or for all). But if it is not the same but [35] different, and it belongs as triangle, you do not know. Does it belong as triangle or as isosceles? And when does it belong in virtue of this as primitive? And of what does the demonstration hold universally? Clearly whenever after abstraction it belongs primitively—e.g. two right angles will belong to bronze isosceles triangle, but also [74b1] when being bronze and being isosceles have been abstracted. But not when figure or limit have been. But they are not the first. Then what is first? If triangle, it is in virtue of this that it also belongs to the others, and it is of this that the demonstration holds universally.
[5] 6 · Now if demonstrative understanding depends on necessary principles (for what one understands cannot be otherwise), and what belongs to the objects in themselves is necessary (for in the one case it belongs in what they are; and in the other they belong in what they are to what is predicated of them, one of which [10] opposites necessarily belongs), it is evident that demonstrative deduction will depend on things of this sort; for everything belongs either in this way or accidentally, and what is accidental is not necessary.
Thus we must either argue like this, or, positing as a principle that demonstration is necessary10 and that if something has been demonstrated it cannot be [15] otherwise—the deduction, therefore, must depend on necessities. For from truths one can deduce without demonstrating, but from necessities one cannot deduce without demonstrating; for this is precisely the mark of demonstration.
There is evidence that demonstration depends on necessities in the fact that this is how we bring our objections against those who think they are demonstrating [20]—saying that it is not necessary, if we think either that it is absolutely possible for it to be otherwise, or at least for the sake of argument.
From this it is clear too that those people are silly who think they get their principles correctly if the proposition is reputable and true (e.g. the sophists who assume that to understand is to have understanding). For it is not what is reputable or not11 that is a principle, but what is primitive in the genus about which the proof [25] is; and not every truth is appropriate.
That the deduction must depend on necessities is evident from this too: if, when there is a demonstration, a man who has not got an account of the reason why does not have understanding, and if it might be that A belongs to C from necessity but that B, the middle term through which it was demonstrated, does not hold from [30] necessity, then he does not know the reason why. For this is not so because of the middle term; for it is possible for that not to be the case, whereas the conclusion is necessary.
Again, if someone does not know now, though he has got the account and is preserved, and the object is preserved, and he has not forgotten, then he did not know earlier either. But the middle term might perish if it is not necessary; so that though, being himself preserved and the object preserved, he will have the account, [35] yet he does not know. Therefore, he did not know earlier either. And if it has not perished but it is possible for it to perish, the result would be capable of occurring and possible; but it is impossible to know when in such a state.
Now when the conclusion is from necessity, nothing prevents the middle term [75a1] through which it was proved from being non-necessary; for one can deduce a necessity from a non-necessity, just as one can deduce a truth from non-truths. But when the middle term is from necessity, the conclusion too is from necessity, just as [5] from truths it is always true; for let A be said of B from necessity, and this of C—then that A belongs to C is also necessary. But when the conclusion is not necessary, the middle term cannot be necessary either; for let A belong to C not from necessity, but to B and this to C from necessity—therefore A will belong to C [10] from necessity too; but it was supposed not to.
Since, then, if a man understands demonstratively, it must belong from necessity, it is clear that he must have his demonstration through a middle term that is necessary too; or else he will not understand either why or that it is necessary for that to be the case, but either he will think but not know it (if he believes to be [15] necessary what is not necessary) or he will not even think it (equally whether he knows the fact through middle terms or the reason why actually through immediates).
Of accidentals which do not belong to things in themselves in the way in which things belonging in themselves were defined, there is no demonstrative understanding. For one cannot prove the conclusion from necessity; for it is possible for what is [20] accidental not to belong—for that is the sort of accidental I am talking about. Yet one might perhaps puzzle about what aim we should have in asking these questions about them, if it is not necessary for the conclusion to be the case; for it makes no difference if one asks chance questions and then says the conclusion. But we must ask not as though the conclusion were necessary because of what was asked, but [25] because it is necessary for anyone who says them to say it, and to say it truly if they truly hold.
Since in each kind what belongs to something in itself and as such belongs to it [30] from necessity, it is evident that scientific demonstrations are about what belongs to things in themselves, and depend on such things. For what is accidental is not necessary, so that you do not necessarily know why the conclusion holds—not even if it should always be the case but not in itself (e.g. deductions through signs). For you will not understand in itself something that holds in itself; nor will you [35] understand why it holds. (To understand why is to understand through the explanation.) Therefore the middle term must belong to the third, and the first to the middle, because of itself.
7 · One cannot, therefore, prove anything by crossing from another genus—e.g. something geometrical by arithmetic. For there are three things in demonstrations: [40] one, what is being demonstrated, the conclusion (this is what belongs to some genus in itself); one, the axioms (axioms are the things on which the demonstration [75b1] depends); third, the underlying genus of which the demonstration makes clear the attributes and what is accidental to it in itself.
Now the things on which the demonstration depends may be the same; but of things whose genus is different—as arithmetic and geometry, one cannot apply [5] arithmetical demonstrations to the accidentals of magnitudes, unless magnitudes are numbers. (How this is possible in some cases will be said later.)12
Arithmetical demonstrations always include the genus about which the demonstration is, and so also do the others; hence it is necessary for the genus to be the same, either simpliciter or in some respect, if the demonstration is going to [10] cross. That it is impossible otherwise is clear; for it is necessary for the extreme and the middle terms to come from the same genus. For if they do not belong in themselves, they will be accidentals.
For this reason one cannot prove by geometry that there is a single science of opposites, nor even that two cubes make a cube; nor can one prove by any other [15] science the theorems of a different one, except such as are so related to one another that the one is under the other—e.g. optics to geometry and harmonics to arithmetic. Nor can one prove by geometry anything that belongs to lines not as lines and as from their proper principles—e.g. whether the straight line is the most beautiful of lines or whether it is contrarily related to the circumference; for that [20] belongs to them not as their proper genus but as something common.
8 · It is evident too that, if the propositions on which the deduction depends are universal, it is necessary for the conclusion of such a demonstration and of a demonstration simpliciter to be eternal too. There is therefore no demonstration of [25] perishable things, nor understanding of them simpliciter but only accidentally, because it does not hold of it universally, but at some time and in some way.
And when there is such a demonstration it is necessary for the one proposition to be non-universal and perishable—perishable because when it is the case the conclusion too will be the case, and non-universal because its subjects will sometimes be and sometimes not be13—so that one cannot deduce universally, but only that it holds now. [30]
The same goes for definitions too, since a definition is either a principle of demonstration or a demonstration differing in position or a sort of conclusion of a demonstration.
Demonstrations and sciences of things that come about often—e.g. eclipses of the moon—clearly hold always in so far as they are of such-and-such a thing, but are particular in so far as they do not hold always. As with the eclipse, so in the [35] other cases.
9 · Since it is evident that one cannot demonstrate anything except from its own principles if what is being proved belongs to it as that thing, understanding is not this—if a thing is proved from what is true and non-demonstrable and immediate. (For one can conduct a proof in this way—as Bryson proved the [40] squaring of the circle.) For such arguments prove in virtue of a common feature which will also belong to something else; that is why the arguments also apply to other things not of the same kind. So you do not understand it as that thing but [76a1] accidentally; for otherwise the demonstration would not apply to another genus too.
We understand a thing non-accidentally when we know it in virtue of that in virtue of which it belongs, from the principles of that thing as that thing—e.g. we [5] understand having angles equal to two right angles when we know it in virtue of that to which what has been said belongs in itself, from the principles of that thing. Hence if that too belongs in itself to what it belongs to, it is necessary for the middle to be in the same genus.
If this is not so, then the theorems are proved as harmonical theorems are proved through arithmetic. Such things are proved in the same way, but they differ; [10] for the fact falls under a different science (for the underlying genus is different), but the reason under the higher science under which fall the attributes that belong in themselves. Hence from this too it is evident that one cannot demonstrate anything simpliciter except from its own principles. But the principles of these sciences have the common feature.[15]
If this is evident, it is evident too that one cannot demonstrate the proper principles of anything; for those will be principles of everything, and understanding of them will be sovereign over everything. For you understand better if you know from the higher explanations; for you know from what is prior when you know from [20] unexplainable explanations. Hence if you know better and best, that understanding too will be better and best. But demonstration does not apply to another genus—except, as has been said, geometrical demonstrations apply to mechanical or optical demonstrations, and arithmetical to harmonical. [25]
It is difficult to be aware of whether one knows or not. For it is difficult to be aware of whether we know from the principles of a thing or not—and that is what knowing is. We think we understand if we have a deduction from some true and primitive propositions. But that is not so, but it must be of the same genus as the [30] primitives.
10 · I call principles in each genus those which it is not possible to prove to be. Now both what the primitives and what the things dependent on them signify is assumed; but that they are must be assumed for the principles and proved for the [35] rest—e.g. we must assume what a unit or what straight and triangle signify, and that the unit and magnitude are; but we must prove that the others are.
Of the things they use in the demonstrative sciences some are proper to each science and others common—but common by analogy, since things are useful in so [40] far as they bear on the genus under the science. Proper: e.g. that a line is such and such, and straight so and so; common: e.g. that if equals are taken from equals, the remainders are equal. But each of these is sufficient in so far as it bears on the [76b1] genus; for it will produce the same result even if it is not assumed as holding of everything but only for the case of magnitudes—or, for the arithmetician, for numbers.
Proper too are the things which are assumed to be, about which the science considers what belongs to them in themselves—as e.g. arithmetic is about units, and [5] geometry is about points and lines. For they assume these to be and to be this. As to what are attributes of these in themselves, they assume what each signifies—e.g. arithmetic assumes what odd or even or quadrangle or cube signifies, and geometry what irrational or inflection or verging signifies and they prove that they are, [10] through the common items and from what has been demonstrated. And astronomy proceeds in the same way.
For every demonstrative science has to do with three things: what it posits to be (these form the genus of what it considers the attributes that belong to it in itself); and what are called the common axioms, the primitives from which it demonstrates. [15] and thirdly the attributes, of which it assumes what each signifies. Nothing, however, prevents some sciences from overlooking some of these—e.g. from not supposing that its genus is, if it is evident that it is (for it is not equally clear that number is and that hot and cold are), and from not assuming what the attributes [20] signify, if they are clear—just as in the case of the common items it does not assume what to take equals from equals signifies, because it is familiar. But none the less there are by nature these three things, that about which the science proves, what it proves, and the things from which it proves.
What necessarily is the case because of itself and necessarily seems to be the case is not a supposition or a postulate. For demonstration is not addressed to [25] external argument—but to argument in the soul—since deduction is not either. For one can always object to external argument, but not always to internal argument.
Whatever a man assumes without proving it himself although it is provable—if he assumes something that seems to be the case to the learner, he supposes it (and it is a supposition not simpliciter but only in relation to the learner); but if he assumes the same thing when there is either no opinion present in the learner or actually a [30] contrary one present, he postulates it. And it is in this that suppositions and postulates differ; for a postulate is what is contrary to the opinion of the learner, which14 though it is demonstrable is assumed and used without being proved.
Now terms are not suppositions (for they are not said to be or not be [35] anything),15 but suppositions are among the propositions, whereas one need only grasp the terms; and suppositions are not that (unless someone will say that hearing is a supposition), but rather propositions such that, if they are the case, then by their being the case the conclusion comes about.
Nor does the geometer suppose falsehoods, as some have said, stating that one [40] should not use a falsehood but that the geometer speaks falsely when he says that the line which is not a foot long is a foot long or that the drawn line which is not straight is straight. But the geometer does not conclude anything from there being [77a1] this line which he himself has described, but from what is made clear through them.
Again, every postulate and supposition is either universal or particular; but terms are neither of these.
11 · For there to be forms or some one thing apart from the many is not [5] necessary if there is to be demonstration; however, for it to be true to say that one thing holds of many is necessary. For there will be no universal if this is not the case; and if there is no universal, there will be no middle term, and so no demonstration either. There must, therefore, be some one and the same thing, non-homonymous, holding of several cases.
That it is not possible to affirm and deny at the same time is assumed by no [10] demonstration—unless the conclusion too is to be proved in this form. It is proved by assuming that the first term is true of the middle and that it is not true to deny it. It makes no difference if one assumes that the middle term is and is not; and the same holds of the third term too. For if it is granted that that of which it is true to [15] say man, even if not-man is also true of it—but provided only that it is true to say that a16 man is an animal and not not an animal—for17 it will be true to say that Callias, even if not Callias, is nevertheless an animal and not not an animal. The explanation is that the first term is said not only of the middle but also of something else, because it holds of several cases; so that even if the middle both is it and is not [20] it, that makes no difference with regard to the conclusion.
That everything is affirmed or denied truly is assumed by demonstration per impossibile, and that not always universally but as far as is sufficient in so far as it bears on the genus (I say on the genus—i.e. the genus about which one is bringing the demonstrations), as has been said earlier too. [25]
All the sciences associate with one another in respect of the common items (I call common those which they use as demonstrating from them—not those about which they prove nor what they prove); and dialectic associates with them all, and so would any science that attempted to prove universally the common items—e.g. [30] that everything is affirmed or denied, or that equals from equals leave equals, or any things of the sort. But dialectic is not in this way concerned with any determined set of things, nor with any one genus. For then it would not ask questions; for one cannot ask questions when demonstrating because when opposites are the case the [35] same thing is not proved. This has been proved in the account of dedication.18
12 · If a deductive question and a proposition of a contradiction are the same thing, and there are propositions in each science on which the deductions in each depend, then there will be a sort of scientific question from which the [40] deduction appropriate to each science comes about. It is clear, therefore, that not every question will be geometrical (or medical—and similarly in the other cases too), but only those from which either there is proved one of the things about which [77b1] geometry is concerned, or19 something which is proved from the same things as geometry, such as optical matters. And similarly in the other cases too.
And for those one should indeed supply an argument from the principles and conclusions of geometry; but for the principles, the geometer as geometer should not [5] supply an argument; and similarly for the other sciences too. We should not, therefore, ask each scientist every question, nor should he answer everything he is asked about anything, but only those determined by the scope of this science. If one [10] argues in this way with a geometer as geometer it is evident that one will do so correctly, if one proves something from these things; but otherwise, not correctly. And it is clear that one does not refute the geometer either, except incidentally; so that one should not argue about geometry among non-geometers—for the man who [15] argues badly will escape notice. And the same goes for the other sciences too.
Since there are geometrical questions, are there also nongeometrical ones? And in each science which sort of ignorance is it in regard to which they are, say, geometrical? And is a deduction of ignorance a deduction from the opposites (or a [20] paralogism, though a geometrical one)? Or is it a deduction from another art? e.g. a musical question is non-geometrical about geometry, but thinking that parallels meet is geometrical in a sense and non-geometrical in another way. For this is twofold (like being non-rhythmical), and one way of being non-geometrical is by [25] not having geometrical skill (like being non-rhythmical) and the other by having it badly; and it is this ignorance and ignorance depending on such principles that is contrary to understanding.
In mathematics paralogism does not occur in the same way, because the twofold term is always the middle term; for something is said of all this, and this [30] again is said of all something else (of what is predicated one does not say all), and one can as it were see these by thought, though they escape notice in arguments. Is every circle a shape? If you draw one it is clear. Well, is the epic a circle? It is evident that it is not.
One should not bring an objection against it if the proposition is inductive. For just as there is no proposition which does not hold of several cases (for otherwise it [35] will not hold of all cases; but deduction depends on universals), it is clear that there is no objection either. For propositions and objectives are the same thing; for what one brings as an objection might become a proposition, either demonstrative or dialectical.
It happens that some people argue non-deductively because they assume what follows both terms—e.g. Caeneus does when he says that fire consists in multiple [78a1] analogy,20 for fire, he says, is generated quickly, and so is this analogy. In this way there is no deduction; but there is if multiple analogy follows fastest analogy and the fastest changing analogy follows fire.
Now sometimes it is not possible to make a deduction from the assumptions; [5] and sometimes it is possible, but it is not seen.
If it were impossible to prove truth from falsehood, it would be easy to make an analysis; for they would convert from necessity. For let A be something that is the case; and if this is the case, then these are the case (things which I know to be the case, call them B). From these, therefore, I shall prove that the former is the case. [10] (In mathematics things convert more because they assume nothing accidental—and in this too they differ from argumentations—but only definitions.)
A science increases not through the middle terms but by additional assumption—e.g. A of B, this of C, this again of D, and so on ad infinitum; and [15] laterally—e.g. A both of C and of E (e.g. A is definite—or even indefinite—number; B is definite odd number; C odd number; therefore A holds of C. And D [20] is definite even number; E is even number: therefore A holds of E).
13 · Understanding the fact and the reason why differ, first in the same science—and in that in two ways: in one way, if the deduction does not come about through immediates (for the primitive explanation is not assumed, but understanding [25] of the reason why occurs in virtue of the primitive explanation); in another, if it is through immediates but not through the explanation but through the more familiar of the converting terms. For nothing prevents the nonexplanatory one of the counterpredicated terms from sometimes being more familiar, so that the demonstration will occur through this.
E.g. that the planets are near, through their not twinkling: let C be the planets, [30] B not twinkling, A being near. Thus it is true to say B of C; for the planets do not twinkle. But also to say A of B; for what does not twinkle is near (let this be got through induction or through perception). So it is necessary that A belongs to C; so [35] that it has been demonstrated that the planets are near. Now this deduction is not of the reason why but of the fact; for it is not because they do not twinkle that they are near, but because they are near they do not twinkle.
But it is also possible for the latter to be proved through the former, and the demonstration will be of the reason why—e.g. let C be the planets, B being near, A [78b1] not twinkling. Thus B belongs to C and A to B; so that A belongs to C. And the deduction is of the reason why; for the primitive explanation has been assumed.
Again, take the way they prove that the moon is spherical through its [5] increases—for if what increases in this way is spherical and the moon increases, it is evident that it is spherical. Now in this way the deduction of the fact comes about; but if the middle term is posited the other way about, we get the deduction of the [10] reason why; for it is not because of the increases that it is spherical, but because it is spherical it gets increases of this sort. Moon, C; spherical, B; increases, A.
But in cases in which the middle terms do not convert and the non-explanatory term is more familiar, the fact is proved but the reason why is not.
Again, in cases in which the middle is positioned outside—for in these too the demonstration is of the fact and not of the reason why; for the explanation is not [15] mentioned. E.g. why does the wall not breathe? Because it is not an animal. For if this were explanatory of breathing—i.e. if the denial is explanatory of something’s not belonging, the affirmation is explanatory of its belonging (e.g. if imbalance in the hot and cold elements is explanatory of not being healthy, their balance is [20] explanatory of being healthy), and similarly too if the affirmation is explanatory of something’s belonging, the denial is of its not belonging. But when things are set out in this way what we have said does not result; for not every animal breathes. The deduction of such an explanation comes about in the middle figure. E.g. let A be [25] animal, B breathing, C wall: then A belongs to every B (for everything breathing is an animal), but to no C, so that B too belongs to no C—therefore the wall does not breathe.
Explanations of this sort resemble those which are extravagantly stated (that [30] consists in arguing by setting the middle term too far away)—e.g. Anacharsis’ argument that there are no flute-girls among the Scyths, for there are no vines.
Thus with regard to the same science (and with regard to the position of the middle terms) there are these differences between the deduction of the fact and that of the reason why.
The reason why differs from the fact in another fashion, when each is [35] considered by means of a different science. And such are those which are related to each other in such a way that the one is under the other, e.g. optics to geometry, and [79a1] mechanics to solid geometry, and harmonics to arithmetic, and star-gazing to astronomy. Some of these sciences bear almost the same name—e.g. mathematical and nautical astronomy, and mathematical and acoustical harmonics. For here it is for the empirical scientists to know the fact and for the mathematical to know the [5] reason why; for the latter have the demonstrations of the explanations, and often they do not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation.
These are those which, being something different in substance, make use of forms. For mathematics is about forms, for its objects are not said of any underlying subject—for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied.
Related to optics as this is related to geometry, there is another science related [10] to it—viz. the study of the rainbow; for it is for the natural scientist to know that fact, and for the student of optics—either simpliciter or mathematical—to know the reason why. And many even of those sciences which are not under one another are related like this—e.g. medicine to geometry; for it is for the doctor to know the fact that circular wounds heal more slowly, and for the geometer to know the reason [15] why.
14 · Of the figures the first is especially scientific. For the mathematical sciences carry out their demonstrations through it—e.g. arithmetic and geometry and optics—and so do almost all those which make inquiry after the reason why; [20] for the deduction of the reason why occurs, either in general or for the most part and in most cases, through this figure. Hence for this reason too it will be especially scientific; for consideration of the reason why has most importance for knowledge.
Next, it is possible to hunt for understanding of what a thing is through this [25] figure alone. For in the middle figure no affirmative deduction comes about; but understanding what a thing is is understanding an affirmation. And in the last figure an affirmative deduction does come about, but it is not universal; but what a thing is is something universal—for it is not in a certain respect that man is a two-footed animal.
Again, this figure has no need of the others, but they are thickened and [30] increased through it until they come to the immediates.
So it is evident that the first figure is most important for understanding.
15 · Just as it was possible for A to belong to B atomically, so it is also possible for it to not belong in this way. By belonging or not [35] belonging atomically I mean that there is no middle term for them; for in this way their belonging or not belonging will no longer be in virtue of something else.
Now when either A or B is in some whole or both are, it is not possible for A to belong to B primitively. For let A be in the whole C. Now if B is not in the whole C (for it is possible that A is in some whole and B is not in it), there will be a deduction that A does not belong to B; for if C belongs to every A and to no B, A belongs to not [79b1] B. And similarly too, if B is in some whole, e.g. in D; for D belongs to every B and A to no D, so that A will belong to no B through a deduction. And it will be proved in the same way again if both are in some whole. [5]
That it is possible for B not to be in a whole that A is in, or again for A not to be in a whole that B is in, is evident from those chains of predicates which do not overlap one another. For if nothing in the chain A,C,D is predicated of anything in the chain B,E,F, and A is in the whole H (which is in the same chain as it), it is [10] evident that B will not be in H; for otherwise the chains will overlap. And similarly too if B is in some whole.
If neither is in any whole and A does not belong to B, it is necessary for it to not belong atomically. For if there is to be a middle term, it is necessary for one of them to be in some whole. For the deduction will be either in the first or in the middle [15] figure. Now if it is in the first, B will be in some whole (for the proposition with it as subject must be affirmative); and if it is in the middle, one or other of them will be in some whole (for a deduction comes about if the negative is assumed with either as [20] subject—but with both negative there will not be one).
So it is evident that it is possible for one thing to not belong to another atomically, and we have said when it is possible, and how.
16 · Ignorance—what is called ignorance not in virtue of a negation but in virtue of a disposition—is error coming about through deduction. In the case of [25] what belongs or does not belong primitively this comes about in two ways: either when one believes simpliciter that something belongs or does not belong, or when one gets the belief through deduction. Now for simple belief the error is simple, but when it is through deduction there are several ways of erring.
For let A belong to no B atomically: now if you deduce that A belongs to B, [30] assuming C as a middle term, you will have erred through deduction. Now it is possible for both the propositions to be false, and it is possible for only one to be. For if neither A belongs to any of the C’s nor C to any of the B’s, and each has been [35] assumed the other way about, both will be false. And it is possible that C is so related to A and B that it neither is under A nor holds universally of B. For it is impossible for B to be in any whole (for A was said to not belong to it primitively), and it is not necessary that A holds universally of everything there is; hence both will be false.
But it is also possible to assume one truly—not, however, whichever you like, [80a1] but only AC; for the proposition CB with always be false because B is not in anything, but AC may be true—e.g. if A belongs atomically both to C and to B (for when the same thing is predicated primitively of several things neither will be in the [5] other). But it makes no difference even if it belongs non-atomically.
Now error about belonging comes about by these means and in this way only; for in no other figure was there a deduction of belonging. But error about not belonging comes about both in first and in the middle figure.
Now first let us say in how many ways and under what characterization of the [10] propositions it comes about in the first figure. Now it is possible when both premisses are false, e.g. if A belongs atomically both to C and to B; for if A is assumed to belong to no C and C to every B, the propositions are false. It is also [15] possible when one is false, and that whichever you like. For it is possible for AC to be true and CB false—AC true because A does not belong to everything there is; CB false because it is impossible for C, to none of which A belongs, to belong to B (for the proposition AC will no longer be true, and at the same time if they are both true [20] the conclusion too will be true). But it is also possible for CB to be true while the other is false, i.e. if B is both in C and in A; for it is necessary for the one to be under the other, so that if you assume that A belongs to no C, the proposition will be false.
[25] So it is evident that both when one proposition is false and when both are the deduction will be false.
In the middle figure it is not possible for both propositions to be false as wholes; for when A belongs to every B, one cannot assume anything which will belong to all the one and none of the other; but it is necessary to assume the propositions in such a [30] way that something belongs to one and does not belong to the other if there is to be a deduction. So if when assumed in such a way they are false, clearly they will be the other way about when assumed in the contrary way; but this is impossible.
But nothing prevents each being partially false i.e. if C were to belong both to some A and to some B. For if it is assumed to belong to every A and to no B, both [35] propositions will be false—not, however, as wholes but partially. And if the negative is posited the other way about, the same holds.
It is possible for one of them to be false, and that whichever you like. For what belongs to every A also belongs to B; so if C is assumed to belong to the whole of A and to not belong to the whole of B, CA will be true and CB false. Again, what [80b1] belongs to no B will not belong to every A; for if to A, then to B too—but it did not belong to B. So if C is assumed to belong to the whole of A and to no B, the proposition CB is true and the other false. [5]
Similarly too if the negative is transposed. For what belongs to no A will not belong to any B either; so if C is assumed to not belong to the whole of A and to belong to the whole of B, the proposition AC21 will be true and the other false. And again, what belongs to every B it is false to assume belongs to no A. For it is [10] necessary, if it belongs to every B, for it also to belong to some A; so if C is assumed to belong to every B and to no A, CB will be true and CA false.
So it is evident that both when both are false and when only one is there will be [15] an erroneous deduction in the case of atomic propositions.
17 · In the case of what belongs non-atomically, when the deduction of the falsehood comes about through the appropriate middle term, it is not possible for both propositions to be false but only the one relating to the major extreme. (I call [20] appropriate a middle term through which the deduction of the contradictory comes about.) For let A belong to B through a middle term C. Now since it is necessary for CB to be assumed as an affirmative if a deduction comes about, it is clear that this will always be true; for it does not convert. And AC is false; for if this converted the [25] contrary deduction comes about.
Similarly too if the middle term is taken from another chain—e.g. D, if it is both in the whole of A and predicated of every B; for it is necessary for the proposition DB to stand and for the other to be converted, so that the one is always [30] true and the other always false. And this sort of error is much the same as that through the appropriate middle.
But if the deduction comes about not through the appropriate middle, then when the middle term is under A and belongs to no B, it is necessary for both to be false. For the propositions must be assumed with the contrary character to that [35] which they actually have if there is going to be a deduction; and so assumed both come out false. I.e. if A belongs to the whole of D and D to none of the B’s; for when these are converted there will be a deduction and the propositions will both be false.
[81a1] But when the middle term, i.e. D, is not under A, AD will be true and DB false. For AD is true because D was not in A; and DB false because if it were true the conclusion too would be true, but it was false.
[5] When the error comes about through the middle figure, it is not possible for both propositions to be false as wholes; for when B is under A it is not possible for anything to belong to all the one and none of the other, as was said earlier. But it is possible for one to be false as a whole, and that whichever you like.
[10] For if C belongs both to A and to B, if it is assumed to belong to A and not to belong to B, then AC22 will be true and the other false. And again, if C were assumed to belong to B and to no A, CB will be true and the other false.
[15] Now if the deduction of the error is negative, we have said when and by what means the error will occur. If it is affirmative, then when it is through the appropriate middle term it is impossible for both to be false; for it is necessary for CB to stand if there is to be a deduction, as was said earlier. Hence CA23 will always [20] be false; for this is the proposition that converts.
Similarly too if the middle term were taken from another chain, as was said in the case of negative error too; for it is necessary for DB to stand and AD to convert. And the error is the same as the earlier one.
[25] When it is not through the appropriate middle term, then if D is under A, this will be true and the other false; for it is possible for A to belong to several things which are not under one another. And if D is not under A, this clearly will always be false (for it is assumed as an affirmative), but it is possible for DB both to be true [30] and false. For nothing prevents A from belonging to no D and D to every B, e.g. animal to knowledge, knowledge to music; nor again A from belonging to none of the D’s and D to none of the B’s.
So it is evident that if the middle term is not under A it is possible both for both propositions to be false and for whichever you like to be.24
[35] So it is evident in how many ways and by what means errors in virtue of deduction may come about, both in the case of the immediates and in the case of what is established through demonstration.
18 · It is evident too that if some perception is wanting, it is necessary for some understanding to be wanting too—which it is impossible to get if we learn either by induction or by demonstration, and demonstration depends on universals [81b1] and induction on particulars, and it is impossible to consider universals except through induction (since even in the case of what are called abstractions one will be able to make familiar through induction that some things belong to each genus, [5] even if they are not separable, in so far as each thing is such and such), and it is impossible to get an induction without perception—for of particulars there is perception; for it is not possible to get understanding of them; for it can be got neither from universals without induction nor through induction without perception.
19 · Every deduction is through three terms; and the one type is capable of [10] proving that A belongs to C because it belongs to B and that to C, while the other is negative, having one proposition to the effect that one thing belongs to another and the other to the effect that something does not belong. So it is evident that the principles and what are called the suppositions are these; for it is necessary to [15] assume these and prove in this way—e.g. that A belongs to C through B, and again that A belongs to B through another middle term, and that B belongs to C in the same way.
Now those who are deducing with regard to opinion and only dialectically clearly need only inquire whether their deduction comes about from the most reputable propositions possible; so that even if there is not in truth any middle term [20] for AB but there seems to be, anyone who deduces through this has deduced dialectically. But with regard to truth one must inquire on the basis of what actually holds. It is like this: since there is something which itself is predicated of something else nonaccidentally (I mean by accidentally—e.g. we sometimes say that that [25] white thing is a man, not speaking in the same way as when we say that the man is a white thing;25 for it is not the case that, being something different, he is a white thing, whereas the white thing is a man because the man was accidentally white)—now there are some things such as to be predicated in themselves.
Well, let C be such that it itself no longer belongs to anything else and B [30] belongs to it primitively and there is nothing else between. And again let E belong to F in the same way and this to B. Now is it necessary for this to come to a stop, or is it possible for it to go on ad infinitum?
And again, if nothing is predicated of A in itself and A belongs to H primitively and to nothing prior in between, and H belongs to G and this to B, is it necessary for [35] this to come to a stop or is it possible for this to go on ad infinitum? This differs from the earlier question to this extent, that the one is: Is it possible, beginning from something such that it belongs to nothing else and something else belongs to it, to go upwards ad infinitum? while the other has us begin from something such that it is predicated of something else and nothing is predicated of it and consider if it is [82a1] possible to go downwards ad infinitum.
Again, is it possible for the terms in between to be indefinitely many if the extremes are determined? I mean, e.g., if A belong to C, and B is a middle term for them, and for B and A there are other middle terms, and for these others, is it [5] possible for these to go on ad infinitum, or impossible?
This is the same as to inquire whether demonstrations go on ad infinitum and whether there is demonstration of everything, or whether some terms are bounded by one another.
I say the same in the case of negative deductions and propositions too; i.e. if A [10] does not belong to any B, either it will not belong to B primitively, or there will be something prior in between to which it does not belong (e.g. G, which belongs to all B), and again another still prior to this (e.g. H, which belongs to every G). For in these cases too either the prior terms it belongs to are indefinitely many or they come to a stop.
[15] (The same does not go for terms that convert. For among counterpredicated terms there is none of which any is predicated primitively or finally (for in this respect at least every term is related to every other in a similar way), and if26 its predicates are indefinitely many, then the things we are puzzling over are indefinitely many in both directions—unless it is possible that they convert not [20] similarly but the one as an accidental, the other as a predicate.)
20 · Now it is clear that it is not possible for the terms in between to be indefinitely many if the predications come to a stop downwards and upwards—I mean by upwards, towards the more universal; and by downwards, towards the [25] particular. For if when A is predicated of F the terms in between—the B’s—are indefinitely many, it is clear that it would be possible both that from A downwards one thing should be predicated of another ad infinitum (for before F is reached the terms in between are indefinitely many) and that from F upwards there are indefinitely many before A is reached. Hence if these things are impossible, it is also impossible for there to be indefinitely many terms between A and F.
[30] For if someone were to say that some of A,B,F are next to one another so that there are none between them, and that the others cannot be grasped, that makes no difference; for whichever of the B’s I take, the terms in between in the direction of A or in the direction of F will either be indefinitely many or not. Well, it makes no difference which is the first term from which they are indefinitely many—whether [35] at once or not at once—for the terms after these are indefinitely many.
21 · It is evident too that in the case of negative demonstration it will come to a stop if it comes to a stop in both directions in the affirmative case. For let it be possible neither to go upwards from the last term ad infinitum (I call last that which [82b1] itself belongs to nothing else while something else belongs to it, e.g. F), nor from the first to the last (I call first that which itself holds of another while nothing holds of it). Well, if this is so, it will come to a stop in the case of negation too.
For a thing is proved not to belong in three ways. For either B belongs to [5] everything to which C does and A to nothing to which B does—in the case of BC, then and in general of the second premiss it is necessary to come to immediates; for this premiss is affirmative. And clearly if the other term does not belong to something else that is prior, e.g. to D, this will have to belong to every B; and if again it does not belong to something else prior to D, that will have to belong to every D. [10] Hence since the way upwards comes to a stop, the way to A will come to a stop too, and there will be some first thing to which it does not belong.
Again, if B belongs to every A and to no C, A belongs to none of the C’s. Again, [15] if one has to prove this, clearly it will be proved either in the above fashion, or in this or the third. Now the first has been described, and the second will now be proved.
You might prove it in this way—e.g. that D belongs to every B and to no C—if it is necessary for something to belong to B. And again, if this is not to belong to C, something else belongs to D which does not belong to C. So since belonging to ever [20] higher terms comes to a stop, not belonging will come to a stop too.
The third way was: if A belongs to every B and C does not belong to it, C does not belong to everything to which A does. Again, this will be proved either in the ways described above or similarly. Well, if the former, it comes to a stop; and if the [25] latter, one will again assume that B belongs to E, to not all of which C belongs. And this again similarly. Since it is supposed that it comes to a stop in the downward direction too, it is clear that C’s not belonging will also come to a stop.
It is evident that even if it is proved not in one way but in all—sometimes from the first figure, sometimes from the second or third—that it will come to a stop even [30] so; for the ways are finite, and necessarily anything finite taken a finite number of times is finite.
So it is clear that it comes to a stop in the case of negation if it does in the case of belonging. That it comes to a stop in the latter case is evident if we consider it [35] generally, as follows.
22 · Now in the case of things predicated in what something is, it is clear; for if it is possible to define, or if what it is to be something is knowable, but one cannot go through indefinitely many things, it is necessary that the things predicated in what something is are finite.
We argue universally, as follows: one can say truly that the white thing is [83a1] walking, and that that large thing is a log, and again that the log is large and that the man is walking. Well, speaking in the latter and in the former ways are different. For when I say that the white thing is a log, then I say that that which is [5] accidentally white is a log; and not that the white thing is the underlying subject for the log; for it is not the case that, being white or just what is some white, it came to be a log, so that it is not a log except accidentally. But when I say that the log is white, I do not say that something else is white and that that is accidentally a log, as [10] when I say that the musical thing is white (for then I say that the man, who is accidentally musical, is white); but the log is the underlying subject which did come to be white without being something other than just what is a log or a particular log.
Well, if we must legislate, let speaking in the latter way be predicating, and in [15] the former way either no predicating at all, or else not predicating simpliciter but predicating accidentally. (What is predicated is like the white, and that of which it is predicated is like the log.) Thus let it be supposed that what is predicated is always predicated simpliciter of what it is predicated of, and not accidentally; for [20] this is the way in which demonstrations demonstrate. Hence when one thing is predicated of one, either it is predicated in what a thing is or it says that it has some quality or quantity or relation or is doing something or undergoing something or is at some place of time.
Again, the things signifying a substance signify of what they are predicated of [25] just what is that thing or just what is a particular sort of it; but the things which do not signify a substance but are said of some other underlying subject which is neither just what is that thing nor just what is a particular sort of it, are accidental, e.g. white of the man. For the man is neither just what is white nor just what is some [30] white—but presumably animal; for a man is just what is an animal. But the things that do not signify a substance must be predicated of some underlying subject, and there cannot be anything white which is not white through being something different. (For we can say goodbye to the forms; for they are nonny-noes, and if there are any they are nothing to the argument; for demonstrations are about things [35] of this type.)
Again, if it cannot be the case that this is a quality of that and the latter of the former—a quality of a quality—it is impossible for them to be counterpredicated of one another in this way—it is possible to say it truly, but it is not possible to counterpredicate truly. Now either it will be predicated as a substance, i.e. either [83b1] being the genus or the difference of what is predicated—but it has been proved that these will not be infinitely many, either downwards or upwards (e.g. man is two-footed, that is animal, that is something else; nor animal of man, that of Callias, [5] and that of another thing in what it is); for one can define every substance of that kind, but one cannot go through infinitely many things in thought. Hence they are not infinitely many either upwards or downwards; for one cannot define that of which infinitely many things are predicated. Thus they will not be counterpredicated [10] of one another as genera; for a thing will itself be just what is some of itself.
But neither will any case of quality or the other kinds of predication be counterpredicated unless it is predicated accidentally; for all these are accidental and are predicated of substances.
But it is clear that they will not be infinitely many upwards either; for of each is predicated whatever signifies either a quality or a quantity or one of those things, [15] or what is in its substance; but these are finite, and the genera of predications are finite—for they are either quality or relation or doing or undergoing or place or time.
It is supposed that one thing is predicated of one thing, and that things which are not what something is are not predicated of themselves. For they are all [20] accidental (though some in themselves and some in another fashion) and we say that all of them are predicated of some underlying subject, and that what is accidental is not an underlying subject; for we posit nothing of this type which is not called what it is called through being something different, and itself belongs to other things.27
[25] Neither upwards, therefore, nor downwards will one thing be said to belong to one thing. For the things of which the accidentals are said are whatever is in the substance of each thing; and these are not infinitely many. And upwards there are both these and their accidentals, and neither are infinitely many. It is necessary, therefore, for there to be something of which something is predicated primitively, and something else of that; and for this to come to a stop, and for there to be [30] something which is no longer predicated of anything prior and of which nothing else prior is predicated.
Now this is one way of demonstration; but there is still another, if there is demonstration of that of which some prior things are predicated, and it is not possible either to be more happily related to the things of which there is demonstration than by knowing them or to know them without demonstration, and [35] if this is familiar through these and we neither know these nor are more happily related to them than by knowing them, we shall not understand what is familiar through them either.
So if one can know something through demonstration—simpliciter, and not dependent on something, nor on a supposition—it is necessary for the predications in between to come to a stop. For if they do not come to a stop but there is always [84a1] something above what has been taken, there will be demonstration of everything; hence if it is not possible to go through infinitely many things, we shall not know through demonstration the things of which there is demonstration. So if we are not more happily related to them than by knowing them, we will be able to understand nothing through demonstration simpliciter but only on a supposition. [5]
Now generally, one might be convinced of what we said by this; but analytically, it is evident more concisely from the following facts that neither upwards nor downwards can the terms predicated be infinitely many in the demonstrative sciences with which our inquiry is concerned. [10]
For demonstration is of what belongs to the objects in themselves—in themselves in two ways: both what belongs in them in what they are, and the things which have what they themselves belong to belonging in what they are (e.g. odd to number—odd belongs to number and number itself inheres in its account; and again [15] plurality or divisibility inheres in the account of number). And it is not possible for either of these sorts of term to be infinitely many—either as odd of number (for then there would again be something else belonging to odd in which odd inhered; and if this is prime, number will inhere in what belongs to it; so if it is not possible [20] for infinitely many such things to belong in the one thing, they will not be infinitely many in the upward direction; but it is necessary that everything belongs to the primitive term, i.e. to number, and number to them, so that they will be convertible and will not exceed it). Nor yet can the terms inhering in what something is be infinitely many; for then it would not be possible to define. [25]
Hence if all the terms predicated are said in themselves, and there are not infinitely many, then the terms leading upward will come to a stop. Hence they will come to a stop in the downward direction too. And if this is so, the terms in between two terms will also always be finite.
And if this is the case, it is now clear too that of necessity there are principles of [30] demonstrations and there is not demonstration of everything (which, as we said at the beginning, some men assert). For if there are principles, neither is everything demonstrable, nor is it possible to go on ad infinitum; for for either of these to be the case is nothing other than for there to be no immediate and indivisible proposition [35] but for all to be divisible. For it is by interpolating a term inside and not by taking an additional one that what is demonstrated is demonstrated; hence if it is possible for this to go on ad infinitum, it would be possible for there to be infinitely many middle terms in between two terms. But this is impossible if the predications come to a stop [84b1] upwards and downwards. And that they do come to a stop has been proved generally before and analytically now.
23 · Now that this has been proved, it is evident that if one and the same thing belongs to two things—e.g. A both to C and to D—which are not predicated [5] one of the other (either not at all or not in every case), that it will not always belong in virtue of something common. E.g. having angles equal to two right angles belongs to isosceles and to scalene in virtue of something common (for it belongs to them as figures of a certain sort and not as different things); but this is not always so.
[10] For let B be that in virtue of which A belongs to C, D. It is clear, then, that B too will belong to C and D in virtue of some other common feature, and that in virtue of another; so that infinitely many terms would fall between two terms. But that is impossible.
It is not necessary, then, that when one and the same thing belongs to several things it should always do so in virtue of something common, since there are [15] immediate propositions. Yet it is necessary for the terms to be in the same genus and dependent on the same atoms, if the common feature is to be something belonging in itself; for it turned out impossible that what is proved should cross from one genus to another.
It is evident too that when A belongs to B, then if there is some middle term you [20] can prove that A belongs to B, and the elements of this are28 as many as the middle terms (for the immediate propositions are the elements, either all of them or the universal ones); but if there is no middle term, there is no longer a demonstration, but this is the path to the principles.
Similarly, too, if A does not belong to B, then if there is either a middle or a [25] prior term to which it does not belong, there is a demonstration; and if not, there is not, but it is a principle. And there are as many elements as terms; for the propositions containing these are principles of the demonstration. And just as there are some non-demonstrable principles to the effect that this is this and this belongs to this, so too there are some to the effect that this is not this and this does not [30] belong to this; so that there will be principles some to the effect that something is, and others to the effect that something is not.
When you have to prove something, you should assume what is predicated primitively of B. Let it be C; and let D be predicated similarly of this. And if you always proceed in this way no proposition and nothing belonging outside A will ever [35] be assumed in the proof, but the middle term will always be thickened, until they become indivisible and single. It is single when it becomes immediate; and a single proposition simpliciter is an immediate one. And just as in other cases the principle is simple, though it is not the same everywhere—but in weight it is the ounce, in song the semitone, and in other cases other things—so in deduction it is the unit29 [85a1] and in demonstration and understanding it is comprehension.
So, in deductions proving something to belong, nothing falls outside; but in deductions, in one case nothing falls outside the term which must belong—i.e. if A does not belong to B through C (if C belongs to every B, and A to no C), then if [5] again you have to prove that A belongs to no C, you should assume a middle term for A and C; and it will always proceed in this way.
If you have to prove that D does not belong to E by the fact that C belongs to every D and to no E it will never fall outside E (this is the term to which it must belong).
In the case of the third way, it will never pass outside either the term of which [10] it must be denied or that which must be denied of it.
24 · Some demonstrations are universal, others particular, and some are affirmative, others negative; and it is disputed which are better. And similarly too [15] for those which are said to demonstrate and those which lead to the impossible. Now first let us inquire about universal and particular demonstrations; and when we have made this clear, let us speak about those which are said to prove and those which lead to the impossible
Now it might perhaps seem to some, inquiring as follows, that particular [20] demonstration is better: if a demonstration in virtue of which we understand better is a better demonstration (for this is the excellence of demonstration), and we understand a thing better when we know it in itself than when we know it in virtue of something else (e.g. we know musical Coriscus better when we know that Coriscus [25] is musical than when we know that a man is musical; and similarly in the other cases too), and the universal demonstration shows that something else and not that the thing itself is in fact so and so (e.g. of the isosceles,30 it shows not that the isosceles but that the triangle has two right angles), while the particular demonstration shows that the thing itself has in fact two right angles—well, if a demonstration of something in itself is better, and the particular rather than the universal is of that [30] type, then the particular demonstration will be better.
Again, if the universal is not a thing apart from the particulars, and demonstration instils an opinion that that in virtue of which it demonstrates is some thing, and that this belongs as a sort of natural object among the things there are (e.g. a triangle apart from the individual triangles, and a figure apart from the individual figures and a number apart from the individual numbers), and a [35] demonstration about something there is is better than one about something that is not, and one by which we will not be led into error is better than one by which we will be, and universal demonstration is of this type (for as they go on they prove as in the case of proportion, e.g. that whatever is of such a type—neither line nor number nor solid nor plane but something apart from these—will be proportional)—so, if [85b1] this is more universal and is less about something there is than the particular demonstration, and instils a false opinion, then the universal will be worse than the particular.