by Aristotle
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 h
e 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.