Posterior Analytics · Book I

Aristotle · a new plain-English translation from the original language

1| and generous, how shall we know which of the signs that accompany each peculiarly is the sign of which? Perhaps if both belong to some other genus, not as a whole, but each belongs to those in which it does not hold as a whole, when one has it and the other does not: for if something is courageous but not generous, and of the two attributes it has this one, clearly this is also, in the lion's case, the sign of courage. Physiognomy works by the middle term, in the first figure, converting with the first extreme but extending beyond the third and not converting with it — for instance let A be courage, B, on which the large extremities stand, and C, lion. Then B belongs to everything C belongs to, but also to others; while A belongs to everything B belongs to, and to no more, but converts with it; otherwise one thing would not be the sign of one thing. POSTERIOR ANALYTICS, Book 1.

2| All teaching and all intellectual learning come about from already existing knowledge. This is clear if one looks at every case: for it is in this way that the mathematical sciences arise, and likewise each of the other crafts. And the same holds of arguments too, whether they proceed through syllogisms or through induction; for both produce their teaching through things already known, the one taking them as from those who understand, the other showing the universal through the particular's being evident. In the same way rhetorical arguments too produce persuasion, either through examples, which is induction, or through enthymemes, which is a syllogism. And there must be prior knowledge of two kinds: of some things one must already assume that they are, of others one must understand what the thing said is, and of others both — for instance, of the claim that everything is either truly affirmed or truly denied, one must know that it is so; 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. Now it is possible to come to know some things by having known them beforehand, and others by grasping the knowledge at the very same time — namely, whatever happens to fall under the universal of which one already has knowledge.

3| For that every triangle has angles equal to two right angles he knew beforehand; but that this figure in the semicircle is a triangle he came to know at the very moment he was led on by induction. (For the learning of some things occurs in this way, and the last term is not made known by way of a middle term — namely, whatever things happen already to be particulars and not said of some further subject.) And before being led on by induction, or before grasping a syllogism, one must presumably say that in one way one knows, but in another way one does not. For how could one who did not know without qualification whether the thing exists have known without qualification that it has two right angles? But clearly he knows it in this way — he knows it universally — but does not know it without qualification. Otherwise the puzzle in the Meno will result: either one will learn nothing, or only what one already knows. For one must not put it the way some people try to resolve it. Do you know that every two is even, or not? When the person says yes, they bring forward some two which he did not think was a two, and so not even. They resolve it by denying that they know that every two is even, but only what they know to be a two. Yet they know that of which they have the demonstration and that which they took as premise, and what they took was not everything they knew to be a triangle or a number, but simply of every number and triangle without qualification. For no premise is taken of the form 'of the number you know' or 'of the rectilinear figure you know,' but of every one without qualification. But nothing, I think, prevents what one is learning from being in one way known and in another way not known; for the strange thing is not that one should in some way know what one is learning, but that one should know it in this precise way — namely, in the way and the respect in which one is learning it.

4| We think we know each thing without qualification, and not in the sophistical, accidental way, when we think we know the cause through which the thing is, that it is the cause of that thing, and that it is not possible for this to be otherwise. It is clear, then, that knowing is something of this sort; for both those who do not know and those who know — the former think they themselves are in such a state, while those who know actually are in it — so that that of which there is knowledge without qualification is incapable of being otherwise. Whether there is also another way of knowing we shall say later; but we say that we also know through demonstration. By demonstration I mean a scientific syllogism; and by scientific I mean one in virtue of having which we know. If, then, knowing is what we have posited it to be, demonstrative knowledge in particular must proceed from things that are true, primary, immediate, more familiar than, prior to, and causes of the conclusion; for on these terms the starting points will also be proper to what is being proved. For there can be a syllogism even without these conditions, but there cannot be a demonstration, since it will not produce knowledge. Now they must be true, because what is not the case cannot be known — for instance, that the diagonal is commensurable. And they must proceed from things primary and indemonstrable, because otherwise one will not know them without having a demonstration of them; for to know, without qualification, those things of which there is a demonstration, is to have a demonstration of them. And they must be causes, and more familiar, and prior: causes, because we know a thing only when we know its cause; and prior, if indeed they are causes; and known beforehand not only in the sense of understanding them, but also in the sense of knowing that they are. Things are prior and more familiar in two ways;

5| for the same thing is not prior by nature and prior in relation to us, nor is the same thing more familiar and more familiar to us. By prior and more familiar in relation to us I mean what is nearer to perception, and by prior and more familiar without qualification I mean what is farther from it. The most universal things are farthest away, and particulars are nearest; and these are opposed to one another. What proceeds from things primary is what proceeds from starting points that are its own; for I use 'primary' and 'starting point' to mean the same thing. A starting point of demonstration is an immediate premise, and a premise is immediate when there is no other premise prior to it. A premise is one of the two parts of an assertion, one thing said of one thing; it is dialectical when it takes either part alike, demonstrative when it determinately takes the one, on the ground that it is true. An assertion is either part of a contradiction; a contradiction is an opposition which of itself admits nothing in between; and of a contradiction, the part that affirms something of something is an affirmation, and the part that denies something of something is a denial. By an immediate starting point of a syllogism I mean, when it cannot be proved, and when it is not necessary for one who is to learn anything to have it, a thesis; and when it is necessary for anyone who is to learn anything whatever to have it, an axiom — for there are some things of this sort, since it is especially in the case of things like these that we are accustomed to use this name. Of a thesis, the kind that takes either part of a contradiction — I mean, for instance, that something is or that it is not — is a hypothesis; the kind that does not do this is a definition.

6| For a definition is a positing — the arithmetician posits that a unit is what is indivisible in respect of quantity — but it is not a hypothesis, since what a unit is and that a unit exists are not the same thing. Since we must trust and know the matter by having the kind of syllogism we call a demonstration, and this depends on the things from which the syllogism proceeds actually being so, it is necessary not only to know the primary things beforehand — either all of them or some — but to know them better. For that through which each thing belongs always belongs to that thing more, as that through which we love something is itself more an object of love. So if we know and trust something because of the primary things, we know and trust those primary things more, since it is through them that the later things are known too. But one cannot trust, more than what he knows, things he does not happen to know at all and toward which he is no better disposed than if he did happen to know them. But this will result unless someone first knows the things through which he comes to trust by demonstration; for it is necessary to trust the principles — either all of them or some — more than the conclusion. Anyone who is to have scientific knowledge through demonstration must not only know the principles better and trust them more than what is being proved, but nothing else must be more trustworthy or better known to him than the things opposed to the principles from which a syllogism of the contrary error could be constructed — since one who has knowledge in the unqualified sense must be unshakeable in his conviction.

7| Some, then, because of the requirement that the primary things be known, think there is no scientific knowledge at all; others think there is, but hold that there is demonstration of everything. Neither of these views is true or necessary. Those who suppose there is no knowing at all hold that one must be led back to infinity, on the ground that we could not know the later things through the earlier ones if there were no first things — and in this they speak rightly, for it is impossible to traverse an infinite series. And they hold that if the regress does stop and there are principles, these are unknowable, since there is no demonstration of them, which they say is the only thing knowing is; and if the primary things cannot be known, neither can the things that come from them be known in the unqualified or proper sense, but only on the hypothesis that those primary things exist. The others agree about what knowing is — that it comes only through demonstration — but hold that nothing prevents there being demonstration of everything, since demonstration can proceed in a circle, each thing being demonstrated from the others. We, however, say that not all knowledge is demonstrative, but that knowledge of the immediate premises is without demonstration (and that this is necessary is clear: for if it is necessary to know the prior things, the things from which the demonstration proceeds, and the immediate premises are at some point reached, these must be undemonstrable). This, then, is what we say; and we hold not only that there is scientific knowledge but also that there is some principle of knowledge by which we come to know the terms. And it is clear that it is impossible to demonstrate in the unqualified sense by going in a circle,

8| if indeed a demonstration must proceed from things that are prior and better known; for it is impossible for the same things to be at once prior and posterior to the same things, except in the other sense — one thing prior relative to us, another prior in an unqualified sense — the way induction makes things known. But if this is so, then knowing in the unqualified sense would not be well defined but would be twofold; or else the second kind of demonstration is not demonstration in the unqualified sense, since it proceeds only from things better known to us. Now for those who say demonstration goes in a circle, not only does what has just been said follow, but they end up saying nothing more than that this is the case if this is the case — and in that way it is easy to prove anything. It is clear that this results when three terms are posited. For it makes no difference whether one says the circling back happens through many terms or through few, or through few or two. For whenever, if A is, B is of necessity, and if B is, C is, then if A is, C will be. If, then, when A is, B is necessarily, and when B is, A is (for this is what going in a circle means), let C be substituted for A. So to say that, when B is, A is, is to say that C is; and this in turn means that, when A is, C is.

9| But C is the same as A. So it follows that those who claim demonstration goes in a circle are saying nothing other than that, when A is, A is — and in this way it is easy to prove anything. But in fact not even this is possible, except in the case of things that follow reciprocally from one another, as properties peculiar to a subject do. Now it has been shown that when a single thing is posited, nothing else is ever necessarily so (I mean by "a single thing" that neither a single term nor a single premise has been posited); but from two premises, the first and fewest possible, it is possible, provided a syllogism can also be formed from them. If, then, A follows both B and C, and these follow each other and A, then in this way it is possible to prove all the terms assumed from one another in the first figure, as has been shown in the discussion of the syllogism; and it has also been shown that in the other figures either no syllogism results, or none concerning the terms assumed. But things that are not counter-predicated of one another cannot be proved circularly at all; so since few things in demonstrations are of this kind, it is clear that it is empty and impossible to say that demonstration proceeds from things that reciprocate with one another, and that for this reason demonstration of everything is possible.

10| Since it is impossible for that of which there is knowledge in the unqualified sense to be otherwise, what is knowable in accordance with demonstrative knowledge would have to be necessary; and demonstrative knowledge is the knowledge we have by having a demonstration. Demonstration, then, is a syllogism from necessary premises. We must therefore grasp from what things, and of what kind, demonstrations proceed. But first let us define what we mean by "of every case," by "in its own right," and by "universal." By "of every case," then, I mean whatever holds not of some instances and not of others, nor at some times and not at others — for example, if animal holds of every man, then if it is true to call this thing a man, it is also true to call it an animal, and if one of these holds now, so does the other; and likewise if point holds of every line. A sign of this: it is in just this way that we raise objections when asked whether something holds of every case — either that it does not hold in some instance, or that it does not hold at some time. "In its own right" applies to whatever belongs in the what-it-is, as line belongs to triangle and point to line (for their substance is composed of these, and these are present in the account that states what they are); and also to whatever belongs to things such that the things themselves are present in the account that makes clear what those things are — as straight and curved belong to line, and odd and even, prime and composite, equilateral and oblong belong to number.

11| For in all these cases, line or number is present in the account that states what the thing is. Similarly in other cases too, I call things of this sort "in their own right" for each subject; whatever belongs in neither of these ways is accidental — as musical or pale belongs to an animal. Again, "in its own right" applies to whatever is not said of some other underlying subject — as the walking thing, being something else, is walking, and the pale thing likewise is pale; but substance, and whatever signifies a this-something, is what it is without being something else that it merely happens to be. Things not said of an underlying subject I call "in their own right"; things said of an underlying subject I call accidents. Again, in another sense, whatever belongs to a thing because of that thing itself belongs to it in its own right, while whatever does not belong because of the thing itself is accidental — as, if it lightened while someone was walking, that is accidental, for it was not because of his walking

12| ...it lightninged; but this, we say, was coincidental. But if it holds on account of the thing itself, then it holds in its own right. For example, if something died while being slaughtered, then it also died in accordance with the slaughtering, because it died on account of being slaughtered, and it was not a coincidence that, while being slaughtered, it died. So the things said to belong in this way, in their own right, to the objects of unqualified scientific knowledge -- as either inhering in the things predicated, or having these things inhering in them -- hold both on account of the things themselves and of necessity. For it is not possible for them not to belong, either without qualification or as one of a pair of opposites -- for example, straight or curved to a line, and odd or even to a number. For the contrary is either a privation or a contradictory within the same genus -- for example, even is what-is-not-odd among numbers, insofar as this follows. So if it is necessary either to affirm or to deny, it is necessary also that what belongs in its own right should belong. Let 'holding of every case' and 'in its own right,' then, be defined in this way. By universal I mean whatever belongs to every case, and in its own right, and as itself. It is clear, then, that whatever is universal belongs of necessity to the things in question. 'In its own right' and 'as itself' are the same thing -- for example, point and straight belong to a line in its own right (for they belong to it qua line), and having angles equal to two right angles belongs to a triangle qua triangle (for a triangle in its own right is equal to two right angles).

13| A thing belongs universally when it is demonstrated of an arbitrary instance and a primary one. For example, having angles equal to two right angles is universal neither to figure (although one can prove of a figure that it has angles equal to two right angles, but not of an arbitrary figure, nor does the proof use an arbitrary figure in demonstrating it -- for a square is a figure, but it does not have angles equal to two right angles) -- nor to isosceles: the isosceles does have, as any arbitrary instance of it, angles equal to two right angles, but not as the primary case; rather, the triangle is prior. So whatever is the first thing of which, taking an arbitrary instance, having angles equal to two right angles -- or whatever other attribute -- is demonstrated, to that thing, as primary, the attribute belongs universally, and the demonstration is, in its own right, a demonstration of this as universal; of the other things it holds in a way, but not in their own right -- nor indeed is it universal of the isosceles either, but extends more widely than that. We must not fail to notice that it often happens that we go wrong, and that what is demonstrated does not in fact belong to the primary universal, in the way it seems to be demonstrated as belonging to the primary universal. We fall into this error when either there is nothing higher to take beyond the particular,

14| or there is something, but it is nameless as covering things that differ in species, or the term of which the demonstration is given happens to be a whole that is really only a part of some larger whole. For the demonstration will indeed hold of the things within that part, and will hold of every one of them, but nevertheless the demonstration will not be of this as the primary universal. By 'demonstration of this as primary, insofar as it is this,' I mean a demonstration that is of the primary universal. If, then, someone were to prove that perpendiculars do not meet, the demonstration might seem to be of this, because it holds of all perpendiculars. But it is not, since this result comes about not because they are equal in this particular way, but because they are equal in any way at all. And if there were no triangle other than the isosceles, the attribute would seem to belong to it qua isosceles. So too with the fact that proportionals alternate: it used to be demonstrated of numbers, and of lines, and of solids, and of times, each separately, just as it once was proved apart in each case, though it is possible for it to be demonstrated of all of them by a single demonstration. But because there was no single named thing for all of these -- numbers, lengths, times, solids -- and because they differ from one another in species, it was taken up separately. But now it is demonstrated universally; for it did not belong to them qua lines or qua numbers, but qua this very thing, which they suppose to belong to them universally.

15| For this reason, even if someone should prove, for each triangle taken separately -- the equilateral apart, and the scalene, and the isosceles -- by one demonstration or by different ones, that each has angles equal to two right angles, he does not yet know that the triangle has angles equal to two right angles, except in the sophistical way, nor does he know it of triangle as a whole, not even if there is no triangle besides these. For he does not know it qua triangle, nor of every triangle -- except in respect of number; in respect of species he does not know it of every triangle, even if there is none that he fails to know. When, then, does he not know it universally, and when does he know it without qualification? Clearly, he would know it universally if being a triangle and being equilateral were the same -- either for each or for all together. But if they are not the same but different, and the attribute belongs to the equilateral qua triangle, then he does not know it. Does the attribute belong qua triangle, then, or qua isosceles? And when does it belong to this as primary? And of what is the demonstration universal? Clearly, when, upon the successive removals, it still belongs to the first thing left. For example, having two right angles will belong to a bronze isosceles triangle, but it will still belong once both 'being bronze' and 'isosceles' have been removed. But not once figure or boundary have been removed -- though these are not the primary things either. Of what, then, is it primary? If, then, of triangle, then it belongs to the others in virtue of this, and the demonstration is universal of this.

16| If, then, demonstrative knowledge proceeds from necessary principles (for what one knows scientifically cannot be otherwise), and what belongs to things in their own right is necessary to them (for either the attributes belong within the what-it-is, or the things themselves belong within the what-it-is of the predicates said of them, one of a pair of opposites necessarily belonging to them), it is clear that the demonstrative syllogism would proceed from premises of some such kind. For everything belongs either in this way or accidentally, and accidental attributes are not necessary. Either this is how it must be put, or else, laying it down as a starting point that demonstration is of necessary things, and that if something has been demonstrated it cannot be otherwise -- the syllogism, then, must proceed from necessary premises. For it is possible to reason from true premises without demonstrating, but it is not possible to reason from necessary premises except by demonstrating; for this is already the mark of demonstration. A sign that demonstration proceeds from necessary premises is that we bring our objections in just this way against those who think they are demonstrating something that is not in fact necessary, whenever we suppose either that it can be otherwise altogether, or at least for the sake of argument that it can. It is clear from this, too, that those who think they are taking their starting points properly, provided the premise is reputable and true, are naive -- the sophists, for example, who suppose that knowing scientifically is simply having knowledge.

17| For what is reputable is not, for us, a starting point; rather, the starting point is what is primary in the genus about which the demonstration proceeds, and not everything true is proper to it. That the syllogism must proceed from necessary premises is clear from the following as well. For if someone who has no account of the reason why -- when a demonstration is available -- does not have scientific knowledge, and it might be that A belongs to C of necessity while B, the middle term through which it was demonstrated, does not belong of necessity, then he does not know the reason why. For the conclusion does not hold on account of the middle term in this case: the middle term admits of not being so, while the conclusion is necessary. Further, if someone does not have scientific knowledge now, while he still has the account and is himself unimpaired, the thing itself being unchanged, and he has not forgotten, then he did not have scientific knowledge before either. But the middle term could have perished, if it is not

18| necessary, so that he will have the account while both he and the thing are preserved, and yet not have scientific knowledge. So he did not have scientific knowledge before either. And if it has not perished, but admits of perishing, then what would result would be something merely possible and contingent. But it is impossible to have scientific knowledge while in such a condition. So then, when the conclusion is of necessity, nothing prevents the middle term through which it was proved from not being necessary (for it is possible to reason to a necessary conclusion even from premises that are not necessary, just as it is also possible to reason to a true conclusion from premises that are not true); but when the middle term is of necessity, the conclusion too is of necessity, just as a conclusion from true premises is always true (let A belong to B of necessity, and this belong to C; then it is necessary that A too belongs to C); but when the conclusion is not of necessity, the middle term cannot be necessary either (for let A belong to C not of necessity, but let it belong to B, and let this belong to C of necessity; then A too will belong to C of necessity -- but this was not what was supposed). Since, then, if one has demonstrative scientific knowledge, the attribute must belong of necessity, it is clear that one must also have the demonstration through a necessary middle term;

19| Or else he will know neither the reason why nor that it is necessary for that to be so, but will either think he knows without knowing — if he supposes what is not necessary to be necessary — or will not even think he knows; and this holds alike whether he knows the fact through middle terms or the reason why through terms that are immediate. Of accidents that are not in their own right — in the sense in which we defined things in their own right — there is no demonstrative knowledge. For it is not possible to derive the conclusion necessarily; for what is accidental admits of not holding, and it is this sort of thing I mean by accident. Yet one might perhaps raise the difficulty of what point there is in asking these questions about such things, if the conclusion need not be necessary; for it makes no difference if someone, having asked whatever chance premises, then simply states the conclusion. But one must ask them not on the ground that the conclusion is necessary because of what was asked, but because anyone who states those premises is bound to state the conclusion too, and to state it truly, if the premises truly hold. Since whatever holds of each genus in its own right, and in the respect in which each thing is what it is, holds of it necessarily, it is clear that scientific demonstrations concern things that hold in their own right, and proceed from things of that sort. For accidents are not necessary, so that one need not know the reason why the conclusion holds — not even if it always held, provided it does not hold in its own right, as with syllogisms drawn from signs. For what holds in its own right will not thereby be known as holding in its own right, nor will the reason why be known (and to know the reason why is to know through the cause). The middle term, therefore, must belong to the third term, and the first term to the middle, in virtue of the thing itself.

20| It is not possible, then, to prove something by crossing over from another genus — for instance, to prove a geometrical fact by arithmetic. For there are three things involved in demonstrations: one is what is being demonstrated, the conclusion (this is what belongs to some genus in its own right); another is the axioms (axioms are the things a demonstration proceeds from); and third is the underlying genus, whose attributes and whose accidents that hold in their own right the demonstration makes clear. Now the things a demonstration proceeds from may be the same across sciences; but where the genus is different, as with arithmetic and geometry, it is not possible to apply an arithmetical demonstration to attributes that hold of magnitudes, unless magnitudes are numbers — how this is possible in some cases will be said later. Arithmetical demonstration always keeps to the genus the demonstration is about, and the same holds for the others. So a demonstration, if it is to cross over, requires the genus to be the same either without qualification or in some respect. That it is impossible otherwise is clear; for the extreme terms and the middle terms must come from the same genus, since if they did not hold in their own right, they would be accidental. This is why geometry cannot prove that there is a single science of contraries, nor even that two cubes make a cube; nor can any science prove what belongs to another, except where two sciences stand to one another such that one falls under the other — as optics stands to geometry and harmonics to arithmetic. Nor again can it prove whatever holds of lines not insofar as they are lines and from their own proper principles — for instance, whether the straight line is the finest of lines, or whether it stands opposite to the circular; for this does not hold of them insofar as their own proper genus is concerned, but insofar as something common is concerned.

21| It is also clear that, if the premises from which the syllogism proceeds are universal, the conclusion of such a demonstration — of demonstration in the strict sense — must likewise be eternal. There is, then, no demonstration of perishable things, nor knowledge of them in the strict sense, but only in the way one might speak of it accidentally, since it does not hold of the thing as a whole but only at some time and in some respect. When there is such a demonstration, one of the premises must be non-universal and perishable — perishable, because the conclusion too will exist only while that premise exists; non-universal, because among the things it ranges over it will hold of some and not of others — so that one cannot draw a universal conclusion, but only that the thing holds now. The same holds of definitions too, since a definition is either a starting point of demonstration, or a demonstration differing only in arrangement, or in some way a conclusion of a demonstration. As for demonstrations and knowledge of things that happen often, such as an eclipse of the moon, it is clear that insofar as they are of that kind they always hold, but insofar as they do not always hold, they hold only in part. And as with eclipse, so likewise with the other cases.

22| Since it is clear that it is not possible to demonstrate any given thing except from the principles proper to it — assuming what is proved holds of the thing precisely as that thing — knowing it is not simply a matter of its being proved from true, undemonstrated, and immediate premises. For it is possible to prove something in that way and still not know it, as Bryson proved the squaring of the circle. For arguments of that sort prove by means of something common, which will also belong to something else — which is why such arguments fit other cases as well, ones not of the same kind. So one does not thereby know the thing itself, but only accidentally; for otherwise the demonstration would not fit another genus too. We know each thing non-accidentally when we know it in respect of that in virtue of which it holds, from the principles of that thing precisely as that thing — for instance, having angles equal to two right angles, which belongs in its own right to the thing of which it is said, known from the principles of that very thing. So that if what holds, holds of its subject in its own right, the middle term must belong to the same family. If not, then it is as with harmonics through arithmetic. Such cases are proved in a similar way, but differ: the fact belongs to one science (for the underlying genus is different), while the reason why belongs to the higher science, the one whose own attributes these are. So from this too it is clear that it is not possible to demonstrate any given thing without qualification except from the principles proper to it. But the principles of these sciences have what is common between them.

23| If this is clear, it is also clear that it is not possible to demonstrate the principles proper to each thing; for then those principles would be the principles of everything, and the knowledge of them would be authoritative over everything. For indeed, one who knows from higher causes knows more fully; for he knows from prior things, when he knows from causes that are not themselves caused. So that if he knows more, and knows most of all, that knowledge too would be knowledge in a greater and in the highest degree. But demonstration does not fit another genus, except as has been said — geometrical demonstrations applying to mechanics or optics, and arithmetical demonstrations to harmonics. It is difficult to know whether one knows or not. For it is difficult to know whether we know from the principles proper to each thing or not — and this is precisely what knowing is. We think we know when we have a syllogism from certain true and primary premises. But this is not so; the premises must also be akin to the primary things of that subject.

24| By principles in each genus I mean those things whose existence it is not possible to prove. Now what both the primary things and the things derived from them signify is taken as given; but that they exist, the principles must be assumed, while the rest must be proved. For example, what a unit is, or what the straight and the triangle are, is taken as given; but the existence of the unit and of magnitude must be assumed, while everything else must be proved. Of the things used in the demonstrative sciences, some are proper to each science and some are common — common by analogy, since each is useful only within the genus falling under that science. Proper, for example, is that a line is of such-and-such a kind, and the straight; common, for example, that if equals are taken from equals, the remainders are equal. Each of these common items is sufficient so far as it applies within the genus; for it will have the same effect even if it is not assumed to hold of all things but only of magnitudes, and for the arithmetician, of numbers. Proper, too, are the things whose existence is assumed and about which the science studies the attributes that hold of them in their own right — as arithmetic assumes units, and geometry points and lines. For these sciences assume both that these things exist and that they are of such-and-such a kind. As for the attributes that belong to these in their own right, they take as given what each one signifies — as arithmetic takes as given what odd or even or square or cube is,

25| Geometry, for its part, does not prove what the irrational is, or what it is to be bent or to incline, but only that it is, and it proves this by means of the common principles and the things already demonstrated. Astronomy proceeds in the same way. For every demonstrative science is concerned with three things: the things it posits as existing (these constitute the genus, whose per se attributes it studies), the so-called common axioms, from which as primary premises it demonstrates, and third, the attributes, of which it takes what each signifies. There is, however, nothing to stop some sciences from passing over some of these — for instance, not positing that the genus exists, if it is evident that it does (for it is not equally clear that number exists as that hot and cold exist), and not taking what the attributes signify, if that is clear — just as it does not take what 'taking equals from equals' signifies among the common principles, since that is well known. But none the less these three things exist by nature all the same: that about which the science proves, what it proves, and that from which it proves. What is not a hypothesis, nor a postulate, is that which must exist through itself and must be believed by necessity. For demonstration is not directed at the discourse outside, but at the discourse within the soul, since neither is deduction so directed.

26| For it is always possible to raise an objection against the discourse outside, but not always against the discourse within. As for the things that are demonstrable but which the person takes without demonstrating them himself, these — if he takes them as things the learner believes — he is hypothesizing, and this is not a hypothesis without qualification but only relative to that person; whereas if he takes the same thing when the learner holds no opinion on it, or even holds a contrary opinion, then he is postulating. And this is the difference between a hypothesis and a postulate: a postulate is what is contrary to the learner's opinion, or whatever someone takes and uses, though it is demonstrable, without demonstrating it. Terms, then, are not hypotheses (for nothing is said to be or not to be), but the hypotheses lie among the premises, while terms need only be understood; and this is not a hypothesis (unless someone will say that hearing too is a hypothesis), but rather it consists in whatever things, given that they exist, the conclusion comes to be. (Nor does the geometer posit anything false, as some have said, claiming that one must not make use of falsehood, and that the geometer speaks falsely when he calls a line a foot long when it is not a foot long, or straight when the line drawn is not straight. But the geometer draws no conclusion from the fact that this particular line, the one he has spoken of, is such-and-such, but rather from what is made clear through these lines.) Further, the postulate, and every hypothesis, is either as a whole or as a part, whereas terms are neither of these.

27| It is not necessary, then, that there be forms, or some one thing apart from the many, if there is to be demonstration; it is necessary, however, that it be true to assert one thing of many. For there will be no universal unless this holds; and if there is no universal, there will be no middle term, and so no demonstration either. There must, then, be some one and the same thing holding of several things, and not homonymously. No demonstration takes as a premise that it is not possible at once to affirm and deny — except when it is necessary to prove the conclusion too in this form. And it is proved by taking the first term as true of the middle, and its denial as not true. It makes no difference whether one takes the middle term as existing or not existing, and the same holds for the third term. For if it were granted that it is true to say of something that it is a man, even if it is not true that it is not a man — so long as it is only granted that man is an animal, and not an animal is false of it — then it will be true to say that Callias, even if it is not true that he is not-Callias, is nevertheless an animal, and not an animal is false of him. The reason is that the first term is said not only of the middle but also of something else, because it holds of more things; so that it makes no difference to the conclusion whether the middle term both is and is not itself.

28| The demonstration that proceeds to the impossible takes it that everything must be either affirmed or denied, and this too not always universally, but only so far as is sufficient, and it is sufficient with respect to the genus. By 'with respect to the genus' I mean the genus about which one brings one's demonstrations, as has also been said before. All the sciences share with one another through the common principles (by 'common' I mean those which they use as premises from which to demonstrate, but not those about which they demonstrate, nor those which they demonstrate), and dialectic shares them with all, and so would anyone who tried to demonstrate universally the common principles themselves — for instance, that everything must be either affirmed or denied, or that equals taken from equals leave equals, or some other such things. But dialectic does not deal with things thus determined, nor with any single genus. For otherwise it would not proceed by questioning; for one who is demonstrating cannot ask questions, because the same thing cannot be proved from opposite premises — and this has been shown in the discussion of deduction.

29| If a question suited to deduction is the same thing as a premise stating one side of a contradiction, and the premises proper to each science are those from which the deduction proper to that science is built, then there would be some scientific question — namely, one from which the deduction proper to that science arises. It is clear, then, that not every question would be a geometrical one, nor a medical one, and likewise for the others; rather, the geometrical questions are those from which something is proved about the subjects geometry deals with, or which are proved from the same principles as geometry, as with optics. And likewise for the others. And concerning these one must be prepared to give an account starting from geometrical principles and conclusions, but concerning the principles themselves the geometer, qua geometer, need not give an account; and likewise for the other sciences. So then, not every question is to be put to each man of science, nor is every question put to him to be answered concerning his subject, but only what is marked off as belonging to that science. And if someone argues with a geometer, qua geometer, along these lines, it is plain that he argues well if he proves something from these principles, but not well otherwise. And it is plain too that he does not even refute the geometer except accidentally; so that one should not carry on discussion about geometry among those ignorant of geometry, for a poor argument will pass unnoticed. The same holds for the other sciences as well. Since there are geometrical questions, are there also questions not pertaining to geometry? And for each science, what sort of questions, arising from ignorance of it, count as belonging to that science? And is the deduction that proceeds from ignorance the deduction from opposite premises, or is it rather the fallacy that, while conforming to geometry, is drawn from another craft — for instance, a musical question is a question about geometry not pertaining to geometry, while thinking that the parallels meet is in one way geometrical and in another way not pertaining to geometry? For this is twofold, like being unrhythmical: one sort fails to pertain to geometry by not having it at all,

30| the other by having it badly; and this latter ignorance, the one arising from such principles, is the contrary kind. In the mathematical sciences the fallacy does not work in the same way, because the middle term is always twofold: it is said of the whole of one thing, and this in turn is said of the whole of some further thing (whereas what is predicated is not said of the whole) — and these things can, so to speak, be seen by the intellect, though in verbal arguments they go unnoticed. Is every circle a figure? If one draws it, this is plain. But what about this: are epic verses a circle? Clearly they are not. One should not bring an objection against a premise if it is inductive. For just as there is no premise that does not hold of several things (for then it would not hold of all, and deduction proceeds from universals), so plainly there is no objection either. For premises and objections are the same thing: whatever objection one brings could become a premise, either demonstrative or dialectical. It happens that some people argue without deducing anything, because they take what follows from both terms — as, for instance, Caeneus does, arguing that fire increases in a multiplied ratio, since fire, he says, is generated quickly, and this too is that same ratio.

31| In this way there is no syllogism—unless the fastest ratio follows the manifold ratio, and the fastest ratio in locomotion belongs to fire. So sometimes it is not possible to draw a syllogism from the premises taken, and sometimes it is possible but is not seen. Now if it were impossible to prove something true from something false, resolving an argument into its premises would be easy, for the conclusion would convert of necessity. For let A be something that is—given that this is, then these things are, which I know to be, for instance B; from these, then, I shall show that that other thing is. Now the propositions in mathematics convert more readily, because they take nothing accidental (and in this too they differ from those in dialectical arguments) but only definitions. And a proof grows not through middle terms, but by taking on additional terms—A of B, this of C, this again of D, and so on to infinity; and also sideways, as when A holds both of C and of E. For example, let A stand for being a determinate quantity of number, or even an infinite one; let B stand for the odd number's being a determinate quantity; let C stand for odd number. Then A holds of C. And let D stand for the even number's being a determinate quantity, E for even number; then A holds of E.

32| Knowing the fact and knowing the reason why differ—first, within one and the same science, and here in two ways: in one way, when the syllogism is not built through immediate premises (for then the primary cause is not grasped, and knowledge of the reason why goes by way of the primary cause); in another way, when it is built through immediate premises, but not through the cause, but through whichever of a pair of convertible terms is the more familiar. For nothing prevents the non-cause from sometimes being more familiar than the cause, among terms that are predicated reciprocally of each other, so that the demonstration will proceed through this term—for instance, that the planets are near because they do not twinkle. Let C be planets, B not twinkling, A being near. It is true to say B of C, for the planets do not twinkle; but it is also true to say A of B, for what does not twinkle is near (this has been grasped by induction or by perception). It is necessary, then, that A belong to C, so that it has been demonstrated that the planets are near. This syllogism, then, 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 that they do not twinkle.

33| But it is also possible to show the one through the other, and then the demonstration will be of the reason why—for instance, let C be planets, B being near, A not twinkling; then B belongs to C and A to B, so that A also belongs to C. And this syllogism is of the reason why, since the primary cause has been grasped. Again, take the way people prove that the moon is spherical, from its phases of waxing—for if what waxes in this manner is spherical, and the moon waxes, it is clear that it is spherical: put this way, the syllogism is of the fact; but if the middle term is transposed, it is of the reason why, for it is not on account of its waxings that the moon is spherical, but because it is spherical that it undergoes such waxings. Let moon stand for C, spherical for B, waxing for A.

34| Waxing stands for A. But in cases where the middle terms do not convert, and the non-causal term is the more familiar, the fact is shown but not the reason why. Again, the same holds in cases where the middle term is placed outside the other two. For in these too the demonstration is of the fact and not of the reason why, since the cause is not stated. For instance: why does the wall not breathe? Because it is not an animal. For if this were the cause of not breathing, then being an animal would have to be the cause of breathing—just as, if the negation is the cause of something's not belonging, the affirmation is the cause of its belonging; for instance, if the imbalance of hot and cold is the cause of not being healthy, then their balance is the cause of being healthy—and likewise, if the affirmation is the cause of belonging, the negation is the cause of not belonging. But in cases set out this way, what has just been said does not hold, for not every animal breathes. The syllogism of such a cause comes about in the middle figure. For instance, let A be animal, B breathing, C wall. Then A belongs to every B (for everything that breathes is an animal), but to no C, so that B too belongs to no C: therefore the wall does not breathe.

35| Causes of this sort resemble things said by way of exaggeration—which is to say, taking the middle term at too great a remove, as in Anacharsis' remark, that among the Scythians there are no flute-girls, since there are no vines either. These, then, are the differences between the syllogism of the fact and that of the reason why within one and the same science, and by reference to the position of the middle terms. But there is another way in which the reason why differs from the fact—when each is studied through a different science. Such are the cases where two sciences stand to one another so that one is subordinate to the other—as optics to geometry, mechanics to solid geometry, harmonics to arithmetic, and the study of appearances to astronomy. Some of these sciences are practically synonymous with one another—for instance there is a mathematical astronomy and a nautical astronomy, and a mathematical harmonics

36| and one based on hearing. For here, knowing the fact belongs to those concerned with perception, knowing the reason why to the mathematicians; for the latter possess the demonstrations of the causes, and often they do not know the fact, just as those who study the universal often do not know some of the particulars, for lack of observation. These are cases in which things of a different substance make use of forms belonging to another science. For mathematics is concerned with forms: its objects are not said of any underlying subject—for even if geometrical objects are in fact said of some underlying subject, it is not qua said of a subject that geometry considers them. And as optics stands to geometry, so another science stands to optics—for instance the study of the rainbow: knowing the fact belongs to the natural scientist, knowing the reason why to the student of optics, either without qualification or as applied to that particular branch of mathematics. And many of the sciences that are not subordinate to one another stand in this same relation—for instance medicine to geometry: that circular wounds heal more slowly is for the doctor to know, but why, is for the geometer.

37| Of the figures, the first is the most scientific. For the mathematical sciences carry their demonstrations through this figure—arithmetic, geometry, and optics—and practically all the sciences that make the reason why their object of inquiry; for either altogether, or for the most part and in the majority of cases, the syllogism of the reason why proceeds through this figure. So on this ground too it would be the most scientific, since grasping the reason why is the most authoritative way of knowing. Next, it is possible to pursue knowledge of the essence through this figure alone. For in the middle figure no affirmative syllogism comes about, and knowledge of the essence is a matter of affirmation; and in the last figure an affirmative syllogism does come about, but not a universal one, whereas the essence belongs among universals—man is not in a certain respect a two-footed animal. Further, this figure has no need of the others, while they are condensed and built up through it, until they arrive at immediate premises. It is clear, then, that the first figure is the most authoritative for scientific knowledge. Just as it was possible for A to belong to B atomically, so too it is possible for it not to belong. By belonging or not belonging atomically I mean that there is no middle term between them; for in that case the belonging or not belonging will no longer be in virtue of something else. Now whenever either the

38| or B is contained wholly within some term, or both are, then it is not possible for A to fail to belong to B primitively. For let A be contained wholly in C. Then if B is not contained wholly in C (for it is possible for A to be within some whole while B is not in that same whole), there will be a syllogism that A does not belong to B: for if C belongs to all A but to no B, then A belongs to no B. Likewise too if B is contained wholly in some term, say D: for D belongs to all B, but A belongs to no D, so that A will belong to no B through a syllogism. The same method will show it too if both are contained wholly in some term. That it is possible for B not to be in the whole in which A is, or again for A not to be in that in which B is, is clear from the co-ordinate series, as many as do not overlap one another. For if none of the members of the C-D series is predicated of any of the members of the B-Z series, and A is contained wholly in Θ, which belongs to that same series, it is clear that B will not be in Θ:

39| for otherwise the series would overlap one another. And likewise too if B is contained wholly in some term. But if neither is contained wholly in anything, and A does not belong to B, then it is necessary that it not belong immediately. For if there were to be some middle term, one of the two would have to be contained wholly in some term. For the syllogism will occur either in the first figure or in the middle figure. Now if in the first, B will be contained wholly in some term (for the premise concerning this must be affirmative); but if in the middle figure, either of the two indifferently (for a syllogism comes about when the negative premise is taken with respect to either one, but if both premises are negative there will be no syllogism). It is clear, then, that it is possible for one thing to fail to belong to another immediately, and we have stated when this is possible and how. Ignorance that is spoken of not as a negation but as a positive state is the deception that comes about through a syllogism. This occurs, in the case of things that belong or do not belong primitively, in two ways: either when one simply supposes that something belongs or does not belong, or when one arrives at the supposition through a syllogism. Now for the simple supposition the deception is simple, but for that

40| which comes through a syllogism, there are several kinds. For let A belong to no B immediately. Then if one draws the syllogistic conclusion that A belongs to B, taking C as middle term, one will be deceived through the syllogism. Now it is possible for both premises to be false, and it is also possible for only one of them to be. For if it is neither true that A belongs to no C nor that C belongs to no B, but each has been taken the other way round, both will be false. But it is possible for C to stand in relation to A and B in such a way that it is neither subordinate to A nor universal to B. For it is impossible for B to be contained wholly in anything (since A was said to belong to it not at all, and primitively), while A need not be universal to everything there is, so that both premises are false. But it is also possible to take one of the two as true — not, however, whichever one happens, but the A-C premise. For the C-B premise will always be false, because B is contained in nothing, while the A-C premise admits of being true — for instance if A belongs immediately to both C and B (for when the same thing is predicated primitively of several things, neither of those things will be in the other). And it makes no difference either if it does not belong immediately.

41| The deception concerning belonging, then, comes about only through these premises and in this way (for there was no syllogism concerning belonging in another figure), while that concerning not-belonging occurs both in the first figure and in the middle figure. Let us first say in how many ways it occurs in the first figure, and how the premises then stand. It is possible, then, when both premises are false — for instance if A belongs immediately to both C and B: for if it is taken that A belongs to no C, but C to all B, the premises are false. It is also possible when only one is false, and this whichever one happens. For it is possible for the A-C premise to be true and the C-B false — the A-C true because A does not belong to everything there is, the C-B false because it is impossible for C to belong to B, to which A belongs not at all: for then the C-B premise would no longer be true; and at the same time, if both premises were true, the conclusion too would be true. But it is also possible for the C-B premise to be true while the other is

42| false — for instance if B is contained both in C and in A: for one of the two must be subordinate to the other, so that if one takes A to belong to no C, that premise will be false. It is clear, then, that whether only one premise is false or both are, the syllogism will be deceptive in the case of immediate terms. In the middle figure it is not possible for both premises to be wholly false: for whenever A belongs to all B, there will be nothing one can take that belongs to all of the one and to none of the other; but the premises must be taken in such a way that the middle belongs to the one and not to the other, if there is to be a syllogism at all. If, then, taken in this way they are false, it is clear that they must actually stand contrariwise, the other way round; but this is impossible. Nothing, however, prevents each from being false in part — for instance if C belongs to some of both A and B: for if it is taken to belong to all A but to no B, both premises are false, yet not wholly, only in part. And likewise too if the negative is placed the other way round. It is also possible for only one of the two to be false, whichever one. For what belongs to all A also belongs to some B:

43| so if it is taken that C belongs to all A, but to no B at all, the C-A premise will be true, but the C-B false. Again, what belongs to no B will not belong to all A either: for if it belonged to A, it would belong to B too; but it did not belong to B. So if C is taken to belong to all A, but to no B, the C-B premise is true, the other false. And likewise too if the negative is transposed. For what belongs to no A will belong to no B either: so if C is taken to belong to no A at all, but to all B, the C-A premise will be true, the other false. And again, to take what belongs to all B as belonging to no A is false: for necessarily, if it belongs to all B, it also belongs to some A. So if C is taken to belong to all B, but to no A, the C-B premise will be true, the C-A false. It is clear, then, that both when both premises are false and when only one is, there will be a deceptive syllogism in the case of immediate terms.

44| In things that do not belong immediately, whenever the syllogism that produces the falsehood comes about through the proper middle term, it is not possible for both premises to be false, but only the one next to the major extreme. (By "proper middle term" I mean the one through which the syllogism of the contradiction comes about.) For let A belong to B through the middle term C. Since, then, when a syllogism comes about, CB must be taken as affirmative, it is clear that this premise will always be true; for it does not convert. But AC is false; for when this one is converted, the syllogism becomes the contrary one. Similarly too if the middle term were taken from another co-ordinate series — for instance D, if D both falls within the whole and is predicated of all B — for the premise DB must remain, while the other premise converts, so that one is always true and the other always false. And this kind of deception is roughly the same as that through the proper middle term. But if the syllogism does not come about through the proper middle term — when the middle term falls under A but belongs to none of B — both premises must be false.

45| For they must be taken contrary to how they actually stand, if there is to be a syllogism; and when they are taken this way, both come out false. For instance, if it belongs to the whole of D, but D belongs to none of B; for when these are converted there will be a syllogism, and both premises will be false. But when the middle term, say D, does not fall under A, AD will be true, but DB will be false. For AD is true because D was not within A, while DB is false because, if it were true, the conclusion too would have been true — but it was false. But when the deception comes about through the middle figure, it is not possible for both premises to be wholly false (for when B falls under A, nothing can belong to the one wholly and to the other not at all, as was said before too), but it is possible for one of them to be false, whichever it happens to be. For if

46| C belongs both to A and to B, and it is taken that it belongs to A but does not belong to B, then CA will be true, but the other premise false. Again, if it is taken to belong to B, but to none of A, then CB will be true, but the other false. Now if the syllogism of the deception is privative, it has been stated when and through what it will occur. But if it is affirmative: when it comes about through the proper middle term, it is impossible for both premises to be false; for CB must remain, if there is to be a syllogism at all, as was said before too. So AC will always be false; for this is the one that converts. Similarly too if the middle term is taken from another co-ordinate series, as was said also in the case of the privative deception; for DB must remain, while AD converts, and the deception is the same as before. But when it does not come about through the proper middle term: if D falls under A, this premise will be true, but the other false; for it is possible for A to belong to several things that are not subordinate to one another. But if D does not fall under A, this premise will clearly always be false (since it is taken as affirmative), while DB may be either true or false; for nothing prevents A from belonging to none of D while D belongs to all of B — for instance, animal to knowledge, and knowledge to music. Nor again is there anything to prevent A from belonging to none of D and D belonging to none of B.

47| It is clear, then, in how many ways and through what causes deceptions in syllogism can arise, both among immediate things and among things reached through demonstration. It is also clear that if any perception is lacking, some knowledge must be lacking too — knowledge which it is impossible to acquire, if indeed we learn either by induction or by demonstration. Demonstration proceeds from universals, induction from particulars; but it is impossible to study universals except through induction (since even the things spoken of by abstraction can be made known through induction, because certain things belong to each genus, even if they are not separable, insofar as each is such-and-such) — and it is impossible to be led on by induction without having perception. For perception is of particulars; it is not possible to acquire knowledge of them in any other way — not from universals without induction, nor through induction without perception.

48| Every syllogism proceeds through three terms, and one kind of syllogism is able to show that something belongs to C because it belongs to B and this to C, while the other, the privative kind, has one premise stating that one thing belongs to another and the other premise stating that it does not belong. It is clear, then, that these are the starting-points and so-called hypotheses; for taking these, one must show, in this way, for instance, that something belongs to it through B, and again that it belongs to B through another middle term, and likewise that B belongs to C. Now for those reasoning in accordance with opinion, and merely dialectically, it is clear that this alone must be examined: whether the syllogism comes from the most reputable premises available, so that even if there is in truth no middle term for B, but it seems there is one, whoever reasons through it has reasoned dialectically; but with a view to truth one must examine from what actually holds. And the position is this: since there is something that is itself predicated of another thing, not accidentally — by "accidentally" I mean the sort of case where we say that thing, the white thing, is a man, not speaking in the same way as when we say the man is white; for the man, without being anything else, is white, whereas the white thing is called man because it is accidental to the man to be white — there are, then, some things of such a kind as to be predicated in their own right. Let C, then, be of such a kind that it itself no longer belongs to anything else, but B belongs to it as a first thing, with nothing else between. And again let E likewise belong to F, and this to B. Must this, then, necessarily come to a stop, or is it possible to go on to infinity? And again, but A belongs to Θ as a first thing, with nothing prior between, and Θ belongs to H, with nothing prior between, and Θ belongs to

49| H, and this to B, must this too necessarily come to a stop, or is it possible for this too to go on to infinity? This differs from the former question to this extent: the one asks whether, starting from something that belongs to nothing else, but something else belongs to it, it is possible to go upward to infinity; the other, starting from something that is itself said of something else, while nothing is predicated of it, asks whether it is possible to go downward to infinity. Further, can the middle terms be infinite while the extremes are fixed? I mean, for instance, if A belongs to C, and B is the middle term between them, and there are others between B and A, and yet others between these, can these too go on to infinity, or is that impossible? This is the same thing to examine as whether demonstrations go on to infinity, and whether there is demonstration of everything, or whether they are bounded reciprocally by one another. I say the same about privative syllogisms and premises too — for instance, if A belongs to no B, either this holds of it first, or there will be some middle term prior to which it does not belong (for instance, if it does not belong to H, which belongs to all B), and again still another term prior to this, for instance if it does not belong to Θ, which belongs to all H. For in these cases too, either the prior terms to which it belongs are infinite in number, or the series comes to a stop. But with convertible terms the matter does not stand the same way; for among things reciprocally predicated there is no first or last term of which something is predicated, since all stand alike with respect to all in this way, whether the things predicated of it are infinite, or both of the difficulties raised are infinite — except that it is not possible for them to convert in the same way, but the one as accident, the other as predicate.

50| That the intermediates cannot be infinite, if the predications come to a stop going downward and upward, is clear. By "upward" I mean the direction toward the more universal, and by "downward" the direction toward the particular. For if, when A is predicated of Z, the intermediates are infinite (call them the B's), it is clear that it would then be possible, going downward from A, for one term to be predicated of another without limit (for before reaching Z the intermediates are infinite), and also, going upward from Z, for the intermediates to be infinite before reaching A. So if these things are impossible, it is also impossible for the intermediates between A and Z to be infinite. Nor does it make any difference if someone should say that some of the A's, B's, and Z's are contiguous with one another, so that there is no intermediate between them, while others cannot be so taken — it makes no difference at all. For whichever of the B's I take, it will stand to A or to Z either with infinite intermediates between them or not; and it makes no difference from which point the intermediates first become infinite, whether immediately or not immediately, since the terms after that point are infinite in either case.

51| It is evident also in the case of privative demonstration that the series will come to a stop, if indeed it comes to a stop in both directions in the case of affirmative demonstration. Let it be supposed that it is not possible to proceed to infinity either upward from the last term (by "last" I mean a term that does not itself belong to anything else, though something else belongs to it, for instance Z) or from the first term to the last (by "first" I mean a term that is itself said of something else, while nothing else is said of it). If this is so, then the series will also come to a stop in the case of negation. For non-belonging is shown in three ways: either B belongs to everything C belongs to, and A belongs to nothing that B belongs to; or, for the interval B-C — and always for the other interval — one must proceed to immediate premises, since this interval is affirmative. As for the other interval, it is clear that if A does not belong to some prior term, say D, then B will have to belong to all of D. And again, if A does not belong to some term prior to D, that term will have to belong to all of D. So, since the upward path comes to a stop, the path toward A will also come to a stop, and there will be some first term to which A does not belong. Again, if B belongs to all of A, and to none of C, then A belongs to none of C. Again, if this must be shown, clearly it will be shown either by the method described above, or by this one, or by the third. The first has been discussed; the second will now be shown. It would be shown in this way: D belongs to all of B and to none of C, if something must belong to B. And again, if this does not belong to C, something else belongs to D that does not belong to C.

52| That belongs to D but not to C. Since, then, belonging always comes to a stop as we go upward, non-belonging will also come to a stop. The third method was this: if a certain term belongs to all of B, but does not belong to C, then that term does not belong to all of what A belongs to. Again, this in turn will be shown either through the methods stated above or in the same way. In the former case it comes to a stop; if in the latter way, then again B will be taken as belonging to some E, to which C does not belong wholly. And this again in the same way. But since it is assumed that the series comes to a stop going downward as well, it is clear that the non-belonging of C will also come to a stop. And it is clear that even if the demonstration proceeds not by one method alone but by all of them — sometimes from the first figure, sometimes from the second or third — it will still come to a stop this way too; for the methods are finite in number, and a finite number of finite things taken a finite number of times must all together be finite. That the series comes to a stop in the case of privation, given that it does in the case of belonging, is clear. That it comes to a stop in those latter cases is evident, when one looks at it dialectically, in this way.

53| In the case of terms predicated in the what-it-is, this is clear: for if it is possible to define, or if the essence is knowable, and infinite things cannot be traversed, then the terms predicated in the what-it-is must be finite. We put the general point this way. One can truly say "the white thing walks" and "that large thing is wood," and again "the wood is large" and "the man walks." But saying it the one way is different from saying it the other way. For when I say that the white thing is wood, I mean that what happens to be white is wood, not that white is the underlying subject for the wood; for the thing did not become wood by being white or by being a certain white thing — so it is wood only accidentally. But when I say that the wood is white, I do not mean that something else is white and that it happens to be wood — as when I say that the musical thing is white (for then I mean that the man, who happens to be musical, is white) — rather, the wood is the underlying subject, the very thing that came to be, and it is not something other than what is properly wood, or a kind of wood. If, then, we are to lay down a rule, let this

54| way of speaking be called predicating, and the other way either not predicating at all, or predicating not in an unqualified sense but predicating accidentally. And the white is the thing predicated, as it were, and the wood is that of which it is predicated. Let it be laid down, then, that the predicate is always predicated of the subject in an unqualified sense, not accidentally; for it is in this way that demonstrations demonstrate. So that, whenever one thing is predicated of one thing, it is predicated either in the what-it-is, or as a quality, quantity, relation, doing, being-affected, place, or time. Further, terms signifying substance signify, of that of which they are predicated, exactly that thing or a kind of that thing; but whatever does not signify substance, but is said of another underlying subject which it is not — either exactly that thing or a kind of that thing — is an accident, as white is said of man. For man is neither exactly white nor a kind of white, but rather, presumably, an animal — for man is exactly an animal. Whatever does not signify substance must be predicated of some underlying subject, and there cannot be something white that is not something else which is white. As for the Forms, let them go their way — they are mere twitterings, and even if they exist, they are irrelevant to our argument,

55| since demonstrations are concerned with things of the sort just described. Further, if this is not a quality of that, and that a quality of this, nor a quality of a quality, then it is impossible for such terms to be counter-predicated of one another — that is, it is possible to say something true, but not possible to counter-predicate truly. For either it will be predicated as substance, that is, as being the genus or the differentia of the thing predicated. But it has been shown that these cannot be infinite, either going downward or going upward (for instance: man, biped; this, animal; this, something else; nor is animal predicated of man, and this of Callias, and this of something else again, within the what-it-is) — for it is possible to define any such substance in its entirety, whereas infinite things cannot be gone through in thought. So there are no infinite series either upward or downward: for it is not possible to define that of which infinite things are predicated.

56| So genera will not be predicated reciprocally of one another; for then each would be exactly what the other is. Nor again will quality, or any of the other categories, be predicated of a substance, unless as an accident; for all these are accidents, and are predicated of substances. But further, there will not be an infinite regress upward either. For of each thing there is predicated whatever signifies either a quality or a quantity or something of that sort, or the elements within its substance; and these are limited in number, and the genera of the categories are limited — for a thing is either a quality, a quantity, a relative, active, passive, a where, or a when. It is assumed, then, that one thing is predicated of one thing, and that things which are not answers to "what is it" are not predicated of themselves as such; for all such things are accidents, some in their own right, others in a different way. All of these, we say, are predicated of some underlying subject, while an accident is not itself an underlying subject; for we posit none of such things as being something that is not, in being said, said to be something else, but rather each of them is said of something else, and that in turn of something else again. So it will not be said that one thing is predicated of one thing without limit, either upward or downward. For the things of which accidents are predicated, namely whatever falls within the substance of each thing, are not infinite;

57| and upward too, both these and the accidents together are not infinite. There must, then, be something of which something is predicated first, and something else of that, and this series must come to a stop, and there must be something which is no longer predicated of anything else prior, nor is anything else prior predicated of it. This, then, is one way in which the proof is stated; and there is a further one besides: if there is demonstration of the things of which certain prior things are predicated, while of the things of which there is demonstration it is not possible either to be in a better position than knowing them, or to know them without demonstration; and if this thing is known through those, while we do not know those and are not in a better position with respect to them than knowing, then we will not have scientific knowledge of what is known through them either. If, then, it is possible to know something through demonstration without qualification, and not from certain things nor from a hypothesis, the intermediate predications must come to a stop. For if they do not stop, but there is always something above what has been taken, there will be demonstration of everything; so that if

58| it is not possible to traverse an infinite series, then the things of which there is demonstration we will not know through demonstration. If, then, we are not in a better position with respect to them than knowing, there will be no scientific knowledge of anything through demonstration without qualification, but only from a hypothesis. This, then, is how one might be persuaded of what has been said by dialectical argument; but analytically it becomes clear more concisely through the following, that it is not possible for the things predicated to be infinite either upward or downward in the demonstrative sciences with which our inquiry is concerned. For demonstration concerns whatever belongs to things in their own right. And "in its own right" is said in two ways: both what belongs to those things within their essence, and those things within whose essence the things themselves belong — for instance, oddness belongs to number, and belongs to it in its own right, while number itself belongs within the definition of oddness; and again, plurality, or the divisible, belongs within the definition of number. Neither of these can be infinite: not oddness in relation to number in the way described (for then oddness would again have something else within which it was present as belonging; but if this is so, number will be present first among the things belonging to it; so if it is not possible for infinitely many such things to belong within the one, there will not be an infinite regress upward either;

59| but rather it is necessary that all these belong to the first thing, number for instance, and that number belong to them, so that they will convert with one another, and not extend beyond); nor again can the things that belong within the essence be infinite, for then definition would not be possible either. So if all the things predicated in their own right are counted as such, and these are not infinite, the series will come to a stop going upward, and so also going downward. And if this is so, the terms lying between two limiting terms are always finite. And if this is so, it is already clear as regards demonstrations too that there must be principles, and that there cannot be demonstration of everything — which is exactly what we said some people claim at the outset. For if there are principles, not everything is demonstrable, nor is it possible to proceed to infinity; for the being of either of these amounts to nothing other than there being no immediate and indivisible interval, but all intervals being divisible. For what is demonstrated is demonstrated by a term being inserted within the interval, not by one being added on outside it; so that if this could proceed to infinity, it would be possible for infinitely many middle terms to fall between two limiting terms. But this is impossible, if the predications come to a stop both upward and downward. And that they do come to a stop has been shown by dialectical argument earlier, and now analytically.

60| Now that these points have been shown, it is clear that if the same thing belongs to two things — say A belongs to both C and D — and neither of these is predicated of the other, either not at all or not of every case, then it will not always belong to them in virtue of something common. For instance, having angles equal to two right angles belongs to the isosceles and the scalene triangle in virtue of something common (it belongs to them insofar as they are a certain figure, and not insofar as they differ from each other); but this is not always the case. For let B be that in virtue of which A belongs to C and D. Then it is clear that B too belongs to C and D in virtue of some other common thing, and that in turn in virtue of yet another, so that infinitely many terms would fall between two limiting terms. But this is impossible. It is not necessary, then, that the same thing always belong to several things in virtue of something common, at least if there are to be immediate intervals. Yet the terms must belong to the same genus and be drawn from the same indivisibles, if the common element is to belong to the things that belong in their own right; for it was not possible for what is proved to cross over from one genus to another. It is also clear that when A belongs to B: if there is some middle term, it is possible to show that it belongs to B, and the elements of this proof are the same in number as the middle terms; for the immediate premises are elements, either all of them or the universal ones. But if there is no middle term, there is no longer demonstration, but this is the road that leads to the principles. Likewise too if A does not belong to B: if there is either a middle term or something prior to which it does not belong, there is demonstration; but if not, there is none, but a principle, and there are as many elements as there are terms; for the premises of these terms are the principles of the demonstration

61| . And just as some principles are undemonstrated to the effect that this is this and this belongs to this, so too there are principles to the effect that this is not this, nor does this belong to this, so that some principles will assert that something is, and others that something is not. When one must prove something, one must take what is predicated first of B. Let this be C, and similarly of C let it be D. And proceeding always in this way, no premise wider than A is ever taken in the course of the proof, nor is anything wider taken as belonging; rather the middle term is always packed more densely, until the terms become indivisible and one. And a term is one when it becomes immediate, and the immediate premise alone is, without qualification, one premise. And just as in other domains the starting principle is something simple, though this is not the same thing everywhere — in weight it is the mina, in melody the quarter-tone, and something different in each different case — so in a syllogism the unit is the immediate premise, while in demonstration and scientific knowledge it is the intellect. In ostensive syllogisms, then, nothing that belongs falls outside the terms reached; but in privative syllogisms, where something must be shown to belong, nothing falls outside this term either — for instance, if A does not belong to B through C (for if C belongs to all B, and A belongs to no C); and again, if it is necessary to show that A belongs to no C, a middle term between A and C must be taken, and the process will always proceed in this way. But if it is necessary to show that D does not belong to E, because C belongs to all D and to no E, the process will never fall outside E; for this is the term to which it must belong. And in the third figure, the process will never go outside either the term from which the denial must proceed or the term that must be denied.

62| Since demonstration is either universal or particular, and either affirmative or privative, it is disputed which kind is better; and likewise concerning the demonstration said to prove directly and the demonstration that leads to the impossible. Let us first consider the universal and the particular; and having made this clear, let us then speak of the demonstration said to prove directly and the one that leads to the impossible. Now to some, looking at it in the following way, it might seem that the particular

63| is better. For if that demonstration is better by which we know more, and this is better by that which we know each thing when we know it in its own right rather than when we know it in virtue of something else (for instance, we know musical Coriscus better when we know that Coriscus is musical than when we know that a man is musical, and likewise in the other cases), and the universal demonstration proves that something else belongs, not that the thing itself happens to belong (for instance, that the isosceles triangle has its angles equal to two right angles not because it is isosceles but because it is a triangle), while the particular demonstration proves that the thing itself has this — if, then, the demonstration that proceeds in virtue of the thing itself is better, and the particular is more of this sort than the universal, the particular demonstration would be the better. Further, if the universal is not something over and above the particulars, while demonstration produces the opinion that there is some such thing in virtue of which it demonstrates, and that some such nature belongs among the things that are — for instance, some nature of triangle over and above the particular triangles, and of figure over and above the particular figures, and of number over and above the particular numbers — while a demonstration about what is is better than one about what is not, and better is that demonstration by which one will not be deceived than one by which one will — and the universal demonstration is of this latter sort (for in proceeding they demonstrate as they do about the proportional, namely that whatever is such-and-such will be proportional, though it is neither line nor number nor solid nor plane, but something over and above these) — if, then, this demonstration is more universal but concerns what is less, and produces a false opinion, the universal demonstration would be worse than the particular.

64| Or is it rather that, first, neither the one account nor the other applies any more to the universal than to the particular? For if having angles equal to two right angles belongs not insofar as it is isosceles but insofar as it is a triangle, then one who knows that it is isosceles knows the fact less well, in respect of why it belongs, than one who knows that it is a triangle. And in general, if a thing is demonstrated of something not insofar as it is a triangle, that would not be a demonstration; but if it is demonstrated insofar as it is, then one who knows each thing insofar as each thing belongs knows it better. If, then, triangle extends more widely, and the account is the same, and triangle is not spoken of by mere homonymy, and having two right angles belongs to every triangle, then it is not the triangle that has the property insofar as it is isosceles, but the isosceles that has it insofar as it is a triangle, that it has its angles thus. So that one who knows universally knows that it belongs better than one who knows the particular. The universal demonstration is therefore better

65| than the particular. Further, if there is some single account and the universal is not spoken of by homonymy, it would exist no less than some of the particulars, but even more so, inasmuch as the imperishable things are found among the universals, while the particulars are more perishable. Further, there is no necessity at all to suppose that this universal is something over and above the particulars, on the ground that it signifies a single thing, any more than in the case of the other things that do not signify a this-something but rather a quality or a relation or a doing. And if it is supposed to be something over and above them, the fault lies not with the demonstration but with the hearer. Further, if demonstration is a deductive syllogism proving the cause and the reason why, and the universal is more of a cause (for that to which something belongs in its own right is itself the cause of itself, and the universal is primary; therefore the universal is the cause) — so that the demonstration is also better, since it more concerns the cause and the reason why. Further, we seek the reason why up to the point at which we think we know, namely when it is no longer the case that this is something else either coming to be or being; for the end and the limit is now the extreme point in this way. For instance, for what purpose did he come? In order to get the money; and this, in order to pay back what he owed; and this, in order not to act unjustly;

66| and going on in this way, when it is no longer for the sake of something else nor for the sake of another thing, we say it is for this as an end that he came and that it is and that it comes to be, and then we know best why he came. If, then, it holds similarly for all causes and reasons why, and among those causes that are of this kind — that is, as that for the sake of which — we know best in this way, then in the other cases too we know best when this no longer belongs because of something else. So when we know that the external angles are equal to four right angles because the figure is isosceles, it still remains to ask why the isosceles has this — because it is a triangle; and this again, because it is a rectilinear figure. And if this is no longer so because of something else, then we know best. And it is then also universal; the universal demonstration is therefore better. Further, the more a demonstration is particular, the more it falls into infinities, while the universal tends toward the simple and the limited. And insofar as things are infinite they are not knowable, but insofar as they are limited they are knowable. So insofar as things are universal they are more knowable than insofar as they are particular. Therefore the universal things are more

67| demonstrable; and of demonstrable things, the more demonstrable ones admit demonstration more, for correlative things increase together. The universal demonstration is therefore better, since it is more truly a demonstration. Further, that demonstration is more choiceworthy by which one knows this thing and also something else, than one by which one knows only this thing; and one who has the universal demonstration knows the particular as well, but one who has the particular does not know the universal; so that on this ground too the universal would be more choiceworthy. Further, consider it this way: to demonstrate more universally is to demonstrate through a middle term that is nearer to the starting point. And nearest of all is the immediate premise; and this is a starting point. If, then, the demonstration from a starting point is more exact than the one not from a starting point, then the demonstration more from a starting point is more exact than the one less so. And the universal demonstration is more of this sort; the universal demonstration would therefore be superior. For instance, if one had to demonstrate A of D: the middle terms are B and C; B is higher up, so that the demonstration through it is more universal. But some of the things said belong to logic; and it is most clearly evident that the universal is the more authoritative demonstration on this ground: that among premises, when we have the prior one we in a way also know the posterior and have it potentially — for instance, if someone knows that every triangle has two right angles, he in a way also knows, potentially, that the isosceles has two right angles, even if he does not know that the isosceles is a triangle; but one who has this latter premise does not know the universal at all, neither potentially nor in actuality. And the universal is an object of intellect, while the particular terminates in perception.

68| Let this much be said to show that the universal demonstration is better than the particular. That the demonstration proving directly is better than the privative one is clear from the following. Let this demonstration be better than another, other things being equal, which proceeds from fewer postulates or hypotheses or premises. For if they are equally well known, knowing more quickly through these will result, and this is more choiceworthy. The account of the premise, that the demonstration from fewer is better, is this, stated universally: if it were equally the case that the middle terms are known, but the prior ones are better known, let the demonstration through middle terms be that of the

69| ...through B, C, D that A belongs to E, and the other through F that A belongs to E. Now it stands the same way with the claim that A belongs to D and that A belongs to E: the claim that A belongs to D is prior and more knowable than the claim about E, since the latter is demonstrated through the former, and what a thing is proved through is more credible than what is proved. So a demonstration through fewer terms is better than the others, other things being equal. Now both kinds are proved through three terms and two premises, but the one takes as a premise only that something is, while the other takes both that something is and that something is not; so it proceeds through more, and is therefore worse. Again, since it has been shown that it is impossible for a syllogism to come about when both premises are negative, but one premise must be of that kind and the other must be affirmative, we must grasp this point as well, in addition. For as a demonstration is extended, the affirmative premises must become more numerous, but the negative premises cannot be more than one in any syllogism whatever. For let A belong to none of the things B belongs to, and let B belong to all of C. If, then, it should be necessary again to extend both premises, a middle term must be inserted.

70| Let D be a middle for B, and E a middle for B. Now E is clearly affirmative, while D is affirmative with respect to B but is set up in relation to the negative conclusion: for D belongs to all of B, but A must belong to none of D. So there comes to be one negative premise, the one about A and D. The same manner holds in the other syllogisms too. For the middle term, when both extreme terms are affirmative, is affirmative in both directions; but where one extreme is negative, the middle must be negative in that one direction, so that this becomes the one premise of that kind, and the others are affirmative. If, then, that through which something is proved is more knowable and more credible, and the negative conclusion is proved through the affirmative, while the affirmative is not proved through the negative, the affirmative — being prior, more knowable, and more credible — would be better. Again, if the principle of a syllogism is the universal immediate premise, and in the ostensive syllogism this universal premise is affirmative while in the negative syllogism it is negative, and the affirmative is prior to and more knowable than the negative (for the negation is known through the affirmation, and the affirmation is prior, just as being is prior to not-being), then the principle of the ostensive syllogism is better than that of the negative one; and the syllogism that uses better principles is better. Further, the ostensive syllogism is more like a principle, since the negative syllogism cannot exist without the one that proves something positively.

71| Since the affirmative syllogism is better than the negative, it is clear that it is also better than the one that leads to the impossible. But we must know what the difference between these two is. Let A belong to none of B, and let B belong to all of C: then A must of necessity belong to none of C. Now when the terms are taken this way, the negative demonstration that A does not belong to C is ostensive. But the reduction to the impossible works like this: if one had to prove that A does not belong to B, one must assume that it does belong, and that B belongs to C, so that it results that A belongs to C. Let this be known and agreed to be impossible. Then it is not possible for A to belong to B. So if it is agreed that B belongs to C, it is impossible for A to belong to B. The terms, then, are arranged in the same way in both cases, but it makes a difference which of the two negative premises is more knowable — whether that A does not belong to B, or that A does not belong to C. So whenever the conclusion — that something is not the case — is more knowable, the demonstration becomes one leading to the impossible; but whenever the premise within the syllogism is more knowable, it is the demonstrative one.

72| But by nature, the claim that A does not belong to B is prior to the claim that A does not belong to C. For the things from which the conclusion comes are prior to the conclusion; and A's not belonging to C is the conclusion, while A's not belonging to B is that from which the conclusion comes. For it is not the case that, whenever something happens to be done away with, this is the conclusion and those other things are what it comes from; rather, that from which a syllogism comes is whatever stands in such a relation that it is either whole to part or part to whole, and the premises 'A to C' and 'B to C' do not stand in this relation to each other. If, then, the demonstration from more knowable and prior things is superior, and both kinds of demonstration are credible by way of something's not being the case, but the one is from what is prior and the other from what is posterior, then the negative demonstration would be, without qualification, better than the one leading to the impossible; so that the affirmative demonstration, being better than the negative, is clearly also better than the one leading to the impossible. One science is more exact than, and prior to, another science: when it is at once a science of the fact and of the reason why, and not a science of the fact separately from one of the reason why; and when it is not a science of an underlying subject, as arithmetic is more exact than harmonics; and when it proceeds from fewer things than one that proceeds by addition, as arithmetic is more exact than geometry. By 'from addition' I mean this: a unit is a substance without position, while a point is a substance with position — the latter is arrived at by addition.

73| A science is one science when it belongs to one genus — as many things as are composed out of the primary items and are parts of these, or are attributes of these in their own right. One science is different from another when their principles come neither from the same source nor some of them from the others. A sign of this is when one arrives at the undemonstrables, for these must be in the same genus as the things demonstrated. And it is also a sign of this when the things proved by way of them are in the same genus and akin to one another. It is possible for there to be several demonstrations of the same conclusion, not only by taking, from the same co-ordinate series, a middle term that is not the immediately connecting one — as, among the terms falling under B, taking C or D or F — but also by taking a middle from a different series. For example, let A be 'change,' let D be 'being in motion,' let B be 'feeling pleasure,' and again let G be 'coming to rest.' Then it is true to predicate D of B and A of D: for what feels pleasure is in motion, and what is in motion changes. Again, it is true to predicate G of B and A of G: for everyone who feels pleasure comes to rest, and what comes to rest changes. So the syllogism proceeds through different middle terms, and not from the same co-ordinate series. This does not mean, however, that neither middle term is predicated of the other, for both must belong to one and the same thing. One should also examine, working through the other figures, in how many ways it is possible for a syllogism of the same conclusion to come about.

74| There is no scientific knowledge, by way of demonstration, of what comes about by chance. For what happens by chance is neither something that happens of necessity nor something that happens for the most part, but what comes about apart from these; and demonstration is of one or the other of these two. For every syllogism proceeds either through necessary premises or through premises that hold for the most part: and if the premises are necessary, the conclusion too is necessary, while if they hold for the most part, the conclusion is likewise of that kind. So if what happens by chance is neither for the most part nor of necessity, there can be no demonstration of it.

75| Nor can one have scientific knowledge through perception. For even if perception is of a thing of such-and-such a quality and not merely of some particular, still one necessarily perceives a this, and here, and now. But it is impossible to perceive what is universal and holds in all cases; for that is not a this nor now, since if it were it would not be universal — for we say that what is universal is what holds always and everywhere. Since, then, demonstrations are universal, and these cannot be perceived, it is clear that one cannot have scientific knowledge through perception either. Indeed it is plain that even if we could perceive that the triangle has its angles equal to two right angles, we would still be looking for a demonstration, and would not, as some say, already have known it; for perception must be of a particular, whereas scientific knowledge consists in coming to know the universal. That is why, even if we were standing on the moon and saw the earth screening off the sun, we would not know the cause of the eclipse. For we would perceive that it is eclipsing now, but not why in general — since there was no perception of the universal. Not that, from observing this happening often, we could not go on to hunt out the universal and so obtain a demonstration; for the universal becomes evident from many particular cases.

76| The universal is valuable because it makes the cause plain; so that, concerning things whose cause is other than themselves, knowledge of the universal is more valuable than the perceptions and than the direct grasping of the particular — though a different account holds for primary things. It is clear, then, that it is impossible to know any of the demonstrable things by perceiving it, unless one means by 'perceiving' having scientific knowledge through demonstration. There are, however, some problems that get traced back to a failure of perception. Some things, if we saw them, we would not go on inquiring into — not because we would know them by the seeing itself, but because from the seeing we would have the universal. For example, if we saw that the glass was perforated and the light passing through it, it would be clear also why it burns, because we see each case separately, but at the same time understand that this holds in the same way in every such case.

77| It is impossible for the same principles to hold for all syllogisms — first, on general grounds. For some syllogisms are true and some are false. For even if it is possible to reach a true conclusion from false premises, this happens only once — for instance if A holds of C truly, but the middle term B is false: for A does not hold of B, nor B of C. But if middle terms are taken for these premises, they will be false, because every conclusion from false premises is false, while true conclusions come from true premises — the false and the true being different things. Again, false conclusions do not even come from premises identical with one another: false things can be contrary to one another and even impossible to hold together at once — for instance, that justice is injustice or cowardice, that a human being is a horse or an ox, or that the equal is greater or less. From what has been laid down, the point follows in this way too: not even among true things are the principles the same for everything. For the principles of many things differ in genus, and do not fit together — for instance, units do not fit with points, since the one has no position while the other does. And it is necessary that they either fit into the middle, or above, or below, or that some fall inside the terms and some outside.

78| But not even among the common principles can there be some from which everything whatsoever will be demonstrated. By common principles I mean, for example, that of everything one may either affirm or deny it. For the genera of things that are, are different, and some common principles apply only to quantities, others only to qualities, and it is with these that demonstration proceeds through the common principles. Further, the principles are not much fewer than the conclusions; for the premises are principles, and the premises arise either by taking on an additional term or by inserting one. Further, the conclusions are unlimited, but the terms are limited. Further, some principles are necessary, others admit of being otherwise. Looking at it in this way, then, it is impossible for the same principles, being limited in number, to hold for the conclusions, which are unlimited. But if someone should mean it in some other way — that these are the principles of geometry, these of arithmetic, these of medicine — what would this claim amount to except that there are principles belonging to the sciences? But to say that they are the same principles

79| is absurd, on the ground that a thing is the same as itself; for on that basis everything would turn out to be the same. But surely it is not the case either that anything whatsoever is demonstrated from everything, which is what seeking the same principles for everything would amount to; for that is far too simple-minded a view. For this is not what happens in the manifest mathematical sciences, nor is it possible in analysis; for the immediate premises are the principles, and a different conclusion arises when a different immediate premise is added. But if someone should say that it is the primary immediate premises that are the principles, there is one such in each genus. But if it is neither the case that anything whatsoever ought to be demonstrated from all of them together, nor that the principles are so different from one another that each science has wholly different ones, what remains is to ask whether the principles of all things are akin to one another, but that these conclusions come from these principles and those conclusions from those. But it is clear that this too is not possible; for it has been shown that things differing in genus have principles differing in genus. For principles are of two kinds: those from which a demonstration proceeds, and that about which it proceeds. Now the principles from which are common, but those about which are proper to the genus — for instance, number, magnitude.

80| Scientific knowledge and its object differ from opinion and its object in that scientific knowledge is universal and proceeds through necessary things, and the necessary cannot be otherwise. But there are some things that are true and are the case, yet admit of being otherwise. It is clear, then, that scientific knowledge is not concerned with these; for then things capable of being otherwise would turn out incapable of being otherwise. Nor, again, does intellect concern them — by intellect I mean the principle of scientific knowledge — nor does undemonstrated knowledge, which is the grasping of the immediate premise. Now intellect, scientific knowledge, opinion, and what is stated through these, are true; so it remains that opinion is concerned with what is true or false, yet admits of being otherwise. This is the grasping of an immediate but non-necessary premise. And this agrees with how things appear: for opinion is unstable, and such is its nature. Besides this, no one thinks he is holding an opinion when he thinks it impossible for a thing to be otherwise — he thinks he has knowledge; but when he thinks a thing is so, yet nothing prevents its also being otherwise, then he holds an opinion, on the ground that opinion concerns things of that sort, while scientific knowledge concerns the necessary.

81| How then can the same thing be both opined and known, and why will opinion not simply be scientific knowledge, if one supposes that everything one knows admits also of being opined? For the one who knows and the one who opines will each proceed through the middle terms, until he arrives at the immediates, so that if the former has knowledge, the one who opines has knowledge as well. For just as it is possible to opine the fact, so it is possible to opine the reason why — and this is the middle term. Or is it rather this: if one grasps things that do not admit of being otherwise in the way one grasps the definitions through which demonstrations proceed, one will not be opining but will have knowledge; but if one holds that they are true, yet not that these belong to the things in virtue of their substance and form, one will be opining and will not truly have knowledge — both the fact and the reason why, if one opines through the immediates; but if not through the immediates, one will opine the fact alone. Scientific knowledge and opinion of the same thing are not possible together in every case; rather, just as false and true belief can in a sense be of the same thing, so too scientific knowledge and opinion can be of the same thing. For when some say that true and false opinion are of the same thing, absurd consequences follow, among others that one would not be opining what one opines falsely —

82| Since the same thing is spoken of in several ways, there is a sense in which it is possible, and a sense in which it is not. For to have a true opinion that the diagonal is commensurable would be absurd; but because the diagonal, which the opinions are about, is the same, in that way they are opinions of the same thing, though what it is to be each of them, by definition, is not the same. Similarly too, knowledge and opinion are of the same thing. For knowledge is of the animal in such a way that it is not possible for it not to be an animal, while opinion is of it in such a way that it is possible for it not to be — for instance, if one grasps the one as precisely what man is, and the other as man, but not as precisely what man is. For it is the same thing because both grasp man, but the manner is not the same. It is clear from this that it is not possible to have opinion and knowledge of the same thing at the same time. For then one would at the same time hold the supposition that the same thing could be otherwise and could not be otherwise, which is not possible. For each of the two can belong to a different subject of the same thing, as has been said, but in the same subject not even in this way is it possible; for one would hold the supposition at the same time — for instance, that man is precisely animal (for this was what it is for it not to be possible not to be an animal) and also not precisely animal; for let this be what 'possible' means. As for how the remaining notions should be distributed among thought, intellect, knowledge, craft, practical wisdom, and wisdom, some belong more to the study of nature, others more to the study of ethics.

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