by Aristotle
Whatever a man assumes without proving it himself although it is provable—if he assumes something that seems to be the case to the learner, he supposes it (and it is a supposition not simpliciter but only in relation to the learner); but if he assumes the same thing when there is either no opinion present in the learner or actually a [30] contrary one present, he postulates it. And it is in this that suppositions and postulates differ; for a postulate is what is contrary to the opinion of the learner, which14 though it is demonstrable is assumed and used without being proved.
Now terms are not suppositions (for they are not said to be or not be [35] anything),15 but suppositions are among the propositions, whereas one need only grasp the terms; and suppositions are not that (unless someone will say that hearing is a supposition), but rather propositions such that, if they are the case, then by their being the case the conclusion comes about.
Nor does the geometer suppose falsehoods, as some have said, stating that one [40] should not use a falsehood but that the geometer speaks falsely when he says that the line which is not a foot long is a foot long or that the drawn line which is not straight is straight. But the geometer does not conclude anything from there being [77a1] this line which he himself has described, but from what is made clear through them.
Again, every postulate and supposition is either universal or particular; but terms are neither of these.
11 · For there to be forms or some one thing apart from the many is not [5] necessary if there is to be demonstration; however, for it to be true to say that one thing holds of many is necessary. For there will be no universal if this is not the case; and if there is no universal, there will be no middle term, and so no demonstration either. There must, therefore, be some one and the same thing, non-homonymous, holding of several cases.
That it is not possible to affirm and deny at the same time is assumed by no [10] demonstration—unless the conclusion too is to be proved in this form. It is proved by assuming that the first term is true of the middle and that it is not true to deny it. It makes no difference if one assumes that the middle term is and is not; and the same holds of the third term too. For if it is granted that that of which it is true to [15] say man, even if not-man is also true of it—but provided only that it is true to say that a16 man is an animal and not not an animal—for17 it will be true to say that Callias, even if not Callias, is nevertheless an animal and not not an animal. The explanation is that the first term is said not only of the middle but also of something else, because it holds of several cases; so that even if the middle both is it and is not [20] it, that makes no difference with regard to the conclusion.
That everything is affirmed or denied truly is assumed by demonstration per impossibile, and that not always universally but as far as is sufficient in so far as it bears on the genus (I say on the genus—i.e. the genus about which one is bringing the demonstrations), as has been said earlier too. [25]
All the sciences associate with one another in respect of the common items (I call common those which they use as demonstrating from them—not those about which they prove nor what they prove); and dialectic associates with them all, and so would any science that attempted to prove universally the common items—e.g. [30] that everything is affirmed or denied, or that equals from equals leave equals, or any things of the sort. But dialectic is not in this way concerned with any determined set of things, nor with any one genus. For then it would not ask questions; for one cannot ask questions when demonstrating because when opposites are the case the [35] same thing is not proved. This has been proved in the account of dedication.18
12 · If a deductive question and a proposition of a contradiction are the same thing, and there are propositions in each science on which the deductions in each depend, then there will be a sort of scientific question from which the [40] deduction appropriate to each science comes about. It is clear, therefore, that not every question will be geometrical (or medical—and similarly in the other cases too), but only those from which either there is proved one of the things about which [77b1] geometry is concerned, or19 something which is proved from the same things as geometry, such as optical matters. And similarly in the other cases too.
And for those one should indeed supply an argument from the principles and conclusions of geometry; but for the principles, the geometer as geometer should not [5] supply an argument; and similarly for the other sciences too. We should not, therefore, ask each scientist every question, nor should he answer everything he is asked about anything, but only those determined by the scope of this science. If one [10] argues in this way with a geometer as geometer it is evident that one will do so correctly, if one proves something from these things; but otherwise, not correctly. And it is clear that one does not refute the geometer either, except incidentally; so that one should not argue about geometry among non-geometers—for the man who [15] argues badly will escape notice. And the same goes for the other sciences too.
Since there are geometrical questions, are there also nongeometrical ones? And in each science which sort of ignorance is it in regard to which they are, say, geometrical? And is a deduction of ignorance a deduction from the opposites (or a [20] paralogism, though a geometrical one)? Or is it a deduction from another art? e.g. a musical question is non-geometrical about geometry, but thinking that parallels meet is geometrical in a sense and non-geometrical in another way. For this is twofold (like being non-rhythmical), and one way of being non-geometrical is by [25] not having geometrical skill (like being non-rhythmical) and the other by having it badly; and it is this ignorance and ignorance depending on such principles that is contrary to understanding.
In mathematics paralogism does not occur in the same way, because the twofold term is always the middle term; for something is said of all this, and this [30] again is said of all something else (of what is predicated one does not say all), and one can as it were see these by thought, though they escape notice in arguments. Is every circle a shape? If you draw one it is clear. Well, is the epic a circle? It is evident that it is not.
One should not bring an objection against it if the proposition is inductive. For just as there is no proposition which does not hold of several cases (for otherwise it [35] will not hold of all cases; but deduction depends on universals), it is clear that there is no objection either. For propositions and objectives are the same thing; for what one brings as an objection might become a proposition, either demonstrative or dialectical.
It happens that some people argue non-deductively because they assume what follows both terms—e.g. Caeneus does when he says that fire consists in multiple [78a1] analogy,20 for fire, he says, is generated quickly, and so is this analogy. In this way there is no deduction; but there is if multiple analogy follows fastest analogy and the fastest changing analogy follows fire.
Now sometimes it is not possible to make a deduction from the assumptions; [5] and sometimes it is possible, but it is not seen.
If it were impossible to prove truth from falsehood, it would be easy to make an analysis; for they would convert from necessity. For let A be something that is the case; and if this is the case, then these are the case (things which I know to be the case, call them B). From these, therefore, I shall prove that the former is the case. [10] (In mathematics things convert more because they assume nothing accidental—and in this too they differ from argumentations—but only definitions.)
A science increases not through the middle terms but by additional assumption—e.g. A of B, this of C, this again of D, and so on ad infinitum; and [15] laterally—e.g. A both of C and of E (e.g. A is definite—or even indefinite—number; B is definite odd number; C odd number; therefore A holds of C. And D [20] is definite even number; E is even number: therefore A holds of E).
13 · Understanding the fact and the reason why differ, first in the same science—and in that in two ways: in one way, if the deduction does not come about through immediates (for the primitive explanation is not assumed, but understandi
ng [25] of the reason why occurs in virtue of the primitive explanation); in another, if it is through immediates but not through the explanation but through the more familiar of the converting terms. For nothing prevents the nonexplanatory one of the counterpredicated terms from sometimes being more familiar, so that the demonstration will occur through this.
E.g. that the planets are near, through their not twinkling: let C be the planets, [30] B not twinkling, A being near. Thus it is true to say B of C; for the planets do not twinkle. But also to say A of B; for what does not twinkle is near (let this be got through induction or through perception). So it is necessary that A belongs to C; so [35] that it has been demonstrated that the planets are near. Now this deduction is not of the reason why but of the fact; for it is not because they do not twinkle that they are near, but because they are near they do not twinkle.
But it is also possible for the latter to be proved through the former, and the demonstration will be of the reason why—e.g. let C be the planets, B being near, A [78b1] not twinkling. Thus B belongs to C and A to B; so that A belongs to C. And the deduction is of the reason why; for the primitive explanation has been assumed.
Again, take the way they prove that the moon is spherical through its [5] increases—for if what increases in this way is spherical and the moon increases, it is evident that it is spherical. Now in this way the deduction of the fact comes about; but if the middle term is posited the other way about, we get the deduction of the [10] reason why; for it is not because of the increases that it is spherical, but because it is spherical it gets increases of this sort. Moon, C; spherical, B; increases, A.
But in cases in which the middle terms do not convert and the non-explanatory term is more familiar, the fact is proved but the reason why is not.
Again, in cases in which the middle is positioned outside—for in these too the demonstration is of the fact and not of the reason why; for the explanation is not [15] mentioned. E.g. why does the wall not breathe? Because it is not an animal. For if this were explanatory of breathing—i.e. if the denial is explanatory of something’s not belonging, the affirmation is explanatory of its belonging (e.g. if imbalance in the hot and cold elements is explanatory of not being healthy, their balance is [20] explanatory of being healthy), and similarly too if the affirmation is explanatory of something’s belonging, the denial is of its not belonging. But when things are set out in this way what we have said does not result; for not every animal breathes. The deduction of such an explanation comes about in the middle figure. E.g. let A be [25] animal, B breathing, C wall: then A belongs to every B (for everything breathing is an animal), but to no C, so that B too belongs to no C—therefore the wall does not breathe.
Explanations of this sort resemble those which are extravagantly stated (that [30] consists in arguing by setting the middle term too far away)—e.g. Anacharsis’ argument that there are no flute-girls among the Scyths, for there are no vines.
Thus with regard to the same science (and with regard to the position of the middle terms) there are these differences between the deduction of the fact and that of the reason why.
The reason why differs from the fact in another fashion, when each is [35] considered by means of a different science. And such are those which are related to each other in such a way that the one is under the other, e.g. optics to geometry, and [79a1] mechanics to solid geometry, and harmonics to arithmetic, and star-gazing to astronomy. Some of these sciences bear almost the same name—e.g. mathematical and nautical astronomy, and mathematical and acoustical harmonics. For here it is for the empirical scientists to know the fact and for the mathematical to know the [5] reason why; for the latter have the demonstrations of the explanations, and often they do not know the fact, just as those who consider the universal often do not know some of the particulars through lack of observation.
These are those which, being something different in substance, make use of forms. For mathematics is about forms, for its objects are not said of any underlying subject—for even if geometrical objects are said of some underlying subject, still it is not as being said of an underlying subject that they are studied.
Related to optics as this is related to geometry, there is another science related [10] to it—viz. the study of the rainbow; for it is for the natural scientist to know that fact, and for the student of optics—either simpliciter or mathematical—to know the reason why. And many even of those sciences which are not under one another are related like this—e.g. medicine to geometry; for it is for the doctor to know the fact that circular wounds heal more slowly, and for the geometer to know the reason [15] why.
14 · Of the figures the first is especially scientific. For the mathematical sciences carry out their demonstrations through it—e.g. arithmetic and geometry and optics—and so do almost all those which make inquiry after the reason why; [20] for the deduction of the reason why occurs, either in general or for the most part and in most cases, through this figure. Hence for this reason too it will be especially scientific; for consideration of the reason why has most importance for knowledge.
Next, it is possible to hunt for understanding of what a thing is through this [25] figure alone. For in the middle figure no affirmative deduction comes about; but understanding what a thing is is understanding an affirmation. And in the last figure an affirmative deduction does come about, but it is not universal; but what a thing is is something universal—for it is not in a certain respect that man is a two-footed animal.
Again, this figure has no need of the others, but they are thickened and [30] increased through it until they come to the immediates.
So it is evident that the first figure is most important for understanding.
15 · Just as it was possible for A to belong to B atomically, so it is also possible for it to not belong in this way. By belonging or not [35] belonging atomically I mean that there is no middle term for them; for in this way their belonging or not belonging will no longer be in virtue of something else.
Now when either A or B is in some whole or both are, it is not possible for A to belong to B primitively. For let A be in the whole C. Now if B is not in the whole C (for it is possible that A is in some whole and B is not in it), there will be a deduction that A does not belong to B; for if C belongs to every A and to no B, A belongs to not [79b1] B. And similarly too, if B is in some whole, e.g. in D; for D belongs to every B and A to no D, so that A will belong to no B through a deduction. And it will be proved in the same way again if both are in some whole. [5]
That it is possible for B not to be in a whole that A is in, or again for A not to be in a whole that B is in, is evident from those chains of predicates which do not overlap one another. For if nothing in the chain A,C,D is predicated of anything in the chain B,E,F, and A is in the whole H (which is in the same chain as it), it is [10] evident that B will not be in H; for otherwise the chains will overlap. And similarly too if B is in some whole.
If neither is in any whole and A does not belong to B, it is necessary for it to not belong atomically. For if there is to be a middle term, it is necessary for one of them to be in some whole. For the deduction will be either in the first or in the middle [15] figure. Now if it is in the first, B will be in some whole (for the proposition with it as subject must be affirmative); and if it is in the middle, one or other of them will be in some whole (for a deduction comes about if the negative is assumed with either as [20] subject—but with both negative there will not be one).
So it is evident that it is possible for one thing to not belong to another atomically, and we have said when it is possible, and how.
16 · Ignorance—what is called ignorance not in virtue of a negation but in virtue of a disposition—is error coming about through deduction. In the case of [25] what belongs or does not belong primitively this comes about in two ways: either when one believes simpliciter that something belongs or does not belong, or when one gets the belief through deduction. Now for simple belief the error is simpl
e, but when it is through deduction there are several ways of erring.
For let A belong to no B atomically: now if you deduce that A belongs to B, [30] assuming C as a middle term, you will have erred through deduction. Now it is possible for both the propositions to be false, and it is possible for only one to be. For if neither A belongs to any of the C’s nor C to any of the B’s, and each has been [35] assumed the other way about, both will be false. And it is possible that C is so related to A and B that it neither is under A nor holds universally of B. For it is impossible for B to be in any whole (for A was said to not belong to it primitively), and it is not necessary that A holds universally of everything there is; hence both will be false.
But it is also possible to assume one truly—not, however, whichever you like, [80a1] but only AC; for the proposition CB with always be false because B is not in anything, but AC may be true—e.g. if A belongs atomically both to C and to B (for when the same thing is predicated primitively of several things neither will be in the [5] other). But it makes no difference even if it belongs non-atomically.
Now error about belonging comes about by these means and in this way only; for in no other figure was there a deduction of belonging. But error about not belonging comes about both in first and in the middle figure.
Now first let us say in how many ways and under what characterization of the [10] propositions it comes about in the first figure. Now it is possible when both premisses are false, e.g. if A belongs atomically both to C and to B; for if A is assumed to belong to no C and C to every B, the propositions are false. It is also [15] possible when one is false, and that whichever you like. For it is possible for AC to be true and CB false—AC true because A does not belong to everything there is; CB false because it is impossible for C, to none of which A belongs, to belong to B (for the proposition AC will no longer be true, and at the same time if they are both true [20] the conclusion too will be true). But it is also possible for CB to be true while the other is false, i.e. if B is both in C and in A; for it is necessary for the one to be under the other, so that if you assume that A belongs to no C, the proposition will be false.