by Aristotle
[25] Neither upwards, therefore, nor downwards will one thing be said to belong to one thing. For the things of which the accidentals are said are whatever is in the substance of each thing; and these are not infinitely many. And upwards there are both these and their accidentals, and neither are infinitely many. It is necessary, therefore, for there to be something of which something is predicated primitively, and something else of that; and for this to come to a stop, and for there to be [30] something which is no longer predicated of anything prior and of which nothing else prior is predicated.
Now this is one way of demonstration; but there is still another, if there is demonstration of that of which some prior things are predicated, and it is not possible either to be more happily related to the things of which there is demonstration than by knowing them or to know them without demonstration, and [35] if this is familiar through these and we neither know these nor are more happily related to them than by knowing them, we shall not understand what is familiar through them either.
So if one can know something through demonstration—simpliciter, and not dependent on something, nor on a supposition—it is necessary for the predications in between to come to a stop. For if they do not come to a stop but there is always [84a1] something above what has been taken, there will be demonstration of everything; hence if it is not possible to go through infinitely many things, we shall not know through demonstration the things of which there is demonstration. So if we are not more happily related to them than by knowing them, we will be able to understand nothing through demonstration simpliciter but only on a supposition. [5]
Now generally, one might be convinced of what we said by this; but analytically, it is evident more concisely from the following facts that neither upwards nor downwards can the terms predicated be infinitely many in the demonstrative sciences with which our inquiry is concerned. [10]
For demonstration is of what belongs to the objects in themselves—in themselves in two ways: both what belongs in them in what they are, and the things which have what they themselves belong to belonging in what they are (e.g. odd to number—odd belongs to number and number itself inheres in its account; and again [15] plurality or divisibility inheres in the account of number). And it is not possible for either of these sorts of term to be infinitely many—either as odd of number (for then there would again be something else belonging to odd in which odd inhered; and if this is prime, number will inhere in what belongs to it; so if it is not possible [20] for infinitely many such things to belong in the one thing, they will not be infinitely many in the upward direction; but it is necessary that everything belongs to the primitive term, i.e. to number, and number to them, so that they will be convertible and will not exceed it). Nor yet can the terms inhering in what something is be infinitely many; for then it would not be possible to define. [25]
Hence if all the terms predicated are said in themselves, and there are not infinitely many, then the terms leading upward will come to a stop. Hence they will come to a stop in the downward direction too. And if this is so, the terms in between two terms will also always be finite.
And if this is the case, it is now clear too that of necessity there are principles of [30] demonstrations and there is not demonstration of everything (which, as we said at the beginning, some men assert). For if there are principles, neither is everything demonstrable, nor is it possible to go on ad infinitum; for for either of these to be the case is nothing other than for there to be no immediate and indivisible proposition [35] but for all to be divisible. For it is by interpolating a term inside and not by taking an additional one that what is demonstrated is demonstrated; hence if it is possible for this to go on ad infinitum, it would be possible for there to be infinitely many middle terms in between two terms. But this is impossible if the predications come to a stop [84b1] upwards and downwards. And that they do come to a stop has been proved generally before and analytically now.
23 · Now that this has been proved, it is evident that if one and the same thing belongs to two things—e.g. A both to C and to D—which are not predicated [5] one of the other (either not at all or not in every case), that it will not always belong in virtue of something common. E.g. having angles equal to two right angles belongs to isosceles and to scalene in virtue of something common (for it belongs to them as figures of a certain sort and not as different things); but this is not always so.
[10] For let B be that in virtue of which A belongs to C, D. It is clear, then, that B too will belong to C and D in virtue of some other common feature, and that in virtue of another; so that infinitely many terms would fall between two terms. But that is impossible.
It is not necessary, then, that when one and the same thing belongs to several things it should always do so in virtue of something common, since there are [15] immediate propositions. Yet it is necessary for the terms to be in the same genus and dependent on the same atoms, if the common feature is to be something belonging in itself; for it turned out impossible that what is proved should cross from one genus to another.
It is evident too that when A belongs to B, then if there is some middle term you [20] can prove that A belongs to B, and the elements of this are28 as many as the middle terms (for the immediate propositions are the elements, either all of them or the universal ones); but if there is no middle term, there is no longer a demonstration, but this is the path to the principles.
Similarly, too, if A does not belong to B, then if there is either a middle or a [25] prior term to which it does not belong, there is a demonstration; and if not, there is not, but it is a principle. And there are as many elements as terms; for the propositions containing these are principles of the demonstration. And just as there are some non-demonstrable principles to the effect that this is this and this belongs to this, so too there are some to the effect that this is not this and this does not [30] belong to this; so that there will be principles some to the effect that something is, and others to the effect that something is not.
When you have to prove something, you should assume what is predicated primitively of B. Let it be C; and let D be predicated similarly of this. And if you always proceed in this way no proposition and nothing belonging outside A will ever [35] be assumed in the proof, but the middle term will always be thickened, until they become indivisible and single. It is single when it becomes immediate; and a single proposition simpliciter is an immediate one. And just as in other cases the principle is simple, though it is not the same everywhere—but in weight it is the ounce, in song the semitone, and in other cases other things—so in deduction it is the unit29 [85a1] and in demonstration and understanding it is comprehension.
So, in deductions proving something to belong, nothing falls outside; but in deductions, in one case nothing falls outside the term which must belong—i.e. if A does not belong to B through C (if C belongs to every B, and A to no C), then if [5] again you have to prove that A belongs to no C, you should assume a middle term for A and C; and it will always proceed in this way.
If you have to prove that D does not belong to E by the fact that C belongs to every D and to no E it will never fall outside E (this is the term to which it must belong).
In the case of the third way, it will never pass outside either the term of which [10] it must be denied or that which must be denied of it.
24 · Some demonstrations are universal, others particular, and some are affirmative, others negative; and it is disputed which are better. And similarly too [15] for those which are said to demonstrate and those which lead to the impossible. Now first let us inquire about universal and particular demonstrations; and when we have made this clear, let us speak about those which are said to prove and those which lead to the impossible
Now it might perhaps seem to some, inquiring as follows, that particular [20] demonstration is better: if a demonstration in virtue of which we understand better is a better demonstration (for this is the excellence of demonstration), and we understand a thing better when
we know it in itself than when we know it in virtue of something else (e.g. we know musical Coriscus better when we know that Coriscus [25] is musical than when we know that a man is musical; and similarly in the other cases too), and the universal demonstration shows that something else and not that the thing itself is in fact so and so (e.g. of the isosceles,30 it shows not that the isosceles but that the triangle has two right angles), while the particular demonstration shows that the thing itself has in fact two right angles—well, if a demonstration of something in itself is better, and the particular rather than the universal is of that [30] type, then the particular demonstration will be better.
Again, if the universal is not a thing apart from the particulars, and demonstration instils an opinion that that in virtue of which it demonstrates is some thing, and that this belongs as a sort of natural object among the things there are (e.g. a triangle apart from the individual triangles, and a figure apart from the individual figures and a number apart from the individual numbers), and a [35] demonstration about something there is is better than one about something that is not, and one by which we will not be led into error is better than one by which we will be, and universal demonstration is of this type (for as they go on they prove as in the case of proportion, e.g. that whatever is of such a type—neither line nor number nor solid nor plane but something apart from these—will be proportional)—so, if [85b1] this is more universal and is less about something there is than the particular demonstration, and instils a false opinion, then the universal will be worse than the particular.
[5] Or, first, is the other argument any better fitted to the universal than to the particular case? For if two right angles belong not as isosceles but as triangle, one who knows that the isosceles has two right angles will know it less well as such than one who knows that a triangle has two right angles.
And in general, if it does not hold as triangle and yet someone proves it, this will not be a demonstration; and if it does, it is the man who knows a thing as it [10] belongs who knows it better. Thus if triangle extends further, and there is the same account and triangles are not so called in virtue of a homonymy, and two right angles belong to every triangle, it will not be that the triangle has two right angles as isosceles but that the isosceles has such angles as triangle. Hence one who knows universally knows it better as it belongs than one who knows it particularly. [15] Therefore the universal demonstration is better than the particular.
Again, if there is some single account and the universal is not a homonymy, it will be some thing no less than some of the particulars, but actually more so, inasmuch as what is imperishable is among the former and it is rather the particulars that are perishable. And again, there is no necessity to believe that this is [20] a thing apart from the particulars on the grounds that it makes one thing clear, any more than in the case of the other things which do not signify an individual but either quality or quantity or relation or doing. If, therefore, this is believed, it is not the demonstration but the audience which is responsible.
Again, if demonstration is a probative deduction of an explanation and the reason why, and the universal is more explanatory (for that to which something [25] belongs in itself, is itself explanatory for itself; and the universal is primitive: therefore the universal is explanatory); hence the universal demonstration is better; for it is more a demonstration of the explanation and the reason why it is the case.
Again, we seek the reason why up to this point, and it is then we think we know, when it is not the case that this either comes about or is because something else does; [30] for the last term is in this way an end and a limit. E.g. with what aim did he come? So as to get the money—and that so as to give back what he owed; and that so as not to be dishonest. And going on in this way, when it is no longer because of something else or with some other aim, we say it is because of this as an end that he came (and that it is and that it came about) and that then we best know why he came. Thus if [35] the same goes for all explanations and reasons why, and in the case of explanations in terms of aim we know best in this way—in the other cases too, therefore, we then know best when this no longer belongs to it because it is something else. So when we are aware that the external angles are equal to four right angles because it is isosceles, it still remains to ask why the isosceles is so because it is a triangle, and [86a1] that because it is a rectilineal figure. And if this is no longer the case because it is something else, it is then we know best. And it is then too that it is universal; therefore the universal demonstration is better.
Again, the more particular a demonstration is, the more it falls into what is [5] indefinite, while the universal tends to the simple and the limit. And as indefinite, things are not understandable; but as finite they are understandable. Therefore they are more understandable as universal than as particular. Therefore universals are more demonstrable. And of more demonstrable things there is more of a demonstration; for correlatives vary in degree together. Therefore the universal demonstration is better, since it is more of a demonstration. [10]
Again, if a demonstration in virtue of which one knows this and something else is preferable to one in virtue of which one knows this alone; and one who has the universal demonstration knows the particular fact too, but the latter does not know the universal fact31—hence in this way too it will be preferable.
Again, as follows: to prove more universally is to prove through a middle term that is nearer to the principle. The immediate is nearest, and this is a principle. So if [15] a demonstration depending on a principle is more precise than one not depending on a principle, a demonstration more dependent on a principle is more precise than one less so; and the more universal demonstration is of such a type; therefore the universal will be superior. E.g. if you had to demonstrate A of D; the middle terms are B, C: well, B is higher, so that the demonstration through it is more [20] universal.
But some of the things we have said are general. It is most clear that universal demonstration is more important from the fact that grasping the prior of the propositions we have in a sense the posterior one too and we grasp it potentially. E.g. if someone knows that every triangle has two right angles, he knows in a sense of the [25] isosceles too that it has two right angles—potentially—even if he does not know of the isosceles that it is a triangle. But one who grasps the latter proposition does not know the universal in any sense, neither potentially nor actually.
And the universal proposition is comprehensible, while the particular terminates in perceptions. [30]
25 · So much, then, for the view that universal demonstration is better than particular; that probative is better than negative is clear from what follows.
Let that demonstration be better which, other things being equal, depends on fewer postulates or suppositions or propositions. For if they are equally familiar, [35] knowing will come about more quickly in this way; and that is preferable.
The argument for the proposition that the one depending on fewer things is better is, put universally, this: if it is the case that the middle terms are equally familiar, and the prior terms are more familiar, let the one demonstration show that A belongs to E through middle terms B, C, D, and the other that A belongs to E through F, G. Thus that A belongs to D and that A belongs to E are similar. But that [86b1] A belongs to D is prior to and more familiar than the proposition that A belongs to E; for the latter is demonstrated through the former, and that through which a thing [5] is demonstrated is more convincing. Therefore the demonstration through the fewer items is better, other things being equal.
Now both are proved through three terms and two propositions, but the one assumes that something is the case and the other both that something is and that something is not the case; therefore it is through more items, so that it is worse.
[10] Again, since it has been proved that it is impossible for a deduction to come about when both propositions are negative, but that one must be so and the other to the effect that something belongs,
in addition to that one must assume this: the affirmative propositions, as the demonstration increases, necessarily become more numerous, whereas it is impossible for the negatives to be more than one in any [15] deduction.
For let A belong to none of the B’s and B belong to every C. Well, if we must again increase both propositions, a middle term must be interpolated. Let it be D for A B, and E for B C. Well, it is evident that E is affirmative, and that D is affirmative [20] of B but lies as negative towards A. For D holds of every B, and A must belong to none of the D’s. So a single negative proposition, A D, comes about.
The same holds of the other deductions too. For the middle for the affirmative [25] terms is always affirmative both ways; but for the negative it is necessarily negative in one way, so that this comes to be the single such proposition and the others are affirmative.
Thus if that through which something is proved is more familiar and more convincing, and the negative demonstration is proved through the affirmative while the latter is not proved through the former, then, being prior and more familiar and more convincing, the affirmative will be better.
[30] Again, if the universal immediate proposition is a principle of deduction, and the universal proposition is affirmative in the probative demonstration and negative in the negative, and the affirmative is prior to and more familiar than the negative [35] (for the negation is familiar because of the affirmation, and the affirmation is prior, just as being the case is prior to not being the case)—hence the principle of the probative is better than that of the negative; and the one which uses better principles is better.