by Aristotle
[25] So it is evident that both when one proposition is false and when both are the deduction will be false.
In the middle figure it is not possible for both propositions to be false as wholes; for when A belongs to every B, one cannot assume anything which will belong to all the one and none of the other; but it is necessary to assume the propositions in such a [30] way that something belongs to one and does not belong to the other if there is to be a deduction. So if when assumed in such a way they are false, clearly they will be the other way about when assumed in the contrary way; but this is impossible.
But nothing prevents each being partially false i.e. if C were to belong both to some A and to some B. For if it is assumed to belong to every A and to no B, both [35] propositions will be false—not, however, as wholes but partially. And if the negative is posited the other way about, the same holds.
It is possible for one of them to be false, and that whichever you like. For what belongs to every A also belongs to B; so if C is assumed to belong to the whole of A and to not belong to the whole of B, CA will be true and CB false. Again, what [80b1] belongs to no B will not belong to every A; for if to A, then to B too—but it did not belong to B. So if C is assumed to belong to the whole of A and to no B, the proposition CB is true and the other false. [5]
Similarly too if the negative is transposed. For what belongs to no A will not belong to any B either; so if C is assumed to not belong to the whole of A and to belong to the whole of B, the proposition AC21 will be true and the other false. And again, what belongs to every B it is false to assume belongs to no A. For it is [10] necessary, if it belongs to every B, for it also to belong to some A; so if C is assumed to belong to every B and to no A, CB will be true and CA false.
So it is evident that both when both are false and when only one is there will be [15] an erroneous deduction in the case of atomic propositions.
17 · In the case of what belongs non-atomically, when the deduction of the falsehood comes about through the appropriate middle term, it is not possible for both propositions to be false but only the one relating to the major extreme. (I call [20] appropriate a middle term through which the deduction of the contradictory comes about.) For let A belong to B through a middle term C. Now since it is necessary for CB to be assumed as an affirmative if a deduction comes about, it is clear that this will always be true; for it does not convert. And AC is false; for if this converted the [25] contrary deduction comes about.
Similarly too if the middle term is taken from another chain—e.g. D, if it is both in the whole of A and predicated of every B; for it is necessary for the proposition DB to stand and for the other to be converted, so that the one is always [30] true and the other always false. And this sort of error is much the same as that through the appropriate middle.
But if the deduction comes about not through the appropriate middle, then when the middle term is under A and belongs to no B, it is necessary for both to be false. For the propositions must be assumed with the contrary character to that [35] which they actually have if there is going to be a deduction; and so assumed both come out false. I.e. if A belongs to the whole of D and D to none of the B’s; for when these are converted there will be a deduction and the propositions will both be false.
[81a1] But when the middle term, i.e. D, is not under A, AD will be true and DB false. For AD is true because D was not in A; and DB false because if it were true the conclusion too would be true, but it was false.
[5] When the error comes about through the middle figure, it is not possible for both propositions to be false as wholes; for when B is under A it is not possible for anything to belong to all the one and none of the other, as was said earlier. But it is possible for one to be false as a whole, and that whichever you like.
[10] For if C belongs both to A and to B, if it is assumed to belong to A and not to belong to B, then AC22 will be true and the other false. And again, if C were assumed to belong to B and to no A, CB will be true and the other false.
[15] Now if the deduction of the error is negative, we have said when and by what means the error will occur. If it is affirmative, then when it is through the appropriate middle term it is impossible for both to be false; for it is necessary for CB to stand if there is to be a deduction, as was said earlier. Hence CA23 will always [20] be false; for this is the proposition that converts.
Similarly too if the middle term were taken from another chain, as was said in the case of negative error too; for it is necessary for DB to stand and AD to convert. And the error is the same as the earlier one.
[25] When it is not through the appropriate middle term, then if D is under A, this will be true and the other false; for it is possible for A to belong to several things which are not under one another. And if D is not under A, this clearly will always be false (for it is assumed as an affirmative), but it is possible for DB both to be true [30] and false. For nothing prevents A from belonging to no D and D to every B, e.g. animal to knowledge, knowledge to music; nor again A from belonging to none of the D’s and D to none of the B’s.
So it is evident that if the middle term is not under A it is possible both for both propositions to be false and for whichever you like to be.24
[35] So it is evident in how many ways and by what means errors in virtue of deduction may come about, both in the case of the immediates and in the case of what is established through demonstration.
18 · It is evident too that if some perception is wanting, it is necessary for some understanding to be wanting too—which it is impossible to get if we learn either by induction or by demonstration, and demonstration depends on universals [81b1] and induction on particulars, and it is impossible to consider universals except through induction (since even in the case of what are called abstractions one will be able to make familiar through induction that some things belong to each genus, [5] even if they are not separable, in so far as each thing is such and such), and it is impossible to get an induction without perception—for of particulars there is perception; for it is not possible to get understanding of them; for it can be got neither from universals without induction nor through induction without perception.
19 · Every deduction is through three terms; and the one type is capable of [10] proving that A belongs to C because it belongs to B and that to C, while the other is negative, having one proposition to the effect that one thing belongs to another and the other to the effect that something does not belong. So it is evident that the principles and what are called the suppositions are these; for it is necessary to [15] assume these and prove in this way—e.g. that A belongs to C through B, and again that A belongs to B through another middle term, and that B belongs to C in the same way.
Now those who are deducing with regard to opinion and only dialectically clearly need only inquire whether their deduction comes about from the most reputable propositions possible; so that even if there is not in truth any middle term [20] for AB but there seems to be, anyone who deduces through this has deduced dialectically. But with regard to truth one must inquire on the basis of what actually holds. It is like this: since there is something which itself is predicated of something else nonaccidentally (I mean by accidentally—e.g. we sometimes say that that [25] white thing is a man, not speaking in the same way as when we say that the man is a white thing;25 for it is not the case that, being something different, he is a white thing, whereas the white thing is a man because the man was accidentally white)—now there are some things such as to be predicated in themselves.
Well, let C be such that it itself no longer belongs to anything else and B [30] belongs to it primitively and there is nothing else between. And again let E belong to F in the same way and this to B. Now is it necessary for this to come to a stop, or is it possible for it to go on ad infinitum?
And again, if nothing is predicated of A in itself and A belongs to H primitively and to nothing prior in between, and H belongs to G and this to B, is it necessary for [35] this to come to
a stop or is it possible for this to go on ad infinitum? This differs from the earlier question to this extent, that the one is: Is it possible, beginning from something such that it belongs to nothing else and something else belongs to it, to go upwards ad infinitum? while the other has us begin from something such that it is predicated of something else and nothing is predicated of it and consider if it is [82a1] possible to go downwards ad infinitum.
Again, is it possible for the terms in between to be indefinitely many if the extremes are determined? I mean, e.g., if A belong to C, and B is a middle term for them, and for B and A there are other middle terms, and for these others, is it [5] possible for these to go on ad infinitum, or impossible?
This is the same as to inquire whether demonstrations go on ad infinitum and whether there is demonstration of everything, or whether some terms are bounded by one another.
I say the same in the case of negative deductions and propositions too; i.e. if A [10] does not belong to any B, either it will not belong to B primitively, or there will be something prior in between to which it does not belong (e.g. G, which belongs to all B), and again another still prior to this (e.g. H, which belongs to every G). For in these cases too either the prior terms it belongs to are indefinitely many or they come to a stop.
[15] (The same does not go for terms that convert. For among counterpredicated terms there is none of which any is predicated primitively or finally (for in this respect at least every term is related to every other in a similar way), and if26 its predicates are indefinitely many, then the things we are puzzling over are indefinitely many in both directions—unless it is possible that they convert not [20] similarly but the one as an accidental, the other as a predicate.)
20 · Now it is clear that it is not possible for the terms in between to be indefinitely many if the predications come to a stop downwards and upwards—I mean by upwards, towards the more universal; and by downwards, towards the [25] particular. For if when A is predicated of F the terms in between—the B’s—are indefinitely many, it is clear that it would be possible both that from A downwards one thing should be predicated of another ad infinitum (for before F is reached the terms in between are indefinitely many) and that from F upwards there are indefinitely many before A is reached. Hence if these things are impossible, it is also impossible for there to be indefinitely many terms between A and F.
[30] For if someone were to say that some of A,B,F are next to one another so that there are none between them, and that the others cannot be grasped, that makes no difference; for whichever of the B’s I take, the terms in between in the direction of A or in the direction of F will either be indefinitely many or not. Well, it makes no difference which is the first term from which they are indefinitely many—whether [35] at once or not at once—for the terms after these are indefinitely many.
21 · It is evident too that in the case of negative demonstration it will come to a stop if it comes to a stop in both directions in the affirmative case. For let it be possible neither to go upwards from the last term ad infinitum (I call last that which [82b1] itself belongs to nothing else while something else belongs to it, e.g. F), nor from the first to the last (I call first that which itself holds of another while nothing holds of it). Well, if this is so, it will come to a stop in the case of negation too.
For a thing is proved not to belong in three ways. For either B belongs to [5] everything to which C does and A to nothing to which B does—in the case of BC, then and in general of the second premiss it is necessary to come to immediates; for this premiss is affirmative. And clearly if the other term does not belong to something else that is prior, e.g. to D, this will have to belong to every B; and if again it does not belong to something else prior to D, that will have to belong to every D. [10] Hence since the way upwards comes to a stop, the way to A will come to a stop too, and there will be some first thing to which it does not belong.
Again, if B belongs to every A and to no C, A belongs to none of the C’s. Again, [15] if one has to prove this, clearly it will be proved either in the above fashion, or in this or the third. Now the first has been described, and the second will now be proved.
You might prove it in this way—e.g. that D belongs to every B and to no C—if it is necessary for something to belong to B. And again, if this is not to belong to C, something else belongs to D which does not belong to C. So since belonging to ever [20] higher terms comes to a stop, not belonging will come to a stop too.
The third way was: if A belongs to every B and C does not belong to it, C does not belong to everything to which A does. Again, this will be proved either in the ways described above or similarly. Well, if the former, it comes to a stop; and if the [25] latter, one will again assume that B belongs to E, to not all of which C belongs. And this again similarly. Since it is supposed that it comes to a stop in the downward direction too, it is clear that C’s not belonging will also come to a stop.
It is evident that even if it is proved not in one way but in all—sometimes from the first figure, sometimes from the second or third—that it will come to a stop even [30] so; for the ways are finite, and necessarily anything finite taken a finite number of times is finite.
So it is clear that it comes to a stop in the case of negation if it does in the case of belonging. That it comes to a stop in the latter case is evident if we consider it [35] generally, as follows.
22 · Now in the case of things predicated in what something is, it is clear; for if it is possible to define, or if what it is to be something is knowable, but one cannot go through indefinitely many things, it is necessary that the things predicated in what something is are finite.
We argue universally, as follows: one can say truly that the white thing is [83a1] walking, and that that large thing is a log, and again that the log is large and that the man is walking. Well, speaking in the latter and in the former ways are different. For when I say that the white thing is a log, then I say that that which is [5] accidentally white is a log; and not that the white thing is the underlying subject for the log; for it is not the case that, being white or just what is some white, it came to be a log, so that it is not a log except accidentally. But when I say that the log is white, I do not say that something else is white and that that is accidentally a log, as [10] when I say that the musical thing is white (for then I say that the man, who is accidentally musical, is white); but the log is the underlying subject which did come to be white without being something other than just what is a log or a particular log.
Well, if we must legislate, let speaking in the latter way be predicating, and in [15] the former way either no predicating at all, or else not predicating simpliciter but predicating accidentally. (What is predicated is like the white, and that of which it is predicated is like the log.) Thus let it be supposed that what is predicated is always predicated simpliciter of what it is predicated of, and not accidentally; for [20] this is the way in which demonstrations demonstrate. Hence when one thing is predicated of one, either it is predicated in what a thing is or it says that it has some quality or quantity or relation or is doing something or undergoing something or is at some place of time.
Again, the things signifying a substance signify of what they are predicated of [25] just what is that thing or just what is a particular sort of it; but the things which do not signify a substance but are said of some other underlying subject which is neither just what is that thing nor just what is a particular sort of it, are accidental, e.g. white of the man. For the man is neither just what is white nor just what is some [30] white—but presumably animal; for a man is just what is an animal. But the things that do not signify a substance must be predicated of some underlying subject, and there cannot be anything white which is not white through being something different. (For we can say goodbye to the forms; for they are nonny-noes, and if there are any they are nothing to the argument; for demonstrations are about things [35] of this type.)
Again, if it cannot be the case that this is a
quality of that and the latter of the former—a quality of a quality—it is impossible for them to be counterpredicated of one another in this way—it is possible to say it truly, but it is not possible to counterpredicate truly. Now either it will be predicated as a substance, i.e. either [83b1] being the genus or the difference of what is predicated—but it has been proved that these will not be infinitely many, either downwards or upwards (e.g. man is two-footed, that is animal, that is something else; nor animal of man, that of Callias, [5] and that of another thing in what it is); for one can define every substance of that kind, but one cannot go through infinitely many things in thought. Hence they are not infinitely many either upwards or downwards; for one cannot define that of which infinitely many things are predicated. Thus they will not be counterpredicated [10] of one another as genera; for a thing will itself be just what is some of itself.
But neither will any case of quality or the other kinds of predication be counterpredicated unless it is predicated accidentally; for all these are accidental and are predicated of substances.
But it is clear that they will not be infinitely many upwards either; for of each is predicated whatever signifies either a quality or a quantity or one of those things, [15] or what is in its substance; but these are finite, and the genera of predications are finite—for they are either quality or relation or doing or undergoing or place or time.
It is supposed that one thing is predicated of one thing, and that things which are not what something is are not predicated of themselves. For they are all [20] accidental (though some in themselves and some in another fashion) and we say that all of them are predicated of some underlying subject, and that what is accidental is not an underlying subject; for we posit nothing of this type which is not called what it is called through being something different, and itself belongs to other things.27