Prior Analytics · Book II

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

1| Book 2. In how many figures, then, and through what kind and number of premises, and when and how a syllogism comes about, and further what one must look to in refuting and in establishing a point, and how one must inquire concerning the matter proposed by any method whatever, and further by what route we shall grasp the principles proper to each subject — we have already gone through. Since some syllogisms are universal and others particular, all the universal ones always establish more than one conclusion, while among the particular ones the affirmative ones establish more, but the negative ones only the conclusion itself. For the other premises convert, but the privative one does not convert. And the conclusion is something predicated of something, so that the other syllogisms establish more,

2| For example, if A has been proved to belong to all B or to some B, then B must belong necessarily to some A; and if A belongs to no B, then B belongs to no A either — this is a different case from the one before. But if A does not belong to some B, it is not necessary that B does not belong to some A; for it is possible for B to belong to all A. This, then, is a cause common to all cases, both universal and particular; but about the universal cases one can speak in another way as well. For whatever falls under the middle term or under the conclusion, the same syllogism will apply to all of them, if some are put in place of the middle term and others in place of the conclusion. For example, if AB is the conclusion reached through C, then A must be predicated of everything that falls under B or under C. For if D is wholly contained in B, and B is contained in A, then D too will be contained in A; again, if E is wholly contained in C, and C is contained in A, then E too will be contained in A. And likewise if the syllogism is negative.

3| In the second figure, it will be possible to draw a syllogism only about what falls under the conclusion. For example, if A belongs to no B but to all C, the conclusion is that B belongs to no C. If, then, D falls under C, it is clear that B does not belong to it; but for the things that fall under A, that B does not belong to them is not made clear through this syllogism. Yet B does not in fact belong to E, if E falls under A — but this, that B belongs to no C, has been proved through the syllogism, whereas that it does not belong to what falls under A is taken without proof, so that B's not belonging to E does not result from the syllogism. In the case of particular premises, for the things that fall under the conclusion there will be no necessity (for no syllogism results when this premise is taken as particular), but for all the things that fall under the middle term there will be necessity, though not because of the syllogism. For example, if A belongs to all B, and B belongs to some C: if something is placed under C there will be no syllogism, but if it is placed under B there will be — only not because of the syllogism already produced. And the same holds in the other figures.

4| For with respect to what falls under the conclusion there will be no necessity, but with respect to the other term there will be, only not because of the syllogism — just as, in the universal cases too, what falls under the middle term was shown from a premise that was itself undemonstrated. So either there will be no necessity there either, or there will be also in these cases. It is possible, then, for things to be such that the premises through which the syllogism proceeds are true,

5| and possible for them to be such that they are false, and possible for one to be true and the other false. But the conclusion is necessarily either true or false. Now from true premises it is not possible to deduce something false, but from false premises it is possible to deduce something true — except not the reason why, but only that it is so; for there is no demonstration of the reason why from false premises. The cause of this will be stated in what follows. First, then, that it is not possible to deduce something false from true premises is clear from this. If, when A is, B must be, then, when B is not, A must not be. If, then, A is true, B must be true, or else the same thing will both be and not be at the same time — and that is impossible. And one should not suppose, just because a single term A is posited, that something can result of necessity from a single thing's being the case — for that is not possible; what results of necessity is the conclusion, and the fewest terms through which this comes about are three, with two intervals, that is, two premises. If, then, it is true that A belongs to everything that B belongs to, and that B belongs to everything that C belongs to, then A must belong to whatever C belongs to, and it is not possible for this to be false;

6| for otherwise the same thing will at once belong and not belong. A, then, is posited as if it were one thing, though it is really two premises taken together. And the same holds for negative syllogisms too — for it is not possible to prove something false from true premises. But from false premises it is possible to deduce something true, both when both premises are false and when only one is — and if only one, not whichever one happens to be false, but the second, provided it is taken as wholly false; if it is not taken as wholly false, either one will do. For let A belong to all of C, and to no B, and let B belong to no C either. This is possible — for example, animal belongs to no stone, and stone belongs to no man. If, then, it is assumed that A belongs to all B and B belongs to all C, A will belong to all C, so that the conclusion is true though drawn from two false premises; for every man is an animal. The negative case works the same way. For it is possible for neither A nor B to belong to any C, while A belongs to all B — for example, if, with the same terms taken, man is put as the middle term; for animal belongs to no stone, and neither does man, while animal belongs to every man. So that if

7| whatever A actually belongs to is taken as something it belongs to none of, and whatever it does not belong to is taken as something it belongs to all of, the conclusion will be true even though drawn from two false premises. And the same will be shown to hold if each premise is taken as false only in part. But if only one premise is posited as false: if it is the first that is taken as wholly false — say AB — the conclusion will not be true, but if it is BC that is wholly false, it will be. By 'wholly false' I mean the contrary of the truth — for instance, if what belongs to none is taken as belonging to all, or what belongs to all as belonging to none. For let A belong to no B, and let B belong to all C. If, then, I take the premise BC as true, but the premise AB as wholly false — that is, take A as belonging to all B — it is impossible for the conclusion to be true; for A belonged to none of the C's, since if A belongs to none of what B belongs to, and B belongs to all C, then A belongs to no C. Likewise it is not true either if A belongs to all B and B belongs to all C, and the premise BC is taken as true while AB is taken as wholly false — that is, A belonging to none of what B belongs to — the conclusion will be false;

8| for A will in fact belong to all C, since A belongs to all of what B belongs to, and B belongs to all C. It is clear, then, that when the first premise is taken as wholly false, whether affirmative or negative, and the other premise is true, the conclusion does not come out true. But if the first premise is not taken as wholly false, it will. For if A belongs to all C and to some B, and B belongs to all C — for example, animal belongs to all swan and to some white thing, and white belongs to all swan — then if it is assumed that A belongs to all B and B belongs to all C, A will belong to all C truly; for every swan is an animal. Likewise if AB is negative; for it is possible for A to belong to some B and to no C, while B belongs to all C — for example, animal belongs to some white thing but to no snow, while white belongs to all snow. If, then, it is assumed that A belongs to no B, and B belongs to all C, A will belong to no C. And if the premise AB is taken as wholly true, but BC as wholly false, the syllogism will be true;

9| For nothing prevents a term from belonging to all of B and all of C, while B belongs to none of C — for instance, species of the same genus that are not subordinate to one another: animal belongs to both horse and man, but horse belongs to no man. If then A is assumed to belong to all B, and B to all C, the conclusion will be true, though the premise BC is wholly false. Similarly too when the premise AB is privative. For it is possible for A to belong neither to B nor to C, and yet for B to belong to no C — for instance, a genus in relation to species from another genus: animal belongs neither to music nor to medicine, nor does music belong to medicine. If then it is assumed that A belongs to no B, and that B belongs to all C, the conclusion will be true. And even if C is not wholly false but only false in part, the conclusion will likewise be true. For nothing prevents A from belonging to the whole of both B and C, while B belongs only to some C — for instance, a genus in relation to a species and a differentia: animal belongs to every man and to every footed thing, but man belongs to some footed thing, not to all.

10| If then A is assumed to belong to all B, and B to all C, A will belong to all C — which was true. Similarly too when the premise B is privative. For it is possible for A to belong neither to B nor to C, while B belongs to some C — for instance, a genus in relation to a species and differentia from another genus: animal belongs to no practical wisdom and to no theoretical science, but practical wisdom belongs to some theoretical science. If then it is assumed that A belongs to no B, and B to all C, A will belong to no C — which was true. In the case of particular syllogisms, it is possible for the conclusion to be true even when the first premise is wholly false and the other true, and when the first is false in part and the other true, and when the one is true and the other false in part, and when both are false. For nothing prevents A from belonging to no B but to some C, and B from belonging to some C — for instance, animal belongs to no snow but to some white thing, and snow belongs to some white thing.

11| If then snow is posited as the middle term, and animal as the first, and it is assumed that A belongs to the whole of B, and B to some C, the premise AB will be wholly false, but BC true, and the conclusion true. Similarly too when the premise AB is privative. For it is possible for A to belong to the whole of B, yet not belong to some C, while B belongs to some C — for instance, animal belongs to every man, but does not follow upon everything white, while man belongs to some white thing; so that if man is posited as middle term and it is assumed that A belongs to no B, and B to some C, the conclusion will be true, though the premise AB is wholly false. And if the premise B is false in part, the conclusion will likewise be true. For nothing prevents A from belonging to some of both B and C, and B from belonging to some C — for instance, animal belongs to something fine and to something large, and the fine belongs to something large. If then it is assumed that A belongs to all B and B to some C, the premise AB will be false in part, but BC true, and the conclusion true. Similarly too when the premise B is privative —

12| for the same terms will serve, posited in the same way, for the demonstration. Again, if AB is true and BC is false, the conclusion will be true. For nothing prevents A from belonging to the whole of B but to some C, while B belongs to none of C — for instance, animal belongs to every swan but to some black thing, while swan belongs to no black thing. So that if it is assumed that A belongs to all B, and B to some C, the conclusion will be true, though BC is false. Similarly too when the premise B is taken as privative. For it is possible for A to belong to no B and not to belong to some C, while B belongs to no C — for instance, a genus in relation to a species from another genus and to an accident of its own species: animal belongs to no number, but to some white thing, and number belongs to no white thing. If then number is posited as middle term, and it is assumed that A belongs to no B, and B to some C, A will not belong to some C — which was true; and the premise AB is true, but BC is false.

13| And if AB is false in part, and BC is also false, the conclusion will be true. For nothing prevents A from belonging to some of B and to some of C, each of them,

14| while B belongs to no C — for instance, if B is contrary to C, and both are accidents of the same genus: animal belongs to something white and to something black, but white belongs to no black thing. If then it is assumed that A belongs to all B, and B to some C, the conclusion will be true. And similarly when the premise B is taken as privative; for the same terms, posited in the same way, will serve for the demonstration. And even when both premises are false, the conclusion will be true. For it is possible for A to belong to no B but to some C, while B belongs to no C — for instance, a genus in relation to a species from another genus and to an accident of its own species: animal belongs to no number, but to some white thing, and number belongs to no white thing. If then it is assumed that A belongs to all B and B to some C, the conclusion is true, but both premises are false. Similarly too when the premise B is privative. For nothing prevents A from belonging to the whole of B, but not belonging to some C, while B belongs to none of C — for instance, animal belongs to every swan, but does not belong to some black thing, while swan belongs to no black thing. So that if it is assumed that A belongs to no B, and B to some C, A will not belong to some C. The conclusion, then, is true, but the premises are false.

15| In the middle figure it is in every way possible to reach a true conclusion through falsehoods — both when both premises are taken as wholly false, and when each is false in part, and when one is true and the other false, whichever of the two is posited as false, and both in universal and in particular syllogisms. For if A belongs to no B but to all C — for instance, animal belongs to no stone but to every horse — then if the premises are posited in the contrary way, and it is assumed that A belongs to all B and to no C, the conclusion will be true from premises that are both wholly false. Similarly too if A belongs to all B and to no C: for the syllogism will be the same. Again, if one premise is wholly false and the other

16| ...wholly true. For nothing prevents A from belonging to both B and to all of C, while B belongs to none of C — as, for instance, a genus in relation to species that do not fall under one another. For animal belongs to every horse and to every man, and no man is a horse. If, then, it is assumed that A belongs to all of the one and to none of the other, one premise will be wholly false and the other wholly true, and the conclusion will be true whichever term the privative is set against. And likewise if one premise is false in part while the other is wholly true. For it is possible for A to belong to some B and to all of C, while B belongs to none of C — as, for instance, animal belongs to some white thing and to every raven, and white belongs to no raven. If, then, it is assumed that A belongs to none of B and to the whole of C, the premise AB is false in part, while the premise AC is wholly true, and the conclusion is true. And it is the same when the privative is transposed, for the demonstration proceeds through the same terms. And likewise if the affirmative premise is false in part while the privative is wholly true.

17| For nothing prevents A from belonging to some B and from not belonging to all of C, while B belongs to none of C — as, for instance, animal belongs to some white thing but to no pitch, and white belongs to no pitch. So if it is assumed that A belongs to all of B and to none of C, the premise AB is false in part, while the premise AC is wholly true, and the conclusion is true. And if both premises are false in part, the conclusion will still be true. For it is possible for A to belong to some B and to some C, while B belongs to none of C — as, for instance, animal belongs to some white thing and to some black thing, and white belongs to no black thing. If, then, it is assumed that A belongs to all of B and to none of C, both premises are false in part, but the conclusion is true. And likewise when the privative is transposed, through the same terms. This is clear also in the case of particular syllogisms. For nothing prevents A from belonging to all of B and to some C, while B does not belong to some C — as, for instance, animal belongs to every man and to some white thing, but man will not belong to some white thing.

18| If, then, it is assumed that A belongs to none of B and to some C, the universal premise is wholly false, the particular one true, and the conclusion true. And likewise when the premise BC is taken as affirmative. For it is possible for A to belong to none of B and not to belong to some C, while B does not belong to some C — as, for instance, animal belongs to no inanimate thing, but to some white thing, and the inanimate will not belong to some white thing. If, then, it is assumed that A belongs to all of B and does not belong to some C, the premise AB, the universal one, is wholly false, while AC is true, and the conclusion is true. And likewise when the universal is taken as true and the particular as false. For nothing prevents A from following neither B nor C at all, while B does not belong to some C — as, for instance, animal follows no number and no inanimate thing, and number does not follow some inanimate thing. If, then, it is assumed that A belongs to none of B and to some C, the conclusion will be true, and so will the universal premise, while the particular one is false. And likewise when the universal is taken as affirmative. For it is possible for A to belong to both the whole of B and the whole of C, while B does not follow some C — as, for instance, a genus in relation to a species and a differentia. For animal follows every man and the whole of the footed, but man does not follow every footed thing.

19| So if it is assumed that A belongs to the whole of B but does not belong to some C, the universal premise is true, the particular one false, and the conclusion true. It is also clear that the conclusion will be true from two false premises, if indeed it is possible for A to belong to the whole of both B and C, while B does not follow some C. For when A is assumed to belong to none of B and to some C, both premises are false, but the conclusion is true. And likewise when the universal premise is affirmative and the particular one privative. For it is possible for A to follow none of B and all of C, while B does not belong to some C — as, for instance, animal follows no knowledge but every man, and knowledge does not belong to every man. If, then, it is assumed that A belongs to the whole of B but does not follow some C, the premises are false, but the conclusion is true. In the last figure too there will be truth through falsehoods — both when both premises are wholly false, and when each is false in part, and when one is wholly true and the other false,

20| and when one is false in part and the other wholly true, and the reverse, and however many other ways it is possible for the premises to be varied. For nothing prevents neither A nor B from belonging to any C, while A nonetheless belongs to some B — as, for instance, neither man nor footed follows any inanimate thing, yet man belongs to some footed thing. If, then, it is assumed that A and B both belong to all of C, both premises are wholly false, but the conclusion is true. And likewise when one premise is privative and the other affirmative. For it is possible for B to belong to none of C while A belongs to all of C, and for A not to belong to some B — as, for instance, black belongs to no swan, but animal belongs to every swan, and animal does not belong to every black thing. So if it is assumed that B belongs to all of C and A to none, A will not belong to some B; and the conclusion is true, but the premises are false. And if each premise is false in part, the conclusion will still be true. For nothing prevents both A and B from belonging to some C, and A from belonging to some

21| B, as, for instance, both white and fine belong to some animal, and white belongs to some fine thing. If, then, it is assumed that A and B both belong to all of C, the premises are false in part, but the conclusion is true. And likewise when C is taken as privative. For nothing prevents A from not belonging to some C, while B belongs to some C, and A from not belonging to all of B — as, for instance, white does not belong to some animal, but fine belongs to some animal, and white does not belong to every fine thing. So if it is assumed that A belongs to none of C and B to all of C, both premises are false in part, but the conclusion is true. And likewise when one premise is taken as wholly false and the other as wholly true. For it is possible for both A and B to follow all of C, while A does not belong to some B — as, for instance, both animal and white follow every swan, yet animal does not belong to every white thing. With terms of this kind set down, then, if it is assumed that B belongs to the whole of C, and A does not belong to the whole of it,

22| the whole premise B-C will be true, and the whole premise A-C will be false, and the conclusion will be true. And likewise if B-C is false and A-C is true; for the same terms serve for the demonstration. But this also holds if both premises are taken as affirmative. For nothing prevents B from following every C, while it does not belong to the whole of that thing, and belonging to some of B, as for instance animal belongs to every swan, but black belongs to no swan, while black belongs to some animal. So if it is assumed that A and B both belong to every C, the whole premise B-C will be true, while the whole premise A-C is false, and the conclusion will be true. And likewise if the premise A-C is taken as true; for the demonstration proceeds through the same terms. Again, suppose the one premise is wholly true and the other partly false. For it is possible for B to belong to every C but A only to some C, and A to some B, as for instance two-footed belongs to every man, but fine does not belong to every man, while fine belongs to some two-footed thing. So if it is assumed that both A and B belong to the whole of C, the premise B-C will be wholly true, while A-C is partly false, but the conclusion will be true. And likewise if A-C is taken as true and B-C as partly false;

23| for by transposing the same terms the demonstration will work. And likewise when one premise is privative and the other affirmative. For since it is possible for B to belong to the whole of C but to only some of it in another respect, and when things stand thus, A does not belong to every B, then if it is assumed that B belongs to the whole of C, but to none of it, the privative premise will be partly false, while the other is wholly true, and so will the conclusion. Again, since it has been shown that when A belongs to no C, and B belongs to some C, it is possible for A not to belong to some B, it is clear that even when the premise A-C is wholly true and B-C is partly false, it is possible for the conclusion to be true. For if it is assumed that A belongs to no C, and B to every C, the premise A-C will be wholly true and B-C partly false. It is also clear in the case of particular syllogisms that in every way a true conclusion can come through false premises. For the same terms must be taken, both when the premises are universal — affirmative terms in the affirmative syllogisms, and privative terms in the privative ones.

24| For it makes no difference, with respect to the setting-out of the terms, whether one takes what belongs to none as belonging to every,

25| or takes what belongs to some as belonging universally; and likewise with the privative syllogisms. It is clear, then, that if the conclusion is false, the things from which the argument proceeds must be false, either all or some of them; but when the conclusion is true, it is not necessary that anything be true, either some part or the whole — rather it is possible, with nothing in the syllogism being true, for the conclusion nonetheless to be true, though not of necessity. The reason is that when two things stand to one another such that if one exists the other exists of necessity, then if the second does not exist, neither will the first, but if the second does exist, the first need not exist; but it is impossible for the same thing to be and not be of necessity because of the same thing. I mean, for example, that if A being white entails of necessity that B is large, and A not being white entails of necessity that C is not white — for when this, A, is white, that, B, must be large, and if B is large, C is not white, then it is necessary, if A is white, that C is not white. And when, of two things, if one exists the other must exist, then if the second does not exist, the first must not exist. So if B is not large, A cannot be white. But if it is necessary that, when this is not white, B is large, it follows of necessity that when B is not large, B itself is large — which is impossible. For if B is not large, that (namely A being white) will not be true of necessity. So if, when this is not white, B is going to be large, it follows that if B is not large, B is large — as through three terms.

26| Demonstrating in a circle and from one another means demonstrating the remaining premise — the one assumed in the other syllogism — by taking, through the conclusion and the converse of the predication, the other premise. For instance, if it were necessary to show that A belongs to every C, and this was shown through B, then if one should again show that A belongs to B, by assuming that A belongs to C and that C belongs to B — but earlier B was assumed, conversely, to belong to C. Or if it is necessary to show that B belongs to C, if one should assume that A holds of C, which was the conclusion,

27| and that B holds of A — whereas earlier A was assumed, conversely, to hold of B. In no other way is it possible to demonstrate from one another. For if a different middle term is taken, it is not circular, since none of the same terms is used; and if one of these same terms is taken, only one of the two can be so used — for if both were used, the conclusion would be the same, whereas it must be different. Now in cases where the terms do not convert, the syllogism comes from one premise that is undemonstrated; for it is not possible to demonstrate, through these terms, that the third belongs to the middle, or the middle to the first. But in cases where the terms do convert, it is possible to demonstrate everything through one another, as for instance if A, B, and C all convert with one another. Let A-C have been shown through the middle term B; and again let A-B be shown both through the conclusion and through the premise B-C converted, and likewise let B-C be shown both through the conclusion and through the premise A-B converted. But it is necessary to demonstrate both the premise C-B and the premise B-A; for these are the only ones we have used undemonstrated. So if it is assumed that B belongs to every C and C belongs to every A, there will be a syllogism of B in relation to A.

28| Again, if it is assumed that C belongs to every A, and A to every B, C must belong to every B. In both these syllogisms, then, the premise C-A has been taken undemonstrated; for the others had been demonstrated. So if we demonstrate this one too, all will have been demonstrated through one another. So if it is assumed that C belongs to every B, and B to every A, both premises are taken as demonstrated, and C must belong to A. It is clear, then, that only among terms that convert is it possible for demonstrations to come about in a circle and through one another; in other cases it is as we said before. But it turns out that in these cases too one uses, for the demonstration, the very thing being demonstrated; for C is shown to hold of B, and B of A, by assuming that C is said of A, while C is shown to hold of A through these premises — so that we are using the conclusion for the demonstration. In the case of privative syllogisms, this is how demonstration from one another comes about. Let B belong to every C, and let A belong to none

29| ...to B; the conclusion is that A belongs to no C. If, again, one must conclude that A belongs to no B, which was assumed earlier, let A belong to no C, and C to all B; for this is how the premise is converted the other way. But if it is B belonging to C that must be concluded, A and B are no longer to be converted in the same way (for it is the same premise to say that B belongs to no A and that A belongs to no B); instead one must take: whatever A belongs to none of, B belongs to all of. Let A belong to no C — this was the conclusion; and let it be assumed that B belongs to all of whatever A belongs to none of. Then B must belong to all C. So, there being three propositions, each has become a conclusion in turn, and this is what demonstrating in a circle is: taking the conclusion together with one of the premises converted, and deducing the remaining premise. In the case of particular syllogisms, the universal premise cannot be demonstrated through the others, though the particular one can. That the universal cannot be demonstrated is clear: for the universal is shown through universals, but the conclusion is not universal, and one must show it from the conclusion together with the other premise.

30| Further, no syllogism results at all once the premise is converted, since in that case 58b both premises become particular. But the particular premise can be shown. For let A have been proved of some C through B. If then B is taken to belong to all A, and the conclusion stands, B will belong to some C — for this becomes the first figure, with A as the middle term. But if the syllogism is negative, the universal premise cannot be shown, for the reason already stated; but the particular one can, if A and B are converted in the same way as in the universal case — for example, taking that whatever A does not belong to some of, B belongs to some of; otherwise no syllogism results, because the particular premise is negative. In the second figure, the affirmative conclusion cannot be shown by this method, but the negative can. The affirmative

31| is not demonstrated, because both premises would not be affirmative — for the conclusion is negative, whereas the affirmative conclusion was shown from two affirmative premises. But the negative is shown as follows. Let A belong to all B, and to no C; the conclusion is that B belongs to no C. If then it is assumed that B belongs to all A, A must belong to no C — for this becomes the second figure, with B as middle term. But if B is taken negatively and the other premise affirmatively, it will be the first figure. For C belongs to all A, and B to no C, so that B belongs to no A; therefore A does not belong to B either. So from the conclusion and one premise alone no syllogism results, but once the other premise is added, there will be one. But if the syllogism is not universal, the premise holding of the whole is not proved, for the same reason already stated; but the particular one is proved, when the universal premise is affirmative. For let A belong to all B, and not to all C; the conclusion concerns B and C — namely, that B does not belong to some C. If then it is assumed that B belongs to all A, but not to all C, A will not belong to some C; B is the middle term. But if the universal premise is negative, the premise A-C will not be proved by converting A-B; for it turns out that either both premises, or the other one, become negative, so that there will be no syllogism. But it will be proved in the same way as in the universal cases, if it is assumed that, whatever B does not belong to some of, A belongs to some of.

32| In the third figure, when both premises are taken as universal, it is not possible to prove them through one another: for the universal is proved through universals, but the conclusion in this figure is always particular, so it is clear that it is altogether impossible to prove the universal premise through this figure. But if one premise is universal and the other particular, sometimes it will be possible, sometimes not. When both premises are taken affirmatively and the universal one is next to the minor term, it will be possible; but when it is next to the other term, it will not be possible. For let A belong to all C, and B to some C; the conclusion is

33| A-B. If then it is assumed that C belongs to all A, C has been shown to belong to some B, but B has not been shown to belong to some C. And yet it is necessary, if C belongs to some B, that B also belongs to some C. But it is not the same thing for this to belong to that as for that to belong to this; rather, one must further assume that if this belongs to some of that, the other also belongs to some of this. Once this is assumed, the syllogism no longer comes from the conclusion and the other premise alone. But if B belongs to all C, and A to some C, it will be possible to show A-C, when it is assumed that C belongs to all B and A to some B. For if C belongs to all B, and A to some B, A must belong to some C; B is the middle term. And when one premise is affirmative and the other negative, with the affirmative one universal, the other will be shown. For let B belong to all C, and A not belong to some C; the conclusion is that A does not belong to some B. If then it is further assumed that C belongs to all B, A must not belong to some C; B is the middle term. But when the negative premise becomes the universal one, the other is not shown, except as in the previous cases, if it is assumed that, whatever this does not belong to some of, the other belongs to some of — for example, if A belongs to no C, and B to some C; the conclusion is that A does not belong to some B. If then it is assumed that C belongs to some of whatever A does not belong to, C must belong to some B. In no other way is it possible, by converting the universal premise, to show the other; for in no case will there then be a syllogism.

34| To convert is to transpose the conclusion and construct the syllogism so that either the extreme term will not belong to the middle, or the middle will not belong to the last term. For once the conclusion is converted and the other premise remains as it was, the remaining premise must be destroyed; for if it held, the conclusion would also hold. But it makes a difference whether one converts the conclusion oppositely or contrarily; for the syllogism that results is not the same in each case of conversion — this will become clear from what follows. By 'opposite' I mean 'belongs to all' set against 'does not belong to all,' and 'belongs to some' against 'belongs to none'; by 'contrary' I mean 'belongs to all' against 'belongs to none,' and 'belongs to some' against 'does not belong to some.' Let A have been shown of C through the middle term B. If, then, A is assumed to belong to no C, but to all B, B will belong to no C. And if A belongs to no C, and B to all C, A will not belong to all B, but not simply to none either; for the universal was not proved through the last figure. And in general, the premise concerning the major term cannot be universally overturned by conversion —

35| For it is always destroyed through the third figure; for it is necessary to take both premises in relation to the last term. And if the syllogism is negative, the same holds. For let it have been proved, through B, that A belongs to no C. Then if it is assumed that A belongs to all C, while A belongs to no B, B will belong to no C. And if A and B both belong to all C, A will belong to some B; but it belonged to none. But if the conclusion is converted contradictorily, the resulting syllogisms will be contradictory and not universal. For one of the premises becomes particular, so that the conclusion too will be particular. For let the syllogism be affirmative, and let it be converted in this way. Then if A does not belong to all C, while A belongs to all B, B will not belong to all C; and if A does not belong to all C, while B belongs to all C, A will not belong to all B. And likewise if the syllogism is negative. For if A belongs to some C, and to no B, B will not belong to some C — not simply to none of it;

36| and if A belongs to some C, and B to all C, as was assumed at the start, A will belong to some B. In the case of particular syllogisms, when the conclusion is converted contradictorily, both premises are destroyed, but when contrarily, neither. For it no longer follows, as it did with universal syllogisms, that destruction occurs even though the conclusion falls short in the conversion — rather, there is no destruction at all. For let it have been proved that A holds of some C. Then if it is assumed that A belongs to no C, while B belongs to some C, A will not belong to some B; and if A belongs to no C, while A belongs to all B, B will belong to no C. So both are destroyed. But if it is converted contrarily, neither. For if A does not belong to some C, while A belongs to all B, B will not belong to some C, but the original premise is not yet destroyed; for it is possible for it to belong to some and not belong to some. And of the universal premise, the B premise, no syllogism results at all; for if A does not belong to some C, and B belongs to some C, neither of the premises is universal. And likewise if the syllogism is negative; for if it is assumed that A belongs to all C, both are destroyed, but if to some, neither. The proof is the same.

37| In the second figure the premise involving the major term cannot be destroyed contrarily, however the conversion is made; for the conclusion will always fall in the third figure, and there is no universal syllogism in that figure. But the other premise we shall destroy in the same manner as the conversion. By 'in the same manner' I mean: if the conversion is contrary, contrarily; if contradictory, contradictorily. For let A belong to all B, and to no C; the conclusion holds between B and C. Then if it is assumed that B belongs to all C, and A-B remains as it was, A will belong to all C; for this becomes the first figure. But if B belongs to all C, and A belongs to no C, A will not belong to all B; this is the last figure. But if B-C is converted contradictorily, the B premise will be shown in the same way, but the C premise contradictorily. For if B belongs to some C, and A to no C, A will not belong to some B. Again, if B belongs to some C, and A belongs to all

38| B, A will belong to some C, so that the resulting syllogism is contradictory. And it will be shown in the same way if the premises are reversed. But if the syllogism is particular, when the conclusion is converted contrarily neither of the premises is destroyed, just as in the first figure; but when contradictorily, both are. For let A be posited as belonging to no B, and to some C; the conclusion holds between B and C. Then if it is assumed that B belongs to some C, and the premise A-B remains, the conclusion will be that A does not belong to some C, but the original premise has not been destroyed; for it is possible for a thing to belong to some and not belong to some. Again, if B belongs to some C and A belongs to some C, there will be no syllogism; for neither of the premises taken is universal, so that the B premise is not destroyed. But if it is converted contradictorily, both are destroyed. For if B belongs to all C, and A to no B, A will belong to no C; but it belonged to some. Again, if B belongs to all C, and A to some C, A will belong to some B. And the proof is the same if the universal premise is affirmative.

39| In the third figure, when the conclusion is converted contrarily, neither of the premises is destroyed in any of the syllogisms; but when contradictorily, both are, and in all of them. For let it have been proved that A belongs to some B, with C taken as the middle term, and let the premises be universal. Then if it is assumed that A does not belong to some B, while B belongs to all C, no syllogism results connecting A and C. Nor, if A does not belong to some B, while A belongs to all C, will there be a syllogism connecting B and C. And it will be shown in the same way if the premises are not universal. For either both premises must become particular as a result of the conversion, or the universal premise must fall with the minor term; and in that arrangement there was no syllogism, either in the first figure or in the second. But if it is converted contradictorily, both premises are destroyed. For if A belongs to no B, and B belongs to all C, A will belong to no C. Again, if A belongs to no B, while A belongs to all C, B will belong to no C. And if one of the premises is not universal, likewise. For if A belongs to no B, and B

40| belongs to some C, A will not belong to some C; but if A belongs to no B, while A belongs to all C, B will belong to no C. And likewise if the syllogism is negative. For let it have been proved that A does not belong to some B, and let B-C be affirmative, and A-C negative; for it was in this way that the syllogism arose. Now when the contrary of the conclusion is assumed, there will be no syllogism. For if A belongs to some B, and B belongs to all C, there was no syllogism connecting A and C. Nor, if A belongs to some B, and to no C, was there a syllogism connecting B and C. So the premises are not destroyed. But when the contradictory is assumed, they are destroyed. For if A belongs to all B and B belongs to all C, A will belong to all C; but it belonged to none. Again, if A belongs to all B, and to no C, B will belong to no C; but it belonged to all. And it is shown in the same way if the premises are not universal. For the C premise becomes both universal and negative, while the other is particular and affirmative.

41| If then A belongs to all B, and B belongs to some C, it follows that A belongs to some C; but it was assumed to belong to none. Again, if A belongs to all B, but to none of C, then B belongs to none of C; but it was assumed to belong to some. But if A belongs to some B, and B belongs to some C, no syllogism results; nor if A belongs to some B, but to none of C, does one result either. So the premises are destroyed in the first way, but not destroyed in this way. It is clear, then, from what has been said, how, when the conclusion is converted, a syllogism comes about in each figure, and when it is contrary to the premise and when it is its opposite; and that in the first figure the syllogisms come about through the middle and the last terms, and the premise next to the minor extreme is always destroyed through the middle term, while the one next to the major is destroyed through the last term; in the second figure they come about through the first and the last terms, the premise next to the minor extreme always through the first figure, and the one next to the major through the last term; and in the third figure they come about through the first and through the middle term, and the premise next to the major always through the first, while the one next to the minor through the middle.

42| What conversion is, then, and how a syllogism comes about in each figure, and what sort it is, is clear. The syllogism through the impossible is demonstrated when the contradictory of the conclusion is posited and another premise is added, and it comes about in all the figures; for it is similar to conversion, except that it differs in this respect: conversion happens once a syllogism has come about and both premises have been taken, whereas reduction to the impossible does not depend on the opposite having been agreed to beforehand, but on its being evident that it is true. The terms are the same in both, and the assumption is the same for both. For example, if A belongs to all B, and C is the middle term, then if it is supposed that A belongs either not to all B or to none of B, while it belongs to all C — which was true — C must belong either to none of B or not to all of B. But this is impossible, so that what was supposed is false; therefore the opposite is true. Similarly too in the other figures; for whatever admits of conversion also admits of the syllogism through the impossible. Now all the other problems are demonstrated through the impossible in all the figures, but the universal affirmative is demonstrated in the middle and the third figure, but not demonstrated in the first.

43| For let it be supposed that A does not belong to all B or belongs to none, and let another premise be added either way — either that C belongs to all A, or that B belongs to all D; for in this way it would be the first figure. If then it is supposed that A does not belong to all B, no syllogism results whichever premise is taken; but if it is supposed that A belongs to none of B, then when BD is added, there will be a syllogism of a falsehood, but the thing proposed is not demonstrated. For if A belongs to none of B, and B belongs to all D, A belongs to none of D. Let this be impossible; then it is false that A belongs to none of B. But it does not follow that if belonging to none is false, belonging to all is true. And if C is added, no syllogism results, nor when it is supposed that A does not belong to all B. So it is clear that belonging to all is not demonstrated in the first figure through the impossible. But belonging to some, and belonging to none, and not belonging to all are demonstrated. For let it be supposed that A belongs to none of B, and let B be taken to belong to all or to some C.

44| It is then necessary that A belongs to none, or not to all, of C. But this is impossible — for let it be true and evident that A belongs to all of that term — so that if this is false, it is necessary that A belongs to some B. But if the other premise is taken with A, there will be no syllogism; nor when the contrary of the conclusion is supposed, for instance that it belongs to some but not to some. It is clear, then, that the opposite must be supposed. Again, let it be supposed that A belongs to some B, and let C be taken to belong to all A. Then C must belong to some B. Let this be impossible, so that what was supposed is false. If so, it is true that it belongs to none. Likewise too if CA was taken as privative. But if the premise next to B is taken, there will be no syllogism. And if the contrary is supposed, there will be a syllogism, and the impossibility, but the thing proposed is not demonstrated. For let it be supposed that A belongs to all B, and let C be taken to belong to all A. Then C must belong to all B. But this is impossible, so that it is false that A belongs to all B.

45| But it is not yet necessary, if it does not belong to all, that it belongs to none. Likewise too if the other premise were taken with B; for there will be a syllogism and the impossibility, but the supposition is not destroyed; so that the opposite must be supposed. To show that A does not belong to all B, one must suppose that it belongs to all; for if A belongs to all B, and C belongs to all A, then C belongs to all B, so that if this is impossible, what was supposed is false. Likewise too if the other premise were taken with B. And if CA was privative, the same holds; for a syllogism comes about that way too. But if the privative is next to B, nothing is demonstrated. And if it is supposed not that it belongs to all but to some, it is not demonstrated that it does not belong to all, but that it belongs to none. For if A belongs to some B, and C belongs to all A, C will belong to some B. If then this is impossible, it is false that A belongs to some B, so that it is true that it belongs to none. But if this is demonstrated, the truth is destroyed along with it; for A belonged

46| to some B, but did not belong to some. Further, nothing impossible follows from the supposition; for it would have to be false, if indeed a falsehood cannot be deduced from truths — but as it is, it is true; for A belongs to some B. So one must not suppose that it belongs to some, but that it belongs to all. Likewise too if we were showing that A does not belong to some B; for if belonging to some, not, and not belonging to all are the same, the demonstration of both is the same. It is clear, then, that in all the syllogisms one must suppose not the contrary but the opposite; for in this way both the necessity will hold and the claim will be reputable. For if of everything either the affirmation or the negation holds, then once it is shown that the negation does not hold, the affirmation must be true. Again, if one does not posit that the affirmation is true, it is reputable to claim the negation. But the contrary fits neither claim: for it is neither necessary, if belonging to none is false, that belonging to all is true; nor is it reputable, that if one is false the other is true.

47| It is clear, then, that in the first figure all the other problems are proved through the impossible, but the universal affirmative is not proved. In the middle and the last figure this too is proved. For let it be assumed that A does not belong to all B, and let it be taken that A belongs to all C. Then if A does not belong to all B, but belongs to all C, C does not belong to all B. But this is impossible; for let it be evident that C belongs to all B, so that the assumption is false. Therefore it is true that A belongs to all B. But if the contrary is assumed, there will be a syllogism and the impossible result too, but the point proposed will not be proved. For if A belongs to no B, and A belongs to all C, then C belongs to no B. But this is impossible, so that it is false that A belongs to no B. But it does not follow that, if this is false, it is true that A belongs to all B. To prove that A belongs to some B, let it be assumed that A belongs to no B, and let A belong to all C. Then C must belong to no B. So if this is impossible, A must belong to some B. But if it is assumed that A does not belong to some B, the same result will follow as in the first figure. Again, let it be assumed that A belongs to some

48| B, and let A belong to no C. Then C must not belong to some B. But C belonged to all B, so that what was assumed is false; therefore A will belong to no B. To prove that A does not belong to all B, let it be assumed that A belongs to all B, and that A belongs to no C. Then C must belong to no B. But this is impossible, so that it is true that A does not belong to all B. It is clear, then, that all these syllogisms come about through the middle figure.

49| Similarly also through the last figure. For let it be assumed that A does not belong to some B, and that C belongs to all B; then A does not belong to some C. If, then, this is impossible, it is false that A does not belong to some B, so that it is true that A belongs to all B. But if it is assumed that A belongs to no B, there will be a syllogism and the impossible result, but what was proposed will not be proved; for if the contrary is assumed, the same result will follow as in the earlier cases. Rather, this assumption must be taken to prove that A belongs to some B. For if A belongs to no B, and C belongs to some B, then A does not belong to all C. If, then, this is false, it is true that A belongs to some B. To prove that A belongs to no B, let it be assumed that A belongs to some B, and let it also be taken that C belongs to all B. Then A must belong to some C. But A belonged to no C, so that it is false that A belongs to some B. But if it is assumed that A belongs to all B, what was proposed is not proved; rather, this assumption must be taken to prove that A does not belong to all B. For if A belongs to all B, and C belongs to all B, then A belongs to some C. But this was not so; so that it is false that A belongs to all B. And if so, it is true that A does not belong to all B. But if it is assumed that A belongs to some B, the same result will follow as in the cases already stated. It is clear, then, that in all arguments through the impossible one must assume the opposite. And it is clear too that in the middle figure the affirmative is in some way proved, and in the last figure the universal.

50| Demonstration through the impossible differs from direct demonstration in that it posits what it wishes to refute, and leads it away to an agreed falsehood, whereas direct demonstration starts from agreed positions. Both, then, take two agreed premises; but the one takes the premises from which the syllogism proceeds, while the other takes one of these, and, as the other, the contradictory of the conclusion. And in the one case it is not necessary that the conclusion be known in advance, nor to assume beforehand whether it holds or not; in the other case it is necessary to assume in advance that it does not hold. It makes no difference whether the conclusion is an affirmation or a negation; the case is the same for both. Everything concluded directly can also be proved through the impossible, and everything proved through the impossible can also be proved directly, through the same terms. For whenever the syllogism comes about in the first figure, the true result will be in the middle or the last figure — the negative in the middle, the affirmative in the last. And whenever the syllogism is in the middle figure, the true result is in the first figure, for all the problems. And whenever the syllogism is in the last figure, the true result is in the first and the middle — the affirmative ones in the first, the negative ones in the middle. For let it be supposed that it has been proved, through the first figure, that A belongs to no B or not to all B.

51| Then the assumption was that A belongs to some B, and C was taken to belong to all A, but to no B; for it was in this way that the syllogism, and the impossible result, came about. But this is the middle figure, if C belongs to all A but to no B. And it is clear from this that A belongs to no B. The case is similar too if it has been proved that A does not belong to all B. For the assumption is that A belongs to all B, and C was taken to belong to all A, but not to all B. And it is the same too if the premise C-A is taken as negative; for in this way too the middle figure comes about. Again, let it have been proved that A belongs to some B. Then the assumption was that A belongs to no B, and B was taken to belong to all C, and A either to all or to some C; for it is in this way that the impossible result will come about. But this is the last figure, if A too, and B, belong to all C. And it is clear from this that A must belong to some B. The case is similar too if B or A were taken to belong to some C.

52| Again, let it have been proved, in the middle figure, that A belongs to all B. Then the assumption was that A does not belong to all B, and it was taken that A belongs to all C and C belongs to all B; for it is in this way that the impossible result will come about. But this is the first figure — A belonging to all C, and C to all B. The case is similar too if it has been proved that A belongs to some B; for the assumption was that A belongs to no B, and it was taken that A belongs to all C and C belongs to some B. But if the syllogism is negative, the assumption was that A belongs to some B, and it was taken that A belongs to no C and C belongs to all B, so that the first figure results. And if the syllogism is not universal, but it has been proved that A does not belong to some B, the case is the same. For the assumption was that A belongs to all B, and it was taken that A belongs to no C and C belongs to some B; for it is in this way that the first figure results. Again, let it have been proved, in the third figure, that A belongs to all B. Then the assumption was that A does not belong to all B, and it was taken that C belongs to all B and A belongs to all C.

53| For that is how the impossibility will result. And this is the first figure. Likewise too if the demonstration concerns a particular case: for the hypothesis is that A belongs to no B, and it has been assumed that C belongs to some B and A to all C. But if the syllogism is privative, the hypothesis is that A belongs to some B, and it has been assumed that C belongs to no A and to all B — and this is the middle figure. Likewise too if the demonstration is not universal: for the hypothesis will be that A belongs to all B, and it has been assumed that C belongs to no A and to some B — and this is the middle figure. It is clear, then, that each of the problems can be proved ostensively as well, through the same terms. Similarly, when the syllogisms are ostensive, it will be possible to reduce them to the impossible using the terms assumed, whenever the premise opposite to the conclusion is taken. For the same syllogisms result as those obtained through conversion, so that we have at once the figures through which each will come about as well. It is clear, then, that every problem can be proved in both ways,

54| both through the impossible and ostensively, and it is not possible for the one way to occur without the other. In which figure it is possible to draw a syllogism from opposite premises, and in which it is not, will be clear as follows. I call premises opposite, in respect of wording, in four ways — namely 'belongs to all' as against 'belongs to none,' 'belongs to all' as against 'does not belong to all,' 'belongs to some' as against 'belongs to none,' and 'belongs to some' as against 'does not belong to some' — but in respect of truth, in three ways; for 'belongs to some' and 'does not belong to some' are opposed only in wording. Of these, the universal ones are contraries — 'belongs to all' as against 'belongs to none,' for example 'every science is worthwhile' as against 'no science is worthwhile' — while the rest are opposites. In the first figure, then, there cannot be a syllogism from opposite premises, neither an affirmative one nor a negative one — not an affirmative one, because both premises must be affirmative, while opposite premises consist of an assertion and a denial; nor a privative one, because opposite premises affirm and deny the same thing of the same thing, whereas the middle term in the first figure is not said of both extremes in that way — rather it is denied of the one, and it is itself predicated of the other, and these are not opposed. But in the middle figure it is possible for a syllogism to come about both from opposites and from contraries.

55| For let A stand for the good, and B and C for knowledge. If, then, one has assumed that every kind of knowledge is worthwhile and also that none is, A belongs to all B and to no B, so that B belongs to no C; therefore no knowledge is knowledge. Likewise too if, having assumed that every kind of knowledge is worthwhile, one has assumed that medicine is not worthwhile: for A belongs to all B and to no C, so that some knowledge will not be knowledge. Also if A belongs to all C and to no B, and B stands for knowledge, C for medicine, and A for conception — for then, having assumed that no knowledge is a conception, one has assumed that some thing, namely medicine, is a conception. This differs from the previous case in that the terms are converted: before, the affirmative premise stood next to B, but now it stands next to C. And if one of the two premises is not universal, the result is the same; for the middle term is always

56| the term that is denied of the one extreme and affirmed of the other. So it is possible to conclude to opposites, but not always and not in every case, but only when the terms falling under the middle term are related in such a way as to be either the same or as whole to part. Otherwise it is impossible: for then the premises will in no way be either contraries or opposites. In the third figure, an affirmative syllogism will never come about from opposite premises, for the reason already stated in the case of the first figure as well, but a negative one will, whether the terms are taken universally or not universally. For let B and C stand for knowledge, and A for medicine. If, then, one assumes that all medicine is knowledge and that no medicine is knowledge, B has been assumed to belong to all A and to none, so that some knowledge will not be knowledge. Likewise too if the premise B-A is not taken universally: for if some medicine is knowledge, and again no medicine is knowledge, it results that some knowledge is not knowledge. When the terms are taken universally, the premises are contraries; but if one of the two is taken as particular, they are opposites. One must notice that it is possible to take the opposites in the way we said — that every kind of knowledge is worthwhile, and again that none is, or that some is not — a mistake that does not usually go unnoticed.

57| But it is possible to get the other premise conceded through different questions, or else to obtain it as was said one should in the Topics. And since the oppositions of affirmations are three, it turns out there are six ways of taking the opposites — either 'to all' and 'to none,' or 'to all' and 'not to all,' or 'to some' and 'to none' — and this may also be converted with respect to the terms, for example A belonging to all B and to no C, or to all C and to no B, or to all of the one and not to all of the other, and this again converted with respect to the terms. Likewise too in the third figure. So it is clear in how many ways, and in which figures, it is possible for a syllogism to come about from opposite premises. It is also clear that from false premises one can draw a true conclusion by syllogism, as was said before, but from opposite premises

58| one cannot; for the conclusion is always contrary to the fact — for example that what is good is not good, or that what is an animal is not an animal — because the syllogism proceeds from a contradiction, and the underlying terms are either the same or one is whole and the other part. It is also clear that in fallacious reasonings nothing prevents the contradiction of the hypothesis from coming about, for example that what is odd is not odd. For the syllogism from opposite premises was contrary to the fact; if, then, one takes premises of that sort, the contradiction of the hypothesis will result. One must notice, however, that it is not possible in this way to conclude contraries from a single syllogism, so that the conclusion is that the very thing which is not good is good, or something else of that sort, unless the premise is taken in that form straightaway (for example that every animal is white and not white, and that man is an animal); rather, one must either add the contradiction — for example, that every kind of knowledge is a conception, and then take that medicine is a kind of knowledge but no conception at all, the way refutations are made — or draw it from two syllogisms. For the assumptions to be contraries in point of truth, there is no other way than this, as has been said before.

59| Begging the question and assuming it consists, taken generically, in failing to demonstrate the thing proposed; and this happens in many ways — if the argument does not reason at all, and if it reasons through things less known or equally unknown, and if it demonstrates what is prior through what is posterior; for demonstration proceeds from things more credible and prior. None of these, then, is what begging the original question is. But since some things are naturally known through themselves, and others through other things — for principles are known through themselves, while what falls under the principles is known through other things — whenever someone attempts to prove what is not known through itself by means of itself, then he is begging the original question. This can be done by straightaway claiming the very thing proposed, but it is also possible, by shifting to certain other things which are naturally proved through that very thing, to demonstrate the original point through them — for example if A were to be proved through B, and B through C, and C were naturally such as to be proved

60| through A; for it turns out that those who reason in this way are proving A through A itself. This is what people do who think they are proving parallel lines parallel: they fail to notice that they are assuming things that cannot be demonstrated unless the parallels exist. So it turns out that for those who reason this way, each thing is asserted on the ground that each thing is so; and in that way everything will be known through itself, which is impossible. If, then, it is unclear whether A holds of C, and equally unclear whether it holds of B, and someone asks that A be granted to hold of B, it is not yet clear whether he is begging the point at issue, but it is clear that he is not demonstrating anything; for a premise equally unclear cannot be a starting point of demonstration. If, however, B stands to C in such a way that they are the same, or it is clear that they convert, or one belongs to the other, then he is begging the point at issue. For he could show that A holds of B through those very terms, if they converted (as it is, this is prevented not by the method but by the fact that they do not convert). But if he did make them convert, he would be doing what has been described, and converting through three terms. Likewise too if he took it that B holds of C, since it is equally unclear whether A does, this is not yet begging the original point, but it still fails to demonstrate anything.

61| But if A and B are the same, either because they convert or because A follows B, then he is begging the original point, for the same reason. For we have said what begging the original point amounts to: proving through itself what is not clear through itself. If, then, begging the point at issue is proving through itself what is not clear through itself, and this is failing to prove, occurring whenever the thing being proved and that through which it is proved are equally unclear, either because the same things hold of the same thing or because the same thing holds of the same things, then in the middle figure and the third this could happen either way, while in a categorical syllogism it can happen in the third figure and the first. When the syllogism is negative, it happens when the same things are denied of the same thing; and the two premises are not alike in this respect (and likewise in the middle figure), because in negative syllogisms the terms do not convert. Begging the point at issue occurs, in demonstrations, with respect to things that actually stand so, and in dialectical arguments, with respect to things that are so according to opinion.

62| As for the falsehood not coming about on account of this thesis — something we often say in arguments — this occurs, in the first place, in reductions to the impossible, when the conclusion is the contradictory of what was being proved by the reduction to the impossible. For someone who has not asserted the contradictory will not say 'not on account of this,' but rather that some falsehood was assumed among the earlier premises; nor will he say it in the case of a direct demonstration, since in that case he is not positing the thing that would be contradicted. Further, when something is done away with directly through B and C, one cannot say that the syllogism did not come about on account of the thesis laid down. For we say that a conclusion does not come about on account of a thesis when, that thesis being removed, the syllogism is completed nonetheless — which cannot happen in direct demonstrations, since if the position is removed, the syllogism directed against it will not exist either. It is clear, then, that 'not on account of this' is said in reductions to the impossible, and when the original hypothesis stands to the impossible conclusion in such a way that the impossibility results whether the hypothesis holds or not. The clearest way for the falsehood to be not on account of the thesis is when the syllogism from the middle terms to the impossibility is unconnected with the hypothesis, as has also been said in the topical treatises.

63| For that is exactly what it is to set down the non-cause as cause: for instance, if someone wanting to show that the diagonal is incommensurable were to try Zeno's argument that motion is impossible, and reduce that to the impossible; for the falsehood in Zeno's argument has no continuity whatsoever with the original claim. Another way is when the impossibility is continuous with the hypothesis but does not come about because of it. For this can happen whether one takes the continuity going upward or going downward: for instance, if A is posited as holding of B, B of C, and C of D, and it turns out to be false that B holds of D. For if, when A is removed, B still holds of C and C of D just the same, then the falsehood would not be on account of the original hypothesis. Or again, if one takes the continuity going upward, as when A holds of B, E holds of A, and Z holds of E,

64| and it turns out to be false that Z holds of A: for in this case too the impossibility would result no less even with the original hypothesis removed. But the impossibility must connect to the original terms; only then will it be because of the hypothesis — for instance, when the continuity is taken downward, in relation to the term that is predicated (for if it is impossible for A to hold of D, then once A is removed the falsehood will no longer exist); and when taken upward, in relation to the term of which the other is predicated (for if it is not possible for Z to hold of B, then once B is removed the impossibility will no longer exist). The same holds for syllogisms that are negative. It is clear, then, that when the impossibility does not bear on the original terms, the falsehood does not come about on account of the thesis. Or is it not even so that the falsehood will always be on account of the hypothesis? For suppose A was posited to hold not of B but of K, and K of C, and this of D; then the impossibility still stands (and likewise if the terms are taken going upward); so that, since the impossibility results whether this term exists or not, it would not be on account of the thesis. Or rather, that the falsehood comes about no less when this term does not exist must not be understood as meaning that the impossibility results when some other term is posited, but rather that, when this term is removed, the same impossibility is brought about through the remaining premises — since it is perhaps not at all strange that the same falsehood should result from several different hypotheses, as, for example, that parallel lines meet follows both from supposing that the interior angle is less than the exterior, and from supposing that a triangle has more than two right angles.

65| A false argument comes about on account of a first falsehood. For every syllogism is from two premises or from more. If, then, it is from two, one or both of these must be false; for a false syllogism could not come from true premises. But if it is from more, say C is proved through A and B, and these through D, E, Z, and H, then one of these higher premises will be false, and the argument is false on account of that; for A and B are concluded through them. So it is from those premises that the conclusion, and the falsehood, results.

66| To avoid being reasoned against, one should be on watch, when someone puts his argument without stating the conclusions, so as not to grant the same thing twice among the premises, since we know that no syllogism comes about without a middle term, and the middle term is the one stated more than once. How one must watch the middle term with respect to each conclusion is clear from knowing which term is proved in each figure; and this will not escape us, because we know how we are sustaining the argument. But what we recommend guarding against as answerers, we should ourselves try to bring about unnoticed as questioners. This will happen, first, if the conclusions are not drawn in advance, but remain unclear once the necessary premises have been secured; and further, if one asks not for the terms that are close together, but rather those that are as far as possible from being immediate. For example, suppose A must be concluded of Z, with B, C, D, E as middle terms. One should ask whether A holds of B, and then not whether B holds of C, but whether D holds of E, and then whether B holds of C, and so on with the rest. And if the syllogism comes about through a single middle term, one should begin from the middle term; for in that way the answerer would be least likely to notice.

67| Now that we have established when and how the terms must stand for a syllogism to come about, it is clear also when there will be a refutation and when there will not be. For if everything is granted, or the answers are given alternately -- one negative, one affirmative -- it is possible for a refutation to come about; for a syllogism resulted whether the terms stood this way or that, so that if the position laid down is contrary to the conclusion, a refutation must come about, since a refutation is a syllogism of a contradiction. But if nothing is granted, it is impossible for a refutation to come about; for there was no syllogism when all the terms are privative, and so no refutation either -- for if there is a refutation, a syllogism must exist, but if a syllogism exists, a refutation need not. Likewise, too, if nothing is posited as holding of the whole in the answer, since the same determination will apply to refutation as to syllogism. It sometimes happens that, just as we are deceived in the position of the terms, so too deception comes about with respect to supposition,

68| for instance, if it is possible for the same thing to hold of a plurality of primary subjects, and someone may fail to notice this and suppose that it holds of none, while at the same time knowing that it holds. Let A hold of B and of C in their own right, and let both B and C hold of every D in the same way. Now if someone supposes that A holds of every B, and this of every D, but that A holds of no C, and this of every D, he will have both knowledge and ignorance of the same thing in the same respect. Again, suppose someone were deceived about terms drawn from the same series -- for instance, if A holds of B, this of C, and C of D, but he supposes that it holds of all of the one and again of none of C; for then he will at once know and fail to suppose that it holds. Is it, then, the case that from these he claims nothing else than that what he knows he does not suppose? For he knows in a way that A holds of C through B, as the particular is known through the universal, so that what he knows in a way, he claims not to suppose at all -- which is impossible. As for the case mentioned earlier, if the middle term is not from the same series, it is not possible to suppose both premises with respect to either middle term -- for instance, that A holds of every B but of no C, while both B and C hold of every D.

69| For then it turns out that the first premise is taken as contrary, either simply or in part. For if he supposes that A holds of everything of which B holds, and knows that B holds of D, then he knows that A holds of D. 67a So that if, in turn, he supposes that A holds of nothing of which C holds, and B holds of some of that of which C holds, he does not suppose that A holds of that thing. But to suppose that A holds of everything of which B holds, and again to suppose, of something of which B holds, that A does not hold, is contrary, either simply or in part. In this way, then, it is not possible to suppose both; but with respect to each of the two middle terms taking one premise, or with respect to one of them taking both, nothing prevents it -- for instance, that A holds of every B and B of every D, and again that A holds of no C. For this sort of deception is like the way we are deceived about particular cases -- for instance, if A holds of everything of which B holds, and B holds of every C, then A will hold of every C. If, then, someone knows that A holds of everything of which B holds,

70| he also knows that it holds of every C. But nothing prevents his being ignorant that C exists -- for instance, let A be having angles equal to two right angles, B be triangle, and C be perceptible triangle. For someone might suppose that C does not exist, while knowing that every triangle has two right angles, so that he will at once know and be ignorant of the same thing. For knowing that every triangle has two right angles is not a single thing, but one sort consists in having the universal knowledge, the other in having the knowledge of the particular. In this way, then, he knows as by the universal that C has two right angles, but as by the particular he does not know it, so that he will not hold the contraries. And the argument in the Meno is similar, that learning is recollection. For it never turns out that one has prior knowledge of the particular, but rather one grasps the knowledge of particulars together with the induction, as though recognizing them. For some things we know at once -- for instance, that a figure has angles equal to two right angles, if we see that it is a triangle. And likewise in the other cases. By the universal, then, we contemplate particulars, but we do not know them by the knowledge proper to them, so that it is possible to be deceived about them too, though not contrarily, but rather to have the universal knowledge while being deceived about the particular.

71| So too, likewise, with the cases discussed before: the deception concerning the middle term is not contrary to the knowledge that comes by the syllogism, nor is the supposition made with respect to either middle term. And nothing prevents someone who knows both that A holds of the whole of B and that this in turn holds of C from supposing that A does not hold of C -- for instance, knowing that every mule is barren, and yet supposing that this particular mule, being a mule, is pregnant; for he does not know that A holds of C, since he is not contemplating both together. So it is clear that even if he knows the one but not the other, he will be deceived -- which is just the relation universal kinds of knowledge have to knowledge of particulars. For we know none of the perceptible things once they have passed outside perception, not even if we have actually perceived them, except as by the universal and by having the knowledge proper to the case, but not as by actually exercising it. For 'knowing' is said in three ways: either as by the universal, or as by the knowledge proper to the case, or as by actually exercising it, so that being deceived is said in just as many ways. So nothing prevents someone from both knowing and being deceived about the same thing, only not

72| contrarily. This is what happens too to someone who knows each of the two premises but has not previously considered them together. For in supposing that the mule is pregnant, he does not have the knowledge that consists in actually exercising it, nor again does he, through this supposition, have a deception contrary to his knowledge; for the contrary deception to the universal knowledge would be a syllogism. Now someone who supposes that what it is to be good is what it is to be bad will suppose the same thing to be both what it is to be good and what it is to be bad. For let what it is to be good be A, what it is to be bad be B, and again let what it is to be good be C. Since, then, he supposes B and C to be the same, he will suppose that C is B, and again, likewise, that B is A, so that he will also suppose that C is A. For just as, if it were true that whatever C is, B is, and whatever B is, A is, then the truth would also hold of C, so it is too in the case of supposing. And likewise in the case of being: since C and B are the same, and again B and A are the same, C was also the same as A; so that it is likewise, too, in the case of holding an opinion. Is this, then, necessary, if one grants the first point? But perhaps that is false -- that anyone should suppose what it is to be bad to be what it is to be good, except accidentally; for it is possible to suppose this in many ways. But this must be examined more closely.

73| When the extremes are convertible, the middle term must also be convertible with both. For if A holds of C through B, then, if it converts and holds -- that is, if every C, whatever A holds of, holds -- then B too will convert with A, and A will hold of every B through the middle term C; and C will convert with B through the middle term A. And it is the same in the case of not holding -- for instance, if B holds of C, but A does not hold of B, then A will not hold of C either. If, then, B converts with A, C will also convert with A. For let B not hold of A; then C will not hold of A either, for B held of every C. And if C converts with B, A too will convert with it; for of everything of which B holds, C also holds. And if C converts with A as well, B too converts. For of whatever B holds,

74| That to which A belongs, C does not belong. And only this conclusion begins from the conclusion; the others are not related in the same way, nor is it so in the case of the affirmative syllogism. Again, if A and B convert with each other, and likewise C and D, and it is necessary that either A or C belong to whatever lacks the other, then B and D will also be so related that one or the other belongs to everything. For since that to which A belongs, B belongs, and that to which C belongs, D belongs, and to everything either A or C belongs, and not both together, it is clear that either B or D belongs to everything, and not both together; for two syllogisms are combined here. Again, if to everything either A or B belongs, and either C or D, but not together, then, if A and C convert, B and D also convert. For if B does not belong to something to which D belongs, it is clear that A belongs to it. But if A belongs, so does C, since they convert; so C and D would belong together. But this is impossible. For example: if the ungenerated is imperishable and the imperishable is ungenerated, then what has come to be must be perishable, and what is perishable must have come to be.

75| When A belongs to the whole of B and of C, and is predicated of nothing else, and B also belongs to all of C, then A and B must convert. For since A is said only of B and C, and B is predicated both of itself and of C, it is clear that of everything of which A is said, B will also be said, except of A itself. Again, when A and B both belong to the whole of C, and C converts with B, then A must belong to all of B. For since A belongs to all of C, and C belongs to B because they convert, A will also belong to all of B. When, there being two things, A is more desirable than B, these being opposites, and likewise D more desirable than C, then, if A and C together are more desirable than B and D, A is more desirable than D. For A is to be pursued and B is to be avoided to the same degree, since they are opposites, and likewise C and D, for these too are opposed. If, then, A is desirable to the same degree as D, then B is also to be avoided to the same degree as C;

76| for each stands to the other in the same relation, the avoidable to the pursued. So both pairs stand toward each other in the same way. But since A and C together are more desirable, they cannot be equal to B and D; for otherwise B and D would be equal too. But if D is more desirable than A, then B is less to be avoided than C; for the lesser is opposed to the lesser. And the greater good together with the lesser evil is more desirable than the lesser good together with the greater evil; so the whole, B and D together, would then be more desirable than A and C together — but this is not the case. A, then, is more desirable than D, and C is therefore less to be avoided than B. Now if every lover, in accordance with love, would choose A — being in such a state that the beloved grants favor — over not having the beloved grant favor, which is C, and would choose the beloved's granting favor, which is D, over not being in such a state as to have favor granted, which is B, then it is clear that A, being in such a state, is more desirable than the granting of favor itself. Being loved, therefore, is more desirable than intercourse, so far as love is concerned. Love, then, is more a matter of being loved than of being together. And if it is above all concerned with this, then this is its end. Being together, therefore, either is not an end at all, or exists for the sake of being loved; for the other appetites and crafts stand in the same relation to their ends.

77| How the terms stand with respect to conversions, and with respect to being more or less desirable or to be avoided, is now clear. It should now be said that not only dialectical and demonstrative syllogisms come about through the figures already discussed, but rhetorical syllogisms too, and in general any proof whatever, by whatever method it is reached. For we come to believe everything either through syllogism or from induction. Induction, then — that is, the syllogism that proceeds from induction — is the inferring, through one extreme, that the other extreme belongs to the middle term; for example, if B is the middle term between A and C, showing through C that A belongs to B. For this is how we carry out inductions. For example, let A be long-lived, let B be that which has no gall-bladder, and let C be the particular long-lived things, such as man and

78| horse and mule. Now A belongs to the whole of C, for every C is long-lived; but B too, not having a gall-bladder, belongs to all of C. If, then, C converts with B, and the middle term does not extend beyond it, A must belong to B. For it has been shown earlier that if two things belong to the same subject, and the extreme converts with one of them, then the other of the predicated terms will also belong to the one that converts. And C must be understood as the whole made up of all the particular cases; for induction proceeds through all of them. A syllogism of this kind establishes the first and immediate premise; for where there is a middle term, the syllogism proceeds through the middle term, and where there is none, it proceeds through induction. And in a way induction is opposed to syllogism; for the one shows the extreme belonging to the third term through the middle term, while the other shows the extreme belonging to the middle term through the third term. By nature, then, the syllogism through the middle term is prior and more knowable, but to us the one through induction is more perspicuous.

79| There is argument from example when it is shown that the extreme belongs to the middle term by means of a term similar to the third. And the middle term must be known to belong to the third, and the first term must be known to belong to the similar term. For example, let A be bad, let B be to undertake war against neighbors, let C be the Athenians against the Thebans, and let D be the Thebans against the 69a Phocians. If, then, we want to show that making war on the Thebans is bad, we must assume that making war on neighbors is bad. And the proof of this comes from similar cases — for instance, that the war of the Thebans against the Phocians was bad. Since, then, war against neighbors is bad, and war against the Thebans is a case of war against neighbors, it is clear that making war on the Thebans is bad. That B belongs to both C and D is clear, for both are cases of undertaking war against neighbors, and that A belongs to D is also clear, for the war against the Phocians did not turn out well for the Thebans; but that A belongs to B will be shown through D. The same holds even if the proof of the middle term's relation to the extreme should come about through several similar cases. It is clear, then, that the example

80| is related neither as part to whole nor as whole to part, but as part to part, when both fall under the same term and one of the two is already known. And it differs from induction in that induction shows the extreme belonging to the middle term on the basis of all the individual cases, and does not connect the syllogism to the extreme, whereas the example both makes that connection and does not show it on the basis of all cases.

81| Reduction occurs when it is clear that the first term belongs to the middle term, but it is unclear whether the middle term belongs to the last term, though this is equally credible, or more credible, than the conclusion; or again, when there are few middle terms between the last term and the middle term — for in every such case the result is that we come closer to knowledge. For example: let A be teachable, let B be knowledge, and let C be justice. That knowledge is teachable is evident; but whether virtue is knowledge is unclear. If then BC is as credible as AC, or more so, this is reduction; for we are closer to knowing, since we have gained the knowledge that A belongs to B, which we did not have before. Or again, if the middle terms between B and C are few; for in this way too we are closer to knowing. For example, let D be 'being squared', E be 'rectilinear figure', and F be 'circle'; if there is only one middle term between E and F — that the circle together with lunes becomes equal to a rectilinear figure — then we would be close to knowing. But when BC is neither more credible than AC, nor are the middle terms few, I do not call this reduction. Nor when BC is immediate; for that is already knowledge.

82| An objection is a premise contrary to a premise. It differs from a premise in that an objection can be particular, while a premise either cannot be particular at all, or not in universal syllogisms. An objection is brought in two ways and 69b through two figures: in two ways, because every objection is either universal or particular; through two figures, because objections are brought as opposites to the premise, and opposite conclusions are reached only in the first and third figures. For when someone claims that something belongs to every case, we object either that it belongs to none, or that it does not belong to some; and of these, 'to none' comes from the first figure, and 'not to some' from the last figure. For example, let A be 'there is a single knowledge',

83| and let B be 'contraries'. When someone puts forward that there is a single knowledge of contraries, we object either that in general there is not a single knowledge of opposites — and contraries are opposites, so that this becomes the first figure — or that there is not a single knowledge of the knowable and the unknowable; and this is the third, for of C, the knowable and the unknowable, it is true that they are contraries, but false that there is a single knowledge of them. Again, likewise with the privative premise. For when someone claims that there is not a single knowledge of contraries, we say either that there is a single knowledge of all opposites, or that there is a single knowledge of some contraries — for example, of the healthy and the diseased. The former comes from the first figure, the latter from the third. For in general, in all cases, one who objects universally must state the contradiction in relation to the universal of what is being asserted — for example, if someone denies that there is a single knowledge of contraries, by saying that there is a single knowledge of all opposites. In this way it is necessarily the first figure; for the universal becomes the middle term in relation to the original terms. But if the objection is particular, it is directed to that of which the premise, taken universally, is said — for example, saying that there is not a single knowledge of the knowable and the unknowable; for contraries are universal in relation to these.

84| And this becomes the third figure; for what is taken as particular becomes the middle term — for example, the knowable and the unknowable. For it is from the terms out of which we can draw the opposite syllogistic conclusion that we also try to state our objections. This is why we bring objections only from these figures; for it is only in these that opposed syllogisms occur, since through the middle figure there was no affirmative conclusion. Further, an objection through the middle figure would also require a longer argument — for example, if someone should refuse to grant that A belongs to B because C does not follow it; for this becomes clear only through other premises. But an objection ought not to turn aside to other matters, but should have its other premise immediately evident. We must also examine the other kinds of objections — those from the contrary, from the similar, and from reputable opinion — and whether it is possible to take a particular objection from the first figure, or a privative one from the middle 70a figure.

85| An enthymeme is a syllogism from probabilities or signs. A probability and a sign are not the same thing: a probability is a reputable premise; for what people know happens or does not happen, or is or is not, for the most part in a certain way — that is a probability, for example, that people who envy hate, or that those who are loved feel affection in return. A sign, by contrast, is meant to be a demonstrative premise, either necessary or reputable; for that whose presence entails a thing's existing, or whose occurrence has come to be either before or after the thing, is a sign of the thing's having occurred or existing. A sign is taken in three ways, the same number of ways the middle term is taken in the figures: either as in the first, as in the middle, or as in the third. For example, showing that a woman is pregnant because she has milk is from the first figure; for 'having milk' is the middle term. Let A be 'being pregnant', B 'having milk', and C 'woman'. That the wise are good, since Pittacus is good, is proved through the last figure. Let A be 'good', B 'the wise', and C 'Pittacus'. It is true, then, to predicate both A and B of C;

86| except that people do not state the one premise, because they know it, but they do assume the other. That a woman is pregnant, because she is pale, is meant to be through the middle figure; for since paleness follows on pregnant women, and it also follows on this woman, they think it has been shown that she is pregnant. Let 'pale' be A, 'being pregnant' be B, and 'woman' be C. Now if only the one premise is stated, the result is merely a sign; but if the other is also added, it is a syllogism — for example, that Pittacus is generous, since ambitious men are generous, and Pittacus is ambitious. Or again, that the wise are good, since Pittacus is good, but he is also wise. This, then, is how syllogisms come about; except that the one through the first figure is irrefutable, if it is true (for it is universal), while the one through the last figure

87| is refutable, even if the conclusion is true, because the syllogism is not universal, nor relevant to the matter; for it is not the case that, if Pittacus is good, the others must therefore also be wise. But the one through the middle figure is always and altogether refutable; for no syllogism ever results when the terms stand this way — for it is not the case that, if a pregnant woman is pale, and this woman too is pale, this woman must be pregnant. Truth, then, may hold in all these signs, but they have the differences stated. Either, then, the sign must be divided in this way, and of these the one through the middle term must be taken as a proof (for they say that a proof is what produces knowledge, and it is above all the one through the middle term that is such) — or else the ones from the extreme terms must be called sign, and the one from the middle term must be called proof; for the one through the first figure is the most reputable and the truest. Physiognomy is possible, if one grants that body and soul change together in all affections that are natural; for perhaps someone who has learned music has changed his soul in some respect, but this affection is not one of those that belong to us by nature — rather, what I mean are such things as fits of anger and appetites, which are among the natural motions. If then this is granted, and also that there is one sign for one affection, and we are able to grasp the affection and the sign proper to each kind, we will be able to practice physiognomy.

88| For if some affection belongs peculiarly to some one indivisible genus, as courage belongs to lions, there must also be some sign of it; for it is assumed that they are affected together. And let this be having large extremities — which can also belong to other genera, not as a whole. For a sign is peculiar in this sense, that it is peculiar to the whole genus, and not peculiar to that genus alone, as we are accustomed to say. This will then hold in another genus too, and there will be a courageous man and some other courageous animal as well. Each will then have the sign; for one thing was sign of one thing. If this is so, then, we shall also be able to collect such signs in the case of those animals which have only one affection that is peculiar to them, and each has a sign, since it must have one; and so we shall be able to practice physiognomy. But if the whole genus has two peculiar attributes, as the lion is both courageous

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