odds become equal to evens if it is assumed commensurate. The equality of the odds and evens, then, we conclude by a syllogism; but the incommensurability of the diameter we prove by means of a hypothesis, since a falsehood follows through the contradictory (of the conclusion). For, to conclude per impossibile was to prove some impossibility through the original hypothesis. So that, since a demonstrative syllogism is formed with a false conclusion in inferences which are reduced to the impossible-and since the required conclusion is proved by means of a hypothesis--and since we said above that demonstrative syllogisms are inferred by these figures, it is obvious that the syllogisms per impossibile, will also be through these figures. And in like manner all the others which proceed from a hypothesis; for in all of them the syllogism is formed in reference to what is assumed different (from the original premisses), but the required conclusion is inferred by concession, or some other kind of hypothesis. But if this be true, every demonstration and every syllogism must be formed by the aforesaid figures, three in number. And if this be proved, it is evident that every syllogism is both perfected by the first figure, and is reduced to the universal syllogisms in that figure.

.. n2 must be an even number (for since m is an even

number, ()mu
'must be an integer);

.. n is even.

But it was proved to be odd: which is impossible.

Prior Analytics I. 24, § 1.

And further, in every syllogism, one of the terms 51. must be affirmative, and there must be an universal predication ; for, without the universality, either there will be no syllogism, or not with reference to the proposed subject, or else the question will be begged. For let it be required to prove that the pleasure arising from music is good. If one were to claim as a premiss that 'pleasure is good,' without adding 'all,' there will be no syllogism; but if one means some pleasure, if it be a different pleasure, that has nothing to do with the proposed conclusion; and if one means this particular pleasure, one is begging the question.

Prior Analytics I. 25, § 1.

And it is clear also that every demonstration will be by three terms, and not more, unless the same proof be made through separate sets of premisses; as, if E be proved through A and B, as well as through C and D, or through A and B and through B and C. For nothing prevents there being several middles to the same terms. But when there are these, the syllogisms are not one but several.

For these rules vid. Jevons' Elementary Logic, p. 128, &c. It is clear that if both premisses are negative—that is, if both major and minor term are said to disagree in some way with the middle-we can make no inference whatever as to the agreement or disagreement of the two extremes. If both premisses are particular, there must be an undistributed middle or an illicit process of the major.

52.

53.

Prior Analytics I. 25, § 8.

1

Now, since this is manifest, it is clear that they are also formed from two propositions, and not more; for the three terms make two propositions, unless something be assumed in addition-as was said at the outset for the perfecting of the syllogisms.

2

''This' refers to a principle laid down in the preceding section (omitted by Magrath), 'that every demonstration and every syllogism will be by three terms only.'

2 For instance, in reducing a second- or third-figure syllogism, per impossibile, an additional premiss, the contradictory of one of the original premisses, is assumed πρὸς τὴν τελείωσιν τοῦ συλλογισμοῦ.

27

PRIOR ANALYTICS II.

Prior Analytics II. 2, § 1.

The propositions then of which the syllogism is formed may be either true or false, or one may be true and the other false; whilst the conclusion is necessarily either true or false. From true premisses, then, one cannot draw a false conclusion; but from false premisses one may draw a true conclusion, not, however, so as to show the reason, but only the fact; for there is no syllogism of the reason arising from false premisses.1

Prior Analytics II. 4, §§ 13, 14.

It is then manifest that if the conclusion be false, all or some of the elements of which the argument is 1 That is to say, in a syllogism from false premisses it is impossible to show the true reason of the conclusion. The true conclusion may be obtained from premisses, either one or both false :Thus: All animals are quadrupeds,

Or,

All horses are animals;

.. All horses are quadrupeds.

All birds are quadrupeds,

All horses are birds,

.. All horses are quadrupeds.

It is obvious in these instances that though the conclusion is true in both syllogisms, they give no indication of the reason why it is true. Perhaps there is no logical truth which is less recognised in practice than this: that to disprove the premisses is not to disprove the conclusion.

61.

62.

63.

formed must be false; but when the conclusion is true that they need not, either some or all, be true. But it is possible, without any of the elements of the syllogism being true, for the conclusion to be true none the less, though it is not necessarily so. The reason is that when two things are so related, that if the first is, the second must also necessarily be; if the second is not, neither will the first be: but if the second is, the first need not necessarily be.1

Prior Analytics II. 15, § 1.

Now, I say that in language the kinds of opposed propositions are four: (a) all to none; (B) all to not all; (y) some to none; (8) some to not some; but that in reality there are only three; for that of 'some to not some is a merely verbal opposition. Of these, I call the universals contrary (i.e. all to none); for instance, 'All knowledge is good' is contrary to 'No knowledge is good;' the others are (contradictorily) opposed.2

1 If B is a necessary consequence of A, it is evident that if B is not true, A cannot be so either; since if A were true, B would be a necessary consequence of it. But if B is true, it does not follow that A is true also; for B may be true but not in consequence of A. e.g. in the inferences :

(1) No men are animals .. no animals are men ; (2) All men are animals .. some animals are men ; All animals are men.. some men are animals; if the consequent be false, the antecedent must be false: but if the consequent be true, the antecedent may be either true or false.

2 For åvtikeiμévas vide note 2, page 10. Aristotle reckons the opposition of I and O merely verbal, because they may both be true together, although in Pr. Anal. II. 8. 2 he calls them contrary. 'Subaltern opposition' he does not recognise at all. The three kinds of opposition, then, that he admits as real are those between A and E, A and O, and E and I.