To reduce this, convert the major premise by negation, and then transpose the premises. We have:

Therefore

All Y's are Z's,

Some not-X's are Y's;

Some not-X's are Z's.

This conclusion is the converse by negation of the former conclusion, the truth of which is thus proved by reduction to a syllogism in Darii.

Both these moods, Baroko and Bokardo, may however be proved by a peculiar process of indirect reduction, closely analogous to the indirect proofs often employed by Euclid in Geometry. This process consists in supposing the conclusion of the syllogism to be false, and its contradictory therefore true, when a new syllogism can easily be constructed which leads to a conclusion contradictory of one of the original premises. Now it is absurd in logic to call in question the truth of our own premises, for the very purpose of argument or syllogism is to deduce a conclusion which will be true when the premises are true. The syllogism enables us to restate in a new form the information which is contained in the premises, just as a machine may deliver to us in a new form the material which is put into it. The machine, or rather the maker of the machine, is not responsible for the quality of the materials furnished to it, and similarly the logician is not responsible in the least for the truth of his premises, but only for their correct treatment. He must treat them, if he treat them at all, as true; and therefore a conclusion which requires the falsity of one of our premises is altogether absurd.

To apply this method we may take Baroko, as before:

If this conclusion be not true then its contradictory, 'all Z's are X's' must of necessity be regarded as true (pp. 76—79).· Making this the minor premise of a new syllogism with the original major premise we have:

Hence

All X's are Y's

(1)

All Z's are X's.........contradictory of (3)

All Z's are Y's.

Now this conclusion in A, is the contradictory of our old minor premise in O, and we must either admit one of our own premises to be false or allow that our original conclusion is true. The latter is of course the alternative we choose.

We treat Bokardo in a very similar manner;

If this conclusion be not true then 'all Z's are X's' must

be true. Now we can make the syllogism:

All Z's are X's.......... Contradictory of (3)
All Y's are Z's

Hence All Y's are X's.

(2)

This conclusion is the contradictory of (1), the original major premise, and as this cannot be allowed, we must either suppose (2) the original minor premise to be false, which is equally impossible, or allow that our original conclusion is true.

It will be observed that in both these cases of Indirect Reduction or Proof we use a syllogism in Barbara, which fact is indicated by the initial letters of Baroko and Bokardo. The same process of Indirect proof may be applied to any of the other moods, but it is not usual to do so, as the simpler process of direct or as it is often called ostensive reduction is sufficient.

It will be remembered that when in Lesson XV. (p. 135) we considered the rules of the syllogism, there were two supplementary rules, the 7th and 8th, concerning particular premises, which were by no means of a self-evident character, and which require to be proved by the six more fundamental rules. We have now sufficiently advanced to consider this proof with advantage. The 7th rule forbids us to draw any conclusion from two particular premises; now such premises must be either II, 10, 01, or 00. Of these II contain no distributed term at all, so that the 3rd rule, which requires the middle term to be distributed, must be broken. The premises 00 evidently break the 5th rule, against negative premises. The conclusion of the pair 10 must be negative by the 6th rule, because one premise is negative; the major term therefore will be distributed, but as the major premise is a particular affirmative it cannot be distributed without committing the fallacy of illicit process of the major, against rule 4. Lastly the premises OI contain only one distributed term, the predicate of the major premise. But as the conclusion must be negative by rule 6th, the major term must be distributed: we ought to have then in the premises two distributed terms, one for the middle term, the other for the major term; but as the premises contain only a single distributed term, we must commit the fallacy either of undistributed middle or of illicit process of the major term, if we attempt to draw any conclusion at all. We thus see that in no possible case can a pair of particular premises give a valid conclusion.

The 8th rule of the syllogism instructs us that if one premise of a syllogism be particular the conclusion must also be particular. It can only be shown to be true by going over all the possible cases and observing that the six principal rules of the syllogism always require the conclusion to be particular. Suppose for instance the

premises are A and I; then they contain only one distributed term, the subject of A, and this is required for the middle term by rule 3. Hence the minor term cannot be distributed without breaking rule 4, so that the conclusion must be the proposition I. The premises AO would contain two distributed terms, the subject of A and the predicate of 0; but if we were to draw from them the conclusion E, the major and minor terms would require to be distributed, so that the middle term would remain undistributed against rule 3. The reader can easily prove the other cases such as EI by calculating the number of distributed terms in a similar manner: it will always be found that there are insufficient terms distributed in the premises to allow of a universal conclusion.

LESSON XVIII.

IRREGULAR AND COMPOUND SYLLOGISMS.

IT may seem surprising that arguments which are met with in books or conversation are seldom or never thrown into the form of regular syllogisms. Even if a complete syllogism be sometimes met with, it is generally employed in mere affectation of logical precision. In former centuries it was, indeed, the practice for all students at the Universities to take part in public disputations, during which elaborate syllogistic arguments were put forward by one side and confuted by precise syllogisms on the other side. This practice has not been very long discontinued at the University of Oxford, and is said to be still maintained in some continental Universities; but except in such school disputations it must be allowed that perfectly formal syllogism.s are seldom employed,

In truth, however, it is not syllogistic arguments which are wanting; wherever any one of the conjunctions, therefore, because, for, since, hence, inasmuch as, consequently occurs, it is certain that an inference is being drawn, and this will very probably be done by a true syllogism. It is merely the complete statement of the premises and conclusion, which is usually neglected because the reader is generally aware of one or other of the premises, or he can readily divine what is assumed; and it is tedious and even offensive to state at full length what the reader is already aware of. Thus, if I say "atmospheric air must have weight because it is a material substance," I certainly employ a syllogism; but I think it quite needless to state the premise, of which I clearly assume the truth, that "whatever is a material substance has weight." The conclusion of the syllogism is the first proposition, viz. "atmospheric air has weight." The middle term is "material substance," which does not occur in the conclusion; the minor is "atmospheric air," and the major, "having weight." The complete syllogism is evidently:

All material substances have weight,

Atmospheric air is a material substance;
Therefore atmospheric air has weight.

This is in the very common and useful mood Barbara.

A syllogism when incompletely stated is usually called an enthymeme, and this name is often supposed to be derived from two Greek words (èv, in, and Ovμós, mind), so as to signify that some knowledge is held by the mind and is supplied in the form of a tacit, that is a silent or understood premise. Most commonly this will be the major premise, and then the enthymeme may be said to be of the First Order. Less commonly the minor premise is unexpressed, and the enthymeme is of the Second