CHAPTER V.

OF THE REDUCTION OF SYLLOGISMS.

THE four Moods in the first Figure, as they are the clearest and most natural, are called perfect. The Moods of the other Figures are called imperfect, because Aristotle's dictum cannot be immediately applied to them. But, as it is on this dictum that all Reasoning ultimately depends, all the Moods of the other three Figures can be brought, in some way or other, into one of the four Moods of the first Figure. When a syllogism is thus operated upon, it is said to be reduced from an imperfect to a perfect Mood. This has given rise to the Reduction of Syllogisms; and any argument that cannot be so reduced as to be stated legitimately according to one of the four Moods of the first Figure is not valid.

In the Reduction of Syllogisms we are not allowed, of course, to change the terms, or introduce any new proposition. The premises being laid down, and their truth granted, all that is permitted is, that we so convert, or transpose, or otherwise operate on these premises, that they may become subject to the laws of the first Figure. This may be done in two ways, either by Ostensive Reduction, or by Reductio ad impossibile.

SECTION I.

Of Ostensive Reduction.

By Ostensive Reduction we prove in the first Figure, from the premises of the imperfect syllogism originally given, either the very same conclusion, or one that implies it, and from which it may be justly and easily deduced. The truth of any proposition implies the truth of its illative converse. We are, therefore, allowed to convert the major or the minor premise, by the methods of Conversion formerly explained; and, if necessary, to transpose the premises after they have thus been converted; in this way the imperfect Mood may be reduced to

one of the four perfect Moods of the first Figure. Take the following as examples :

Every virtue is praiseworthy :

Injustice is not praiseworthy; therefore,
Injustice is not a virtue.

This is a syllogism in Camestres of the second Figure, and it may be reduced to Clarent of the first, by simply converting the minor, and then transposing the premises; thus,

Again :

That which is praiseworthy is not injustice:

Every virtue is praiseworthy; therefore,

No injustice is a virtue.

All tyrants are cruel :

All tyrants are men; therefore,

Some men are cruel.

This is a syllogism in Darapti of the third Figure; but it may be reduced to Darii of the first, by converting the minor premise per accidens; thus,

Again:

All tyrants are cruel :

Some men are tyrants; therefore,
Some men are cruel.

Some slaves are not discontented :
All slaves are wronged; therefore,

Some who are wronged are not discontented.

This is a syllogism in Disamis of the third Figure, and it may be reduced to Darii of the first, by converting the major by contraposition, and then transposing the premises; thus,

All slaves are wronged:

Some who are not discontented are slaves; therefore,
Some who are not discontented are wronged.

In this case the conclusion is the converse by negation of the original conclusion, and therefore may be inferred from it. By these different methods all the imperfect Moods may be reduced to the four perfect Moods of the first Figure; and

this is called Ostensive Reduction, because either the same conclusion is proved, or one which implies it, and from which it may be justly inferred.

SECTION II.

Of Reductio ad impossibile.

By Reductio ad impossibile we prove, in the first Figure, not directly that the conclusion of the imperfect syllogism is true, but that it cannot be false; or, in other words, that an absurdity would follow on the supposition of its being false. The following will furnish an example :

All truly wise men live virtuously :

Some philosophers do not live virtuously; therefore,
Some philosophers are not truly wise men.

If this conclusion be not true, its contradictory must be true; viz.

All philosophers are truly wise men.

Make this proposition, then, the minor premise of the above syllogism, and a false conclusion will be proved; thus,

All truly wise men live virtuously:

All philosophers are truly wise men; therefore,
All philosophers live virtuously.

This conclusion is the contradictory of the original minor premise; it must therefore be false, because the premises are always supposed to be granted. If this conclusion is false, then one of the premises from which it has been correctly deduced must be false also; but the major premise, being one of those originally granted, must be true; the falsity must, therefore, be in the minor premise. But the minor premise is the contradictory of the original conclusion; hence the original conclusion must be true.

This kind of reduction is a very indirect and obscure mode of reasoning, and is seldom employed except for Baroco and

Bocardo. These two moods, however, can be reduced ostensively by contraposition.

The mnemonic lines formerly quoted are of great service in the Reduction of Syllogisms. The names given to the various Moods in the several Figures, although they may seem harsh and unmeaning, have been framed so as to point out the manner in which each of the imperfect Moods is to be reduced. The initial letters of all the Moods are B, C, D, F. The first letter in every imperfect Mood indicates that it is to be reduced to that Mood of the first Figure which begins with the same letter. If its initial letter be B, it must be reduced to Barbara; if it be C, to Celarent; if D, to Dari ; and if F, to Ferio. This rule has been expressed by the Logicians; thus,

Barbara demonstrat, B; Celarent, C, reducit:

D, redit ad Darii; F, redit ad Ferio.

Besides the initial letters, there are other consonants, found in the middle or end of these names which designate the different Moods, which are also made use of to indicate the kind of reduction that is to be employed. These letters are s, p, m, and c, and their meaning is as follows: s, shews that the proposition denoted by the vowel immediately preceding it, is to be converted simply; p, that the proposition denoted by the vowel immediately preceding it, is to be converted per accidens; and m, shews that the premises are to be transposed. Thus, in Bamarip, the B shews that it must be reduced to Barbara; the m, that the premises must be transposed; and the p, that the conclusion must be converted per accidens. So in Camestres, the C indicates that it must be reduced to Celarent; the m, shews that the premises must be transposed; and the two ss, shew that the minor premise and the conclusion must be converted simply. The other consonant c, points out the reductio ad impossibile. Wherever it occurs it shews that the proposition denoted by the vowel immediately before it must be left out, and the contradictory of the conclusion substituted in its place; consequently in Baroco the contradictory of the conclusion is to be substituted for the minor premise; and in Bocardo it is to be substituted for the major. These rules have been expressed thus,

S, vult simpliciter verti; P, vero per accid.
M, vult transponi; C, per impossibile duci.

CHAPTER VI.

OF HYPOTHETICAL SYLLOGISMS.

WE have hitherto been considering pure Categorical Syllogisms. It is often necessary, however, to introduce into reasoning various kinds of Conditional Propositions; and as the force of the argument sometimes turns on these hypothetical premises, it is necessary that we notice this class of syllogisms, and explain the rules whereby their validity may be ascertained.

A Hypothetical Proposition consists of two or more Categorical propositions, connected by a conjunction so as to make them one proposition. They are denominated, after their respective conjunctions, either Conditional or Disjunctive. Thus : "if A is B, then C is D," is a Conditional proposition; "either A is B, or C is D," is a Disjunctive proposition.

Sometimes a hypothetical conclusion is inferred from a hypothetical premise, while the reasoning process remains, properly speaking, purely categorical. In this case the force of the reasoning does not turn on the hypothesis; the condition expressed is considered as attached to one of the terms; and the reasoning proceeds, and is to be judged of, in the same way as if it were a categorical syllogism. For example :—

Every conqueror is either a hero or a villain :
Cæsar was a conqueror; therefore,

Cæsar was either a hero or a villain.

In this case if "either a hero or a villain" be considered as merely the predicate of the major premise, and the predicate of the conclusion, the syllogism may be considered merely categorical.

But when the reasoning rests on the hypothesis, and a hypo