are arguments precisely similar, and are both ranked in the first mood of the first figure.
The reasons why syllogisms in any of the above forms are legitimate, that is, why, if the premises are true, the conclusion must inevitably be so, and why this is not the case in any other possible mood, (that is, in any other combination of universal and particular, affirmative and negative propositions,) any person taking interest in these inquiries may be presumed to have either learned from the common school books of the syllogistic logic, or to be capable of discovering for himself. The reader may, however, be referred, for every needful explanation, to Archbishop Whately's Elements of Logic, where he will find stated with philosophical precision, and explained with remarkable perspicuity, the whole of the common doctrine of the syllogism.
All valid ratiocination; all reasoning by which, from general propositions previously admitted, other propositions equally or less general are inferred; may be exhibited in some of the above forms. The whole of Euclid, for example, might be thrown without difficulty into a series of syllogisms, regular in mood and figure.
Though a syllogism framed according to any of these formulæ is a valid argument, all correct ratiocination admits of being stated in syllogisms of the first figure alone. The rules for throwing an argument in any of the other figures into the first figure, are called rules for the reduction of syllogisms. It is done by the conversion of one or other, or both, of the premises. Thus an argument in the first mood of the second figure, as—
| No C is B |
| All A is B |
| therefore |
| No A is C, |
may be reduced as follows. The proposition, No C is B, being an universal negative, admits of simple conversion, and may be changed into No B is C, which, as we showed, is the very same assertion in other words—the same fact differently expressed. This transformation having been effected, the argument assumes the following form:—
| No B is C |
| All A is B |
| therefore |
| No A is C, |
which is a good syllogism in the second mood of the first figure. Again, an argument in the first mood of the third figure must resemble the following:—
| All B is C |
| All B is A |
| therefore |
| Some A is C, |
where the minor premise, All B is A, conformably to what was laid down in the last chapter respecting universal affirmatives, does not admit of simple conversion, but may be converted per accidens, thus, Some A is B; which, though it does not express the whole of what is asserted in the proposition All B is A, expresses, as was formerly shown, part of it, and must therefore be true if the whole is true. We have, then, as the result of the reduction, the following syllogism in the third mood of the first figure:—