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 mode and figure.
Although 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 premisses. Thus an argument in the first mode 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 mode of the first figure. Again, an argument in the first mode of the third figure must resemble the following:--
All B is C All B is A therefore Some A is C,
where the minor premiss, 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 mode of the first figure:--
All B is C Some A is B, from which it obviously follows, that Some A is C.
In the same manner, or in a manner on which after these examples it is not necessary to enlarge, every mode of the second, third, and fourth figures may be reduced to some one of the four modes of the first. In other words, every conclusion which can be proved in any of the last three figures, may be proved in the first figure from the same premisses, with a slight alteration in the mere manner of expressing them. Every valid ratiocination, therefore, may be stated in the first figure, that is, in one of the following forms:--