§ 668. Reduction is of two kinds—
(1) Direct or Ostensive.
(2) Indirect or Ad Impossibile.
§ 669. The problem of direct, or ostensive, reduction is this—
Given any mood in one of the imperfect figures (II, III and IV) how to alter the form of the premisses so as to arrive at the same conclusion in the perfect figure, or at one from which it can be immediately inferred. The alteration of the premisses is effected by means of immediate inference and, where necessary, of transposition.
§ 670. The problem of indirect reduction, or reductio (per deductionem) ad impossibile, is this—Given any mood in one of the imperfect figures, to show by means of a syllogism in the perfect figure that its conclusion cannot be false.
§ 671. The object of reduction is to extend the sanction of the Dictum de Omni et Nullo to the imperfect figures, which do not obviously conform to it.
§ 672. The mood required to be reduced is called the Reducend; that to which it conforms, when reduced, is called the Reduct.
Direct or Ostensive Reduction.
§ 673. In the ordinary form of direct reduction, the only kind of immediate inference employed is conversion, either simple or by limitation; but the aid of permutation and of conversion by negation and by contraposition may also be resorted to.