According to the ancient theory, reduction is necessary as a matter of _final_ and _absolute proof_ that the conclusion follows from the given premises. But, as this claim has been satisfactorily refuted by modern logicians, we need not give more space to the process. The meaning of k, as related to “_indirect reduction_,” is explained in most of the earlier works on logic. See Hyslop, page 193. =7. RELATIVE VALUE OF THE FOUR FIGURES.= _The first figure._ The first figure is known as the _perfect_ figure; because it is the only one which proves _all_ of the four logical propositions. Recalling the moods of the first figure makes this evident: A E A E A A I I _A_ _E_ _I_ _O_ It is likewise the more natural figure; because it is the only one which uses both the subject and predicate of the conclusion in the same relative places as they appear in the premises. Symbolizing the figure makes this apparent: M ― _G_ _S_ ― M _S_ ― _G_ The first figure, being the only figure which proves a “universal affirmation” (A), is used most by the _scientist_; as the object of science is to establish _universal affirmative_ truths. _The second figure._ As the second figure conditions negative conclusions only, it is called the figure of _disproof_, or the exclusive figure. It is easy to see how negative conclusions may be used to narrow the inquiry down to one definite theory. For example, suppose it is desired to ascertain which boy of the five broke the window; by a series of deductions the teacher may be able to prove that the culprit is not A, not B, not C and not D; hence the guilty one must be E. This figure is virtually the one used in diagnosing most diseases. _The third figure._ The third figure admits of particular conclusions only, and in consequence is of little value to the scientist. Since, however, the easiest way to contradict a universal affirmative (A) or a universal negative (E), is to prove the truth, respectively, of a particular negative (O) and a particular affirmative (I), it follows that the third figure serves a purpose. _The fourth figure._ This figure is so nearly like the first that it is of little value; in fact, it may be changed to the first by simply interchanging the major and minor premises. Some authorities refuse to recognize the fourth figure. =8. OUTLINE.= FIGURES AND MOODS OF THE SYLLOGISM. (1) The four figures of the syllogism. Definition――symbolization. Illustrations――device for remembering. (2) The moods of the syllogism. _Twenty-four_ valid. (3) Testing the validity of the moods. Application of the general rules of the syllogism. Weakened conclusion――five. _Nineteen_ useful moods. A thought exercise.