Deductive Logic Part 32

All C is B.

.'. Undistributed Middle.

-- 661. Thus we are left with six valid moods, which yield four direct conclusions and five indirect ones, corresponding to the four moods of the original first figure and the five moods of the original fourth, which appear now as indirect moods of the first figure.

-- 662. But why, it maybe asked, should not the moods of the first figure equally well be regarded as indirect moods of the fourth? For this reason-that all the moods of the fourth figure can be elicited out of premisses in which the terms stand in the order of the first, whereas the converse is not the case. If, while retaining the quant.i.ty and quality of the above premisses, i. e. the mood, we were in each case to transpose the terms, we should find that we were left with five valid moods instead of six, since AI in the reverse order of the terms involves undistributed middle; and, though we should have Celarent indirect to Camenes, and Darii to Dimaris, we should never arrive at the conclusion of Barbara or have anything exactly equivalent to Ferio. In place of Barbara, Bramantip would yield as an indirect mood only the subaltern AAI in the first figure. Both Fesapo and Fresison would result in an illicit process, if we attempted to extract the conclusion of Ferio from them as an indirect mood. The nearest approach we could make to Ferio would be the mood EAO in the first figure, which may be elicited indirectly from the premisses of CAMENES, being subaltern to CELARENT. For these reasons the moods of the fourth figure are rightly to be regarded as indirect moods of the first, and not vice versa.

$663. FIGURE II.

_Mood AA._ All A is B.

All C is B.

.'. Undistributed Middle.

_Mood AE._ All A is B.

No C is B.

.'. No C is A, or No A is C, (Camestres & Cesare).

_Mood AI._ All A is B.

Some C is B.

.'. Undistributed Middle.

_Mood AO._ All A is B.

Some C is not B.

.'. Some C is not A, (Baroko), or Illicit Process.

_Mood EA._ No A is B.

All C is B.

.'. No C is A, or No A is C, (Cesare & Carnestres).

_Mood EI_ No A is B.

Some C is B.

.'. Some C is not A, (Festino), or Illicit Process.

_Mood IA._ Some A is B.

All C is B.

.'. Undistributed Middle.

_Mood IE._ Some A is B.

No C is B.

.'. Illicit Process, or Some A is not C, (Festino).

_Mood OA._ Some A is not B.

All C is B.

.'. Illicit Process, or Some A is not C, (Baroko).

-- 664. Here again we have six valid moods, which yield four direct conclusions corresponding to Cesare, CARNESTRES, FESTINO and BAROKO. The same four are repeated in the indirect moods.

-- 665. FIGURE III.

_Mood AA._ All B is A.

All B is C.

.'. Some C is A, or Some A is C, (Darapti).

_Mood AE._ All B is A.

No B is C.

.'. Illicit Process, or Some A is not C, (Felapton).

_Mood AI._ All B is A, Some B is C.

.'. Some C is A, or Some A is C, (Datisi & Disamis).

_Mood AO._ All B is A.

Some B is not C.

.'. Illicit Process, Or Some A is not C, (Bokardo).

_Mood EA._ No B is A.

All B is C.

.'. Some C is not A, (Felapton), or Illicit Process.

_Mood EI._ No B is A.

Some B is C.

.'. Some C is not A, (Ferison), or Illicit Process.

_Mood IA._ Some B is A.

All B is C.

.'. Some C is A, Or Some A is C, (Disamis & Datisi).

_Mood IE._ Some B is A.

No B is C.

.'. Illicit Process, or Some A is not C, (Ferison).

_Mood QA._ Some B is not A.

All B is C.

.'. Some C is not A, (Bokardo), or Illicit Process.

-- 666. In this figure every mood is valid, either directly or indirectly. We have six direct moods, answering to Darapti, Disamis, Datisi, Felapton, Bokardo and Ferison, which are simply repeated by the indirect moods, except in the case of Darapti, which yields a conclusion not provided for in the mnemonic lines. Darapti, though going under one name, has as much right to be considered two moods as Disamis and Datisi.

CHAPTER XVIII.

_Of Reduction._

-- 667. We revert now to the standpoint of the old logicians, who regarded the Dictum de Omni et Nullo as the principle of all syllogistic reasoning. From this point of view the essence of mediate inference consists in showing that a special case, or cla.s.s of cases, comes under a general rule. But a great deal of our ordinary reasoning does not conform to this type. It was therefore judged necessary to show that it might by a little manipulation be brought into conformity with it. This process is called Reduction.

-- 668. Reduction is of two kinds--