Logic: Deductive and Inductive Part 13

For else neither premise will distribute the middle term. This rule eliminates A.A.A., A.A.I., A.I.I., I.A.I.

(b) _The major premise must be universal._

For else, the conclusion being negative, there will be illicit process of the major term. This eliminates I.A.I., O.A.O.; leaving six Moods, including two subalterns.

Fig. III. (a) _The minor premise must be affirmative._

For else, in negative moods there will be illicit process of the major term. This rule eliminates A.E.E., A.E.O., A.O.O.

(b) _The conclusion must be particular._

For, if not, the minor premise being affirmative, there will be illicit process of the minor term. This eliminates A.A.A., A.E.E., E.A.E.; leaving six Moods.

Fig. IV. (a) _When the major premise is affirmative, the minor must be universal._

For else the middle term is undistributed. This eliminates A.I.I., A.O.O.

(b) _When the minor premise is affirmative the conclusion must be particular._

Otherwise there will be illicit process of the minor term. This eliminates A.A.A., E.A.E.

(c) _When either premise is negative, the major must be universal._

For else, the conclusion being negative, there will be illicit process of the major term. This eliminates O.A.O.; leaving six Moods, including one subaltern.

-- 6. Reduction is either--(1) Ostensive or (2) Indirect. Ostensive Reduction consists in showing that an argument given in one Mood can also be stated in another; the process is especially used to show that the Moods of the second, third, and fourth Figures are equivalent to one or another Mood of the first Figure. It thus proves the validity of the former Moods by showing that they also essentially conform to the _Dictum_, and that all Categorical Syllogisms are only superficial varieties of one type of proof.

To facilitate Reduction, the recognised Moods have all had names given them; which names, again, have been strung together into mnemonic verses of great force and pregnancy:

Barbara, Celarent, Darii, Ferioque prioris: Cesare, Camestres, Festino, Baroco, secundae: Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet: Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison.

In the above verses the names of the Moods of Fig. I. begin with the first four consonants B, C, D, F, in alphabetical order; and the names of all other Moods likewise begin with these letters, thus signifying (except in Baroco and Bocardo) the mood of Fig. I., to which each is equivalent, and to which it is to be reduced: as Bramantip to Barbara, Camestres to Celarent, and so forth.

The vowels A, E, I, O, occurring in the several names, give the quant.i.ty and quality of major premise, minor premise, and conclusion in the usual order.

The consonants s and p, occurring after a vowel, show that the proposition which the vowel stands for is to be converted either (s) simply or (p) _per accidens_; except where s or p occurs after the third vowel of a name, the conclusion: then it refers not to the conclusion of the given Mood (say Disamis), but to the conclusion of that Mood of the first Figure to which the given Mood is reduced (Darii).

M (_mutare_, metathesis) means 'transpose the premises' (as of Ca_m_estres).

C means 'subst.i.tute the contradictory of the conclusion for the foregoing premise,' a process of the Indirect Reduction to be presently explained (see Baroco, -- 8).

The other consonants, r, n, t (with b and d, when not initial), occurring here and there, have no mnemonic significance.

What now is the problem of Reduction? The difference of Figures depends upon the position of the Middle Term. To reduce a Mood of any other Figure to the form of the First, then, we must so manipulate its premises that the Middle Term shall be subject of the major premise and predicate of the minor premise.

Now in Fig. II. the Middle Term is predicate of both premises; so that the minor premise may need no alteration, and to convert the major premise may suffice. This is the case with Cesare, which reduces to Celarent by simply converting the major premise; and with Festino, which by the same process becomes Ferio. In Camestres, however, the minor premise is negative; and, as this is impossible in Fig. I., the premises must be transposed, and the new major premise must be simply converted: then, since the transposition of the premises will have transposed the terms of the conclusion (according to the usual reading of syllogisms), the new conclusion must be simply converted in order to prove the validity of the original conclusion. The process may be thus represented (_s.c._ meaning 'simply convert')

Camestres. Celarent.

All P is M; ---- /---> No M is S; c/ / s/ / No S is M: ----/ ---> All P is M:

s.c.

? No S is P. <----------- no="" p="" is="">

The Ostensive Reduction of Baroco also needs special explanation; for as it used to be reduced indirectly, its name gives no indication of the ostensive process. To reduce it ostensively let us call it Faksnoko, where k means 'obvert the foregoing premise.' By thus obverting (k) and simply converting (s) (in sum, contrapositing) the major premise, and obverting the minor premise, we get a syllogism in Ferio, thus:

Baroco or Faksnoko. Ferio.

_contrap_ All P is M; -----------------------> No m (not-M) is P;

_obv_ Some S is not M: -----------------------> Some S is m (not-M): ? Some S is not P. ? Some S is not P.

In Fig. III. the middle term is subject of both premises; so that, to reduce its Moods to the First Figure, it may be enough to convert the minor premise. This is the case with Darapti, Datisi, Felapton, and Ferison. But, with Disamis, since the major premise must in the First Figure be universal, we must transpose the premises, and then simply convert the new minor premise; and, lastly, since the major and minor terms have now changed places, we must simply convert the new conclusion in order to verify the old one. Thus:

Disamis. Darii.

Some M is P; ---- /---> All M is S; s./ / /c.

/ All M is S: ----/ ---> Some P is M:

s.c.

? Some S is P. <------------- some="" p="" is="">

Bocardo, like Baroco, indicates by its name the indirect process. To reduce it ostensively let its name be Doksamrosk, and proceed thus:

Bocardo or Doksamrosk. Darii.

Some M is not P; ---------- /---------> All M is S; / / / _contrap_ / All M is S: ----------/ ---------> Some p (not-P) is M:

_convert & obvert_ ? Some S is not P. <------------------------- some="" p="" (not-p)="" is="">

In Fig. IV. the position of the middle term is, in both premises, the reverse of what it is in the First Figure; we may therefore reduce its Moods either by transposing the premises, as with Bramantip, Camenes, and Dimaris; or by converting both premises, the course pursued with Fesapo and Fresison. It may suffice to ill.u.s.trate by the case of Bramantip:

Bramantip. Barbara.

All P is M; ---------- /------> All M is S; / / All M is S: ----------/ ------> All P is M:

convert per acc.

? Some S is P. <----------------------- all="" p="" is="">

This case shows that a final significant consonant (s, p, or sk) in the name of any Mood refers to the conclusion of the new syllogism in the First Figure; since p in Bramantip cannot refer to that Mood's own conclusion in I.; which, being already particular, cannot be converted _per accidens_.

Finally, in Fig. I., Darii and Ferio differ respectively from Barbara and Celarent only in this, that their minor premises, and consequently their conclusions, are subaltern to the corresponding propositions of the universal Moods; a difference which seems insufficient to give them rank as distinct forms of demonstration. And as for Barbara and Celarent, they are easily reducible to one another by obverting their major premises and the new conclusions, thus:

Barbara. Celarent.

obv.

All M is P; -----------------------> No M is p (not-P);

All S is M: -----------------------> All S is M:

obv.