_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 class 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--
(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.
§ 674. There are two moods, Baroko and Bokardo, which cannot be
reduced ostensively except by the employment of some of the means last
mentioned. Accordingly, before the introduction of permutation into
the scheme of logic, it was necessary to have recourse to some other
expedient, in order to demonstrate the validity of these two
moods. Indirect reduction was therefore devised with a special view to
the requirements of Baroko and Bokardo: but the method, as will be
seen, is equally applicable to all the moods of the imperfect figures.
§ 675. The mnemonic lines, 'Barbara, Celarent, etc., provide complete
directions for the ostensive reduction of all the moods of the second,
third, and fourth figures to the first, with the exception of Baroko
and Bokardo. The application of them is a mere mechanical trick, which
will best be learned by seeing the process performed.
§ 676. Let it be understood that the initial consonant of each name of
a figured mood indicates that the reduct will be that mood which
begins with the same letter. Thus the B of Bramantip indicates that
Bramantip, when reduced, will become Barbara.
§ 677. Where m appears in the name of a reducend, me shall have to
take as major that premiss which before was minor, and vice versa-in
other words, to transpose the premisses, m stands for mutatio or
metathesis.
§ 678. s, when it follows one of the premisses of a reducend,
indicates that the premiss in question must be simply converted; when
it follows the conclusion, as in Disamis, it indicates that the
conclusion arrived at in the first figure is not identical in form
with the original conclusion, but capable of being inferred from it by
simple conversion. Hence s in the middle of a name indicates something
to be done to the original premiss, while s at the end indicates
something to be done to the new conclusion.
§ 679. P indicates conversion per accidens, and what has just been
said of s applies, mutatis mutandis, to p.
§ 680. k may be taken for the present to indicate that Baroko and
Bokardo cannot be reduced ostensively.
§ 681. FIGURE II.
Cesare. \ / Celarent.
No A is B. \ = / No B is A.
All C is B. / \ All C is B.
.'. No C is A. / \ .'. No C is A.
Camestres. \ / Celarent.
All A is B. \ = / No B is C.
No C is B. / \ All A is B.
.'. No C is A. / \ .'. No A is C.
.'. No C is A.
Festino. Ferio.
No A is B. \ / No B is A.
Some C is B. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
[Baroko]
§ 682. FIGURE III.
Darapti. \ / Darii.
All B is A. \ = / All B is A.
All B is C. / \ Some C is B.
.'. Some C is A. / \ Some C is A.
Disamis. \ / Darii.
Some B is A. \ = / All B is C.
All B is C. / \ Some A is B.
.'. Some C is A. / \ .'. Some A is C.
.'. Some C is A.
Datisi. \ / Darii.
All B is A. \ = / All B is A.
Some B is C. / \ Some C is B.
.'. Some C is A. / \ .'. Some C is A.
Felapton. \ / Ferio.
No B is A. \ = / No B is A.
All B is C. / \ Some C is B.
.'. Some C is not-A. / \ .'. Some C is not-A.
[Bokardo].
Ferison. \ / Ferio.
No B is A. \ = / No B is A.
Some B is C. / \ Some C is B
.'. Some C is not A. / \ .'. Some C is not A.
§ 683. FIGURE IV.
Bramantip. \ / Barbara.
All A is B. \ = / All B is C.
All B is C. / \ All A is B.
.. Some C is A. / \ .. All A is C.
.'. Some C is A.
Camenes Celarent
All A is B \ / No B is C.
No B is C. | = | All A is B.
.. No C is A./ \ .'. No A is C.
.'. No C is A.
Dimaris. Darii.
Some A is B. \ / All B is C.
All B is C. | = | Some A is B.
.'. Some C is A./ \ .'. Some A is C.
.'. Some C is A.
Fesapo. Ferio.
No A is B. \ / No B is A.
All B is C. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
Fresison. Ferio.
No A is B. \ / No B is A.
Some B is C. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not A.
§ 684. The reason why Baroko and Bokardo cannot be reduced ostensively
by the aid of mere conversion becomes plain on an inspection of
them. In both it is necessary, if we are to obtain the first figure,
that the position of the middle term should be changed in one
premiss. But the premisses of both consist of A and 0 propositions, of
which A admits only of conversion by limitation, the effect of which
would be to produce two particular premisses, while 0 does not admit
of conversion at all,
It is clear then that the 0 proposition must cease to be 0 before we
can get any further. Here permutation comes to our aid; while
conversion by negation enables us to convert the A proposition,
without loss of quantity, and to elicit the precise conclusion we
require out of the reduct of Boltardo.
(Baroko) Fanoao. Ferio.
All A is B. \ / No not-B is A.
Some C is not-B. | = | Some C is not-B.
.'. Some C is not-A./ \ .'. Some C is not-A.
(Bokardo) Donamon. Darii.
Some B is not-A. \ / All B is C.
All B is C. | = | Some not-A is B
.'. Some C is not-A./ \ .'. Some not-A is C.
.'. Some C is not-A.
§ 685. In the new symbols, Fanoao and Donamon, [pi] has been
adopted as a symbol for permutation; n signifies conversion by
negation. In Donamon the first n stands for a process which resolves
itself into permutation followed by simple conversion, the second for
one which resolves itself into simple conversion followed by
permutation, according to the extended meaning which we have given to
the term 'conversion by negation.' If it be thought desirable to
distinguish these two processes, the ugly symbol Do[pi]samos[pi] may
be adopted in place of Donamon.
§ 686. The foregoing method, which may be called Reduction by
Negation, is no less applicable to the other moods of the second
figure than to Baroko. The symbols which result from providing for its
application would make the second of the mnemonic lines run thus--
Benare[pi], Cane[pi]e, Denilo[pi], Fano[pi]o secundae.
§ 687. The only other combination of mood and figure in which it will
be found available is Camenes, whose name it changes to Canene.
§ 688.
(Cesare) Benarea. Barbara.
No A is B. \ / All B is not-A.
All C is B. | = | All C is B.
.'. No C is A. / \ .'. All C is not-A.
.'. No C is A.
(Camestres) Cane[pi]e. Celarent.
All A is B. \ / No not-B is A.
No C is B. | = | All C is not-B.
.'. No C is A. / \ .'. No C is A.
(Festino) Denilo[pi]. Darii.
No A is B. \ / All B is not-A.
Some C is B. | = | Some C is B.
.'. Some C is not A./ \ .'. Some C is not-A.
.'. Some C is not A.
(Camenes) Canene. Celarent.
All A is B. \ / No not-B is A.
No B is C. | = | All C is not-B.
.'. No C is A. / \ .'. No C is A.
§ 689. The following will serve as a concrete instance of Cane[pi]e
reduced to the first figure.
All things of which we have a perfect idea are perceptions.
A substance is not a perception.
.'. A substance is not a thing of which we have a perfect idea.
When brought into Celarent this becomes--
No not-perception is a thing of which we have a perfect idea.
A substance is a not-perception.
.'. No substance is a thing of which we have a perfect idea.
§ 690. We may also bring it, if we please, into Barbara, by permuting
the major premiss once more, so as to obtain the contrapositive of the
original--
All not-perceptions are things of which we have an imperfect idea.
All substances are not-perceptions.
.'. All substances are things of which we have an imperfect idea.
_Indirect Reduction._
§ 691. We will apply this method to Baroko.
All A is B. All fishes are oviparous.
Some C is not B. Some marine animals are not oviparous.
.'. Some C is not A. .'. Some marine animals are not fishes.
§ 692. The reasoning in such a syllogism is evidently conclusive: but
it does not conform, as it stands, to the first figure, nor
(permutation apart) can its premisses be twisted into conformity with
it. But though we cannot prove the conclusion true in the first
figure, we can employ that figure to prove that it cannot be false, by
showing that the supposition of its falsity would involve a
contradiction of one of the original premisses, which are true ex
hypothesi.
§ 693. If possible, let the conclusion 'Some C is not A' be
false. Then its contradictory 'All C is A' must be true. Combining
this as minor with the original major, we obtain premisses in the
first figure,
All A is B, All fishes are oviparous,
All C is A, All marine animals are fishes,
which lead to the conclusion
All C is B, All marine animals are oviparous.
But this conclusion conflicts with the original minor, 'Some C is not
B,' being its contradictory. But the original minor is ex hypothesi
true. Therefore the new conclusion is false. Therefore it must either
be wrongly drawn or else one or both of its premisses must be false.
But it is not wrongly drawn; since it is drawn in the first figure, to
which the Dictum de Omni et Nullo applies. Therefore the fault must
lie in the premisses. But the major premiss, being the same with that
of the original syllogism, is ex hypothesi true. Therefore the minor
premiss, 'All C is A,' is false. But this being false, its
contradictory must be true. Now its contradictory is the original
conclusion, 'Some C is not A,' which is therefore proved to be true,
since it cannot be false.
§ 694. It is convenient to represent the two syllogisms in
juxtaposition thus--
Baroko. Barbara.
All A is B. All A is B.
Some C is not B. \/ All C is A.
.'. Some C is not A. /\ All C is B.
§ 695. The lines indicate the propositions which conflict with one
another. The initial consonant of the names Baroko and Eokardo
indicates that the indirect reduct will be Barbara. The k indicates
that the O proposition, which it follows, is to be dropped out in the
new syllogism, and its place supplied by the contradictory of the old
conclusion.
§ 696. In Bokardo the two syllogisms will stand thus--
Bokardo. Barbara.
Some B is not A. \ / All C is A.
All B is C. X All B is C.
.'. Some C is not A./ \ .'. All B is A.
§ 697. The method of indirect reduction, though invented with a
special view to Baroko and Bokardo, is applicable to all the moods of
the imperfect figures. The following modification of the mnemonic
lines contains directions for performing the process in every
case:--Barbara, Celarent, Darii, Ferioque prioris; Felake, Dareke,
Celiko, Baroko secundae; Tertia Cakaci, Cikari, Fakini, Bekaco,
Bokardo, Dekilon habet; quarta insuper addit Cakapi, Daseke, Cikasi,
Cepako, Cesïkon.
§ 698. The c which appears in two moods of the third figure, Cakaci
and Bekaco, signifies that the new conclusion is the contrary, instead
of, as usual, the contradictory of the discarded premiss.
§ 699. The letters s and p, which appear only in the fourth figure,
signify that the new conclusion does not conflict directly with the
discarded premiss, but with its converse, either simple or per
accidens, as the case may be.
§ 700. l, n and r are meaningless, as in the original lines.