Deductive Logic Part 34

.'. 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 quant.i.ty, 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.