Deductive Logic Part 35

-- 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, Ceskon.

-- 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.

CHAPTER XIX.

_Of Immediate Inference as applied to Complex Propositions._

-- 701. So far we have treated of inference, or reasoning, whether mediate or immediate, solely as applied to simple propositions. But it will be remembered that we divided propositions into simple and complex. I t becomes inc.u.mbent upon us therefore to consider the laws of inference as applied to complex propositions. Inasmuch however as every complex proposition is reducible to a simple one, it is evident that the same laws of inference must apply to both.

-- 702. We must first make good this initial statement as to the essential ident.i.ty underlying the difference of form between simple and complex propositions.

-- 703. Complex propositions are either Conjunctive or Disjunctive (-- 214).

-- 704. Conjunctive propositions may a.s.sume any of the four forms, A, E, I, O, as follows--

(A) If A is B, C is always D.

(E) If A is B, C is never D.

(I) If A is B, C is sometimes D.

(O) If A is B, C is sometimes not D.

-- 705. These admit of being read in the form of simple propositions, thus--

(A) If A is B, C is always D = All cases of A being B are cases of C being D. (Every AB is a CD.)

(E) If A is B, C is never D = No cases of A being B are cases of C being D. (No AB is a CD.)

(I) If A is B, C is sometimes D = Some cases of A being B are cases of C being D. (Some AB's are CD's.)

(O) If A is B, C is sometimes not D = Some cases of A being B are not cases of C being D. (Some AB's are not CD's.)

-- 706. Or, to take concrete examples,

(A) If kings are ambitious, their subjects always suffer.

= All cases of ambitious kings are cases of subjects suffering.

(E) If the wind is in the south, the river never freezes.

= No cases of wind in the south are cases of the river freezing.

(I) If a man plays recklessly, the luck sometimes goes against him.

= Some cases of reckless playing are cases of going against one.

(O) If a novel has merit, the public sometimes do not buy it.

= Some cases of novels with merit are not cases of the public buying.

-- 707. We have seen already that the disjunctive differs from the conjunctive proposition in this, that in the conjunctive the truth of the antecedent involves the truth of the consequent, whereas in the disjunctive the falsity of the antecedent involves the truth of the consequent. The disjunctive proposition therefore

Either A is B or C is D

may be reduced to a conjunctive

If A is not B, C is D,

and so to a simple proposition with a negative term for subject.

All cases of A not being B are cases of C being D.

(Every not-AB is a CD.)

-- 708. It is true that the disjunctive proposition, more than any other form, except U, seems to convey two statements in one breath. Yet it ought not, any more than the E proposition, to be regarded as conveying both with equal directness. The proposition 'No A is B' is not considered to a.s.sert directly, but only implicitly, that 'No B is A.' In the same way the form 'Either A is B or C is D'

ought to be interpreted as meaning directly no more than this, 'If A is not B, C is D.' It a.s.serts indeed by implication also that 'If C is not D, A is B.' But this is an immediate inference, being, as we shall presently see, the contrapositive of the original. When we say 'So and so is either a knave or a fool,' what we are directly a.s.serting is that, if he be not found to be a knave, he will be found to be a fool. By implication we make the further statement that, if he be not cleared of folly, he will stand condemned of knavery. This inference is so immediate that it seems indistinguishable from the former proposition: but since the two members of a complex proposition play the part of subject and predicate, to say that the two statements are identical would amount to a.s.serting that the same proposition can have two subjects and two predicates. From this point of view it becomes clear that there is no difference but one of expression between the disjunctive and the conjunctive proposition. The disjunctive is merely a peculiar way of stating a conjunctive proposition with a negative antecedent.

-- 709. Conversion of Complex Propositions.

A / If A is B, C is always D.

.'. If C is D, A is sometimes B.

E / If A is B, C is never D.

.'. If C is D, A is never B.

I / If A is S, C is sometimes D.

.'. If C is D, A is sometimes B.

-- 710. Exactly the same rules of conversion apply to conjunctive as to simple propositions.

-- 711. A can only be converted per accidens, as above.