Deductive Logic Part 44

.'. A is not B.

If the barometer falls there will be either wind or rain.

There is neither wind nor rain.

.'. The barometer has not fallen.

What we have here is simply a conjunctive major with the consequent denied in the minor. In the consequent of the major it is a.s.serted that the two propositions, 'C is D' and 'E is F' cannot both be false; and in the minor this is denied by the a.s.sertion that they are both false.

-- 792. A dilemma is said to be reb.u.t.ted or retorted, when another dilemma is made out proving an opposite conclusion. If the dilemma be a sound one, and its premisses true, this is of course impossible, and any appearance of contradiction that may present itself on first sight must vanish on inspection. The most usual mode of reb.u.t.ting a dilemma is by transposing and denying the consequents in the major--

If A is B, C is D; and if E is F, G is H.

Either A is B or E is F.

.'. Either C is D or G is H.

The same reb.u.t.ted--

If A is B, G is not H; and if E is F, C is not D.

Either A is B or E is F.

.'. Either G is not H or C is not D.

= Either C is not D or G is not H.

-- 793. Under this form comes the dilemma addressed by the Athenian mother to her son--'Do not enter public life: for, if you say what is just, men will hate you; and, if you say what is unjust, the G.o.ds will hate you' to which the following retort was made--'I ought to enter public life: for, if 1 say what is just, the G.o.ds will love me; and, if 1 say what is unjust, men will love me.' But the two conclusions here are quite compatible. A man must, on the given premisses, be both hated and loved, whatever course he takes. So far indeed are two propositions of the form

Either C is D or G is H, and Either C is not D or G is not H,

from being incompatible, that they express precisely the same thing when contradictory alternatives have been selected, e.g.--

Either a triangle is equilateral or non-equilateral.

Either a triangle is non-equilateral or equilateral.

-- 794. Equally illusory is the famous instance of reb.u.t.ting a dilemma contained in the story of Protagoras and Euathlus (Aul. Gell. Noct. Alt. v. 10), Euathlus was a pupil of Protagoras in rhetoric. He paid half the fee demanded by his preceptor before receiving lessons, and agreed to pay the remainder when he won his first case. But as he never proceeded to practise at the bar, it became evident that he meant to bilk his tutor. Accordingly Protagoras himself inst.i.tuted a law-suit against him, and in the preliminary proceedings before the jurors propounded to him the following dilemma--'Most foolish young man, whatever be the issue of this suit, you must pay me what I claim: for, if the verdict be given in your favour, you are bound by our bargain; and if it be given against you, you are bound by the decision of the jurors.' The pupil, however, was equal to the occasion, and reb.u.t.ted the dilemma as follows. 'Most sapient master, whatever be the issue of this suit, I shall not pay you what you claim: for, if the verdict be given in my favour, I am absolved by the decision of the jurors; and, if it be given against me, I am absolved by our bargain.' The jurors are said to have been so puzzled by the conflicting plausibility of the arguments that they adjourned the case till the Greek Kalends. It is evident, however, that a grave injustice was thus done to Protagoras. His dilemma was really invincible. In the counter-dilemma of Euathlus we are meant to infer that Protagoras would actually lose his fee, instead of merely getting it in one way rather than another. In either case he would both get and lose his fee, in the sense of getting it on one plea, and not getting it on another: but in neither case would he actually lose it.

-- 795. If a dilemma is correct in form, the conclusion of course rigorously follows: but a material fallacy often underlies this form of argument in the tacit a.s.sumption that the alternatives offered in the minor const.i.tute an exhaustive division. Thus the dilemma 'If pain is severe, it will be brief; and if it last long it will be slight,'

&c., leaves out of sight the unfortunate fact that pain may both be severe and of long continuance. Again the following dilemma--

If students are idle, examinations are unavailing; and, if they are industrious, examinations are superfluous, Students are either idle or industrious, .'. Examinations are either unavailing or superfluous,

is valid enough, so far as the form is concerned. But the person who used it would doubtless mean to imply that students could be exhaustively divided into the idle and the industrious. No deductive conclusion can go further than its premisses; so that all that the above conclusion can in strictness be taken to mean is that examinations are unavailing, when students are idle, and superfluous, when they are industrious--which is simply a rea.s.sertion as a matter of fact of what was previously given as a pure hypothesis.

CHAPTER XXVII.

_Of the Reduction of the Dilemma._

-- 796. As the dilemma is only a peculiar variety of the partly conjunctive syllogism, we should naturally expect to find it reducible in the same way to the form of a simple syllogism. And such is in fact the case. The constructive dilemma conforms to the first figure and the destructive to the second.

1) _Simple Constructive Dilemma_.

Barbara.

If A is B or if E is F, C is D. All cases of either A being B or E being F are cases of C being D.

Either A is B or E is F. All actual cases are cases of either A being B OP E being F.

.'. C is D. .'. All actual cases are cases of C being D.

(2) _Simple Destructive_.

Camstres.

If A is B, C is D and E is F. All cases of A being B are cases of C being D and E being F.

Either C is not D or E is not F. No actual cases are cases of C being D and E being F.

.'. A is not B. .'. No actual cases are cases of A being B.

(3) _Complex Constructive_.

Barbara.

If A is B, C is D; and if E is F, All cases of either A being B or G is H. being F are cases of either C being D or G being H.

Either A is B or E is F. All actual cases are cases of either A being B or E being F.

.'. Either C is D or G is H. .'. All actual cases are cases of either C being D or G being H.

(4) _Complex Destructive_.

If A is B, C is D; and if E is F, All cases of A being B and E being F G is H. are cases of C being D and G being H.

Either C is not D Or G is No actual cases are cases of C being not H D and G being H.

Either A is not B or E is No actual cases are cases of A being not F. B and E being F.

-- 797. There is nothing to prevent our having Darii, instead of Barbara, in the constructive form, and Baroko, instead of Camestres, in the destructive. As in the case of the partly conjunctive syllogism the remaining moods of the first and second figure are obtained by taking a negative proposition as the consequent of the major premiss, e.g.--

_Simple Constructive_. Celarent or Ferio.

If A is B or if E is F, C is not D No cases of either A being B or E being F are cases of C being D.

Either A is B or E is F. All (or some) actual cases are cases of either A being B or E being F .'. C is not D. .'. All (or some) actual cases are not cases of C being D.

CHAPTER XXVIII.

_Of the Dilemma regarded as an Immediate Inference._

-- 798. Like the partly conjunctive syllogism, the dilemma can be expressed under the forms of immediate inference. As before, the conclusion in the constructive type resolves itself into the subalternate of the major itself, and in the destructive type into the subalternate of its contrapositive. The simple constructive dilemma, for instance, may be read as follows--

If either A is B or E is F, C is D, .'. Either A being B or E being F, C is D,

which is equivalent to

Every case of either A being B or E being F is a case of C being D.

.'. Some case of either A being B or E being F is a case of C being D.

The descent here from 'every' to 'some' takes the place of the transition from hypothesis to fact.