This is another combination of two conjunctive syllogisms, both pointing to the same conclusion. The proof is again redundant. In this case we have the consequent denied in both, whereas in the former we had the antecedent affirmed. It is only for convenience that such arguments as these are thrown into the form of a single syllogism. Their real distinctness may be seen from the fact that we here deny each proposition separately, thus making two independent statements—C is not D and E is not F. But in the true instance of the simple destructive dilemma, what we deny is not the truth of the two propositions contained in the consequent, but their compatibility; in other words we make a disjunctive denial.
§ 791. Nor yet is the following a dilemma—
If A is B, either C is D or E is F.
Neither C is D nor E is F.
.'. 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 asserted that the two propositions, 'C is D' and 'E is F' cannot both be false; and in the minor this is denied by the assertion that they are both false.
§ 792. A dilemma is said to be rebutted 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 rebutting 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 rebutted—
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 gods will hate you' to which the following retort was made—'I ought to enter public life: for, if 1 say what is just, the gods 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