If A is B, C is D; and if A is B, E is F:
But either C is not D, or E is not F:
∴ A is not B.
If table-rappers are to be trusted, the departed are spirits; and they also exert mechanical energy
: But either the departed are not spirits, or they do not exert mechanical energy:
∴ Table-rappers are not to be trusted.
4. Complex Destructive—
If A is B, C is D; and if E is F, G is H:
But either C is not D, or G is not H:
∴ Either A is not B, or E is not F.
If poetic justice is observed, virtue is rewarded; and if the mirror is held up to Nature, the villain triumphs:
But either virtue is not rewarded, or the villain does not triumph:
∴ Either poetic justice is not observed, or the mirror is not held up to Nature.
Such are the four Moods of the Dilemma that emerge if we only use affirmative hypotheticals for the major premise; but, certainly, it is often quite as natural to employ two negative hypotheticals (indeed, one might be affirmative and the other negative; but waive that); and then four more moods emerge, all having negative conclusions. It is needless to intimidate the reader by drawing up these four moods in battle array: they always admit of reduction to the foregoing moods by obverting the hypotheticals. Still, by the same process we may greatly decrease the number of moods of the Categorical Syllogism; and just as some Syllogisms are most simply expressed in Celarent or Cesare, so some Dilemmas are most simply stated with negative major premises—e.g., The example of a Simple Constructive Dilemma above given would run more naturally thus: If the Tories win, the Government will not innovate; and if the Whigs, the Lords will not let them: and similarly Demosthenes' Dilemma—If Æschines joined, he is not consistent; and if he did not, he is not patriotic. Moreover, the propriety of recognising Dilemmas with negative major premises, follows from the above analysis of the Dilemma into a combination of Conditional Syllogisms, even if (as in [§ 1] of this chapter) we take account of only four Moods of the Hypothetical Syllogism.
In the rhetorical use of the Dilemma, it may be observed that the disjunction in the minor premise ought to be obvious, or (at any rate) easily acceptable to the audience. Thus, Either the Tories or the Whigs will win; Either Æschines joined in the rejoicings, or he did not; such propositions are not likely to be disputed. But if the orator must stop to prove his minor premise, the smacking effect of this figure (if the expression be allowed) will be lost. Hence the minor premises of other examples given above are only fit for a select audience. That Either ghosts are not spirits, or they do not exert mechanical energy, supposes a knowledge of the principle, generally taught by physical philosophers, that only matter is the vehicle of energy; and that Either appearance is all, or there is substance beyond consciousness, is a doctrine which only metaphysical philosophers could be expected to understand, and upon which they could not be expected to agree. However, the chief danger is that a plausible disjunction may not be really such as to exclude any middle ground: Either the Tories or the Whigs win, is bad, if a tie be possible; though in the above argument this is negligible, seeing that a tie cannot directly cause innovations. Either Æschines joined in the rejoicings, or he did not, does not allow for a decent conformity with the public movement where resistance would be vain; yet such conformity as need not be inconsistent with subsequent condemnation of the proceedings, nor incompatible with patriotic reserve founded on a belief that the rejoicings are premature and ominous.
Another rhetorical consideration is, that the alternatives of the disjunctive conclusion of a Complex Dilemma should both point the same way, should be equally distasteful or paradoxical. 'Either inconsistent or unpatriotic': horrid words to a politician! 'Either no reality or no possible knowledge of it': very disappointing to an anxious inquirer! Thus the disjunctive conclusion is as bad for an opponent as the categorical one in a Simple Dilemma.
Logicians further speak of the Trilemma, with three Hypotheticals and a corresponding triple Disjunction; and of a Polylemma, with any further number of perplexities. But anyone who has a taste for logical forms may have it amply gratified in numerous text-books.