| If A is B, C is D ; | |
| C is not D ; | |
| therefore, | A is not B. |
But the two really differ precisely as the simple constructive dilemma given on page [364] differs from the constructive hypothetical syllogism,—
| If A is B, E is F ; | |
| A is B ; | |
| therefore, | E is F. |
Besides, it is clear that the form under discussion is not merely a destructive hypothetical syllogism such as has been already discussed, since the premiss which is combined with the hypothetical premiss is not categorical but alternative.
The following definition is sometimes given:—“The dilemma (or trilemma or polylemma) is an argument in which a choice is allowed between two (or three or more) alternatives, but it is 366 shewn that whichever alternative is taken the same conclusion follows.” This definition, which no doubt gives point to the expression “the horns of a dilemma,” includes the simple constructive dilemma and the simple destructive dilemma; but it does not allow that either of the complex dilemmas is properly so-called, since in each case we are left with the same number of alternants in the conclusion as are contained in the alternative premiss. On the other hand, it embraces forms that are excluded by both the preceding definitions; for example, the following reasoning—which should rather be classed simply as a destructive hypothetico-categorical syllogism—
| If A is, either B or C is ; | |
| but Neither B nor C is ; | |
| therefore, | A is not.[399] |
[399] Compare Ueberweg, Logic, § 123.
Jevons (Elements of Logic, p. 168) remarks that “dilemmatic arguments are more often fallacious than not, because it is seldom possible to find instances where two alternatives exhaust all the possible cases, unless indeed one of them be the simple negative of the other.” In other words, many dilemmatic arguments will be found to contain a premiss involving a fallacy of incomplete alternation. It should, however, be observed that in strictness an argument is not itself to be called fallacious because it contains a false premiss.
EXERCISES.
319. What can be inferred from the premisses, Either A is B or C is D, Either C is not D or E is F? Exhibit the reasoning (a) in the form of a hypothetical syllogism, (b) in the form of a dilemma. [K.]