(3) If C is D, A is not B.
(4) If C is not D, A is B.
Suppose then that an antagonist has granted you a Disjunctive Proposition, you can, using this as a Major Premiss, extract from him four different Conclusions, if you can get him also to admit the requisite Minors. The Mode of two of these is technically called Modus Ponendo Tollens, the mode that denies the one alternative by granting the other—A is B, therefore C is not D; C is D, therefore A is not B. The other Mode is also twice open, the Modus Tollendo Ponens—A is not B, therefore C is D; C is not D, therefore A is B.
Fallacy is sometimes committed through the Disjunctive form owing to the fact that in common speech there is a tendency to use it in place of a mere hypothetical, when there are not really two incompatible alternatives. Thus it may be said "Either the witness is perjured, or the prisoner is guilty," when the meaning merely is that if the witness is not perjured the prisoner is guilty. But really there is not a valid disjunction and a correct use of the disjunctive form, unless four hypotheticals are implied, that is, unless the concession of either involves the denial of the other, and the denial of either the concession of the other. Now the prisoner may be guilty and yet the witness be perjured; so that two of the four hypotheticals, namely—
If the witness is perjured, the prisoner is not guilty,
If the prisoner is guilty, the witness is not perjured—
do not necessarily hold. If, then, we would guard against fallacy, we must always make sure before assenting to a disjunctive proposition that there is really a complete disjunction or mutual incompatibility between the alternatives.
III.—The Dilemma.
A Dilemma is a combination of Hypothetical and Disjunctive propositions.
The word has passed into common speech, and its ordinary use is a clue to the logical structure. We are said to be in a dilemma when we have only two courses open to us and both of them are attended by unpleasant consequences. In argument we are in this position when we are shut into a choice between two admissions, and either admission leads to a conclusion which we do not like. The statement of the alternatives as the consequences hypothetically of certain conditions is the major premiss of the dilemma: once we admit that the relations of Antecedent and Consequent are as stated, we are in a trap, if trap it is: we are on the horns of the dilemma, ready to be tossed from one to the other.