If A is B, E is F ; and if C is D, G is H ;
but Either E is not F or G is not H ;
therefore, Either A is not B or C is not D.[398]
[397] The following is a simple, not a complex, constructive dilemma:
If A is B, E is F or G is H ; and if C is D, E is F or G is H ;
but Either A is B or C is D ;
therefore, Either E is F or G is H.
The hypotheticals which here constitute the major premiss have a common consequent; but since this is itself alternative, the conclusion appears in the alternative form. This case is analogous to the following,—All M is P or Q, All S is M, therefore, All S is P or Q,—where the conclusion of an intrinsically categorical syllogism also appears in the alternative form. Compare the [note] on page 359.
[398] The following is a simple, not a complex, destructive dilemma:
If both P and Q are true then X is true, and under the same hypothesis Y is true ;
but Either X or Y is not true ;
therefore, Either P or Q is not true.
In the case of dilemmas, as in the case of mixed hypothetical syllogisms, the constructive form may be reduced to the destructive form, and vice versâ. All that has to be done is to contraposit the hypotheticals which constitute the major 365 premiss. One example will suffice. Taking the simple constructive dilemma given above, and contrapositing the major, we have,—
If E is not F, A is not B ; and if E is not F, C is not D ;
but Either A is B or C is D ;
therefore, E is F ;
and this is a dilemma in the simple destructive form.
The definition of the dilemma given above is practically identical with that given by Fowler (Deductive Logic, p. 116). Mansel (Aldrich, p. 108) defines the dilemma as “a syllogism having a conditional (hypothetical) major premiss with more than one antecedent, and a disjunctive minor.” Equivalent definitions are given by Whately and Jevons. According to this view, while the constructive dilemma may be either simple or complex, the destructive dilemma must always be complex, since in the corresponding simple form (as in the example given on page [364]) there is only one antecedent in the major. This exclusion seems arbitrary and is a ground for rejecting the definition in question. Whately, indeed, regards the name dilemma as necessarily implying two antecedents ; but it should rather be regarded as implying two alternatives, either of which being selected a conclusion follows that is unacceptable. Whately goes on to assert that the excluded form is merely a destructive hypothetical syllogism, similar to the following,