Since a disjunctive proposition may be turned into a hypothetical proposition (chap. v. § 4,) a Disjunctive Syllogism may be turned into a Hypothetical Syllogism: _Modus tollendo ponens._ _Modus ponens._ Either A is B, or C is D; If A is not B, C is D; A is not B: A is not B: ∴ C is D. ∴ C is D. Similarly the _Modus ponendo tollens_ is equivalent to that kind of _Modus ponens_ which may be formed with a negative major premise; for if the alternatives of a disjunctive proposition be exclusive, the corresponding hypothetical be affirmative or negative: _Modus ponendo tollens._ _Modus ponens._ Either A is B, or C is D; If A is B, C is not D; A is B: A is B: ∴ C is not D. ∴ C is not D. Hence, finally, a Disjunctive Syllogism being equivalent to a Hypothetical, and a Hypothetical to a Categorical; a Disjunctive Syllogism is equivalent and reducible to a Categorical. It is a form of Mediate Inference in the same sense as the Hypothetical Syllogism is; that is to say, the conclusion depends upon an affirmation, or denial, of the fulfilment of a condition implied in the disjunctive major premise. § 3. The Dilemma is perhaps the most popularly interesting of all forms of proof. It is a favourite weapon of orators and wits; and "impaled upon the horns of a dilemma" is a painful situation in which every one delights to see his adversary. It seems to have been described by Rhetoricians before finding its way into works on Logic; and Logicians, to judge from their diverse ways of defining it, have found some difficulty in making up their minds as to its exact character. There is a famous Dilemma employed by Demosthenes, from which the general nature of the argument may be gathered: If Æschines joined in the public rejoicings, he is inconsistent; if he did not, he is unpatriotic; But either he joined, or he did not join: Therefore he is either inconsistent or unpatriotic. That is, reduced to symbols: If A is B, C is D; and if E is F, G is H: But either A is B, or E is F; ∴ Either C is D or G is H (_Complex Constructive_). This is a compound Conditional Syllogism, which may be analysed as follows: Either A is B or E is F. Suppose that E is not F: Suppose that A is not B: Then A is B. Then E is F. But if A is B, C is D; But if E is F, G is H; (A is B): (E is F): ∴ C is D. ∴ G is H. ∴ Either C is D or G is H.