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. | |
A Dilemma, then, is a compound Conditional Syllogism, having for its Major Premise two Hypothetical Propositions, and for its Minor Premise a Disjunctive Proposition, whose alternative terms either affirm the Antecedents or deny the Consequents of the two Hypothetical Propositions forming the Major Premise.