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.

Sec. 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 AEschines 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.

The hypothetical propositions in the major premise, may have all four terms distinct (as in the above example); and then the conclusion is a disjunctive proposition, and the Dilemma is said to be Complex.  Or the two hypothetical propositions may have a common antecedent or a common consequent; and then the conclusion is a categorical proposition, and the Dilemma is said to be Simple.

Again, the alternatives of the disjunctive minor premise may be affirmative or negative:  if affirmative, the Dilemma is called Constructive; and if negative, Destructive.