Sec. 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.

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.