Sec. 2.  A Disjunctive Syllogism consists of a Disjunctive Major Premise, a Categorical Minor Premise, and a Categorical Conclusion.

How many Moods are to be recognised in this kind of argument depends on whether the alternatives of the Disjunctive Premise are regarded as mutually exclusive or possibly coincident.  In saying ’Either A is B, or C is D,’ do we mean ‘either, but not both,’ or ’either, it may be both’? (See chap. v.  Sec. 4.)

When the alternatives of the Disjunctive are not exclusive, we have only the

Modus tollendo ponens.

Either A is B, or C is D;
A is not B (or C is not D): 
.’.  C is D (or A is B).

Either wages fall, or the weaker hands are dismissed;

Wages do not fall: 
.’.  The weaker hands are dismissed.

But we cannot argue—­

Wages fall: 
.’.  The weaker hands are not dismissed;

since in ‘hard times’ both events may happen together.

Rule of the Modus tollendo ponens:  If one alternative be denied, the other is affirmed.

When, however, the alternatives of the Disjunctive are mutually exclusive, we have also the

Modus ponendo tollens.

Either A is B, or C is D;
A is B (or C is D): 
.’.  C is not D (or A is not B).

Either the Tories or the Whigs win the election;

The Tories win: 
.’.  The Whigs do not win.

We may also, of course, argue as above in the Modus tollendo ponens—­

The Tories do not win: 
.’.  The Whigs do.

But in this example, to make the Modus tollendo ponens materially valid, it must be impossible that the election should result in a tie.  The danger of the Disjunctive Proposition is that the alternatives may not, between them, exhaust the possible cases.  Only contradictory alternatives are sure to cover the whole ground.

Rule of the Modus ponendo tollens: If one alternative be affirmed, the other is denied.

Since a disjunctive proposition may be turned into a hypothetical proposition (chap. v.  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.