Logic Part 17

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. -- 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 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 a.n.a.lysed 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.

Using, then, only affirmative hypothetical propositions in the major premise, there are four Moods:

1. The Simple Constructive--

If A is B, C is D; and if E is F, C is D: But either A is B, or E is F: ? C is D.

If the Tories win the election, the Government will avoid innovation; and if the Whigs win, the House of Lords will prevent them innovating:

But either the Tories or the Whigs will win:

? There will be no innovation.

2. The Complex Constructive--

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.

If appearance is all that exists, reality is a delusion; and if there is a substance beyond consciousness, knowledge of reality is impossible:

But either appearance is all, or there is a substance beyond consciousness: