Again, when the subject matter is simple, the Syllogistic form is not really required for protection against error. In such enthymemes as the following for example:--

She must be clever: she is so uncompromisingly ugly.

Romeo must be in love: for is he not seventeen?

it is plain to the average intelligence without any knowledge of Syllogism that the argument takes for granted a general proposition and what the general proposition is.

Another thing is plain to the average intelligence, perhaps plainer than to a proficient in the use of the Syllogism. Clearly we cannot infer with certainty that a woman is clever because she is ugly, unless it is the case that all ugly women are clever. But a Syllogiser, seeing that no certain conclusion can be drawn except upon this condition, is apt to dismiss the argument as altogether worthless. This may be specified as an error incident to the practice of the Syllogism, that it inclines us to look for necessarily conclusive premisses, and to deny all weight to anything short of this. Now in ordinary life it is comparatively seldom that such premisses can be found. We are obliged to proceed on maxims that are not of universal scope, and which lend only a more or less strong colour of probability to cases that can be brought under them. "A little learning is a dangerous thing;" "Haste makes waste;" "Slowness of speech is a sign of depth of thought;" "Vivacity is a sign of shallowness:" such are the "endoxes" or commonplaces of popular knowledge that men bring to bear in daily life. They are not true for all cases, but some of them are true for most or for a good many, and they may be applied with a certain probability though they are not rigidly conclusive. The plain man's danger is that he apply them unthinkingly as universals: the formal logician's danger is that, seeing them to be inapplicable as universals, he dismisses them as being void of all argumentative force.

It helps to fix the limits of Formal Logic to remember that it lies outside its bounds to determine the degree of probability attaching to the application of approximate truths, such as are the staple of arguments in ordinary affairs. Formal Logic, we may repeat, is not concerned with degrees of truth or falsehood, probability or improbability. It merely shows the interdependency of certain arguments, the consistency of conclusion with premisses.

This, however, is a function that might easily be underrated. Its value is more indirect than direct. In showing what is required for a certain conclusion, it puts us on the road to a more exact estimate of the premisses alleged, a sounder judgment of their worth. Well begun is half done: in undertaking the examination of any argument from authority, a formal syllogism is a good beginning.

CHAPTER VII.

CONDITIONAL ARGUMENTS.--HYPOTHETICAL SYLLOGISM, DISJUNCTIVE SYLLOGISM, AND DILEMMA.

The justification of including these forms of argument in Logic is simply that they are sometimes used in debate, and that confusion may arise unless the precise meaning of the premisses employed is understood. Aristotle did not include them as now given in his exposition of the Syllogism, probably because they have no connexion with the mode of reasoning together to which he appropriated the title. The fallacies connected with them are of such a simple kind that to discuss as a question of method the precise place they should occupy in a logical treatise is a waste of ingenuity.[1]

I.--HYPOTHETICAL SYLLOGISMS.

If A is B, C is D A is B } MODUS [.'.]C is D PONENS.

If A is B, C is D C is not D } MODUS [.'.]A is not B TOLLENS.

A so-called Hypothetical Syllogism is thus seen to be a Syllogism in which the major premiss is a HYPOTHETICAL PROPOSITION, that is to say, a complex proposition in which two propositions are given as so related that the truth of one follows necessarily from the truth of the other.

Two propositions so related are technically called the ANTECEDENT or Reason, and the CONSEQUENT.

The meaning and implication of the form, If A is B, C is D, is expressed in what is known as the LAW OF REASON AND CONSEQUENT:--

"_When two propositions are related as Reason and Consequent, the truth of the Consequent follows from the truth of the Antecedent, and the falsehood of the Antecedent, from the falsehood of the Consequent_".

If A is B, C is D, implies that If C is not D, A is not B. If this subject is educative, it quickens the wits; if it does not quicken the wits, it is not educative.

Admitted, then, that the law of Reason and Consequent holds between two propositions--that If A is B, C is D: admitted also the Antecedent, the truth of the Consequent follows. This is the MODUS PONENS or Positive Mode, where you reach a conclusion by obtaining the admission of the Antecedent. Admit the Antecedent and the truth of the Consequent follows.

With the same Major Premiss, you may also, under the Law of Reason and Consequent reach a conclusion by obtaining the denial of the Consequent. This is the MODUS TOLLENS or Negative Mode. Deny the Consequent and one is bound to deny the Antecedent.

But to guard against the fallacy technically known as FALLACIA CONSEQUENTIS, we must observe what the relation of Reason and Consequent does not imply. The truth of the Consequent does not involve the truth of the Antecedent, and the falsehood of the Antecedent does not involve the falsehood of the Consequent.

"If the harbour is frozen, the ships cannot come in." If the harbour is not frozen, it does not follow that the ships can come in: they may be excluded by other causes. And so, though they cannot come in, it does not follow that the harbour is frozen.

QUESTIONS CONNECTED WITH HYPOTHETICAL SYLLOGISMS.

(1) _Are they properly called Syllogisms?_ This is purely a question of Method and Definition. If we want a separate technical name for forms of argument in which two terms are reasoned together by means of a third, the Hypothetical Syllogism, not being in such a form, is not properly so called. The fact is that for the purposes of the Hypothetical Argument, we do not require an analysis into terms at all: it is superfluous: we are concerned only with the affirmation or denial of the constituent propositions as wholes.

But if we extend the word Syllogism to cover all arguments in which two propositions necessarily involve a third, the Hypothetical Argument is on this understanding properly enough called a Syllogism.

(2) _Is the inference in the Hypothetical Syllogism Mediate or Immediate?_

To answer this question we have to consider whether the Conclusion can be drawn from either of the two premisses without the help of the other. If it is possible immediately, it must be educible directly either from the Major Premiss or from the Minor.

(_a_) Some logicians argue as if the Conclusion were immediately possible from the Major Premiss. The Minor Premiss and the Conclusion, they urge, are simply equivalent to the Major Premiss. But this is a misunderstanding. "If A is B, C is D," is not equivalent to "A is B, _therefore_ C is D". "If the harbour is frozen, the ships cannot come in" is not to say that "the harbour is frozen, and therefore," etc.

The Major Premiss merely affirms the existence of the relation of Reason and Consequent between the two propositions. But we cannot thereupon assert the Conclusion unless the Minor Premiss is also conceded; that is, the inference of the Conclusion is Mediate, as being from two premisses and not from one alone.

(_b_) Similarly with Hamilton's contention that the Conclusion is inferrible immediately from the Minor Premiss, inasmuch as the Consequent is involved in the Reason. True, the Consequent is involved in the Reason: but we cannot infer from "A is B" to "C is D," unless it is conceded that the relation of Reason and Consequent holds between them; that is, unless the Major Premiss is conceded as well as the Minor.

(3) _Can Hypothetical Syllogism be reduced to the Categorical Form?_

To oppose Hypothetical Syllogisms to Categorical is misleading, unless we take note of the precise difference between them. It is only in the form of the Major Premiss that they differ: Minor Premiss and Conclusion are categorical in both. And the meaning of a Hypothetical Major Premiss (unless it is a mere arbitrary convention between two disputants, to the effect that the Consequent will be admitted if the Antecedent is proved, or that the Antecedent will be relinquished if the Consequent is disproved), can always be put in the form of a general proposition, from which, with the Minor Premiss as applying proposition, a conclusion identical with the original can be drawn in regular Categorical form.

Thus:--

If the harbour is frozen, the ships cannot come in.

The harbour is frozen.

[.'.] The ships cannot come in.

This is a Hypothetical Syllogism, _Modus Ponens_. Express the Hypothetical Major in the form of the general proposition which it implies, and you reach a conclusion (in _Barbara_) which is only grammatically different from the original.

All frozen harbours exclude ships.

The harbour is frozen.

[.'.] It excludes ships.

Again, take an example of the _Modus Tollens_--

If rain has fallen, the streets are wet.

The streets are not wet.

[.'.] Rain has not fallen.

This is reducible, by formulating the underlying proposition, to _Camestres_ or _Baroko_ of the Second Figure.

All streets rained upon are wet.

The streets are not wet.

[.'.] They are not streets rained upon.

Hypothetical Syllogisms are thus reducible, by merely grammatical change[2], or by the statement of self-evident implications, to the Categorical form. And, similarly, any Categorical Syllogism may be reduced to the Hypothetical form. Thus:--

All men are mortal.

Socrates is a man.

[.'.] Socrates is mortal.

This argument is not different, except in the expression of the Major and the Conclusion, from the following:--

If Socrates is a man, death will overtake him.

Socrates is a man.