Chapter 31

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. [.'.] Death will overtake him.

The advantage of the Hypothetical form in argument is that it is simpler. It was much used in Mediaeval Disputation, and is still more popular than the Categorical Syllogism. Perhaps the prominence given to Hypothetical Syllogisms as syllogisms in Post-Renaissance text-books is due to the use of them in the formal disputations of graduands in the Universities. It was the custom for the Disputant to expound his argument in this form:--

If so and so is the case, such and such follows. So and so is the case. [.'.] Such and such follows.

To which the Respondent would reply: _Accipio antecedentem, nego consequentiam_, and argue accordingly. Petrus Hispanus does not give the Hypothetical Syllogism as a Syllogism: he merely explains the true law of Reason and Consequent in connexion with the Fallacia Consequentis in the section on Fallacies. (_Summulae. Tractatus Sextus._)

II.--DISJUNCTIVE SYLLOGISMS.

A Disjunctive Syllogism is a syllogism in which the Major Premiss is a DISJUNCTIVE PROPOSITION, _i.e._, one in which two propositions are declared to be mutually incompatible. It is of the form Either A is B, or C is D.[3]

If the disjunction between the alternatives is really complete, the form implies four hypothetical propositions:--

(1) If A is B, C is not D. (2) If A is not B, C is D. (3) If C is D, A is not B. (4) If C is not D, A is B.

Suppose then that an antagonist has granted you a Disjunctive Proposition, you can, using this as a Major Premiss, extract from him four different Conclusions, if you can get him also to admit the requisite Minors. The Mode of two of these is technically called MODUS PONENDO TOLLENS, the mode that denies the one alternative by granting the other--A is B, _therefore_ C is not D; C is D, _therefore_ A is not B. The other Mode is also twice open, the MODUS TOLLENDO PONENS--A is not B, _therefore_ C is D; C is not D, _therefore_ A is B.

Fallacy is sometimes committed through the Disjunctive form owing to the fact that in common speech there is a tendency to use it in place of a mere hypothetical, when there are not really two incompatible alternatives. Thus it may be said "Either the witness is perjured, or the prisoner is guilty," when the meaning merely is that if the witness is not perjured the prisoner is guilty. But really there is not a valid disjunction and a correct use of the disjunctive form, unless four hypotheticals are implied, that is, unless the concession of either involves the denial of the other, and the denial of either the concession of the other. Now the prisoner may be guilty and yet the witness be perjured; so that two of the four hypotheticals, namely--

If the witness is perjured, the prisoner is not guilty, If the prisoner is guilty, the witness is not perjured--

do not necessarily hold. If, then, we would guard against fallacy, we must always make sure before assenting to a disjunctive proposition that there is really a complete disjunction or mutual incompatibility between the alternatives.

III.--THE DILEMMA.

A Dilemma is a combination of Hypothetical and Disjunctive propositions.

The word has passed into common speech, and its ordinary use is a clue to the logical structure. We are said to be in a dilemma when we have only two courses open to us and both of them are attended by unpleasant consequences. In argument we are in this position when we are shut into a choice between two admissions, and either admission leads to a conclusion which we do not like. The statement of the alternatives as the consequences hypothetically of certain conditions is the major premiss of the dilemma: once we admit that the relations of Antecedent and Consequent are as stated, we are in a trap, if trap it is: we are on the horns of the dilemma, ready to be tossed from one to the other.

For example:--

If A is B, A is C, and if A is not B, A is D. But A either is or is not B. Therefore, A either is C or is D.

If A acted of his own motive, he is a knave; if A did not act of his own motive, he is a catspaw. But A either acted of his own motive or he did not. Thereupon A is either a knave or a catspaw.

This is an example of the _Constructive_ Dilemma, the form of it corresponding to the common use of the word as a choice between equally unpleasant alternatives. The standard example is the dilemma in which the custodians of the Alexandrian Library are said to have been put by the Caliph Omar in 640 A.D.

If your books are in conformity with the Koran, they are superfluous; if they are at variance with it, they are pernicious. But they must either be in conformity with the Koran or at variance with it. Therefore they are either superfluous or pernicious.

Where caution has to be exercised is in accepting the clauses of the Major. We must make sure that the asserted relations of Reason and Consequent really hold. It is there that fallacy is apt to creep in and hide its head. The Alexandrian Librarians were rash in accepting the first clause of the conqueror's Major: it does not follow that the books are superfluous unless the doctrines of the Koran are not merely sound but contain all that is worth knowing. The propounder of the dilemma covertly assumes this. It is in the facility that it affords for what is technically known as _Petitio Principii_ that the Dilemma is a useful instrument for the Sophist. We shall illustrate it further under that head.

What is known as the _Destructive_ Dilemma is of a somewhat different form. It proceeds upon the denial of the Consequent as involving the denial of the Antecedent. In the Major you obtain the admission that if a certain thing holds, it must be followed by one or other of two consequences. You then prove by way of Minor that neither of the alternatives is true. The conclusion is that the antecedent is false.

We had an example of this in discussing whether the inference in the Hypothetical Syllogism is Immediate. Our argument was in this form:--

If the inference is immediate, it must be drawn either from the Major alone or from the Minor alone. But it cannot be drawn from the Major alone, neither can it be drawn from the Minor alone. Therefore, it is not immediate.

In this form of Dilemma, which is often serviceable for clearness of exposition, we must as in the other make sure of the truth of the Major: we must take care that the alternatives are really the only two open. Otherwise the imposing form of the argument is a convenient mask for sophistry. Zeno's famous dilemma, directed to prove that motion is impossible, covers a _petitio principii_.

If a body moves, it must move either where it is or where it is not. But a body cannot move where it is: neither can it move where it is not. Conclusion, it cannot move at all, _i.e._, Motion is impossible.

The conclusion is irresistible if we admit the Major, because the Major covertly assumes the point to be proved. In truth, _if_ a body moves, it moves neither where it is nor where it is not, but from where it is to where it is not. Motion consists in change of place: the Major assumes that the place is unchanged, that is, that there is no motion.

[Footnote 1: For the history of Hypothetical Syllogism see Mansel's _Aldrich_, Appendix I.]

[Footnote 2: It may be argued that the change is not merely grammatical, and that the implication of a general proposition in a hypothetical and _vice versa_ is a strictly logical concern. At any rate such an implication exists, whether it is the function of the Grammarian or the Logician to expound it.]

[Footnote 3: Some logicians prefer the form Either A is, or B is. But the two alternatives are propositions, and if "A is" represents a proposition, the "is" is not the Syllogistic copula. If this is understood it does not matter: the analysis of the alternative propositions is unessential.]