14. FOUR FORMS.
The four forms of the dilemma are the simple constructive, the simple destructive, the complex constructive, and the complex destructive. The following symbolizations illustrate these four kinds:
Simple Constructive Dilemma.
If A is B, W is X; and if C is D, W is X,
But either A is B or C is D,
Hence W is X.
This is termed a simple dilemma because there is but one consequent; namely, W is X. The conclusion being affirmative makes it constructive.
Simple Destructive Dilemma.
If A is B, W is X; and if A is B, Y is Z,
But either W is not X or Y is not Z,
Hence A is not B.
This is simple because there is but one antecedent, A is B, and destructive because the conclusion is negative.
Complex Constructive Dilemma.
If A is B, W is X; and if C is D, Y is Z,
But either A is B or C is D,
Hence either W is X or Y is Z.
This is complex because there are two antecedents and two consequents; constructive, inasmuch as the conclusion is affirmative.
Complex Destructive Dilemma.
If A is B, W is X; and if C is D, Y is Z,
But either W is not X or Y is not Z,
Hence either A is not B or C is not D.
This is complex because there are two antecedents as well as two consequents, and destructive because the conclusion is negative. Briefly: (1) A simple dilemma is one where either the antecedent or consequent is repeated; whereas if neither is repeated the dilemma is complex. (2) A constructive dilemma contains an affirmative conclusion; while a destructive dilemma uses a negative conclusion. (3) A simple dilemma has as its conclusion a categorical proposition; whereas the conclusion of a complex dilemma is always disjunctive.
If the number of antecedents and consequents be increased, a trilemma, tetralemma, etc., may result.
ILLUSTRATION—Trilemma.
If A is B, W is X; and if C is D, Y is Z; and if E is F, U is V,
But either A is B, or C is D, or E is F,
Hence either W is X, or Y is Z, or U is V.
Some authorities define a dilemma as a syllogism in which the “major-hypothetical” has more than one antecedent while the “minor” must be disjunctive. This viewpoint necessarily excludes the second form or the simple destructive dilemma. The weight of authority, however, appears to favor the classification here recommended.
15. THE ONE RULE INVOLVED IN DILEMMATIC ARGUMENTS.
Since the major premise of the dilemma is hypothetical, the rule for testing such would of necessity be the hypothetical rule; namely, “The minor premise must either affirm the antecedent or deny the consequent.” As this rule and the fallacies incident to it have been treated in detail, further discussion is unnecessary.
16. ILLUSTRATIVE EXERCISE TESTING DISJUNCTIVE AND DILEMMATIC ARGUMENTS.
(1) If the arithmetic contains useful facts, it will help to good citizenship; and if it trains the powers of reason, it will help to good citizenship,
But the arithmetic either contains useful facts or trains the powers of reason,
Hence it will help to good citizenship.
This is a simple constructive dilemma in which the minor premise affirms the antecedents. The argument is, therefore, valid since it conforms to the rules of the hypothetical syllogism. The fact that the minor premise may not be a perfect disjunctive does not invalidate the conclusion, inasmuch as it is perfectly obvious that if the arithmetic fulfilled both the requirements of the antecedents, the conclusion would still obtain. It may, therefore, be inferred that if the dilemma conforms to the rules of the hypothetical argument, it is valid, though the disjunctive proposition which it contains may not be strictly logical.
(2) A man is either temperate or intemperate; and, as I have seen you drunk several times, I conclude that you are intemperate.
Arranged logically.
A man is either temperate or intemperate,
You are not temperate,
∴ You are intemperate.
It would seem that the major premise is a logical disjunctive, since temperate and intemperate indicate that the alternatives are mutually exclusive and the enumeration complete. And since the minor premise denies one alternative while the conclusion affirms the other, we may infer that the argument is valid.
(3) If a man is honest, he will either pay his debts or explain; but this fellow paid no heed to the repeated notifications.
Arranged logically.
If a man is honest, he will pay his debts; and if he is honest, he will explain in case he cannot pay,
This man neither paid his debt, nor explained,
∴ This man is not honest.
This is a simple destructive dilemma, and since the minor premise denies the consequents it is valid.
(4) A voter must either favor protection or free trade; and since you do not favor protection, you must be a free trader. The disjunctive is not logical as one might believe in universal reciprocity. The argument is, therefore, invalid. Why?
(5) If a man were loyal, he would not be unduly critical; and if he were wise, he would not be too loquacious; but I find this clerk has been both unduly critical and too loquacious; hence I consider that he has been not only unwise but strikingly disloyal.
This complex dilemma is valid since the minor premise denies the two consequents.
17. ORDINARY EXPERIENCES RELATED TO THE DISJUNCTIVE PROPOSITION AND HYPOTHETICAL ARGUMENT.
(1) One desires to take a certain trip which involves various routes; information from time tables reveals the fact that there are three routes A, B, and C. Concerning the conditions of the journey the most important factor is the matter of comfort. Further investigation makes evident that route B will be the most comfortable one, and consequently is the route selected. Putting this ordinary experience in argumentative form gives the following:
The route is to be either A, or B, or C;
I will take route A; if it is the most comfortable; (co-extensive)
A is not the most comfortable route,
Hence I will not take route A.
If B is the most comfortable route, I will take it;
B is the most comfortable route,
Hence I will take route B.
(2) The symptoms suggest either malarial or typhoid fever; the physician is undecided till a blood test makes evident that it is not typhoid.
Considered argumentatively.
This disease is either malarial or typhoid fever;
If it is typhoid, the blood will reveal certain evidences;
But the blood does not reveal these evidences,
Hence the disease is not typhoid.
(3) The natural bent of the youth suggests the profession of either the ministry or teaching. He finally decides to follow the one in which he can best serve his fellows. This, after mature deliberation, appears to him to be the work of the teacher. Thrown into the form of an argument the following results:
I am best fitted for either the pulpit or the schoolroom;
If the schoolroom furnishes the richest field for helping my fellows, I will choose that work;
The schoolroom does appear to furnish such a field,
Hence I will choose the work of the teacher.
It would appear from these ordinary experiences that frequently we are brought face to face with a choice of alternatives which are not unattractive, as in the case of the dilemma. Moreover, some condition suggests itself which, if proved or disproved, will lead to a choice of one of these alternatives. Such circumstances when thrown into the form of an argument present a disjunctive proposition followed by a hypothetical argument. To put it differently: Often in our daily affairs a most prominent limiting condition induces us to select one out of several alternatives. These alternatives are not dilemmatic in nature.
18. OUTLINE.
HYPOTHETICAL ARGUMENTS, AND DISJUNCTIVE ARGUMENTS INCLUDING THE DILEMMA.
(1) Three kinds of arguments
Categorical, hypothetical, disjunctive.
(2) Hypothetical arguments
Defined, illustrated.
(3) Antecedent and consequent.
How determined, illustrations.
(4) Two kinds of hypothetical arguments
Constructive, destructive, illustrations.
(5) Rule and two fallacies of the hypothetical argument.
Illustrations and application of rules.
Fallacy of denying antecedent.
Fallacy of affirming consequent.
Co-extensive hypotheticals.
(6) Hypothetical arguments reduced to the categorical form.
Rule, illustrations.
Hypothetical and categorical arguments compared.
(7) Illustrative exercises testing hypothetical arguments of all kinds.
(8) Disjunctive arguments.
Defined, illustrated.
(9) Two kinds of disjunctive arguments.
By “affirming denies,” by “denying affirms.” Illustration.
(10) First rule.
Stated, illustrated.
(11) Second rule
Stated, illustrated.
(12) Reduction of disjunctive argument
Two steps.
(13) The dilemma
Definition.
(14) Four forms of dilemmatic arguments
Simple constructive, simple destructive,
Complex constructive, complex destructive.
Illustrations.
(15) The rule.
(16) Illustrative exercises testing disjunctive and dilemmatic arguments.
(17) Ordinary experiences related to the disjunctive proposition and hypothetical argument.
19. SUMMARY.
(1) Just as there are three kinds of propositions so there are three kinds of arguments; namely, categorical, hypothetical, disjunctive.
(2) Categorical syllogistic arguments are those in which all of the propositions are categorical.
Hypothetical syllogistic arguments are those in which the major premise is hypothetical.
In contradistinction to disjunctives, hypothetical arguments may be referred to as “conjunctive”.
(3) The hypothetical proposition is composed of antecedent and consequent; the former being the limiting condition; while the latter is the direct assertion. As the words indicate the antecedent usually precedes the consequent. The signs of the antecedent are “if,” “though,” “unless,” “suppose,” “granted that,” “when,” etc.
(4) The two kinds of hypothetical syllogisms are the constructive and destructive; the former is involved when the minor premise affirms the antecedent; the latter when the minor premise denies the consequent. These two kinds are sometimes referred to as “modus ponens” and “modus tollens” respectively.
(5) Out of the four possible hypothetical syllogisms only two are valid as investigation proves this rule: The minor premise must affirm the antecedent or deny the consequent. In the case of the hypothetical proposition being co-extensive, the rule does not apply.
(6) Hypothetical arguments may be reduced to the categorical by contracting the antecedent of the hypothetical proposition to form the subject-term, and by contracting the consequent of the hypothetical proposition to form the predicate-term of the major premise of the categorical syllogism. If it is necessary, supply a new minor term.
Denying the antecedent is a matter of illicit major; whereas affirming the consequent is equivalent to undistributed middle.
(7) Hypothetical arguments may be tested by following this outline:
(1) Arrange logically.
(2) Determine antecedent and consequent.
(3) Apply hypothetical rule.
(4) Reduce to categorical form.
(5) Apply categorical rules.
(8) A disjunctive syllogism is one in which the major premise is a disjunctive proposition.
(9) The two kinds of disjunctives are those which “by affirming deny” and those which “by denying affirm.”
(10) In testing disjunctive arguments there are two rules involved: First, “The major premise must assert a logical disjunction.” This necessitates the two requisites “the alternatives must be mutually exclusive” and the “enumeration must be complete.” The two opinions relative to the nature of an alternative assertion are, first, if one is false, the other must be true and vice versa; and second, if one is false, the other must be true, but both may be true. The first is adopted in this discussion.
Second. The second rule involved is “When the minor premise affirms or denies one of the alternatives of a logical disjunctive the conclusion must deny or affirm all of the others.”
(11) Subjecting the disjunctive arguments to the categorical test gives evidence to the close relation existing between the two forms. A logical disjunctive proves to be logical when reduced to the categorical. The reduction entails the two steps, first, reduce to the hypothetical; second, reduce to the categorical.
(12) The logical meaning of the dilemma is suggested by the popular conception. One is said to be in a dilemma when two courses are open to him, neither of which is specially attractive.
A logical dilemma presents two alternatives either one of which might well be avoided.
The major premise of the dilemma is hypothetical; while the minor is disjunctive.
(13) The four forms of the dilemma are the simple constructive, the simple destructive, the complex constructive and the complex destructive.
(14) The dilemma is subject to the hypothetical rule which is, “The minor premise must either affirm the antecedent or deny the consequent.”
(15) The minor premise need not be a logical disjunctive provided the major conforms to the hypothetical rule.
(16) Frequently when ordinary experiences are reduced to augmentative form they present a disjunctive proposition followed by a hypothetical argument.
20. REVIEW QUESTIONS.
(1) Relate the three kinds of arguments to the three general kinds of propositions.
(2) Define and illustrate the hypothetical argument.
(3) Explain the term conjunctive with reference to hypothetical arguments.
(4) Explain and illustrate antecedent and consequent in hypothetical arguments.
(5) Select from the following the antecedent and consequent:
(1) “I usually succeed when I try.”
(2) “I will not undertake it unless you guarantee half of the sum needed.”
(3) “Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass or a tinkling cymbal.”
(6) Illustrate the two kinds of hypothetical syllogisms which are valid.
(7) State and explain the rule to which hypothetical arguments must conform.
(8) State and exemplify the one exception to the hypothetical rule.
(9) Explain how hypothetical arguments may be reduced to the categorical form. Illustrate.
(10) Show by illustration that denying the antecedent is equivalent to illicit major, while affirming the consequent is equivalent to undistributed middle.
(11) Reduce to the categorical form and test:
“If Napoleon had possessed more of the spirit of Washington, he would have been less famous but a better man than he was; but he did not possess the spirit of the ‘Father of His Country.’”
(12) Test according to outline the following hypothetical arguments:
(1) “If it be a good thing to have faith, then certainly he who believes in the bible of a pagan has faith and must have a good thing.”
(2) “If a 10-inch charge burst inside of a tank, there would be nothing left of the tank. It would be blown into small pieces.”
(3) “If the plate found had been originally on the outside of the ship, I should have judged that there must be green paint on it, but I could not find green paint on that part of the ship.”
(4) “If I mistake not, you are the man who did not pay me for that pair of shoes. I am sure that you are the man as I never forget a face.”
(5) “If the maxim ‘Early to bed and early to rise makes one healthy, wealthy and wise’ were true, I would have been a millionaire long ago.”
(13) Define and illustrate a disjunctive argument.
(14) Exemplify the two kinds of disjunctive arguments.
(15) What is meant by a logical disjunction?
(16) “The alternatives must be mutually exclusive.” Explain this, illustrating fully.
(17) Cite cases where the enumeration is not complete.
(18) State in complete form both of the rules to which all disjunctive arguments must conform.
(19) Show by illustration how the disjunctive syllogism may be reduced to the categorical.
(20) Define and illustrate the dilemma.
(21) Give examples, using symbols, of the four dilemmatic forms. Explain why these forms are so named.
(22) Why does the hypothetical rule apply to the dilemmatic syllogism?
(23) Test the validity of the following: Give reasons.
(1) “If a substance is solid it possesses elasticity and so also it does if it be a liquid or gaseous; but all substances are either solid, liquid or gaseous; therefore, all substances possess elasticity.”
(2) “If men were prudent, they would act morally for their own good; if benevolent, for the good of others. But many men will not act morally, either for their own good or that of others; such men, therefore, are not prudent or benevolent.”
(3) “If the majority of those who use public houses are prepared to close them, legislation is unnecessary; but if they are not prepared for such a measure, then to force it upon them by outside pressure is both dangerous and unjust.”
(4) “The man is either a liar or a fool and in either case he is beneath my attention.”
(5) “Either he is sincere or else he is the most astute impostor the world has ever produced; for me I prefer to think him sincere.”
(24) Explain the relation that many experiences appear to bear toward an argument introduced by a disjunctive proposition and followed by a hypothetical syllogism. Illustrate.
21. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.
(1) May both premises of a hypothetical argument be hypothetical propositions? Explain. See Fowler p. 115.
(2) Which of the two is valid? Explain.
(1) If A is B, C is D
If A is B, E is F
∴ If C is D, E is F
(2) If A is B, C is D
If C is D, E is F
∴ If A is B, E is F
(3) Show by circles that two of the possible four hypothetical arguments are invalid.
(4) What categorical rules does the hypothetical argument seem to violate? Explain.
(5) Originate a hypothetical syllogism whose antecedent and consequent are both negative. Test its validity.
(6) Originate a co-extensive hypothetical argument and show that four valid syllogisms may be derived from it.
(7) Explain by word and illustration the two meanings which may be attached to “either-or.”
(8) If we accepted the opinion that both alternates of a disjunctive may be true, which kind of disjunctive argument would it invalidate?
(9) In a logical disjunction what law of thought is involved? Explain.
(10) Why do the laws of the disjunctive seem to contradict the categorical rules? Explain fully.
(11) Show by drawing on common experience that a logical dilemma is closely related to the popular conception of dilemma.
(12) Illustrate by symbols and then place in good English a pentalemma.
(13) State a definition of a dilemma which excludes the simple destructive form.
(14) Give a common experience which, when thrown into argumentative form, results in a disjunctive proposition followed by a hypothetical syllogism. Coin a name for such a combination.