The proposition, constituting the basic unit of the argument, would of necessity be indicative of the nature of said argument; therefore the three general kinds of propositions, categorical, hypothetical and disjunctive, suggest the three kinds of arguments which are in turn categorical, hypothetical and disjunctive. Categorical arguments are those in which all of the propositions are categorical. Since this kind has been treated, it remains for us to consider the other two.
We have observed that a hypothetical proposition is one in which the assertion depends on a condition; for example, in the proposition, “If it is pleasant, I will call on you to-morrow,” the calling depends on the state of the weather. “I will call on you to-morrow,” is the assertion which is limited by the condition, “If the weather is pleasant.” Definition:
The hypothetical argument or syllogism is one in which the major premise is hypothetical and the minor premise categorical.
ILLUSTRATION:
If the people are right more than half of the time, the world will progress;
And the people are right more than half of the time,
Hence the world will progress.
In contradistinction to disjunctives, hypothetical propositions and hypothetical syllogisms are frequently referred to as “conjunctive.”
Facility in detecting the antecedent and consequent of hypotheticals is required in order to deal intelligently with the argument. The hypothetical proposition has been defined as one in which the assertion is limited by a condition. The consequent is the assertion and usually follows (though not always) the antecedent which is the limiting condition. First the antecedent and then the consequent is the logical order as the derivative meaning of the words antecedent and consequent would indicate. The antecedent is introduced by such words as “if,” “though,” “unless,” “suppose,” “granted that,” “when,” etc.
ILLUSTRATIONS:
| Antecedent. | Consequent. |
| 1. If you study, | you will pass. |
| 2. If it rains, | it is cloudy. |
| 3. If two is added to three, | the result is five. |
| 4. If you are temperate, | you will live to a ripe old age. |
| Consequent. | Antecedent. |
| 5. I will go, | unless you wire me to the contrary. |
| 6. I will pay you, | when you present your bill. |
| 7. I shall make the trip in ten hours, | granted that I have no accidents. |
| 8. My overcoat would not have been stolen, | if the door had been locked. |
The two kinds of hypothetical syllogisms are the constructive and destructive.
A constructive hypothetical syllogism is one in which the minor premise affirms the antecedent.
A destructive hypothetical syllogism is one in which the minor premise denies the consequent.
The constructive hypothetical is sometimes referred to as the “modus ponens”; whereas the destructive hypothetical is called the “modus tollens.”
ILLUSTRATIONS:
| Constructive Hypothetical Syllogisms. | |
| Symbols. | Words. |
| If A is B, C is D | If you are diligent, you will succeed; |
| A is B | And you are diligent, |
| ∴ C is D | Therefore you will succeed. |
| Destructive Hypothetical Syllogisms. | |
| If A is B, C is D | If you had been diligent, you would have succeeded; |
| C is not D | But you did not succeed, |
| ∴ A is not B | Therefore you were not diligent. |
From a given hypothetical proposition it is possible to construct four different hypothetical syllogisms, as the attending illustrations make evident:
Consider the hypothetical proposition “If it has rained, the ground is damp.”
(1) Minor premise affirms antecedent.
If it has rained, the ground is damp;
It has rained,
Therefore the ground is damp.
(2) Minor premise denies antecedent.
If it has rained, the ground is damp;
It has not rained,
Therefore the ground is not damp.
(3) Minor premise affirms consequent.
If it has rained, the ground is damp;
The ground is damp,
Therefore it has rained.
(4) Minor premise denies the consequent.
If it has rained, the ground is damp;
The ground is not damp,
Therefore it has not rained.
Without any knowledge of the rules of the hypothetical syllogism let us strive to determine how many of the foregoing are valid. Relative to the first, it would be impossible for any rain to fall without making the ground somewhat damp; a few drops would be sufficient. In short, if the antecedent happens, the consequent must follow. It seems, therefore, that the first argument is valid. Considering the second: rain is not the only cause for the dampness of the ground, as it might result from the falling of dew, or from a dense fog; no rain does not necessarily mean no dampness. It is clear that if the antecedent does not happen, the consequent may or may not follow. Thus it appears that the second argument is invalid. Attention to the third makes evident a condition similar to the second: the ground may be made damp by agencies other than rain, such as fog and dew. Thus the third argument is likewise invalid. But in the fourth argument it is obvious that if the ground is not damp, then there could have been neither rain, nor fog, nor dew. No dampness shuts out all of the conditions, including the rain. Therefore the fourth argument is valid.
This investigation suggests a rule for hypothetical arguments. Since only the first and fourth arguments are valid, this is the rule which must obtain: The minor premise should either affirm the antecedent or deny the consequent.
Any violation of this rule would result in the fallacies of denying the antecedent or affirming the consequent.
There is one exception to this rule which must not be overlooked; viz.: If the antecedent and consequent of the hypothetical proposition are co-extensive then both may be either affirmed or denied.
ILLUSTRATIONS:
(1) If the rectangle is equilateral, then it is a square;
The rectangle is equilateral,
∴ It is a square.
(2) If the rectangle is equilateral, then it is a square;
The rectangle is not equilateral,
∴ The rectangle is not a square.
(3) If the rectangle is equilateral, then it is a square;
It is a square,
∴ The rectangle is equilateral.
(4) If the rectangle is equilateral, then it is a square;
It is not a square,
∴ The rectangle is not equilateral.
The hypothetical syllogism so closely resembles the categorical that it may be changed to it by a slight alteration in the wording. After testing the hypothetical by its own rule, it may be expedient to reduce the argument to the categorical form, and subject it to a second test in which the categorical rules are applied. This reduction usually necessitates two steps; first, change the propositions which represent the antecedent and consequent to a subject term and a predicate term respectively and then unite them to form the major premise; second, supply a new minor term, if necessary.
Illustrations of Reduction; and Comparison of Hypothetical and Categorical Fallacies:
Hypothetical Form:
(1) If it has rained, the ground is damp;
It has rained,
∴ The ground is damp.
Categorical Form:
A
M
The falling rain
makes the G
ground damp,
A
S
In this case rain
has M
fallen,
A
∴ S
In this case the ground
is G
damp ground.
It is seen that the argument in the hypothetical form is valid as the minor premise affirms the antecedent. Reducing to the categorical gives to the argument the mode
A
A
A
in the first figure which we know to be valid.
Hypothetical:
(2) If one were wise, he would study;
But you will not study,
∴ You are not wise.
Categorical:
A
A
G
wise person
would M
study,
E
∴ S
You
will not M
study,
E
∴ S
You
are not G
wise.
In the hypothetical form the argument is valid since the minor premise denies the consequent. Reducing to the categorical gives mood
A
E
E
in the second figure. This is valid.
Hypothetical:
(3) If the wind blows from the south, it will rain;
But the wind is not blowing from the south,
Hence it is not going to rain.
Categorical:
A
M
South wind
brings G
rain,
E
S
This wind
is not a M
south wind,
E
∴ S
This wind
will not G
bring rain.
Hypothetically considered, the minor premise denies the antecedent and consequently the argument is invalid. Reducing to the categorical form, it is found that the major term is distributed in the conclusion, but is not distributed in the major premise; hence the fallacy of illicit major is committed.
Hypothetical:
(4) If a man is just, he will obey the golden rule;
This judge has obeyed the golden rule,
Hence he is just.
Categorical:
A
A
G
just man
will obey the M
golden rule,
A
This
S
judge
has obeyed the M
golden rule,
A
∴ This
S
judge
is a G
just man.
Hypothetically considered, the minor premise affirms the consequent and thus the argument is fallacious; when changed to the categorical we find the fallacy of undistributed middle. If other examples were taken, it could be proved that the hypothetical fallacy of denying the antecedent is usually equivalent to the categorical fallacy of illicit major; whereas the hypothetical fallacy of affirming the consequent amounts to undistributed middle.
In reducing some hypotheticals it is necessary to make use of such expressions as, “the case of” or “the circumstances that.” The attending argument will illustrate this:
If Jefferson was right, man was created free and equal;
(but) Man was not created free and equal,
∴ Jefferson was not right.
Reduced to the categorical:
The
G
case of Jefferson being right
is the case of man being created
M
free and equal;
S
Man
was not created M
free and equal,
∴ A
Jefferson
(this man) was not G
right.
The argument is valid in both cases.
The following brief outline may be followed in testing hypothetical arguments:
1. Arrange logically.
2. Determine antecedent and consequent.
3. Apply the hypothetical rule; name fallacies giving reasons.
4. Reduce to categorical form.
5. Apply the categorical rules, giving fallacies with reasons.
(1) If a man is properly educated, he will not despise manual labor;
therefore I conclude that you have not been properly educated,
since you dislike to work with your hands.
Arranged logically and antecedent and consequent indicated:
If a man is properly educated (antecedent), he will not despise manual labor (consequent);
You despise manual labor (dislike to work with your hands),
∴ You have not been properly educated.
The minor premise denies the consequent, hence the argument is valid according to the rule, “The minor premise must affirm the antecedent or deny the consequent.” The student should note that the consequent is negative and therefore its denial must be an affirmative proposition.
Reduced to the categorical:
E
A
G
properly educated man
will not M
despise manual labor;
A
S
You
despise M
manual labor,
E
∴ S
You
have not been G
properly educated.
Regarded categorically this is valid. Why?
(2) “If one believes in the tenets of the democratic party, then he should vote for its candidates; and since A does believe in them I have asked him to vote for me.”
Arranged, and antecedent and consequent indicated.
If one believes in the tenets of the democratic party (antecedent), then he should vote for its candidates (consequent);
And A does believe in these tenets,
∴ He should vote for its candidates (I have asked him to vote for me).
The minor premise affirms the antecedent and thus the argument is valid according to rule.
Reduced to the categorical:
A
M
One who believes in the tenets of the democratic party
should vote for its M
candidates,
A
S
A
believes in these M
tenets,
A
∴ S
A
should vote for its G
candidates.
Reduced to the categorical gives mood
A
A
A
in the first figure and this we know to be valid.
(3) “If the weather had not been pleasant, I could not have come; but as the weather is pleasant, here I am.”
Arranged and antecedent and consequent indicated:
If the weather had not been pleasant (antecedent), I could not have come (consequent);
The weather is pleasant,
∴ I have come (here I am).
The minor premise denies the antecedent and consequently the argument is invalid according to the rule. (An affirmative minor premise denies a negative antecedent.)
Reduced to the categorical:
E Unpleasant weather would not permit me to come,
E This weather is not unpleasant,
A ∴ This weather enabled me to come.
Fallacy of two negative premises.
(4) “If one pays his debts, he will not be ‘black-listed’; but since you are ‘black-listed,’ I conclude that you have not paid your debts.”
Arranged logically and antecedent and consequent indicated:
If one pays his debts (antecedent), he will not be “black-listed” (consequent);
You are “black-listed,”
∴ You have not paid your debts.
The minor premise denies the consequent hence the argument is valid.
Reduced to categorical form:
E
No
G
one who pays his debts
is M
black listed,
A
S
You
are M
black listed,
E
∴ S
You
have not G
paid your debts.
The mood
E
A
E
in the second figure is valid.
(5) “Men would do right for the sake of themselves, if they appreciated the law of retribution; but they never think of that.”
Arranged, completed, and tested:
If they appreciated the law of retribution (antecedent), men would do right for the sake of themselves (consequent);
But they do not appreciate the law of retribution (never think of that),
Hence they do not do right for the sake of themselves.
Fallacy of denying the antecedent.
Reduced to the categorical:
A
The
case of M
men appreciating the law of retribution,
is the case of G
men
doing right for the sake of themselves;
E
But
S
men
do not M
appreciate the law of retribution,
E
∴ S
Men
do not do G
right for the sake of themselves.
Fallacy of illicit major.
(6) “If an animal is a vertebrate, then it must have a backbone; but the books say that this animal is not a vertebrate, hence it cannot have a backbone.”
Since the minor premise denies the antecedent it would appear that the argument is invalid; yet common knowledge and common sense dictate that the conclusion is true. Surely no invertebrate can have a backbone. As a matter of fact the antecedent and consequent are co-extensive and therefore the hypothetical rule is not applicable.
Reduced to the categorical:
A
M
Vertebrates
must have G
a backbone
(Co-extensive),
E
This
S
animal
is not a M
vertebrate,
E
∴ This
S
animal
cannot have G
a backbone.
As co-extensive A’s distribute their predicates the possibility of there being a fallacy of illicit major is forestalled.
Categorically considered the argument is likewise valid.
It has been observed that a disjunctive proposition is one which expresses an alternative. A disjunctive syllogism is one in which the major premise is a disjunctive proposition.
ILLUSTRATION:
The boy is either honest or dishonest,
He is honest,
∴ He is not dishonest.
The two forms of disjunctive arguments are the one which by affirming denies and the one which by denying affirms. The former is known by the Latin words “modus ponendo tollens”; while the latter is termed the “modus tollendo ponens.”
ILLUSTRATIONS:
(1) By affirming denies.
The defendant is either guilty or innocent,
He is guilty,
∴ He is not innocent.
or
The defendant is either guilty or innocent,
He is innocent,
∴ He is not guilty.
(2) By denying affirms.
The defendant is either guilty or innocent,
He is not guilty,
∴ He is innocent.
or
The defendant is either guilty or innocent,
He is not innocent,
∴ He is guilty.
It may be said that disjunctive arguments depend on two rules. This is the first: The major premise must assert a logical disjunction. A logical disjunction involves two requisites; first, the alternatives must be mutually exclusive; second, the enumeration must be complete.
Illustrations of illogical major premise.
Terms not mutually exclusive:
This boy is either inattentive or indolent,
He is not inattentive,
∴ He is indolent.
It is obvious that the boy might be both inattentive and indolent. Experience teaches that the qualities are usually concurrent, and to assume that the boy must be either one or the other is a clear case of “begging the question.”
Some logicians maintain that “either—or” signify that both alternatives cannot be false, but that both may be true. If this viewpoint were adopted, the major premise of the illustration would not be a case of begging the question. It is unnecessary to argue the point, if it is made perfectly clear which view is to obtain in this discussion. Briefly stated the two points are these. First opinion: “Either—or” when used logically, mean that if the first alternative is false the second must be true, or if the first alternative is true the second must be false. Second opinion: “Either—or” when used logically mean that if the first alternative is false, the second must be true; but if the first alternative is true, the other may or may not be true. This treatise adopts the first opinion. With us all alternative arguments to be logical must be mutually contradictory; i. e., when one is false, the other must be true and when one is true the other must be false; both cannot be false, neither can both be true. When it is intended that this implication should not obtain, then the expressed alternative will take this form, “The boy is either inattentive or indolent or both.”
Other examples where the terms of the disjunctive may not be mutually exclusive:
(1) “Lord Bacon was either exceedingly studious or phenomenally bright.” (Undoubtedly he was both.)
(2) “This teacher is a graduate either of Harvard or of Yale.” (Perhaps both.)
(3) “The defendant is either a liar or a thief.” (The one often leads to the other.)
(4) “To succeed one must either seize the opportunity as it passes or make his own.” (The best success results from doing both.)
Incomplete enumeration:
The cause of the disease was either the water or the milk,
It was not the milk,
∴ It was the water.
When such an argument as this is advanced, it must be with the knowledge that every other alternative has received satisfactory investigation. Without this assurance one could justly claim that the disease might have been caused by the meat or fish supply. Complete enumeration means that the investigation has narrowed the facts to the boundary of the field covered by the alternatives. The fallacy of incomplete enumeration is also one of “begging the question.”
Other examples of a possible incomplete enumeration:
(1) “Jones lives either in Boston or New York.”
(2) “Mary is studying either algebra or geometry.”
(3) “He either committed suicide or was lynched.”
(4) “Either the Giants or the Boston Americans will win the pennant.”
The second rule is made so self evident by the first that there is little need of a detailed discussion concerning it. The rule is this: When the minor premise affirms or denies one of the alternatives of a logical disjunction, the conclusion must, in order, deny or affirm all of the others. To put it differently: When the “minor” affirms, the conclusion must deny every other alternative, and vice versa. When there are but two alternatives reference to any of the foregoing disjunctive arguments will make the rule clear. There may be, however, more than two alternatives. In such a case, if the first rule is observed then the second becomes applicable.
ILLUSTRATIONS:
(1) John Doe lives either in Boston, Albany, or New York;
He lives in New York,
∴ He does not live in either Boston or Albany.
or
He does not live in New York,
∴ He lives in either Boston or Albany.
(2) The season must have been either summer, or autumn, or winter, or spring;
It was neither autumn, nor winter, nor spring,
∴ It must have been summer.
or
It was either autumn, or winter, or spring,
∴ It could not have been summer.
It would seem that the laws of the disjunctive contradict those of the categorical syllogism; for we apparently derive from two affirmatives a negative conclusion, and we also derive an affirmative conclusion when one premise is negative. This objection is seen to be nugatory when the disjunctive is reduced to the categorical form. The reduction involves the two steps of first changing the disjunctive to the hypothetical form and then to the categorical form. The following illustrations will suffice to make the matter clear:
(1) Disjunctive.
A is either B or C
A is B
∴ A is not C
Hypothetical.
If A is B, then A is not C
A is B
∴ A is not C
Categorical.
The case of A being B is the case of A not being C
In this case A is B
∴ A is not C
(2) Disjunctive.
The defendant is either guilty or innocent;
He is not innocent,
∴ He is guilty.
Hypothetical.
If the defendant is guilty, then he is not innocent;
But he is guilty,
∴ He is not innocent.
Categorical.
The case of the defendant being guilty is the case of the defendant not being innocent,
In this case the defendant is guilty,
∴ In this case the defendant is not innocent.
The majority of us are acquainted with the dilemma as related to the activities of life. One is in a dilemma when there are two courses open to him but neither is particularly enticing. One is placed in a dilemma when he is forced to choose the lesser of two evils. For example, one may, without the proper equipment, be overtaken by a heavy rain storm; he seeks the shelter of a wayside shed; the rain continues so that he is forced either to miss his train, or to endure the discomfort of a drenching. Thus the logical dilemma limits one to a choice between alternatives, either one of which might well be avoided.
Definition.
The dilemma is a syllogism in which the major premise consists of two or more hypothetical propositions, while the minor premise is a disjunctive proposition.
It being a combination of hypothetical and disjunctive propositions the dilemma is sometimes appropriately referred to as the “hypothetico-disjunctive” argument. The order of the premises is indifferent, yet it seems to be more natural to use the hypothetical first; thus the definition.