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** EXERCISES
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- Joseph A. Nyberg
Hyde Park High School - Chicago
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Chapter Page Chapter Page
& s 9. Recessary and Sufficient
1. General Statements. 3 Conditions. 44
2. How General Statements are used. l3 || 10. Inductive, Scientific, and
3. Refinitions, Assumptions, Proofs. 18 Deductive tiethods . 46
4. Geometric Proofs. tº 25 ll. i.istakes in Reasoning. 51
5. Converses • 31 || 12. Problem Solving. © 61
6. The Indirect Method. 34 l3. Euler's Circles. © 67
7. Inverses . 38 || 14. Propaganda © 70
8. Contrapositives. 40 || Miscellaneous Exercises º 76
I N T R O D U C T I O N
If you were beginning to study algebra it would be foolish for you to take an exam-
ination in algebra on the first day. But in all your classes you have learned some-
thing about how to think. Hence you can begin this work with an examination that
will test your ability to think. Later, when you have studied this book, you should
try the same examination again. You will then have some evidence to show how much
you have learned about thinking.
. The word "discuss" in any problem means: State whether or not the argument is
correct, give your reasons, and add any explanations that will show that you can
tell the difference between right and wrong arguments.
After the test is finished you can, if the teacher approves, read some of the ex-
planations in class but do not try to decide what the correct answers are or ask the
teacher to tell you. That would require too much time; you will learn the correct
answers as you study this book. Keep your examination paper until the course is
finished; then you can take the test again to see how much you have improved.
T H E T E S T
l. Discuss : I know Mr. Level belongs to the H-Y-P Club because you must be a
graduate of Harvard, Yale, or Princeton to belong to that club. And Mr. Level is a
graduate of Harvard.
2. Discuss: The pupils in the Lowell High School join more clubs than the pupils
in our school. I have a cousin who goes to Lowell and she says they have 37 Clubs in
the school. We have only 21. Some of our pupils don't join even one club.
3. Let us admit the truth of this statement:
No amateur football player gets a salary for his playing.
It happens that John Stanley, a friend of yours, is not an amateur player. Which of
the following statements are then true:
a) John gets a salary for his playing.
b) John does not get a salary for his playing.
c) From the above information we cannot decide whether or not he gets a salary.
4. Discuss: When running for reelection to office Senator Muncie said, "All men
who make any claim to be liberal minded hold these views which I have just been
explaining. I hold these views. Hence I have a right to say that I am a liberal."
Copyright 1940 -Josepb A. Nyberg
5. May said, "If John were in Chicago, I know he would phone me."
Ruth answered, "I'm sure he would."
If May's statement is true, which of the following are then also true:
(a) If John phones May, then he is in Chicago.
(b) If John is not in Chicago, then he will not phone her.
(c) If John does not phone May, then he is not in Chicago.
6. Jones, Smith, and Adams were dining at the Rose Hotel.
Jones said, "This hotel serves terrible meals; it always has served poor meals."
Smith said, "That is hardly true. If it were, the hotel would soon go out of business .
Adams said, "Your argument, Smith, is no good. You assume that if the hotel does
not go out of business then it must be serving good meals. And that is not the same
as saying that if it does not serve good meals it will go out of business."
Who is right, Jones, Smith, or Adams?
7. Discuss : The Forg post Baseball Team was arguing about whether or not they
should buy new uniforms now or wait a month. After Jim had finished arguing for the
purchase now, Henry said, "Ah, Jim wants us to buy them now so that his uncle can
make a profit selling them to us. Remember how we lost last week's game when Jim
dropped the ball in the ninth inning. What does he know about baseball or uniforms."
8. When Mrs. Arno rented an apartment in the Hoyne Flats she knew that the lease
(the contract with the owners) said that a tenant could keep only domestic pets in
his apartment. Later the owner learned that she kept a pet monkey and ordered her to
move out. Did the owner have a right to ask her to move?
9. Discuss the following statement by a pupil in a New York City high school: "In
the Adams High School the pupils spend two days each month visiting dairy farms. The
pupils in every high school ought to do the same thing. Certainly milk is one of our
most important foods, and we all need to know something about it. "
10. We had been driving west, and in the middle of the afternoon came to a place
where we had a choice of three roads. The road to the left was paved. The road
straight ahead and the road to the right were unpaved and hilly.
While we were discussing which road to take my wife said, "I remember Harry told
us last week to take the unpaved road."
"And", added my daughter, "that night at the dance he said that he had sunburnt his
left arm by resting it on the window ledge of the car while driving there."
Which was the correct road, and why?
ll. Discuss: The application of heat to any metal will cause it to expand because
cooling the metal causes it to contract.
12. Discuss the advertisement: "Do you want to be beautiful? Hollywood demands its
girls stay beautiful. Wondero Facial Cream is a best seller in Hollywood."
13. Discuss: Rare objects are very expensive. Inexpensive houses are rare in any
large city. Hence inexpensive houses are very expensive in any large city.
14. Discuss: The Hill Co. sold bonds valued at $1,000,000 to its customers in 1936;
$1,500,000 in 1937; and $2,000,000 in 1938. The number of people who bought bonds
from this company increased each year.
15. Discuss: Mrs. Ames said to a friend, "The Hitore Shoppe sells only sweaters made
of pure wool. I've always wanted a pure wool sweater because they are so warm. This
sweater is not pure wool. I can't afford to trade at the Hitone Shoppe."
C H A P T E R l
G E N E R A L S T A T E M E R T S
WHY WE BELIEVE SOME THINGS. A tree has leaves. The motor of an airplane makes a
loud noise. Henry's fountain pen is broken. If someone should ask us why we believe
these statements we should feel that the questioner was just trying to annoy us. We
should doubtless answer, "Can't we see? Have n't we eyes and ears?" There are some
statements which we believe because we can see, hear, feel, and so forth.
But we also make a great many other statements whose truth is not so obvious. Why
do we believe the sun is about 93 million miles from us? Have we ever measured the
distance? How do we know that I apoleon lost the battle at Waterloo? We were not
there and have not talked to anyone who was there. How do we know that Melba was a
famous singer? We never heard her sing.
There are statements which we believe for some other reasons besides "Can't we see?
Have n't we eyes?" We shall study what these reasons are.
Let us begin with a simple exam.ple. When coming to class you might have passed
Jim rushing to his locker. If asked for a reason for his great hurry he might have
said, "I must get my notebook for History. I left it in my locker." If he had more
time he might have added, "The tardy bell will ring in half a minute. It takes
almost a minute to get to my locker because it is on the other-side of the building
because T'm a senior, and seniors have their lockers on the other side of the building
because their home room is the assembly hall which is on the east side because its
main entrance had to be on Grand Boulevard because . . . . . because . . . . because . . . . .
Or, Mr. Smith might mention to a friend that r armer Adams is a poor man because he
does not keep the weeds out of his garden. Hir. Smith was thinking as follows:
If there are few weeds in a garden, the corn will grow better.
If the corn grows better, the farmer will earn more money.
If the farmer earns more money, he will not be poor.
So Adams is poor because he does not keep the weeds out of his garden.
In these two examples we notice the following ideas:
A certain statement is true because a certain other statement is true, and this
other statement is true because some other statement is true, which in turn is true
because some other statement is true, and so on, and so on. Finally we must get to
some statement which none of us doubt and hence no reason for it is necessary, or we
reach a statement which our listener will not accept and the whole argument collapses.
When all the statements are gathered and arranged so that we can see how they depend
on each other, we say that we have proved the first statement.
A proof uses two kinds of statements:
l. Statements which are true because we can find other statements as a reason for them.
2. Statements which do not have other statements to support them because we are all
willing to accept them without a reason. These statements are called assumptions.
If the assumptions are false, then our argument proves nothing. In the argument
about Farmer Adams the last statement was . If the farmer earns more money, he will
not be poor. But someone might answer, "Oh, is that so? Do you mean that we are
always richer if we earn more money? Suppose my salary is doubled and the price of
everything is more than doubled. Am I any richer?"
If no reason can be given for some statement, then that statement is an assumption.
Just as a chain is not stronger than its weakest link, so no proof is better than the
assumptions which it uses.
E x e r c i s e s
1. What is the difference between an assumption and other statements?
2. That is meant by proving a statement?
3. *ſhy are assumptions needed in proving a statement?
4. What is the missing word in the sentence: I could not convince Jones because
he said my . . . . . . . . . . . . . . . were Wrong.
5. Which of the following statements should you say are assumptions? (it is not
expected that all members of the class will agree. A statement which seems obvious
to one person may not be accepted by someone else without a reason.)
(a) A democracy is a better form of government than a monarchy.
(b) The square root of 25 is 5.
(c) The capital of Australia is Melbourne .
(d) A good farmer rotates the crops in his fields.
(e) A square has four sides.
(f) The government owes every man a job.
(g) The sum of the angles of a triangle is 180 degrees.
6. Which parts of ex. 5 caused the most discussion or disagreement? Can you find
a reason why there was so little disagreement about some of the statements?
7. Write some examples of reasoning like those on page 3 in which a statement is
supported by a reason which is supported by another reason, and so forth. Do not have
all the thoughts written in one long sentence. Divide the argument into a set of
related sentences. You might award a prize to the pupil who can write the longest
argument, that is, the argument with the most "becauses".
8. As in ex. 7 continue the following arguments :
(a) One evening John's father said, "I intend to buy some life insurance because
some day I'll die. Some day I'll die because . . .
(b) A wooden house should be painted once every three years because . . .
(c) Children should drink a quart of milk every day because . . .
9. The following is a part of a speech at a business men's banquet in 1938:
"Business recovery will get under way when the national income begins to increase.
That will happen when payrolls begin to expand. The payrolls will grow larger when
there is an increase in the manufacture of durable goods; and probably the change
will first be noticed in an increase in the output of iron and steel."
Supply the missing words in the following rearrangement of this speech:
If there is an increase in the output of iron and steel, more durable goods will be
produced. If more durable goods are produced, then . . . . If payrolls
increase then . . . . . If . . . . . . . then business will recover .
A BRIEF OF A DEBATE. Hany schools have debating clubs for discussing such topics as:
All final examinations should be abolished.
Tariff's should be imposed for revenue only.
Wages for men and women should be the same when the same work is performed.
To make other people believe as you do about such questions you attempt to prove
your statements; that is, you make certain statements and give reasons for them. As
an example of a proof, page 5 shows an outline (debaters call it a brief) of a debate
on the topic
The American game of football should be abolished.
In the brief we see that
l. It begins with an explanation, part I, of why the question is important.
2. To make sure that the question is clear, part II defines the word football so
that it will not be confused with other games.
3. In any debate there are certain points on which all the debaters agree (hence
need no discussion) and certain points on which they disagree. Hence part III states
the points that the argument will try to settle. Those who believe the answers to
these questions are "yes" are said to be on the affirmative side of the debate; the
others are on the negative side -
4. The brief next states that football should be abolished for reasons I to IV.
Statement I has reasons marked A and B. Statement A has reasons marked l and 2.
Statement, B has reasons l and 2. Statement l under B has reasons a and b.
Here we ask: When do wie stop giving reasons? ..hy is n't there a reason for item b
under 1 under B under I ? Why are the students eager to have the team win? And if we
give a reason for this reason, should there not be a reason for that reason?
We stop when we reach a statement which everyone will accept without a reason.
Eventually we must reach a point where another reas on is not needed because both
parties to the debate will accept it without a reason. In most cases the last reason
will be a simple fact such as: On Nov. 3 the students celebrated the victory with a
bonfire. (If this is doubted, some witness must be produced.) Or the reason may be
a fact like: The gate receipts were £6,543. (If this is doubted, the business office
must testify.) In other cases the last reason may be a statement which may be open
to dispute but which both parties agree to accept. Such a statement is called an
assumption.
An assumption is a statement which may or may not be true and which does not have a
reas on to support it.
The American Game of Football Should be Abolished
I. The origin and history of the question:
A. Give statistics on the number of deaths and injuries.
B. In 1905 Columbia University abolished the game.
C. Other universities asked for a change in the rules.
D
. Various committees have tried to make the game safer.
II. The following terms require definition:
A. The American game of football is the game played under the rules published in
Spaulding's Annual Football Guide, not association football nor rugby.
III. The question is resolved into the following main issues:
A. Does football detract students and players from other collegiate pursuits to an
extent that warrants abolishing the game?
B. Are the accidents resulting from football sufficiently numerous or serious to
warrant abolishing the game?
C. Must we abolish football to overcome the undesirable commercial features?
D. Does football lower the morals of players and spectators?
Proof for the Affirmative.
I. Football detracts students and players from other school pursuits, for
A. The players are drawn from their studies, for
l. Football demands much training and practice.
2. The player's mind is filled with signals and plays that must be learned.
B. Other students neglect their work, for
l. They are filled with enthusiasm over the coming game, for
a. They are eager to see a contest.
b. They are eager to have their team win.
2. The celebrating of victories is quite common.
II. Injuries and deaths are numerous.
(quote figures)
III. Football has been made a commercial enterprise.
(give the evidence)
IV. Football lowers the morals of players and spectators, for
A. Wrong ideals are created, for
etc.
E x e r c i s e s
1. Prepare a brief like that on page 5 to uphold either side of some resolution
like the following:
(a) The work of the editor-in-chief of your high school paper (or the annual) should
count as credit toward graduation.
(b) A course in mathematics should be required for graduation from high school.
(c) No pupil should be allowed to hold more than one major office.
2. If you were debating the question "The R 0 T C should be abolished" which of
the following statements would you consider as assumptions and which would need some
proof:
(a) The R O T C helps to maintain order in school at firedrills and other events in
which all the pupils participate.
(b) The boys in the R O T C learn to improve their personal appearance.
(c) The boys secure much valuable information about national defense.
(d) The boys who like the kind of drill obtained in the R O T C are those who like
to domineer other people.
3. If you were having a debate on the question whether the C C C (Civilian Conser-
vation Corps) should be abolished, what are some of the agreements that both parties
to the debate should make?
4. If you were debating the question "Labor Unions are helpful to the workers"
which of the following statements would be assumed by both parties to the debate?
(a) Labor Unions cause many strikes.
(b) Without cooperation nothing can be accomplished.
(c) If no maximum working hours are set, some people would be forced to work more
hours a day than their health permits.
In the libraries you can find books which contain briefs of all the important
questions which have been debated by college teams. By reading some of the briefs
you can learn much about "proving" a statement.
WHAT A GENERAL STATEMENT IS. Sometime you must have heard a teacher say
"Dick, you must do your homework every day. "
The teacher was talking only to Dick, just to one boy. But you, and Dick, and other
boys knew that the teacher was thinking that
Boys should do their homework every day.
The first statement referred only to one specific boy – Dick. The second statement
refers to all boys - Dick, Tom, Harry, Bill, and others. The statement can also be
changed to include girls :
Pupils should do their homework every day.
The third statement refers to a still larger group of people. We can change it so
that it will refer to a much larger group of people by saying
Everyone should do his work every day.
A statement which refers or applies only to one object is called a specific state-
ment. A statement which refers to many objects is called a general statement.
A general statement says something about all the objects of a group. The larger the
group is, the more general the statement is.
Generalizing a statement means rewording it so that it applies to more objects.
E x e r c i s e 8
l. Make a specific statement about an automobile, or tree, or house, or some object,
and then generalize it. Then try to make the statement still more general.
2. What makes one statement more general than another statement?
3. Generalize the statements :
(a) My father works for Olds & Co. making Dixie automobiles.
(b) When I studied history I learned about George Washington.
(c) My bicycle has two wheels.
(d) Jim should drink orange juice for breakfast.
GENERAL STATEMENTS IN ARITHMETIC. To simplify the fraction # you divide
the numerator and the denominator by 4. You have often said
To simplify a fraction, divide the numerator and denominator by the same number.
This is a general statement because it applies to all fractions. Such general
statements are also called laws, principles, or rules.
E x e r c i s e 8
1. What general statement is used when 3/8 is changed to 6/16
2. Generalize the statements:
(a) 3 + 4 = 4 + 3 (e) vſ. TE - ſº a ſig
(b) 4 x 5 = 5 x 4 WST
9 -1s-
(c 3/4 does not equal 4/3 (f) - E-
) 2 16 Wis
(d) 5/8 does not equal (5/8)
3. Notice that (7 * 3) (7 – 3) = 49 - 9
(10 + 2) (10 – 2) = 100 - 4 (9 + 5) (9 - 5) = 81 – 25
What do you think (8 + 4) (8 - 4) equals?
Complete the generalization: If the sum of two numbers is multiplied by the
difference of the numbers, the product is . . . . . .
4. Generalize the statements :
(a) The area of a rectangle 5 in. long and 4 in. wide is 5 x 4 sq. in.
(b) If the radius of a circle is 6, its area is pi times the square of 6.
5. What name is used for the generalizations like those in ex. 4 3
GENERAL PROBLEMS. Suppose you have solved the problem
John can mow a lawn in 40 min. but it takes Henry 60 min. to do
the same work. How long will it take if both boys are working?
(The answer is 24 min. John does 1/40 of the lawn each minute, and Henry does l/60
each minute. Together they do 1/40 + 1/60 or 5/120 or 1/24 of the lawn each minute.)
Let us now generalize this problem thus:
One machine can do a job in a minutes; another machine can do the same job
in b minutes. How long will it take if both machines are used?
ab
The answer is a b minutes. For example, if the numbers are 50 and 80, the
º 50 x 80 . ; e 30 x 60
arl SWer LS TOTSO " if the numbers are 30 and 60, the answer is 30 T65 • The
answer to the general problem tells the answer to all problems of this kind. This
explains why the general problem and its answer are valuable.
E x e r c i s e s
These are problems in arithmetic, not algebra. Solve them as stated here. Then
generalize the problem. (Algebra is needed to solve the new general problem.)
l. What is the per cent of profit on the cost if goods are bought for $4 and sold
for $5. What is the per cent of profit on the selling price?
2. A dealer sold 2000 hats at #3 and 1000 hats at #4. What is the average selling
price? (By average selling price is meant that if all the hats were sold at this one
price, the money received would equal what the dealer did get by selling some at $3
and some at #4. The answer is not $3.50.)
3. A farmer mixes 100 lb. of milk containing 8 % butter fats with 300 lb. contain-
ing 6 % fats. What is the per cent of fats in the mixture?
4. In a steel mill 10 tons of iron ore containing 6 % iron is mixed with 2 tons of
scrap iron containing 80 % iron. What is the per cent of iron in the mixture?
5. A chemist mixes 10 gal. of acid whose strength is 20 % with 50 gal. whose
strength is 30 %. What is the strength of the mixture? (A strength of 20 % means
that 20 % of the material is acid, the other 80 % being water or other matter.)
6. A banker invests $4000 at 5 %, and $6000 at 2 %. What is the rate of income on
the entire investment; that is, what is the average income?
**.
GENERALIZING IN ALGEBRA. Many problems that appear difficult to a beginner in
algebra become simple as soon as we grasp the idea of generalizing. For example,
5 times any number + 3 times that number = 8 times the number.
If n is used to represent any number, the above relation can be generalized as
5n + 3n = 8n
But 5 and 3 are definite numbers; and so the relation can be made more general
by writing ar). + bn = (a + b)n In this relation
a, b, and n are any numbers. But an expression like x + 2 is also a number,
and y + 5 is also a number. Using x + 2 for a , and y + 5 for b, we can
write (x + 2)n + (y + 5)n = (x + 2 + y + 5)n.
But n is also a number; and so it could be the number s t t . Hence
(x + 2) (s , t) + (y + 5) (s + t) = (x + 2 + y + 5) (s + t)
Looking at it now you should hardly realize that this is merely one form of the
simple relation an * bn = (a + b)n.
As a second example, consider the equation 6x = 12. We know that x = # ... We
can generalize the equation by writing ax = b. Then x = b / a .
A more general form is
If (a + b)(c + d) = e 4 f then a + b = 2–H
E x e r c i s e s
Write some more complicated generalizations of:
2 2 2 Blººm 8.
l. 5 + = 5" + 2 5 6 + 6 5. - F -
º 6) X C X bn lo
2. x - y? = (x + y) (x - y) 6. ſab = Wa x b
3. If x + a = b then x = b - a 7. * + 9 = ad t bo.
b d bd
X – := +
4. If a T ; then bx ay 8. a + n does not equal #
9. Find a simpler statement for which the following is a more general form:
2
(a) (x + y + z) = (x + y)* + 22(x + y) + z*
(b) (x + 3) (y + 2) E x + 3
(x + 4) (y + 2) x + 4
The solution of the following problems require skill in algebra. You should be
able to generalize them even if you cannot solve the generalizations.
10. If a bushel of corn is worth 75 # and a bushel of oats is worth 60 4, how many
bushels of each should a miller mix to make 80 bushels worth 70 g a bushel?
ll. How many gallons of cream containing 20 % fats and how many gallons of milk
containing 8 % fats should be mixed to make 100 gallons containing 12 % fats?
12. How much water should be added to 10 lb. of a 15 % salt solution to make the
result a 5 % solution?
13. How many pounds of pure salt should be added to 40 lb. of a 30 % solution to
increase the per cent of salt in the result to 50 %
14. A man wishes to invest $3000, part at 6 % and the rest at 3 } so as to get an
average of 4 % on his $3000. How much shall he invest at each rate?
15. How much water should be evaporated from 30 gal. of a 10 % salt solution to
make the remainder a lă 7% solution?
16. Prepare a speech that you can present in class on "The Importance of Generali-
zations". You should explain what is meant by generalizing, give some examples, and
explain why general statements are more useful than specific statements.
The ability to generalize is one of the best tests of a person's intelligence. Much
of what you learn from one problem is useless unless you can apply it to another
problem of the same kind.
GENERALIZATIONS IN GEOMETRY. In the first figure below, 2 BAC = 50". Since DB is
a straight line and since 4. BAC + 4 CAD = 180° we know that 4 CAD = 130'. But EC
is also a straight line, and so 4 EAD + 4 CAD = 180" . Hence 2. EAD = 50°. Therefore
2. EAD = <. BAC

C
We could repeat the same kind of D I
argument for the second figure, in A F
which Z, GFH = 40 , showing that
& JFI = 4. GFH. E B G

No matter what size 4 BAC is we can
repeat the argument. Hence we can make a general statement which will apply not
only to the two figures above but to all figures of this kind. The generalization is :
When two straight lines cross, the angles which are opposite each other
are equal; or, as some pupils say Wertical Angles Are Equal.
In the future, when we see two vertical angles, we need not repeat the argument
stated above. We think "Here are two vertical angles. The argument used on page 9
permits us to say that these angles are equal". E
Which angles in the figures below are equal?

D — C C
E D A E; C
E
A B A B


D
GENERALIZATIONS IN OTHER SUBJECTS. You use generalizations in every subject that
you study. The rule of grammar which says that a plural subject should be followed
by a plural verb is a general statement. You also use rules like these :
French: Werbs whose infinitive end in er belong to the first conjugation.
Spanish: In a question the subject is placed after the verb.
German: Nouns ending in el, en, or er are the same in the plural and singular,
History: A man must be 35 years old to be chosen president of this country.
Geography: Cities that are near large coal and iron fields become industrial centers.
Science: An object floating on water displaces a volume whose weight equals the
weight of the object.
E x e r c i s e. Collect some examples of generalizations from whatever subjects
you are studying in school. What use do you make of these generalizations? What is
an important difference between the generalizations in Science and those in French or
Spanish or any language?
HASTY GENERALIZATIONS. If you had taken a trip through southern England about
August 12, 1938 you could have seen the children throwing snowballs, using sleds, and
wearing their winter clothes. You might have written home to your friends "When you
visit England in the summer, be prepared for winter weather".
This is an example of a hasty generalization based on insufficient evidence. Just
because there was a snow storm in August of one year it would not be right to say that
there is a snow storm in August of every year. It is wrong to make a generalization
from a single observation. If you found a freshman studying algebra you should not
conclude that all freshmen study algebra. Perhaps all the college graduates whom you
know are successful in their work; but that fact does not prove that all college
graduates are successful. In Chapter 10 we shall discuss how we try to make sure that
generalizations are correct; at this time we emphasize that we must not make general-
izations from a study of only a few cases. Further, although thousands of examples
will not by themselves prove that a generalization is correct, one single example is
enough to show that some statement is not true for all cases. Evidently, if a rule
fails to work just once then it does not work all the time.
1O
E. x e r c i s e s
1. Give some examples of hasty generalizations from your own experiences or from
conversations you have heard.
2. Find the area and the perimeter of each of the right triangles shown below.

LO l
12 8 - 2O
Find the area and the perimeter of a right triangle whose hypotenuse is ll lyā,
the other sides being 5 1/3 and 10.
Next, try your conclusion on a right triangle whose sides are 3, 4, and 5.
3. Prime numbers are numbers that are not the product of two other integers (the
number 1 and the number itself being excluded) such as 3, 5, 7, 11, 13, 17, . . . .
Find the value of the quantity x + x + ll when x is 1, 2, 3, 4, 5, . . .
Is the result a prime number? Next find the value of the quantity when x = 10.
4. In the figures below, AD bisects 4. BAC. Does it look as if AD bisects BC in
the first three figures? in the fourth figure?
C A A
D D C
..A
B
B B D C B



EXAGGERATED DRAWINGS. Ex. 4 above suggests a way of detecting some errors in
generalizing. Below are some four-sided figures in which the opposite sides are
equal. The lines joining opposite corners are called diagonals.


X /X7 2-7

From the first two figures we might suspect that the diagonals are perpendicular
to each other. To check or test this idea, draw a third figure making it unusually
long. It shows that our suspicion was wrong.
To test an idea about triangles do not draw one that looks as if all the sides were
equal; make the sides decidedly unequal. To test an idea about isosceles triangles
you must make two sides equal, but the third side should be unusually short or long.
To test an idea about right triangles you must have a right triangle, but make the
acute angles decidedly unequal. When drawing a four-sided figure do not make any
sides equal unless the directions demand it; do not make an angle a right angle unless
the directions demand it.
To test an idea, exaggerate the lines or angles as much as possible.
E x e r c i s e s
Test any generalizations that you might make from the following experiments:
1. Construct the three angle-bisectors in several triangles.
2. Construct a perpendicular from each corner of a triangle to the opposite side.
3. In each of several triangles construct the three medians.
4. Construct the perpendicular bisectors of the sides of a triangle.
5. Draw the line joining the midpoints of two sides of a triangle. Compare the
length of that line with the length of the third side of the triangle.
6. In a certain kind of four-sided figures the diagonals will not only bisect each
other but will also be perpendicular to each other. By experimenting with various
figures, find for what kind of four-sided figures this is true.
ll
HYPOTHESIS AND CONCLUSION. To explain the meaning of these words we shall use
the general statement
If a man lives in Chicago, then he lives in Illinois.
This sentence makes a statement about men living in Chicago. If we ask, "What does
the sentence talk about?" the answer is "About men who live in Chicago". It is
not talking about all the men in the world but only about the group of men who live
in Chicago. The part of the sentence that tells what the sentence is talking about
is called the hypothesis of the statement. The hypothesis mentions the group of
objects that we are discussing. The hypothesis calls our attention to a particular
collection of objects.
If the sentence has a subordinate clause beginning with "if" then that clause is
the hypothesis. The "if" clause may appear at the beginning or at the end of the
sentence . The above example has the same hypothesis when it is stated as
A man lives in Illinois if he lives in Chicago.
Next we ask, "What does the sentence say about this group of men?" The sentence
gives certain additional information about the men in the group. It says "he lives
in Illinois". This part of the sentence is called the conclusion. Notice that
the conclusion is the principal clause in the sentence.
E x e r c i s e s
State the hypothesis and the conclusion in the following sentences:
l. If I had a bicycle I could ride to school.
2. I am a freshman if I am studying General Science.
3. Owning a farm would be more profitable if the price of corn were higher.
4. If Tngland owned no colonies and were not an island it would not need a navy.
5. If my earnings were greater I could afford that expensive car provided that I
did not spend my money for other things.
6. If I were at least 5 ft. 6 in. in height and if I weighed 130 lb., I could be
on the basket ball squad.
STATEMENTS WITHOUT AN "IF". Many statements do not contain a subordinate clause
beginning with if. In that case you can either rewrite the sentence so that it
expresses the same idea and does have an if , or, bear in mind that the hypothesis
tells what we are talking about. For example, the sentence
Sugar is sweet
talks about a substance called sugar, and it says that this substance is sweet. Hence
the idea can be stated: If this is sugar, then it is sweet.
Consider next a longer sentence like
Cities that have crowded areas with wooden buildings covered with
wooden shingles have many fires.
Here we are talking about cities, but not about all cities - just the cities that
have crowded areas. Further, we are not talking about all cities that have crowded
areas - just about those in which there are wooden buildings covered with wooden
shingles. The noun cities is modified by the clause "that have . . . . ." ; the noun
areas is modified by the phrase "with wooden buildings"; the noun buildings is
modified by certain words. Hence we see that
The subject of the sentence with all its modifiers is the hypothesis.
The verb and the predicate of the sentence make the conclusion.
E x e r c i s e s
Rewrite the following sentences so that the same idea will be expressed by using an
if clause. State the hypothesis and the conclusion.
l. Grass is green. 4. No plant can grow without light and heat.
2. January has 31 days. 5. Never will the dead speak.
3. Triangles have 3 sides. 6. The government should operate public utilities.
7. A farmer who does not cultivate his corn fields cannot expect a large crop.
8. He is wise who conquers his prejudices.
12
HYPOTHESIS AND CONCLUSION IN GEOMETRY. If a statement has an "if" clause then that
clause is the hypothesis. The main clause is the conclusion. If there is no "if"
clause, we must first decide what the statement is talking about. The sentence
The base angles of an isosceles triangle are equal
talks about angles since angles is the subject of the sentence. Base is an adjective
modifying angles. "Of an isosceles triangle" is a prepositional phrase modifying
angles. "isosceles" is an adjective modifying triangle.
All these ideas are part of the hypothesis. We have a triangle; the triangle is
isosceles; and we are making a statement about its base angles.
The conclusion is "are equal". We are saying that certain angles are equal.
As another example, consider the sentence
The bisector of the vertex angle of an isosceles triangle bisects the base.
The subject is "bisector"; it is modified by the phrase "of the vertex angle".
The word "angle" is modified by "of an isosceles triangle". "Vertex" is an adjective
modifying "angle". "Isosceles" is an adjective modifying "triangle".
For the figure at the right, C

the hypothesis is : AC = BC, and
CD bisects 4. ACB.
the conclusion is ; AD = DB.
The statement could be worded: D B
If a line bisects the vertex angle of
isosceles triangle, then that line bisects the base of the triangle.
E x e r c i s e s
State the hypothesis and the conclusion of these statements:
l. Vertical angles are equal.
2. Complements of equal angles are equal.
3. Two equal supplementary angles are right angles.
4. If two adjacent angles are supplementary, their exterior sides lie on one line.
5. The lines joining the opposite corners of an equilateral four-sided figure
bisect each other.
6. The bisectors of supplementary adjacent angles are perpendicular to each other.
7. If two triangles are congruent the bisectors of corresponding angles are equal.
8. The bisectors of the base angles of an isosceles triangle are equal.
9. You can test your knowlege of grammar and of the above ideas as follows: In
your geometry text select some theorems (in the last part of the book) in which there
are some words you do not understand. Select the hypothesis and the conclusion by
using the idea that the subject and its modifiers is the hypothesis, and the verb and
predicate make the conclusion. Frequently it is easier to select the conclusion
first; then the rest of the ideas in the sentence form the hypothesis.
13
C H A P T E R 2
H O W, G E N E R A L S T A T E M E N T S A R E U S E D
THE IF-THEN IDEA. As a simple example, consider the statement
If you dive into a lake, then you get wet.
Here, diving into a lake is the cause of getting wet. Or, getting wet is the
result of diving into a lake. The hypothesis is the cause of the conclusion, and
the conclusion is the result of the hypothesis.
You are not compelled to dive into a lake, but if you do, then you must expect
certain results. You are not compelled to put a lighted match into a gasoline tank,
but if you do, then you must not complain about the result. You are not compelled
to drive a car sixty or seventy miles an hour, but if you do, you can expect certain
results to follow. Every triangle does not have two equal sides; but if it does,
then certain results follow > All numerators and denominators are not divisible ex-
actly by 2, but if this is so in some fraction, then certain things can happen.
The general statements which we shall study are often called "If–then" relations
because they state what is the result of an "if". We also say that the hypothesis
"implies" the conclusion. The noun "implication" corresponds to the verb "implies";
and we ask, "What are the implications from the hypothesis?" or "What do you infer
from the hypothesis?" The noun "inference" corresponds to the verb "infer" as in
the sentence "What inference do you make from the hypothesis?"
These ideas are well expressed by a certain slang phrase that was once popular.
When someone said, "It looks like rain to-day" or "I got 100 in that test" the
reply would be "SO WIAT’’: " meaning "iiow would your life be changed if it actually
did rain to-day?" or "If you really did get 100, what happened as a result?"
The "If – then" idea is used thousands of times daily. A chemist thinks: If I mix
these fluids, then what? The engineer thinks : If I increase the weight at this
point, then what? The doctor thinks: If I give the patient this medecine, then
what? You think: If I spend my allowance to-day, then what of to-morrow? This book
can be described as a study of "If - then" relations.
A GEOMETRIC EXAMPLE. The figure below was made by drawing a line AC and bisecting
it at B. Next, perpendiculars to AC were drawn D
through A and C. Finally, any line was drawn *
through B intersecting the perpendiculars.
When this is done, the consequences are: A B | C
BD = BE AD = CE 4. ADB = & CEB.
Did we make BD = BE2 NO! He made AB = BC
and made right angles at A and C.
Did we make AD = CE? NO! Wie made AB
and made right angles at A and C.
Did we make 4 ADB = 4 CEB? No! We made AB = BC and made right angles at #and C.
In other words, we did certain things (which we need not repeat ) and 3:3# e n
certain other things followed as an inevitable consequence. (In your geometry class
you learn how to find what the consequences are.) After we have done certain things
we could not in any possible way prevent certain other things from following. No
law of congress could prevent the inevitable consequences. Nothing could stop it!
In every subject that you study, in all daily work, all through your life, you will
be studying
What is the "then" that follows every "if" 2
E
BC


14
E x e r c i s e s
In each exercise state what conclusions you might draw from the information. Use
the words "implication" and "inference" as much as possible. For example, you can
say, "These facts imply that . . . " or "The implication is ... or "From this data
I infer that ... " or "The inference is that ... "
1. John is a normal boy. A circus comes to town. (In this, and in other exercises,
the results are not inevitable, but for practice you can pretend that they are.)
2. May is a movie fan. A popular movie comes to town. Her mother gives her a dime.
3. Dick is hungry. He sees a cookie on the kitchen table.
4. Henry needs to take either physics or chemistry to graduate. He takes neither.
5. Suppose in ex. 4 that Henry did not take physics but did graduate.
6. The dikes along a river were 10 ft. high . The river rose l? ft.
7. You see a man rush out of a house, carrying a screaming child and drive away
hurriedly in his car.
8. A policeman with a note-book in his hand, talking to the driver of an automobile.
9. A farmer is filling the gasoline tank of his tractor.
USING A GENERAL STATEMENT. One of the rules in a certain school is
If a pupil arrives after 8:30 he is marked tardy.
If John, a pupil, arrived at 8:35 was he tardy?
If Mary, a pupil, arrived at 8:25 was she tardy?
If Miss Adams, a teacher, arrived at 8:40 was she tardy?
The correct answer to the third question is "We do not know". The rule applies
only to pupils; and we do not know the corresponding rule for teachers.
A general statement can be applied only to
those objects that satisfy the hypothesis.
If the word "satisfy" in this sentence bothers you, think of it in this way:
A general statement is a statement about a certain class or group or collection of
objects (in this case, pupils) and we can use the statement only if we are dealing
with an object that belongs to or is a part of the group. The rule above applies only
to pupils. It does not apply to teachers because they do not belong to the group.
For example, a statement about dogs cannot be applied to cats, but a statement
about animals would apply to both dogs and cats. A statement about six-cylinder
engines cannot be used if we are dealing with eight-cylinder engines.
When we use a general statement our thinking involves 3 steps:
1. The general statement: If a pupil arrives after 8:30 he is marked tardy.
2. A specific statement: John is a pupil and he arrived after 8; 30
3. The result: John was tardy.
Step 1 is called the major premise. Step 2 is called the minor premise. We must
be sure that the minor premise deals with one of the objects mentioned in the major
premise. Step 2 must "satisfy" or "affirm" the major premise. Step 3 is called
the conclusion. You notice that the word conclusion is used in many different ways.
The general statement in step 1 has a hypothesis and a conclusion; and the entire
argument, consisting of the 3 steps, also has a conclusion (an end or finish or
result or consequence of steps l and 2).
F. x e r c i s e s
State the conclusion if the first sentence is the major premise and the second is
the minor premise:
l. Boys in the lºorgan Academy wear blue caps. Jim is a pupil in this academy.
2. Adjectives modify nouns. Beautiful is an adjective.
3. Scillas grow in shady spots. Campanulata are a kind of scillas.
4. Rectangles are parallelograms. The figure I have drawn is a rectangle.
5. Iron does not float on water. This toy is made of iron.
6. Children between 6 and ló years must attend school. Henry is nine years old.
15
AWOID THIS ERROR. The rule The Hypothesis àust Be Satisfied is very
simple, but mistakes are made through carelessness.
Suppose a major premise is If the dress is blue, the customers will like it
and the minor premise is : This dress is green.
Here we must not conclude that the customers will not like the dress. All that we
can say is that we do not know if the customers will like the dress or not. Maybe
the customers will like it; maybe the customers will not like it. The major premise
deals with blue dresses; it says nothing about green dresses, nor purple dresses,
nor brown dresses, or about hats, overcoats, or motor cars. It deals only with blue
dresses, with dresses that are blue in color, with blue dresses, with dresses that
are blue. You see the dress must be blue. If it is not a dress and not blue, then
the general statement cannot be used.
The rule The Hypothesis Must Be Satisfied does not say anything about
the conclusion. It is the hypothesis that must be satisfied.
If a major premise is Boys in the Clay School wear black caps
and a specific statement is fienry wears a black cap
we must not conclude that Henry attends the Clay School. He may attend some other
school where the boys also wear black caps. The major premise deals with boys who
attend the Clay School, not with boys who wear black caps.
The minor premise must say that we are dealing with one of
the objects mentioned in the hypothesis of the major premise.
E x e r c i s e s
Supply any missing conclusion. Correct any wrong conclusion. If no conclusion
is possible, tell why.
l. If the street has been sprinkled there is no dust in the air. The street has
not been sprinkled. Hence there is dust in the air.
2. All the marbles in box number 5 aro white. These marbles came from box 6. Hence
they are not white.
3. Pupils having an average of 90 on their weekly test, and 85 on their daily work
will be excused Thursday to go to the zoo. John's average on the weekly test was 94.
Will he be excused?
4. If, as in ex. 3, Tom's average on the test was 90, and on the daily work was 84.
Will he be excused?
5. If it is a Bulo Watch it will keep time correctly. This is a Harvey Watch.
6. Newspapers said that if he fulfills his promises he will be a popular mayor.
Later he became a very popular mayor. Had he kept his promises?
7. In one town all policemen must be 5 ft. 8 in. tall at least. i.r. Jones is not a
policeman, but lºr. Eoran is . That do you know about their heights?
8. Only people born in Iowa are allowed to join the Iowa Club. May Adams was born
in Iowa. ...ay she join the club?
Alice Winters was born in Iowa. Is she a member of the club?
9. Citizens who have lived in the state for at least one year may vote at this
election. Pat Thomas has lived in the state 6 years. Alay he vote?
10. If the hat is small it will please lirs. Hardy. This is a large brown hat with
a small feather on it. Will it please iwirs. Hardy?
ll. All freshmen will be admitted to Friday's Freshie Frolic. Bill Jones attended
the Freshie Frolic last year. Will he be admitted this year?
12. If the switch is turned off the room will be dark. The room is dark.
13. If you do not have 25 ¢ you cannot be admitted. Bill has 50 cents.
l4. If Bill can save £25 and get his parent's permission he will be allowed to attend
the Boy Scout Camp for two weeks. He saved $25. Will he attend the camp?
15. "If you will elect me mayor, your taxes will be reduced" said Maloney when
running for office. Next year the taxes were actually reduced.
16. Examine the answer. that you wrote for ex. 1 on the Test on page l. Do you think
now that you answered it correctly?
Are there any other answers to that test which you would now change?
l6
SYLLOGISMS. (pronounced sil – o – gism). This is a new word for something with
which you are familiar. A syllogism is a certain way of presenting an argument. It
consists of 3 steps:
l. A general statement about a group of objects. This is called the major premise.
2. A specific statement which says that a certain object is a member of the group
mentioned in the major premise. This is called the minor premise.
3. The conclusion. It says that whatever is true about the entire group is also true
about any specific object belonging to the group.
For example:
l. Major premise: All seniors in high school study United States History.
2. Minor premise: John Jones is a senior in high school.
3. Conclusion : John Jones studies United States History.
In conversation we seldom present our arguments in the above form. The conclusion
is often stated first and the major premise last, as in the sentence
Oil lamps are becoming scarcer because they are inefficient
and inefficient articles soon disappear.
In syllogistic form this argument is :
l. Major premise: Inefficient articles disappear
2. Minor premise: Oil lamps are inefficient articles
3. Conclusion : Oil lamps are becoming scarcer or disappear.
By putting an argument in the syllogistic form we can detect errors in reasoning.
For example, John might say
"Elkhart watches are good ones. Mine loses 10 minutes a day. It's a Tracy watch."
Here the major premise is: Elkhart watches are good watches.
The expected minor premise would be: Mine is an Elkhart watch.
The conclusion would then be: Mine is a good watch.
John's statements, as he made them, may be interesting conversation to his friends,
but they do not prove anything.
E x e r c i s e s
Arrange the following statements in syllogistic form. You may not agree with all
the statements, but pretend that you do.
l. Automobiles are becoming more numerous because they are useful, and all useful
things increase in number.
2. Electric clocks are better than other clocks because they do not need winding.
3. Silver is not a compound substance. It is a metal, and metals are not compounds.
4. This man is kind to the poor; hence he is not a thief, Kind people are not thieves
5. Everybody in that house is sick. John is sick. He lives in that house.
6. Ann is clever and of course she is popular. All clever people are popular.
7. Jones is a successful man and industrious. All industrious men are successful.
8. Spanish verbs whose infinitive end in ir belong to the third conjugation. Hence
dormir belongs to the third conjugation.
9. A difference of two squares can be factored. Hence I can factor a? º b%.
10. Write some syllogisms from your own experiences or find some in books.
OMITTED STEPS IN SYLLOGISMS. One of the premises is often omitted when it is
obvious. For example, Johnny's mother may say to him, "Come in quick! Can't you
see it is raining." Her argument is:
1. Major premise: When it is raining, John should come into the house.
2. Minor premise: It is raining now.
3. Conclusion: John should come into the house.
Or, someone may say, "The ll: 30 does n't stop at Weehaken; it's an express."
The argument is : Major premise: Express trains do not stop at Weehaken.
Minor premise: The ll: 50 train is an express.
Conclusion: The ll: 30 train does not stop at Weehaken.
1 7
E x e r c i s e s
Tirite in syllogistic form, supplying any missing steps:
l. To-day will be a cold day. There is a northwest wind.
2. Starch contains carbon since it is an organic substance.
3. John attends the Clay school since he wears a green cap.
4. People who are troubled with cancer seldom live long; I fear Yirs. O'Malley wont
live long.
5. "Don't drink that ." screamed the maid to i.irs. Johnson. "It will kill you."
6. "We will win" said Capt. Brown just before the battle started. "We have twice as
many soldiers as the enemy."
7. John must have had 7 cents this morning when he started for school because he
rode to school instead of walking.
8. J.r. Jones is qualified to be major because he has a legal training.
9. Liss Smith has had experience in selling hats and so she will be able to fill
this position.
lo. A battle must be going on northwest of Verdun because the sound of artillery
can be heard from that direction.
ll. This must be an acid because the paper turned red when dipped in it.
l2. Tleanor is an attractive girl because she uses Buty Face Powder.
PROVING A STATEMENT. Proving a statement means:
Find a syllogism of which the statement is the conclusion.
The syllogisms on previous pages proved such statements as : Oil lamps disappear;
John should come into the house; The ll: 30 train does not stop at Weehaken.
Of course these statements are of slight importance, hardly worth proving, but in
studying any new subject we must begin with simple examples. Further, most proofs
use more than one syllogism, as we shall see in the next chapter.
TRUTH WERSUS PROOF. Compare these two arguments:
i!ajor premise: All birds have wings All birds have wings
! inor premise: Robins are birds Robins have wings
Conclusion: Robins have wings Robins are birds
Hotice first that all the statements used in both syllogisms are true. Further, the
argumont in the lefthand column is correct.
In the righthand column the same three statements are arranged in a different order,
and the conclusion does I; OT follow from the two premises. The minor premise "Robins
have wings" does not affirm the major premise. (See page 15.) Hence the argument in
the right hand column is "ROI.G. Each statement is true, but the argument is wrong.
It is true that 3 x 6 = 18, that Chicago is in Illinois, and that water is wet;
but it is impossible to arrange these items in a syllogism. It is important to
remember that we are studying how to find correct conclusions from two premises.
Of course, mankind is always searching for Truth, but right now we are studying how
to draw correct conclusions and not truth.
E x e r c i s e s
Arrange the sentences in each exercise so as to form a correct syllogism:
l. The "Liberty" is a stream-linod train. It costs a great deal of money. Stream-
lined trains are expensive.
2. Morley is in Kiichigan. Cola County is in Michigan. Morley is in Cola County.
3. The football team will play Akron next Saturday. Henry will be in Akron next
Saturday. Henry is on the football team.
4. I shall stay at home. It is raining. then it rains I stay at home.
5. Take three statements which will form a correct syllogism and arrange them in as
many different orders as possible. In which does the conclusion follow from the
premises?
18
C H A P T E R 3
D E F I N I T I O N S, A S S U M P T I O N S, P R O O F S
THE NEED FOR DEFINITIONS. During July 1938 Hughes flew from NewYork to Paris,
Moscow, Omsk, Yakutsk, Fairbanks, and back to New York, covering about 14,000 miles.
The newspapers had large headlines, saying Hughes had circled the globe in four days.
Since the circumference of the earth is about 25,000 miles, had he really circled
the globe? Evidently the answer depends on what you mean by circling the globe.
Try the following problem on your friends :
A dog was watching a cat. They faced each other ready to fight. The dog cautiously
moved to the right in a circle. The cat turned and turned so that he always faced
the dog. After a while the dog was back to his starting point. Had the dog walked
around the cat?
Some folks will say, "yes" because the dog moved in a circle with the cat at the
center. Other folks say, "no" because the cat and the dog always faced each other;
the dog never faced the back of the cat. The answer depends on what you mean by
walking around an object.
You can make the problem still more complicated by having the cat turn in the oppo-
site direction from that in which the dog walks . Then when the dog has moved one
fourth of a circle he is facing the cat's back; and in another fourth of a circle he
is facing the cat again. Has he gone around the cat when he has walked only half way
around the circle?
The answer to many questions depends on the definitions of the words we use.
If we change the definitions, the answer may be changed. Pupils who are planning
to become lawyers should study the matter of definitions carefully. Many legal cases
depend on such questions as: What is a motor bus? Is a rowboat still a rowboat
if you attach an out-board motor to it? If a family has paying roomers living with
it, is the family running a hotel? When does a drug store become a restaurant?
A motor cycle is an automobile in Indiana, but a traction engine is not; both are
automobiles in New Jersey; neither one is an automobile in New York.
Read "It's Still Bologney" in Readers Digest for Oct. 1938 showing how old ideas
can be dressed up in new words. "The public need not worry about the national debt"
says sir. Lightmind. "But our financial obligations to others is a matter of deep
concern to all of us."
E x e r c i s e s
Tell which words need defining and suggest some possible definitions.
l. Many bills have been considered in congress for encouraging the building of
homes. One such bill furnished money at low rates of interest for homes costing
$6000 or less. Mr. Jones planned to purchase a plot of ground for £800, spend $5000
on the house, £200 for a garage, and £25 for a concrete driveway to the garage.
In the debate in congress, one senator claimed that this home cost $5800; another
senator claimed it cost £6000 since a garage was a part of every home; a third sena-
tor claimed this home cost #6025. How can you decide who is right?
2. In one state the primary elections for a certain office are held on the first
Tuesday of April of the even-numbered years. A law states that if a voter votes for
one party (democratic, republican, or other party) at one primary, then he cannot
change to a different party within two years. One year the primaries came on April 7;
two years later the first Tuesday came on April 5.
Should a voter at that time be allowed to change from one party to another?
3. A man was caught stealing $10 from a cash register. The judge sentenced him to
two weeks in jail. The man's lawyer claimed that the heaviest penalty for petty
theft was only one week in jail. On what does his claim depend?
19
4. In the town of Kent 500 people were arrested for various offenses during one
year; 400 of these were convicted (found guilty) and sent to jail, and 100 were
acquitted (found innocent). In the city of Brilley during the same year, 450 were
arrested. 425 of these were convicted and 25 were acquitted. Kent claims it has
less crime than Brilley. Brilley claims it has less crime and better judges.
5. Suggest definitions for the significant words in the following:
a) Should a school emblem be awarded for distinctive achievement in school in other
than athletic activities?
b) Should school clubs be allowed to raise money by selling refreshments during
school entertainments?
c) Are the habits practiced on "Courtesy Day" really permanent?
d) Does the middle west have a milder climate than the east?
e) Does "Daylight Saving Time" save time?
f) Chain stores force many small stores out of business.
g) The United States should maintain its neutrality when other countries are at war.
h) Only adults will be admitted.
6. The famous "Alabama" case between The United States and England depended on the
meaning of "to equip a ship of war". Prepare a report on this for the class. You
can get the necessary information from histories in your school library.
REQUIREMENTS FOR A GOOD DEFINITION. A school is a building used for teaching
children various things. You may not agree with this definition, but it illustrates
this fact :
A definition should tell to what larger group of objects the object
belongs, and how the object differs from other objects of that group.
In our example, we say first that a school is a building (not an animal, not a food,
not a vehicle, ...) A school is one kind of building. Next we tell how a school
differs from other buildings. It is not used for manufacturing purposes, not for
living purposes, not for growing flowers, but for teaching purposes.
In technical language we say that the definition of an object should tell to what
genus and to what species the object belongs • Genus means the larger group which
includes the object we are defining. Species means some smaller special group which
is a part of the larger group. Imagine a collection of buildings - churches, town-
halls, factories, . . . other buildings. In this large group are also found schools.
We describe the schools so that they can be distinguished from the other buildings.
Thus: A farmer is a workman (the genus) who tills the soil (the species).
A furnace is an apparatus (the genus) for heating a building (the species).
A triangle is a polygon (the genus) with three sides (the species).
E x e r c i s e s
l. State some possible definitions for
Home Automobile Bicycle Fountain-pen Sugar Profit, Cash
2. Discuss the following definitions :
a) A cat is a domestic animal.
b) Rent is the income derived from the ownership of land.
c) A ballad is a narrative poem.
d) A secret society is an organization that chooses its own members.
e) Education is the process of learning useful things from other people.
From your discussion in class you have learned that words do not mean the same thing
to everyone. School means one thing to a pupil, something else to a teacher, another
thing to a tax-payer, and something else to a board of education. For a difficult
problem try to define color. A boy raised on a farm will tell you that blackberries
are red when they are green. The study of the meaning of words is called semantics.
It has become quite a fad in the last ten years. You may be able to find some books
about it in your town library.
2O
º:
ASSUMPTIONS. Suppose we are arbitrators (referees, umpires, judges) in a dispute
between some workmen in a factory and the managers. During a meeting of the arbi-
trators one of them says
(a) Considering the cost of food, rent, clothing, and other necessities, the
workmen need to earn at least $1500 a year to live comfortably.
Another arbitrator might say
(b) The workmen need to earn at least $1800 a year.
And a third arbitrator might say
(c) A salary of $1400 a year is sufficient.
It would be difficult to decide which of these statements is true, but some agree-
ment must be made. We must choose (a) , (b), or (c), or a similar statement. Our
choice will be important because it will influence the settlement of the dispute.
There will be certain inevitable consequences of our choice.
We do not know which statement is true but we shall adopt one of them, and there-
after we must talk, argue, and act as if it were true. When used in this manner, the
statement that we select is called an assumption.
An assumption is an agreement that must be made before a disputed point
can be settled . The assumption may be true or false. If we were sure
of its truth we could call the statement a fact rather than an assumption.
If an assumption is changed, the conclusions may be changed.
The conclusion depends on the assumptions we may adopt.
E x e r c i s e s
l. Consider these possible assumptions :
(a) Children under 16 are not allowed to drive cars.
(b) Children under lö are allowed to drive cars if accompanied by an adult.
(c) Children under 17 are not allowed to drive cars.
John Smith is 15 yr. 6 mo. What is the conclusion using (a) 2 (b) 2 (c)?
Tom Jones is l6 yr. 8 mo. What is the conclusion under each assumption?
2. Consider these possible assumption s :
(a) A basket ball team with a tall center has an advantage over a team with a
short center, other things being equal.
(b) A basket ball team with many substitutes has an advantage over a team with
few substitutes, other things being equal.
(c) A basket ball team has an advantage when playing on its home floor.
Which of the following teams would win if you adopt only assumption (a) : If you
adopt only (b) : If you adopt only (c) : If you adopt both (a) and (b)? If you
adopt both (a) and (c)? If you adopt both (b) and (c)? If you adopt all three?
Team l has a short center and is playing on its home floor, with Team 2 which has
a tall center. Both teams have plenty of substitutes.
Team 3 has a tall center and is playing Team 4 on the grounds of Team 4.
Team 5 has few substitutes and is playing Team 6 on the floor of Team 6.
OMITTED ASSUMPTIONS. Sometimes an assumption is not mentioned when it seems
obvious. For example, what assumption is John making when he decides to repair his
old bicycle rather than buy a new one? There are several possible assumptions:
(a) John does not have enough money to buy a new bicycle.
(b) A small sum spent on repairs will make his bicycle as good as new.
When looking for a missing assumption, ask yourself:
What statement is needed, in addition to the given information,
to make the conclusion correct?
If the truth of that missing statement is doubtful in your mind, you are entitled
to call it an assumption. If you are positive of its truth you can call the missing
statement a fact rather than an assumption .
21
E x e r c i s e s
State the missing assumption in each exercise:
l. Lane beat Austin in football. Austin beat Moline. Lane will beat Moline.
2. Toothero cured my mother's tootache. It will cure my tootache.
3. The teachers have decided to introduce student government in our high school.
4. Mr. Higgins has decided to install a gas furnace rather than a coal burning
furnace in his new house.
5. Bill Smith was out of school four weeks due to a broken collar bone. Tom broke
his collar bone last Saturday. Iie will be out of school for four weeks.
6. In his science book Fred read that an acid will turn litmus paper red. Wishing
to know if the solution in a glass was an acid or not, Fred dipped a piece of litmus.
paper into the solution. It turned red. "That shows", he said, "that this solution
is an acid."
7. The average boy or girl begins school when six years old, and graduates from
eighth grade when he is 14. John did not enter school until he was 7 because of an
illness when he was six. How old was John when he graduated?
8. Mr. Jones suffers from diabetes. He should not eat bread.
MISLEADING OMISSIONS. We learned on page 20 that a conclusion depends on the
assumptions on which it is based. Hence, whenever an assumption is omitted, we
should check the argument carefully. Perhaps the speaker or writer purposely omitted
any mention of the assumption because he is not sure that we would accept it, and he
does not wish to call our attention to it.
Some years ago an advertisement for one brand of gasoline stated
O ur Gasoline Is Yellow
You were then expected to supply an argument like:
l. If gasoline is yellow then it is good gasoline.
2. Our gasoline is yellow .
3. Therefore our gasoline is good gasoline.
The advertisement mentioned only step 2, and you were expected to supply both
steps l and 3. If step l is true, then the conclusion is true. Since we are not
sure that step l is true, it is an assumption. The truth of step 3 depends on the
truth of step l.
E x e r c i s e s
As in the example above, supply the missing assumptions. Also, write the 3 steps
of the argument.
1. ºe use Portland cement in all our work.
2. Our tobacco is toasted before it is sold.
3. The beautiful stars of Hollywood use Cleano Soap.
4. Jimmy Jones was the brightest boy in his algebra class last year. He will be
at the top of his geometry class this year.
5. Be sure to see Bert Tailor in "Shipmates". His work in "The Captains Crew"
filled the theater at every performance last year.
6. This ſurniture was made in Grand Rapids.
7. Every day a million people begin the day with Goodstart Coffee.
8. Last year our sales of radios amounted to $1,000,000. This year our sales are
over £2,000,000. Next year our sales will be over £4,000,000.
9. The temperature in this theater is never over 70 degrees.
19. Mary is a very attractive girl because she uses GretGard Face Powder.
ll. Our Sight Seeing Busses pass Grant's Tomb.
12. Purtex Ice Cream is made from Pure Cream.
13. Bill Hanson is a life-guard at the Jackson Beach. He must be a good swimmer.
14. Jim: I'm giving Bill an alarm clock for a Christmas present.
Joe: An alarm clock! How stupid of you! Bill doesn't need an alarm clock. He
is always complaining about the sun waking him up.
Jim: The Sun waking him! You must be wrong. Bill's room is on the west side
of the building.
Joe: Just the same, Bill complains of the sun waking him each day.
22
PROVING A STATEMENT. In daily talk, proving a statement means convincing someone
of the truth of the statement. Usually we prove a statement by giving a reason why
we believe it. To prove the statement
The dial telephone is a benefit to society
we offer the reason: Any machine that saves labor is a benefit to society.
Our argument may be stated more exactly as follows:
l. Any machine that saves labor is a benefit to society.
2. The dial telephone saves labor.
3. The dial telephone is a benefit to society.
This form of presenting ideas is called a syllogism. It has always three steps.
The first step is called the major premise; it is a general statement.
The second step is called the minor premise; it is a specific statement and says
that we are really talking about a situation like that in the major premise.
The third step is called the conclusion; it states the consequence or result or
what follows from the two premises.
Proving a statement means showing that the
statement is the conclusion of a syllogism.
In any argument if our opponent accepts the premises then he must accept the con-
clusion which follows from those premises. If he refuses to accept the premises, the
the premises must be proved in the same way as any other statement is proved.
Consider next a longer proof which uses more than one syllogism.
An advertisement for a fountain pen said that its ink barrel was made of transparen
material so that the amount of ink in it was visible. Hence the pen would not become
dry suddenly and thus annoy you or interrupt your thoughts just as you were about to
write some brilliant idea.
This argument can be presented as follows:
A. l. If the ink supply is visible, the pen will not become dry suddenly.
2. In the X Pen, the ink supply is visible .
3. The X Pen will not become dry suddenly.
B. l. If a pen does not become dry suddenly, it will not harass the brain.
2. The X Pen does not become dry suddenly.
3. The X Pen will not harass the brain.
C. l. If a pen does not harass my brain, I should buy it.
2. The X Pen does not harass my brain.
3. I should buy the X Pen.
Notice that the conclusion in A is the same as step 2 in B; and the conclusion
in B is the same as step 2 in C. To avoid these repititions we write the proof thus:
In the X Pen the ink supply is visible.
The X Pen does not become dry suddenly. If the ink supply is visible, the
pen does not become dry suddenly.
The X Pen does not harass the brain . If a pen does not become dry suddenly,
it will not harass the brain.
I should buy the X Pen. If it will not harass the brain, I should
buy the pen.
In this arrangement the major premises appear in the righthand column as reasons
for the statements in the lefthand column. Comparing the major premises we see that
the conclusion of each one is the hypothesis of the next one.
In conversation we do not present our arguments in the above form. The chief value
of this arrangement is that it shows us what a proof is . It should make us realize
that each statement has a reason, and that there is a certain correct order of the
statements, beginning with the given information (the ink supply is visible) and
leading to the final conclusion (I should buy this pen).
23
We usually have more than one reason for anything that we do. If we are thinking
of spending a vacation in Florida or in California, we collect the reasons for and
against each place and then decide which place has the most or the best reasons in
its favor. Hence the following exercises really prove nothing; they merely furnish
practice in arranging some statements in logical order.
E x e r c i s e s
1. Rearrange the following statements, writing them in two columns as on page 22
so that they prove : The X overcoat is the one I should buy.
The X overcoat is made of pure wool.
If an overcoat will keep me warm I should buy it.
If an overcoat is made of pure wool it will keep me warm.
The X overcoat will keep me warm.
The X overcoat is the one I should buy.
2. What will the following statements prove? Write the proof as on page 22.
John was at school Friday.
John is on the football team.
All members of the Glee Club were present at the assembly on Friday.
All members of the football team are members of the Glee Club.
3. Dick and Jim were fishing in the lake and caught some perch about five inches
long. Jim said, "Let's go home. We wont catch any bigger perch to-day. Every perch
travels with others his own size. So we will not catch any perch bigger than five
inches to-day."
When they reached home, Dick tried to write a proof of Jim's last statement. His
work is shown below. He put each item in the correct column but in the wrong order.
The reasons for the statements are also mixed. Write a correct proof.
l. The perch in the lake near this spot a) Our measurements show this.
are all five inches long.
2. We cannot catch any bigger perch to-day. b) If they are perch, they are all of
one size .
3. All the perch we will catch to-day will c) If there are no bigger ones here, we
be five inches long. cannot catch bigger ones.
4. The perch we now catch are 5 inches long. d) If they are all of one size, we
cannot catch bigger ones.
ANOTHER ExAMPLE OF A PROOF. In the preceding proofs the argument has been like:
If A is true, then B is true . If B is true, then .C is true. If C
is true, then D is true, and so forth.
But many arguments are like this
If A is true, then B is true . If C is true, then D is true. If both
B and D are true, then E is true. For example: •
Data: I see in a store a blue silk dress costing about £15.
Prove: I should buy this dress.
l. The X dress is silk a) Data.
2. I should buy the X dress. b) If a dress is a silk one, I should buy it.
3. The X dress is blue. c) Data.
4. I should buy the X dress . d) If a dress is blue, I should buy it.
5. The X dress costs $15. e) Data.
6. I can afford the X dress. f) If a dress costs £15, I can afford it.
7. I should buy the K dress. g) If a dress is silk, blue, and I can afford it,
I should buy the dress.
If you have difficulty finding the correct reason for some step, ask yourself, "What
general statement is needed to make the conclusion sensible? That is, what major
premise is needed to connect the minor premise and the conclusion?
24
E x e r c i s e s
Write a proof as on pages 22, 23, of whatever each exercise aims to prove:
l. If a building is vacant and deteriorating it should be torn down. Otherwise
the property will be come more worthless each year, and the property in the neighbor-
hood will also decrease in value .
2. An advertisement for e certain kind of gym shoes states that the shoes have an
arch support. This prevents the wearer from getting fallen arches, and the shoes
can be worn more comfortably at all times.
3. If a fireplace throws smoke into the room, the chimney has not been built high
enough. If the chimney were higher there would be a better draft through the smoke-
stack, the fire would burn better, and the fireplace would be used oftener.
4. An advertisement of a certain brand of butter states that each package has the
date on which the butter was made. In this way you can be sure of getting fresh
butter. Stale butter is not only lacking in flavor but is injurious to the health.
5. Collect and study some advertisements. What arguments are used to prove that
you should buy that article? Write the proof.
6. Prove the statement: If A is true, then G is true by rearranging these
statements :
If F is true , then G is true. If A is true, then B is true.
If B is true, then E is true. If E is true, then F is true.
7. This exercise is like ex. 6 but there are some irrelevant statements. (Irrele-
vant means "having nothing to do with the problem" .)
Prove: If A is true, then D is true by using some of these statements :
(a) If D is true, then H is true. (d) If B is true, then C is true.
(b) If C is true, then G is true. (e) If A is true, then C is true.
(c) If B is true, then A is true. (f) If G is true, then D is true.
S
8. Using the statements (a) to (f) in ex. 7, see how many new statements you can
prove. For example, from (d) and (b) you can prove: If B is true then G is true.
This is a good game to play with your friends. Write other statements like those
in ex. 7 and see how many new statements you can get from them.
POSTULATIONAL SYSTEMS. Exercise 8 above illustrates what is meant by deriving
one statement from other statements. If you prove, or agree to, or assume, or accept
the statements (a) to (f) then certain other statements follow from them. If the
assumptions are wrong, then the statements you get from them are wrong. If you
accept the assumptions then you cannot refuse to accept the conclusions. A collection
of statements of this kind is called a postulational system. The word "postulate"
means the same as "assumption".
Your work in geometry is like a postulational system. You make certain agreements
at the start (such as, two lines can cross at only one point) and then try to discover
what are the inevitable results of your agreements.
When studying social and economic problems we ought first of all to agree on
certain definitions and assumptions, and then we ought to prove the statements which
we think follow from these assumptions. We try to do this, but it is difficult to
make people agree to a set of assumptions from which to start.
25
C H A P T E R 4.
G E O M E T R I, C P R O O F S
"I CAN SEE IT" IS NOT A PROOF. You have often tested geometric statements by
looking at your work or by measuring. If one line seemed to be perpendicular to
another line, you wre satisfied to say it was perpendicular. If lines looked paral-
lel, you said that they were parallel. Your eyes are a good test of many things but
they are not a proof. At times your eyes can mislead you. For example, in this
figure the four equal horizontal lines appear to be longer than the four vertical
lines. Such optical illusions are interesting; they are very import-
ant in some kinds of work such as interior decorating. They can make
a small room seem larger. In dress designing optical illusions can
make a stout person seem slim or a short person seem taller.
See the optical illusions on pages, 29 - 50 of "Scientific Experi-
ments and Amusements by C. E. Gibson. (Seeley, Service & Co., London;
1931; $2.50)
But appearances are of no importance when studying how to reason correctly. In
geometric work "I can see it" is not a proof. When you feel tempted to use that
phrase, remember that a proof is an argument that does not depend on eyesight. This
does not mean that you are not allowed to use your eyes. Often when you are not
certain of the truth of a statement it is wise to draw several figures and look at
them. The drawings may give you valuable inspirations. But it is not a proof.
GEOMETRIC DEFINITIONS. Like other branches of mathematics, geometry begins by
stating the definitions of certain words like triangle, perpendicular, right angle,
and so forth. There are some words, like point, which are so simple that we do not
define them but merely gather them in a list and call them undefined terms. We try
to have as few as possible of these undefined terms, and we must be careful to ex-
plain all new words by using only those words that we have previously defined. A
beginner in geometry might say that a right angle equals ninety degrees and that a
degree is one ninetieth of a right angle. We call this "talking in a circle" because
we get back to the word from which we started. You may have had the same trouble
when using small dictionaries. One word is defined by using a second word; and the
second word is defined by using the first word.
When making a definition of a geometric figure we state the genus and the species
to which the figure belongs.
The genus is the larger group of which the object is a member; but the definition
will not be a good one if we place the object in too large a group. Thus a rectangle
is one kind of a parallelogram which is one kind of quadrilateral. Hence a rectangle
should be considered as one kind of parallelogram.
The species is a smaller special group of the larger group. The species tells how
the special group differs from other objects of the larger group. We state the fewest
number of features that will distinguish the object. A definition does not tell every-
thing that is true about a figure. It tells only enough to identify the figure. It
does not mention any fact that can be proved from other facts.
Thus we should not say that an isosceles triangle is a triangle which has two equal
sides and two equal angles. The second feature (two equal angles) is a Consequence
or result of the first feature (two equal sides).
Our choice of definitions often makes a difference in our work. We could define
isosceles triangles as those which have two equal angles. The question is ; which is
easier to prove: that two sides are equal when two angles are known to be equal, or
that two angles are equal when two sides are known to be equal? Which definition is
used in your geometry?
26
A definition says that if certain things are true about a figure then you may give
it a certain name. Hence every definition can be written as a complex sentence
having a subordinate clause beginning with "if" and a principal clause. Thus
If an angle is half of a straight angle, then it is a right angle.
If the sides of a triangle are equal, then it is an equilateral triangle.
We do not always use this "if-then" form but the fact that every definition can be
stated in this form is important for the following reason:
When the "if" and "then" clauses in a definition are interchanged, the new state-
ment is true. (The new statement formed by interchanging the "if" and "then"
clauses is called the converse of the original statement.) This fact must be con-
sidered when selecting a definition. In the above examples we can say:
If an engle is a right angle, then it is half of a straight angle.
If a triangle is equilateral, then its sides are equal.
E x e r c i s e s
1. Criticize these possible definitions:
(a) A rectangle is a parallelogram with one right angle.
(b) A rectangle is a parallelogram whose angles are right angles.
(c) A rhombus has four equal sides.
(d) A circle has a center, a radius, and circumference and area.
(e) A parallelogram is a four sided figure whose opposite sides are parallel and equal.
2. Review the definition of perpendicular, right angle, and straight angle. Which
was defined first? Must it be defined first or are there other possibilities?
3. Define horizontal and vertical lines. Which is the easier word to define first?
4. If you defined a square as a four-sided figure whose diagonals are equal and
perpendicular to each other, what ought you to prove about a square?
5. Why would it be unwise to define a square as a four-sided figure whose diagonals
are equal?
GEOliFTRIC ASSUiwiPTIONS. After adopting some undefined terms and definitions, we
next adopt certain general statements which we call assumptions. For example,
All straight angles are equal.
One and only one straight line can be drawn through two points.
We should adopt as assumptions only those statements which we cannot explain in any
simpler words. Further, we should adopt as few assumptions as possible because every
assumption is likely to be criticized. But, to avoid much tires ome and uninteresting
work, we adopt as assumptions many statements which could be proved if we cared to
take the time to do so.
After agreeing on the undefined terms, the definitions, and the assumptions, our
object is to discover new general statements (which we call theorems) and to show
that these new statements are consequences or results of our previous agreements.
To discover and to prove the new statements we must know how to use syllogisms,
which you studied in Chapter 2. A simple geometric example is :
If you have agreed to the statement: All right angles are equal
and if, in some figure, Angles x and y are right angles
then you may say that: Angle x equals angle y .
PRACTICE WITH SYLl,00ISãS. Let us agree to statements 1 to 8 on page 27. It will
make no difference whether you call them definitions or assumptions or theorems, but
you must understand what each statement means. It is unlikely that you have studied
all of them in class. They are merely some general statements collected here for
use in exercises. Their order is purposely made peculiar so that you will be com-
pelled to read them carefully when doing the exercises. The object of the exercises
is to give you some practice with syllogisms - You had better review the meaning of
major premise, minor premise, and conclusion. See page 14 or 16.
27
l. Corresponding angles of congruent triangles are equal.
2. Corresponding sides of congruent triangles are equal.
3. If two angles of a triangle are equal, the sides opposite them are equal.
4. If two sides and the included angle of one triangle are equal respectively to two
sides and the included angle of another triangle, the triangles are congruent.
5. The sum of the angles of a triangle is 180 degrees.
6. If two angles and the included side of one triangle are equal respectively to two
angles and the included side of another triangle, the triangles are congruent.
7. The angles opposite the equal sides of a triangle are equal.
8. The sum of the angles of any four-sided figure is 360 degrees.
9. If the sides of one triangle are equal respectively to the sides of another tri-
angle, the triangles are congraent.
10. If two lines are perpendicular to each other they form right angles.
E x e r c i s e s D

Supply the missing items:
l. F E
A Ds. C Ds.
Major premise: Statement 2 above
Iáinor premise:
Conclusion:
2. D
C
A B
E
Major premise: Statement 6 above
Minor premise: 4. A = <- B, AC = CB,
and 4 ACD = 4. ECB
Conclusion:
3. D C
A
E
Major premise:
Minor premise: AB = BC, CB = BE,
and Z ABD = 4. EBC
Conclusion:
4.
E
Major premise: Statement 5 above
Minor premise:
Conclusion:
5.
Major premise: Statement 4 above
Minor premise:
Conclusion: A ACD = A ACB
(There are 3 possible minor premises)
A
5.
B C


Major premise: Statement 3 above
Minor premise:
Conclusion : AC = AB C
!
7.
A D
B


Major premise:
Minor premise: AC = AB, AD = AD,
and Z CAD = 4. BAD
Conclusion:


8. C
A B
Major premise:
Miinor premise: AB = CB
Conclusion:
28
SELECTING THE in AJOR PREMIS.E. In most situations in life we know, or are told, the
minor premise and then must use our brains to select a major premise that will be use-
ful. If you cannot remember the correct major premise, hunt for it by reading your
textbook. Frequent work with exercises, will make the theorems stick in your memory.
Frequent reviews will help. ...riting, collecting and arranging theorems in a notebook
will also help if you have a poor memory.
E x e r c i s e S
What general statements are suggested by these facts : E D
D
l. CD -- AB 4. AB -- BD C
BC -- BE
A B B A
|U C
2. AB bisects & CAD 5. AC = BC
D C
A B.
B
3. AB = CD 6. AD = CD
AB = BC 2TS
t–l. ſ
A B C 5 A D C
ARRANGEMENT OF THE WORK. In geometric work an argument like
major premise: All right angles are equal
minor premise: < l and 4-2 are right angles
conclusion: 4 l = 4, 2
would be written:
4 l and & 2 are right angles
4 l = < 2 All right angles are equal.
In the lefthand column is the minor premise and the conclusion. In the righthand
column, at the side of the conclusion, is the major premise.
ExAMPLE OF A GEOMETRIC PROOF. The figure below was made by drawing a line AC and
then bisectinf it at B. Then through A and C perpendiculars were drawn to AC. Next,
any line was drawn through B. It intersects the perpendiculars at D and E. We are
asked to prove that A ABD # A BCE.
We can see that
l. 4 ABD = <- CBE
2. AB = BC
3. Z. BAD = < BCE
suppose now that someone, who things eur argument is poor, asks us the questions in
the lefthand column below. Our answers appear in the right hand column.
a) Why does / ABD = < CBE? Vertical angles are equal.
b) Why are they vertical angles? (See the definition in your textbook.)
c) Why does AB = BC Ž BC was bisected. Bisected means "cut in half".
d) Why does Z BAD = < BCE? All right angles are equal.
e) Why are they right angles? Perpendiculars form right angles.
f) How do you know there are We were told this when making the figure.
perpendiculars?
On the next page this argument is rewritten in such a way that no one will ask us
these questions. A good salesman anticipates the questions and objections of a pros-
pective customer and has his answers ready. The word Data in the righthand column
means that this statement was part of the information about the figure and was used
in constructing the figure. Hence such an item cannot be questioned.

29
Data.; AB = BC, AD 1. AC, CE l- AC. Prove: A ABD = 4 BCE.
1. AC and DE intersect at B . Jata.
2. 4 ABD and 4- CBE are vertical angles. See definition of vertical angles.
3. Z ABD = 4. CBE Vertical angles are equal.
4. AC was bisected at B Tata.
5. AB = BC See definition of "bisected".
6. AD -i- AC and CE + AC Data.
7. 4. BAD and 4 BCE are right angles. See definition of "right angles".
8. Z. BAD = < BCE Right angles are equal.
9. ABT = CBE; AB = BC; 4. BAD = 4. BCE Proved in steps 3, 5, and 8.
10. 4 ABD St A BCE Triangles are congruent if ... etc.
INotice the many syllogisme in this proof.
Šteps 1 and 2 make the syllogism:
i.ajor premise: If the sides of one angle are the prolongations of the sides of
another angle, then they are vertical angles.
Minor premise: The sides of 4 ABC are prolongations of the sides of 4 CBE.
Conclusion: Angles ABD and CBE are vertical angles.
Steps 2 and 3 make the syllogism:
Major premise: If angles are vertical angles, then they are equal.
Minor premise: Angles ABD and CBE are vertical angles.
Conclusion: Angles ABD and CBE are equal.
Steps 4 and 5 make the syllogism:
!ajor premise: If a line is bisected, it is divided into two equal parts.
Minor premise: The line AC was bisected at B.
Conclusion: The parts AB and BC are equal.
Exercise. Find other syllogisms in the above proof.
THE AMOUNT OF DETAIL IN A PROOF. How much detail should be put in a proof depends
on who is going to read the proof. If you were explaining in a science class how the
clutch of an automobile works you would not need as much detail as if you were ex-
plaining it to a brother in the sixth grade. The exercises in geometry get longer
and longer throughout the year, and more detail will need to be omitted. At the start
it is best to put in all the detail that you can think of, and then gradually omit
steps like l, 2, 4, 6, 7, and 9. In most exercises you can omit the definitions and
write merely DEF. to show that a definition belongs at that place.
WHAT IS A PROOF? The work above illustrates that a geometric proof is a collection
of syllogisms written in a certain form.
Notice the nature of the reasons which appear in the righthand column.
In steps l, 4, and 6 the word Data appears, meaning that this was part of the in-
formation about the figure., Often this is called the hypothesis of the problem.
But the word hypothesis is used in so many different ways that data is a better word
for the specific information in a figure. Sometimes the word "Given" is used in place
of "data" because the data is given information.
In steps 2 and 5 the reason is a definition.
In steps 3, 8, and 10 the reason is a general statement which we have accepted or
proved at some previous time.
Step 9 is often omitted since it is merely a summary of certain previous steps.
3O
HOW ARE PROOFS FOUND? You may understand perfectly the proof on page 29 and still
may wonder how to start a proof. By practicing with short exercises you will soon
acquire skill. Try the following method of studying:
1. Some item in the data should suggest a general statement (theorem, definition,
or assumption) which can be used. The data and the general statement will lead to
some conclusion.
2. Repeat step 1 for each item in the data.
3. The conclusions discovered irn steps l and 2, together with additional theorems,
definitions, and assumptions will lead to more conclusions.
4. Repeat steps l, 2, and 3 till you reach the conclusion you want.
When you start you may not see you way clear to the finish. &hen you drive from
one town to another you cannot see the road for eighty miles ahead. But unless you
start you will never get there.
POSTULATIONAL SYSTEMS. After you have wandered through a strange country, stopping
here and there to examine some interesting spot, a village school or a museum, or a
factory, you can get a better picture of how these places are related to each other
if you can find a high hill from where to look back over the entire landscape. Like-
wise, after two or three month's work in geometry you will understand better what you
have been doing if you look back and study the relation between the various parts.
First, you learned the definitions of certain words and adopted some undefined terms.
Second, you agreed to or accepted certain statements called axioms, postulates, or
assumptions.
Third, you used the definitions, undefined terms, and assumptions to discover and
prove general statements called theorems. By means of these theorems and the previous
agreements you found and proved more theorems.
Fourth, you should have ſhearned what a proof is . A proof is an orderly arrangement
of other statements which show how the one you wish to prove is a result of certain
other statements. Or, a proof of a statement is an arrangement of syllogisms that
show that the statement you wish to prove is the conclusion of a syllogism.
Proving a statement does not mean "Convince me of the truth of this statement". It
means "Show that this statement is a consequence of certain other statements".
Fifth, you should notice that any theorem depends on certain previous theorems,
which depend on certain previous theorems, and so on, which depend on definitions and
assumptions made at the start. Naturally if you change any of the assumptions and
definitions which make up the start, the entire structure may change. In fact, there
are systems of geometry (called Non-Euclidean Geometry) which are collections of
theorems found by starting from different assumptions than those you used. Since
everything depends on what assumptions (called postulates) you adopt at the start,
the entire scheme is called a postulational system.
After you have finished the work on parallel lines, try this exercise:
that was the definition of parallel lines? What assumption was made? What was the
first theorem on parallel lines? On which previous theorems does its proof depend?
In the proof of the theorem about the sum of the angles of a triangle which previous
theorems did you use? Examine each reason in each step of the proof to see which
previous theorems were used. On which still earlier theorems did these theorems de-
pend? Trace each reason and theorem as far hack as possible.
31
C H A P T E R 5
C O Iſ W E R S E S
CONVERSES. Compare the statements
If a man lives in Chicago, If a man lives in Illinois,
then he lives in Illinois. then he lives in Chicago.
The righthand statement is not true. If a man lives in Illinois, he may live in
Springfield, Peoria, Rockford, or some other town in the state of Illinois.
The hypothesis (a man lives in Chicago) of the first statement is the conclusion of
the second statement. The conclusion (he lives in Illinois) of the first statement
is the hypothesis of the second statement. Then two statements are related in this
way, each statement is called the converse of the other statement.
Two statements are converses of each other if the hypothesis of each one is the
conclusion of the other.
The staterrents If a man lives in Detroit A man lives in Michigan
then he lives in Eichigan if he lives in Detroit
are n o t converses. They may seem to be converses because the hypothesis (if a
man lives in Detroit) comes at the beginning of one sentence and at the end of the
other sentence. But the position of the hypothesis is of no importance. The "if"
clause is the hypothesis no matter where it appears.
The examples above show that NOT ALL CONVERSES ARE TRUE.
Notice, however, that the converse of every definition is true. If it were false
it would be an unsatisfactory definition and we should replace it by a better one.
E x e r c i s e s
1. Why are the statements below not converses? State the correct converse.
(a) If a lamp is broken it cannot give any light
(b) A lamp cannot give any light if it is broken.
2. State the converses of the following, and tell whether they are true or false.
(a) If a child's name is Mary, then the child is a girl.
(b) I am your teacher if you are my pupils.
(c) If a triangle is equilateral, its sides are all equal.
(d) If a man lives in Ohio, then he lives in the "nited States.
(e) If two triangles are congruent, the angles of one are equal respectively to
the angles of the other.
(f) If two lines are perpendicular to each other, they are not both parallel to
a third line .
3. Are these converses :
(a) If a dog barks, he does not bite. If a dog bites, then he does not bark.
(b) If all men are honest, then no men steal. If all men steal, then no men are honest.
(c) If he is not injured, we can win. Yie cannot win if he is injured.
STATEMENTS WITH SEVERAL CONVERSES. If a statement has more than one item in the
hypothesis then it has more than one converse. For example, the statement
If a carpenter lives in Chicago then he lives in Illinois
has two items in the hypothesis ; l. The man is a carpenter. 2. He lives in Chicago.
and the conclusion is : 3. He lives in Illinois.
Using items 1 and 3 in the hypothesis and item 2 in the conclusion, we get:
If a carpenter lives in Illinois, then he lives in Chicago.
Using items 2 and 3 in the hypothesis and item l in the conclusion, we get:
If a man lives in Chicago and in Illinois, then he is a carpenter.
When a statement has l, 2, or more items in the hypothesis, and l, 2, or more items
in the conclusion, it is agreed that: A converse is made by interchanging one or
more items of the hypothesis with the same number of items from the conclusion.
32
USING CONVERSES TO like DISCOVERIES. Whoever first thought of converses must have
seen at once the possibility of discovering new theorems by interchanging items
between the hypothesis and the conclusion. Further, investigations of this kind will
help us to realize that conclusions depend on the hypotheses.
Suppose you have proved: The bisector of the vertex angle of an isosceles triangle
bisects the base and is perpendicular to the base.
The hypothesis is ; l.AC = BC, and 2. 4 x = 4 y C
The conclusion is: 3. AD = DB, and 4. & r = & 8
(4., x = 4 y is our way of saying that CD bisects y

4. ACB; and 4 r = 4 s is the same as CD + AB.)
Further, whenever it is known that AC = BC we may
say 4: A = 4, B as this was proved in previous work. A B
E x e r c i s e s
Let us see how many theorems we can make by rearranging items l, 2, 3,4 above .
First, select just one item for the hypothesis :
1. If AC = BC and CD is any line through C, can you prove items 2, 3, 4?
2. If 4 x = 4 y and A ABC is any triangle and CD is any line through C, can you
prove items l, 3, 4?
3. If AD = DB, and 4 ABC is any triangle, can you prove items l, 2, 4?
4. If a r = 2 s, and 4 ABC is any triangle, can you prove items l, 2, 3?
Ex. l to 4 show that you cannot prove much if there is not much hypothesis.
Next, select two items for the hypothesis. What can you prove if:
5. AC = BC and AD = DB . 6. AC = BC and 4 r = 6. s. 7. AD = DB and 4 x = 4 y.
8. 4 x = < y and 4. r = 4 s. 9. AD = DB and 4 r = 4. s.
Next, select three items for the hypothesis.
10. What can you prove if AC = BC, 4 x = < y, and AD = DB 2
Ex. 10 illustrates a new situation: there is too much hypothesis' You do not need
all the hypothesis. From the first two items, AC = BC and 4 x = < y, you can prove
the third item, AD = DB. Which item is unnecessary if the hypothesis is
ll. AC = BC, 4 x = < y, and 4 r = < s 2
12. AC = BC, AD = DB, and 4. r = <- s 7
13. Z. x = < y, AD = DB, and < r = 4 s 2
14. Which of the above exercises are converses of each other?
USING CONVERSES TO TEST ARGUMENTS. Frequently we hear arguments that are wrong
because the speaker has confused a proposition and its converse and assumed that the
converse is true. For example, some one may say
l. "If you sleep with the windows of your room closed, you will have colds often.
2. John has a cold nearly all the time.
3. I suppose he sleeps with the windows closed."
The converse of the sentence l is :
l'. If you have frequent colds, then you sleep with the windows closed.
Items l', 2', and 3 make a correct syllogism; but items l, 2, and 3 do not. Hence,
the conclusion, item 3, is correct only if the converse of item l is true.
For another example, we may hear someone say:
l. "In propperous times the prices of all-goods that we buy are high. Eggs cost
60 g a dozen; now they are 20 g.
2. We could bring back these prosperous times if we only raised the prices of
goods. Carpenters ought to get $20 a day instead of $10."
This argument would be correct if it were true that:
l'. If prices are high, then the country will be prosperous.
But l' is the converse of item 1, and we are not sure that l” is true.
33
E x e r c i s e s
What statement must be substituted for the first step in each exercise to make the
conclusion correct? (The new statement may not be true, but what should it be?)
1. All cats have four feet. This animal has four feet. This ani.ial is a cat.
2. If a man swallows poison he will die. Mr. Smith is dead. He swallowed poison.
3. If you increase your advertising you will increase your sales. The sales of the
Bonnet Stores increased this year. They advertised more.
4. If you have anthritis, your knuckles swell. Mrs. Jones has swollen knuckles.
She has anthritis .
5. If the clock ticks loudly, I can't sleep. I could not sleep last night. The
clock was ticking loudly.
6. What is the error in this argument:
To solve the problem "Find John's age if 3 times his age is 20 more than his age"
we let a = John's age. This leads to the equation 3a = a + 20. Hence a = 10.
Since the number 10 satisfies the equation, John's age is 10.
7. Bring to class some arguments which are faulty because they assume a converse.
CONVERSES OF ALGEBRAIC PROBLEliS. If a shoe dealer buys shoes for "4 a pair and
sells them for £5 a pair, his profit is &l, and the rate of profit based on the cost
is 25 %. This statement involves 4 numbers :
The cost of the shoes, 44. The profit, ... l.
The selling price, & 5. The per cent of profit, based on cost, 25 %.
We can represent these numbers by c, s , p, and r (rate of profit).
Let us think of the numbers c and s as the data or hypothesis of the problem,
and the numbers p and r as the conclusion. After studying converses we should
naturally think:
What are the converses of this problem, and can we solve them?
For example, (a) If c and p are given, can we find s and r 2
(b) If c and r are given, can we find s and p 2
(There are 3 more problems like these. What are they?)
Problem (a) is: If the cost is 34 and the profit is ºl, what is the selling price,
and what is the per cent of profit on the cost?
Problem (b) is: If the cost is $4 and the dealer makes a profit of 25 % on the cost,
what is the profit, and the selling price?
There are, in all, six problems that can be made from the numbers c, s , p, and r.
From your study of algebra you have learned that all six problems can be solved in one
way provided you know the relations between the numbers in the problem. Here
Cost + Profit = Selling Price. Rate of Profit, x Cost, - Profit, .
In algebraic language, c + p = s and r c = p
Sometimes it is convenient to use the relation C + r C = S
Any of the problems can be solved by substituting in these equations the numbers
from the data, and solving the equations to find the conclusion. Try it!
Let us try a more difficult problem.
A farmer mixes 40 bu. of wheat worth 90 % a bu . with 60 bu. of corn worth 50 g a bu.
to make some chicken feed. What is the mixture worth per bushel?
The wheat is worth 40 x 90 or .36; the corn is worth 60 x 50 or $30. Hence the
(40 + 60) bu. of feed is worth $66 or 66 £ a bushel.
The data contains 4 numbers: w = no. of bu . of wheat, a = value of wheat per bu .
c = no. of bu . of corn, b = value of corn per bu.
The conclusion has 2 numbers : f = no. of bu . of feed, d = value of feed per bu.
The problem involves 6 numbers. An interesting problem to investigate is : How
many of these numbers must be known in order to find the remaining ones?
34
Of the possible problems mentioned on page 33, two are important: the previous one
and the one for which the data is a, b, f, and f, and the conclusion is w and c. Thus:
Tf wheat is worth 95 g a bul, and corn is worth 55 & a bu. how many bushels of each
should be mixed to make 80 bu . of feed worth 65 & a bushel?
To solve these problems we must know the relations between the numbers. In this case
they are: No. of bu . of wheat + no. of bu . of corn = no. of bu. of feed.
Value of wheat + value of corn = value of feed.
Algebraicly: w - C = f' and aw - bc = dif'
By substituting the data and solving for the conclusion we solve the problem.
E x e r c i s e s
Study some of the problems in your algebra in the way shown above. Do not merely
solve the problem, but tell (a) how many numbers the problem involves, (b) which of
the numbers are in the data and which in the conclusion, (c) what converses can be
made by interchanging the numbers , (d) how many different kinds of problems can be
made, (e) what are the relations between the numbers.
C H A P T E R 6
T H E I N D I R E C T M E T H O D
l. Mr. Jones was sitting comfortably on the front porch when he decided he would
like to finish the novel he had left in the living room. He asked his son Dicky to
get it for him. Dicky looked on the table and said, "Ther's three books here".
Dicky was only four years old and could not read; but his father knew that Dicky had
learned to recognize colors. He said, "Not the blue book, Sicky; and not the green
one." Dicky selected the right book.
2. One day Mrs. Jones wanted to telephone her husband to come home early for dinner
as some cousins from out of town had arrived. She knew that at that time of the day
Mr. Jones was either in his office or on the way to the golf club. She phoned his
office and the clerk answered, "He's not here. I don't know where he is ." So she
phoned his gold club, and left the message for him there. He got the message.
3. Mr. Jones has a set of four keys, one of which will unlock the cupboard in his
office. When he handed the keys to his secretary and told her to unlock the cupboard
she tried one key and it failed; she tried a second key, and it also failed; the
third key also failed. Will the fourth key unlock the cupboard?
The indirect method is based on this idea: If one of two things is correct and one
of the two is wrong, then the other is right. If one of three things is correct and
two of the three are wrong, then the third one is right.
If o n e of n things is correct, and if (n − 1) of
these are shown to be wrong, then the remaining one is correct.
Here are some examples of indirect methods which pupils have written:
l. Two boys were discussing whether a certain tree was an elm or an oak. "Don't you
know" said one of the boys, "that oak trees have acorns? Do you see any acorns around
this tree? If it's either an oak or an elm, then it's an elm."
2. During a spelling test I had to spell the word receive. T could not remember
whether to write ie or ei. I wrote ie. When the paper was returned, the word
was marked "wrong". So now I know it should be ei.
3. Mother asked me to bring the geranium from the window. I did not know what a
geranium looked like. But I could see that one of the plants was an ivy and another
was a marigold. So I picked up the third plant and brought it to mother.
3 5
The indirect method of proving a statement is based on these ideas:
l. If the conclusion is wrong, some other conclusion is right.
2. Investigate all the other possible conclusions.
3. If all the other conclusions are proved wrong, the first conclusion is right.
An indirect proof is really the politest form of argument because you begin by pre-
tending that you are wrong and that the other fellow is right.
For example, you believe that a certain tune heard over the radio was "Old Black Joe".
Your friend who is visiting with you says it was "My Old Kentucky Home". Hence you
say to your friend, "Hell, I suppose I am wrong and you are right. You know so much
more about music than I do. Let's call up the radio station and ask them if they have
played "My Old Kentucky IIome'. " If the radio station answers that they have not
played that tune any time this year, then your friend is wrong. And if you are sure
that the tune was one of these two, then you are right.
To use an indirect proof, proceed as follows:
. Begin by supposing that what you are trying to prove is wrong.
. Decide how many other possible conclusions there could be.
. Prove that all the other possible conclusions of step 2 are wrong.
. Then you can say that the statement you are trying to prove is correct.
:
In step 2 it is important to consider all the possibilities.
In step 3 the possibilities are usually proved wrong by showing that they lead to
contradictions of some previous theorem or agreement. In the example on page 34 the
boys have agreed on three things:
(a) The tree is either an oak or an elm.
(b) Oak trees are the only trees that have acorns.
(c) The tree about which they are arguing has acorns.
The proof that the tree is an oak would then be:
1. Suppose that the tree is n o t an oak.
2. The only other possibility, according to (a), is that it is an elm.
3. This elm has acorn, according to (c)
4. But step 3 contradicts item (b).
5. Hence the supposition in step l is wrong.
6. The tree is an oak since the other supposition was wrong.
A GEOMETRIC EXAMPLE. Suppose we have proved that
An exterior angle of a triangle is greater than either remote interior angle,
and wish now to prove that
If alternate interior angles are equal, the two lines are parallel.
C/ C
V
V
Ul Ul
/A /A N __T /A
There are two possibilities: either (l) the lines are parallel, or (2) the lines
are not parallel. Since we wish to prove (l) we begin by supposing (2) is true.
Suppose the lines are n o t parallel.
In that case the lines meet at some point, either at the right of AC as in the second
figure or at the left of AC as in the third figure. In both of these figures
4, u 4. V because "an exterior angle of a triangle is greater. . . . . . "
But 4 u = & V according to the hypothesis.
Since these two statements contradict each other, our supposition is wrong.
The conclusion (l) is right because conclusion (2) is wrong.
C


E x e r c i s e s
. Supply the missing words in this proof: 8.
Data: a ll c, b ll c, Prove: a ll b. b
l. There are two possibilities:
either a ll b, or a is not to b. C
. Suppose a is not ll to b. Then a and b meet at some point.
. Through that point we have two lines parallel to c.
. Step 3 . . . . . . . . . . . . . . . . the assumption which says . . . . . . . .
. Hence the supposition . . . . . . . . . .
:
2. Prove that through a point there cannot be two perpendiculars to a line.
l. Suppose there are two such perpendiculars, as . . . . . . P
2. Then . . . . . . . . .
3. But . . . . . . . .

4. Steps 2 and 3 contradict each other. Hence, . . . . . A B
t C D
3. In the figure, AD-L AB, AC 1- BD, and it is known
that 4, l is equal to 4. 2, or 4. 3, or 4- 4. D C
Prove 4 l = 4-4, by the indirect method. 6 5

Begin by supposing that 4- 1 = 4- 2. What can you
then prove in the figure? Is the conclusion reasonable?
Suppose next that 4 l = <- 3. What can you then
prove? Is the conclusion reasonable? AV & B
If these conclusions are wrong, then . . . . . .
4. State policeman O'Reilly dashed to the scene of the accident on his motorcycle.
A witness there said, "I almost got the driver's number. It began with 432, and the
last three figures were sixes or sevens. But I've forgotten the exact number. They
were sixes and sevens like 676 or 766 or something like that."
On investigation O'Reilly found that the owners of licenses 432–766, 432-676, and
432–767 were never near the accident. And so he arrested the owner of the license
432-667. Do you think he got the right one?
5. Mr. Jones replaced the old "buzzer bell" in his home with some chimes. He had
three wires numbered l, 2, and 3, which were to be connected to three plugs, lettered
A, B, and C. When he connected l with A, 2 with B, and 3 with C, the chimes did
not ring. How many different arrangements must he try before all the possible ways
have been investigated?
6. Mr. Jones is planning to buy a new car. He wants a car that has three features
which we shall call a, b, and c. Also, the car must not have the feature d, while it
is unimportant whether the car has feature e or not.
Which of the following cars satisfy his requirements:
Car 1 has a, b, d, not c, not e- Car 4 has a, d, not b, not c, not e.
Car 2 has a , b, c, d, not e. Car 5 has a, b, c, e, not, d.
Car 3 has a, b, c, not d, not e - Car 6 has a, b, not c, not d, not e.
7. One of the characters in the senior play "Why Be A Junior" requires a thin boy
about six feet tall, with a deep voice. If the choice depends only on this data,
who of the following boys should play the part;
Tom weighs 120 lb., while Bob is 5 lb., heavier. Both Tom and Bob are 5 ft. ll in.
Henry is 6 ft. tall but weighs 40 lb. more than Tom. John is really a light-weight,
weighing only 130 lb - although he is 6 ft - tall. iienry and Bob both sing base in the
Glee Club, while John and Tom sing tenor.
37
SOME PUZZLES. ... any puzzles are solved by the indirect method. For example,
John, Dick, and Henry are on the athletic teams of the ſorely High School. One of
the boys is on the football team, one plays basketball, and one plays baseball. Use
the information below to deciſie which boy plays which game:
(a) The baseball player often practices with his father.
(b) Henry once owed the basketball player a dime.
(c) John's father has been an invalid for many years.
(d) The basketball player's father plays golf with the father of one of the boys.
The argument would proceed thus:
1. John is not the baseball player. John's father is an invalid by (c), and so
John could not practice with his father (a).
2. John is not the basketball player. John's father is an invalid by (c), and so
could not play golf (d).
3. John is the football player. There are 3 possibilities, and 2 were wrong.
4. Henry is not the football player. Step 3 says John is the football player.
5. Henry is not the basketball player. He could not owe money to himself. See (b).
6. Henry is the baseball player. There are 3 possibilities, and 2 were wrong.
7. Dick is the basketball player. Gee steps 3 and 6. There are 3 possibilities . . . .
E x e r c i s e s
l. ...ary, Jane, and Helen have various hobbies. One sews, another likes tap dancing,
and the remaining one plays the violin. Find each girl's hobby from this data:
(a) l ary's chum is the girl who dances.
(b) Jane thinks that keys are always used to open doors.
(c) Jane applauds whenever the dancer performs.
b 6 a
2. At the right is a problem in multiplication. Some of the numbers c 3
were erased and letters written in their places. Find what numbers 5 d 1
should take the places of the letters. f’ e 4
By the indirect method, a is either l, or 2, or 3, or 4, etc. i h g 1
3. "rite the indirect argument that Johnny is using when he says, "I could n't have
broken that window. I was at the movie yesterday afternoon when it happenned."
4. Frieda is planning to spend her vacation on a dude ranch in Wyoming. She likes
outdoor life and is much interested in sports; she plays bridge well, and does not
read any books. She does not wish to go alone and wants one of the following girls
to go with her. Who would make the best companion?
Angela. She likes tennis and swimming but not horseback riding. She plays bridge
well, and has read most of the latest novels.
Marion likes horseback riding and skating, and has won several cups in golf tourna-
ments. She never reads a book and could not sit still long enough to play any
indoor game.
Tāna likes to take long walks but she is not the athletic type. She sees a movie
only occasionally. She does not care for bridge but will play when asked because she
always tries to be agreeable.
Ruth. This girl loves to cook and entertain her friends with dinners that she
prepares herself. She sews her own clothes which are copied after the latest styles
from Paris. She has never read anything but a cook book or a style book since she
left college, but she plays bridge very well.
38
C H A P T E R 7
I N V E R S E S
INVERSES OF A PROPOSITION. Compare the statements
l. If a man lives in Chicago 2. If a man does not live in Chicago
then he lives in Illinois. then he does not live in Illinois.
The hypothesis of (1) is : a men lives in Chicago
(2) is : a men does not live in Chicago
The conclusion of (l) is: he lives in Illinois
(2) is: he does not live in Illinois
The hypotheses contradict each other, and the conclusions contradict each other.
Each statement is called the inverse of the other.
The inverse of a statement is made by contradicting the hypothesis
and contradicting the conclusion. By contradicting, or denying, is
meant: inserting the word not, or omitting the word not if it is there.
NOT ALL INVERSES ARE TRUE. For example,
the statement If two angles are right angles, then they are equal.
has the false inverse If two angles are not right angles, then they are not equal.
liotice, however, that the inverse of every definition is true. If it were false, it
would be an unsatisfactory definition and we would replace it by a better one.
E x e r c i s e s
State the inverses. Tell whether you think the y are true or false.
. If a quadrilateral is a square, all its sides are equal.
. If two angles are not complementary, their sum is not 90 degrees.
If two lines are parallel, they do not meet when prolonged.
If it rains to-morrow I shall not go to town.
. If you use a Harris watch you will always know the correct time.
. If you do not use "...axo Coal you are wasting money.
. I shall go to the game if the weather is fine .
:
C.
7
If there is more than one item in the hypothesis, an inverse is formed by denying
any one item in the hypothesis and denying the conclusion. For example,
If a man is married and is more than 40 years old, he needs insurance
has two inverses :
l. If a man is not married and is more than 40 years old, he does not need insurance.
2. If a man is married and not more than 40 years old, he does not need insurance.
If a statement has two items in the conclusion, it can be written as two separate
statements, each of which has inverses . Thus, the statement
If a man is married he should have accident insurance and life insurance
can be written as the two statements
If a man is married he should have accident insurance
Tf a man is married he should have life insurance
and each of these statements has its inverse.
ASSUMING THE INVERSE. When a company advertises its fountain pens by saying
In a Tandy Fountain Pen the ink flows freely
it wants you to think: If you pen is not a Dandy, the ink does not flow freely.
When a salesman says to a housewife : The Homey Sweeper picks up 95 % of the dirt
he wants her to think: If the sweeper is not a Homey, it does not pick up 95 % of the dirt.
Almost any candidate for public office will say "I will reduce your taxes". Per-
haps he will, but he wants you to think that the other candidates will not reduce
your taxes. If an inverse of a statement were always true, people would not be mis-
lead so easily. Of course, if a man makes one statement and you think of the inverse,
you cannot blame anyone but yourself if you think that the inverse is true.
39
E x e r c i s e s
The following statements are samples of what you might read in advertisements or
sales talks. What does the advertiser want you to believe in addition to what he says?
Are the inverses true?
1. Blackit, Ink will dry as quickly as you can write.
2. This glue will resist a pull of 50 pounds.
3. Liberty Glass in windows will not distort the vision.
4. The World-Wide Radios do not distort the sound of the high notes.
5. "lirs. Jones" said the salesman, "this machine will last you a life-time".
6. "Of course our price is a little higher than for lost furnaces. Cut if the price
is 10 % higher, you are getting a 20 % better article."
7. Collect and examine some advertisements. Discuss what they say, and what they
want you to believe.
8. Collect some propositions and write the converse and inverse of each. Tse only
simple statements that have only one converse and one inverse. Shoose them so that
some are true and some are false, some of the converses being true and some not, and
some inverses being true and some not. The diagram below shows the possible ways in
which you might have true and false statements.
Column 1 means that all three statements are l 2 3 4 5 6 7 8
true; column 2 means that the statement is true proposition T T T T F F F F
its converse is true, and its inverse is false. Converse T T F F T T F F
You will have difficulty finding examples to inverse T F T F T F T F
fit some of the columns.
Try to find some rule that will tell how the truth of the converse and the inverse
are related to each other and to the truth of the proposition itself.
COkiPARING THE CONVERSE AND INVERSE. In books on logic you will find proofs of the
statements:
If the converse of a statement is true , then the inverse is also true.
If the inverse of a statement is true, then the converse is also true.
!,ikewise, The converse is false if the inverse is false.
The inverse is false if the converse is false .
The Converse and the Tnverse are either Both True or Both False.
You can see at once how useful the above rule is . If you have proved the converse
of a theorem you know that the inverse is true. If you aave proved the inverse, you
know that the converse is true. When you need to examine the converse, it may be
easier to examine the inverse. To prove the inverse, you may prove the converse.
E x e r c i s e s
1. Why is the inverse of every definition true?
2. For which of the columns in ex. 8 above do no examples exist?
3. Is the converse of the inverse of a statement the same as the inverse of the
converse of the statement?
4. If you wished to prove the converse of the following statements (not the state-
ments themselves, but their converse) what other statement might you prove :
(a) If a point is equidistant from the ends of a line, it is on the perpendicular
bisector of the line.
(b) If a point is not on the bisector of an angle, it is not equidistant from the
sides of the angle .
(c) Tf a point is on a circle, its distance from the center is equal to a radius.
(d) If two lines are parallel, the alternate interior angles are equal.
(e) If it rains I shall not go.
(f) If the wages of the workmen are raised, the price of the goods is increased.
(g) If more is spent for advertising, more goods will be sold.
40.
C H A P T E R 8
C O N T R A P O S I T I W E S
(In this chapter we consider only statements having one
item in the hypothesis and one item in the conclusion.)
Some time you must have looked at a tree and said, "It can't be windy to-day; I do
not see any leaves blowing." You were thinking as follows:
l. If the wind were blowing, then the leaves would be moving
2. If the leaves are not moving, then the wind is not blowing.
These two statements are called contrapositives of each other.
The contrapositive of a statement is formed by saying:
If the conclusion is not true, then the hypothesis is not true.
A proposition: If a man lives in Akron, then he lives in Ohio.
Its contrapositive: If a man does not live in Ohio, then he does not live in Akron.
A proposition: All right angles are equal.
Its contrapositive : If angles are not equal, then they are not right angles.
Do not think that the contrapositive always contains "not" twice. If a hypothesis
or conclusion already contains the word "not", we deny it by omitting the "not".
A proposition and its contrapositive are either both true or both false.
In the example at the top of the page, you were using this idea when you assumed
that statement 2 was as good as statement l. When you say, "This triangle does not
have two equal sides; hence it is not isosceles" you were using the contrapositive
of "If a triangle is isosceles, it has two equal sides".
In books on logic you can find a proof of the statement that a proposition and its
contrapositive are either both true or both false. We can explain the idea thus:
Consider the statements :
l. If you are a freshman, you take English l.
2. If you are not taking English l, you are not a freshman.
Imagine all the pupils who are taking English l collected in
a circle. Inside that circle are freshman, some sophomores,
and perhaps some not-very-bright juniors. But you will not
find in the circle anyone who is taking Fmglish 2.
Then the next two statements are either both true or both false:
l. If a pupil is in the inner circle, then he is inside the larger circle.
2. If a pupil is not inside the larger circle, he is not in the smaller circle.
Freshmen
E x e r c i s e s
State the contrapositives of the following:
l. If it is an automobile, then it has four wheels.
2. If a number is divisible exactly by 2, it is an even number.
3. Windows are made of glass.
4. If Mir. Jones is elected mayor, our taxes will be reduced.
5. If a figure is a trapezoid, then all its sides are not equal.
6. If two lines are not parallel, the corresponding angles are not equal.
7.Write a proposition. Then write its converse. Next write the inverse of the con-
verse. Compare it with the contrapositive of the first proposition.
8. Write some proposition. Then write its inverse. Next write the converse of
the inverse. Compare it with the contrapositive of the first proposition.


4l
USING THE CONTRAPOSITIVE. When in doubt about the correctness of an argument, we
can often settle the argument by writing the contrapositive of the statements. For
example, supose the question is
l. All students who may not smoke are freshmen.
2. If John is a senior, may he smoke?
At first glance it seems as if statement l cannot decide the question because prop-
osition l applies to students who may not smoke, and the minor premise is that John
is a senior. However, the contrapositive of proposition l is:
l'. If a student is not a freshman, then he may smoke.
is ow we can rea'ily answer: Since John is not a freshman, he may smoke.
{} x e r c i s e 8
l. Assume that the given statement below is true. Write its converse, inverse,
and contrapositive, and tell whether they are true or false or may be either:
(a) Seniors are permitted to belong to the High - Ilo Club.
(b) Seniors belong to the High - Ho Club.
2. Are the conclusions below correct if the first sentence is the major premise
an... the second is the minor premise?
(a) If he were well, he would write. He has not written. He is not well.
(b) If he were well, he would write. He has written. Hence he is well.
(c) If he were well, he would write. Ile is not well. Hence he has not written.
3. If "only democrats voted for Jones" which of these statements are true:
(a) All who voted for Jones were democrats.
(b) All democrats voted for Jones.
(c) Those who did not vote for Jones were not democrats.
(d) Those who were not democrats did not vote for Jones.
4. T f it is true that "all the parents who were present at the meeting favored
kiss Smith" oes it follow that "parents who did not favor , iss Smith were absent." ?
MISUSE OF THE CONTRAPOSITIVE. ...any advertise...ents and public speakers rely for
their effect on an unconscious use of the contrapositive. When an advertisement says
The aristocrats of the world drive Blixo cars
it wants you to think: If you do not urive a Blixo car, you are not an aristocrat.
These statements are contra positives. If one is true, then the other is true . But
do you know that either one is true?
When a candidate for public office says
"Citizens who are interested in the welfare of their country should vote for me"
he wants you to think
"If you do not vote for me, you are not interested in the welfare of your country"
The mere fact that the speaker does not use the second form, but permits you to
supply it, makes such statements all the more dangerous. You will naturally be more
likely to believe a statement which you yourself have made than a statement made by
someone else. By letting you supply the contrapositive, you have become an authority
for the statement, and "you would n't lie; would you?"
E x e r c i s e s
"...hen the following statements are used in advertisements or as slogans what are you
expected to believe in addition to what the statement actually says?
l. Good pupils write their essays or, an Excello Typewriter.
2. Good citizens of a school do not throw scraps of paper on the floor.
3. The best golfers in your club use Far-Go Golf Balls.
4. T f you are neutral you are helping the dictators.
5. "If I am elected, the parks in this city will be enlargel."
6. Collect some advertisements and slogans. "rite the contrapositives of some of
the statements. Co the latter seem more or less likely to be true than the statements
themselves?
42
SYMBOLIC FORM OF A PROPOSITION. To prove a statement about all numbers that are
divisible by 3, it will not do to write a proof which uses the numbers 3, 6, 9, 12, ..
You must have an algebraic way of writing "all numbers exactly divisible by 3". (The
symbol 3n represents all such hº If you wish to prove a statement about
squares, you may use a drawing of a square but you must not say anything about your
drawing that is not true about all squares. Likewise, if you wish to make a state-
ments about all theorems, you must have sone way of writing a theorem so that it will
represent all theorems. One such method is to write: &
If a = b then c = d.
Here, a , b, c, and d are not numbers. The item a = b is just a way of saying
that the statement has a hypothesis. In one proposition the symbol "a = b" may
mean "all sides are equal"; in another, it may mean "the angles are bisected". Like-
wise, the symbol "c = d" is just a way of saying that the proposition has a conclu-
sion. In one theorem, "c = d" may mean "the lines are parallel"; in another it may
mean "the triangles are congruent".
Other ways of writing a proposition are:
Tf (hyp) then (con) ; If H then C ; or H implies C
When we wish to say that the hypothesis is not true we use the symbol # , an
= sign with a stroke through it. Thus, a # b means "a does not equal b'.
Likewise, the symbol c # d means that the conclusion is not true.
Using this symbolic language, we write
l. A proposition: If a = b then c = d
2. Its converse: If c = d then a = b
3. Its inverse : If a # b then c # d
4. Its contrapositive: If c + d then a # b
RELATIONS BETWEEN THE PROPOSITIONS. A proposition and the three that can be
derived from it may be groupe i thus :
l. A proposition rn v-
G I Both F
roup { 4. Its contrapositive } are True or Both are False
Group II { . Its converse } Both are True or Both are False
3. Its inverse
From this arrangement you can see that
To prove all four statements all that you need to do is to prove one statement from
each group. To test all four statements, you need to investigate only two of them
provided you select one statement from each group.
E x e r c i s e S
1. In how many different ways can you prove the four propositions by proving only
two of them?
2. A pupil remembered a rule like this: If the numbers l,2,3,4 (each representing
one of the above propositions) were written at the four corners of a square in a cer-
tain order, then all four propositions are true (or are false) if any two consecutive
ones are true (or are false). What should be the arrangement of the numbers at the
corners of the square?
3. Since you may forget whether the number 3 stands for the converse or the inverse,
can you improve the scheme in ex- 3 by using letters in place of numbers?
4. Suppose you found that the following statement was true
All two-toed animals have three ears.
(The statement is nonsense, of course; but it is purposely made foolish so that you
may not be influenced by any knowlege of zoology or other subjects.) Write the three
propositions related to this one. What can you say about the truth or falsity of the
related propositions?
43
5. The definition of timber line is "On all very high mountains there is a line
above which atmospheric conditions will not allow trees to grow". ''}rite the converse,
inverse, and contrapositive of this definition. (Suggestion: first write the defini-
tion as: If this is the timber 1 ine, then atmospheric conditions will not allow trees
to grow above this line.) Are the inverse, ... true?
Next, change the statement to: If this is the timber line, trees will not grow above it.
It is now not a definition. Write the three related statements, and discuss whether
or not they are true.
6. Correct any of the following statements which may be wrong:
(a) The converse of the contrapositive of a proposition is the proposition itself.
(b) The contrapositive of the inverse of a proposition is the proposition itself.
(c) The inverse of the contrapositive of a proposition is the converse.
(d) The contrapositive of the converse of a proposition is the inverse.
7. If you assume that a proposition and its contrapositive are either both true or
both false, why does it follow that the converse and the inverse are either both true
or both false.
8. If a statement contains two items in its hypothesis, the rule for forming its
contrapositive is : Deny the conclusion and deny one of the items in the hypothesis,
and interchange these two items. For example, one contrapositive of
If a = b and c = d then e = f' is If a = b and e # f then c # d.
What is the other contrapositive?
*hat are the contrapositives of: If oxygen and water are present, oxidation takes
place.
GEOMETRIC APPLICATIONS. Suppose we wish to prove
l. If alternate interior angles are equal, the lines are parallel
and also wish to prove the three related theorems, namely,
2. If lines are parallel, the alternate interior angles are equal.
3. If alternate interior angles are not equal, the lines are not parallel.
4. If lines are not parallel, the alternate interior angles are not equal.
According to the rule on page 42 we can prove all four theorems by proving only two
of them provided we select the two correctly.
E x e r c i s e s
1. Make a list of the different ways (there are four) in which the above theorems
can be proved by proving only two of them. Which way did you actually use in your
geometry class? To test your ability try one of the other ways.
2. Suppose you wish to prove the following theorems and also their converses.
State the different ways in which you can do this.
(a) A point on the bisector of an angle is equidistant from the sides of the angle.
(b) A point on the perpendicular bisector of a line is equidistant from the ends of
the line.
(c) If a point is on a circle its distance from the center is equal to a radius.
3. Select some theorem that you have proved recently in your geometry class. Write
the three related theorems. In what different ways can you test all four?
4. (Do this exercise when your class is studying Loci.) To prove that a certain
curve or line is the locus of a point you must prove two things:
(a) If a point satisfies the given condition, the point is on the locus.
(b) If a point is on the locus, it satisfies the given condition.
Here we have a proposition and its converse • State the inverse and the contraposi-
tive of (a). What are the four ways in which the four propositions can be proved by
proving only certain two of them?
C H A P T E R 9
N E C E S S A R Y A N D S U F F I C I E M T. C O N D I T I O N S
Ted Jones saw a group of pupils in room 123 having an entertainment. Outside the
door was a sign "Admission 5 ¢ ". Ted bought a ticket but the doorman would not
admit him.
"What's the matter?" asked Ted. "I got a ticket."
"That's necessary", said the doorman, "but not sufficient."
"What d'you mean - not sufficient?" asked Ted. "I got a ticket, have n't I?"
"Yes", was the answer. "But this play is open only to seniors."
(Evidently the sign should have read "Admission 5 2. Only Seniors Admitted." If
Ted did get it, what do you infer?)
This story illustrates the use of the words "necessary" and "sufficient". To get
in, it was necessary to have a nickel ; but having a nickel was not sufficient. You
must also be a senior. Being a senior was necessary; but being a senior was not
sufficient. You must have a nickel.
Some pupils will object to what has been said in the preceding paragraph. They
feel it is wrong to say "a nickel is necessary" if "being a senior" is also necessary.
One pupil may believe that the word necessary should not be used unless we mention
all the things that are necessary. Another pupil may believe that "necessary" may
be used to describe any one of the several things that are needed. Evidently we need
to agree on the meaning of the words necessary and sufficient.
First, are the words "requirements" "requisite" "condition" "satisfy" a part
of your vocabulary? We say that English l is a requisite (some say "prerequisite")
for taking English 2. There is a certain condition you must fulfill or meet or sat-
isfy before you can take English 2; that is , you must have credit in English l.
Being 5 ft. 10 in. tall is a condition required of policemen in some towns. Being
three-sided is a condition satisfied by all triangles. Being 30 years old is a con-
dition that all members of the Senate must satisfy.
NECESSARY CONDITIONS. When we say that food is necessary for life we mean: if we
do not have food we will not live. We also need other things to live, but the state-
ment "Food is necessary for life" means: If we do not have food we can not live.
When we say that water is necessary for life we mean: if we do not have water, we
can not live.
When we say "A" is a necessary condition for "B" we mean:
If "A" is not true, then "B" is not true (l)
The contrapositive of (1) is
If "B" is true, then "A" is true. (2)
Hence, our definition of necessary is:
"A" is necessary for "B" if either statement (1) or (2) is true.
For example, to prove that it is necessary to use Cleano Coap to be beautiful,
prove either (1) If you do not use Cleano Soap then you are not beautiful
or (2) If you are beautiful, you use Cleano Soap.
E x e r c i s e s
What must you prove to prove that
• "Being yellow in color" is a necessary condition for good gasoline.
"Electing Jones for mayor" is necessary for good government.
Hard work is necessary for becoming rich.
• Wearing the latest style in clothes is necessary to be popular.
• Going to college is necessary for getting an easy job.
i
O
45
SUFFICIENT CONDITIONS. In the story on page 44 Ted could enter the room if he were
a senior and had a ticket. Those two conditions were sufficient.
When we say "being 21, a citizen, and having lived in your precint for 30 days" is
sufficient to allow you to vote, we mean: if you satisfy these conditions then you
may vote.
"A" is sufficient for "B" means: If "A" is true, then "B" is true. Or, using
the contrapositive: ff "B" is not true, then "A" is not true.
If "A" is sufficient for "B" then having "A" is enough to insure "B". You do not
need more than "A" to get "B". If "A" is the hypothesis of a theorem, then by
using "A" you can get "B".
For example, to prove that "having a Bolo Watch" is sufficient to give you the
correct time, prove
either (1) If you have a Bolo Watch, you have the correct time
Or (2) If you do, not have the correct time you do not have a Bolo Watch.
NECESSARY AND SUFFICIENT. In most problems it is easier to find the sufficient
conditions than to find the necessary ones. To remove ink from cloth you could pro-
ceed as follows: First soak the spot with lemon juice. Then over the spot spread
a paste formed by mixing milk and corn meal. After this paste has been on the spot
a few minutes scrape it off and add more lemon juice. *hat you would like to know
is whether all this work is necessary. It was evidently sufficient since the spot
disappeared. To find what is necessary you must try the lemon juice alone. If it
removes the spot, then the lemon juice is sufficient. Perhaps the lemon juice is not
necessary; perhaps milk and corn meal alone will remove the spot. In that case, milk
and corn meal are sufficient, but not noc essary.
To say that "A" is necessary and sufficient for "B" means that "A" will produce "B"
and "B" will produce "A". It means that "A" is enough, not too much, and that you
cannot have "B" without "A" or have "A" without "B".
!º X e r c i s e s
l. Complete the sentences: To prove that
(a) Water is sufficient to make plants grow, prove . . . . . . . . . . . . . or prove . . . . .
(b) Being a senior is sufficient to make a pupil important in school, prove. . . . . . . .
2. The next two sentences are wrong. CORRECT THFli.
(a) To prove that the conditions in the hypothesis of a theorem are necessary, prove
the theorem itself.
(b) To prove that the conditions in the hypothesis of a theorem are sufficient,
prove the inverse of the theorem.
3. Is the condition "being right angles" necessary or sufficient or both for the
conclusion "they are equal"? You can answer such questions as this thus:
"Being right angles" is necessary for "being equal" if it is true that
If angles are equal then they are right angles. (Is this true?)
"Being right angles" is sufficient for "being equal" if it is true that
If angles are right angles, then they are equal. (Is this true?)
4. Examine the conditions in the hypothesis of the following statements to see if
these conditions are necessary or sufficient for the conclusion:
(a) If the opposite sides of a quadrilateral are equal, then they are parallel.
(b) If the diagonals of a quadrilateral are equal, then the opposite sides are equal.
(c) Tf the diagonals of a quadrilateral are perpendicular to each other, then all the
sides of the figure are equal.
(d) If a point is inside a circle its distance from the center is less than a radius.
(e) If the diagonals of a quadrilateral are equal, the figure is a parallelogram.
(f) If the angles of a quadrilateral are right angles, the figure is a parallelogram.
(g) If a figure is a rectangle it can be inscribed in a circle.
46
5. If the price of admission to a show is lo é and you have a quarter, have you
sufficient money according to our mathematical definition of sitſficient? You can
answer this question by trying the definition on page 45. Thus,
"Having a quarter" is sufficient if it is true that
If you have a quarter, then you will be admitted.
6. State the following sentence in the "if-then" form and then tell whether the
conditions in the hypothesis are necessary or sufficient for the conclusion.
To be a success in the theater one must have acting ability.
7. If a match is wet it cannot be used to light a fire. Is the given cause in the
hypothesis necessary or sufficient for the conclusion?
8. Is the approach of autumn a necessary or a sufficient condition for the falling
of the leaves from the trees?
9. Last June Air. Jones got a raise in salary , i.irs. Jones hired a maid to do the
work around their home. She began using tieinos Hand Lotion to improve the looks of
her hands. How she endorses the lotion very highly. "My hands have improved a
thousand times since I began using this lotion", she says . Is the use of the lotion
a necessary or a sufficient condition for the improvement of her hands?
10. Bill Smith could never remember just what he must prove when he wanted to show
that a certain condition was necessary or what he must prove to show that something
was sufficient. His friend Jim said, "Oh, that's easy. I have memorized the phrase:
"Don't forget the necessary converse and the sufficient proposition'." Explain
what Jim's phrase means.
ll. Are the following statements correct:
(a) If a theorem is not true, the hypothesis is not sufficient.
(b) Tf a theorem is true, all the items of the hypothesis are necessary.
C H A P T E R 10
N D U C T I W E S C I E N T I F I C A N D D E D U C T I W E Mí E T H O D S
THE INDUCTIVE METHOD, In your study of algebra, you learned that x? ;3 --- **
because x" = x • X and x” = x : x < x; hence x* x* = x • x < x < x < x gr X •
X x4 7
:= X 9
You could write a similar argument to show that x* x* = x and
and other examples like these. Finally you decided that
xany number some number = , the sum of the exponents
X
In mathematical symbols you expressed this idea as x° x” = x* + b
By observing or studying certain situations you found a rule which you could there-
after apply to all situations of that kind . This process is called generalizing,
and the conclusion is called a generalization; this kind of reasoning is called
inductive reasoning .
As another example of inductive reasoning, you might collect sets of equal fractions
(called proportions) such as 2 = + E = 19. 4 = 12. You would soon
3 6 8 16 LO 30
notice that in each proportion the product of certain numbers equals the product of
certain other numbers. Thus 2 x 6 = 3 x 4 5 x 16 = 8 x 10 etc. After you
feel sure that this is true in all the proportions that you might try, you decide to
state your discovery as a generalization. You might say: The first number times
the last one equals the product of the other two numbers.
Inductive reasoning is the process of finding a general
statement by a study of particular cases or examples.
47
E x e r c i s e s
1. You can multiply 25 by 25 thus: first find 2 x 3 = 6; then write C25.
To multiply 65 by 65, first find 6 x 7 = 42; then write 4225. To multiply 75 by 75,
first find 7 x 8 = 56; then write 5625. From a study of these cases and assuming
that the rule works for all numbers ending in 5, find the rule.
2. In the list of numbers below, supply the missing fifth number:
(a) 1,4,9,16, . . . (c) 1, 2, 3, 1/3, ... (e) 2, 3, 5, £, . . .
(b) 2, 8, 14, 20, . . . (d) 0, 3, 8, 15, . . . (f) l, 3, 4, 7, . . .
3. Using the equation y = 2x + 3 and lotting x = l, 2, 3, 4, . . . you can make
a table of corresponding values as x = 1 2 3 4 5 6
y = 5 7 9 ll la là
Suppose now that we are given the table of corresponding values showm below. Find
the equations from which they were made:
(a) x = 1 2 3 4 5 (d) a = 5 10 lă 20 25
y = 6 ll 16 21 26 b = 60 30 20 15 12
(b) x = 1 3 5 7 9 (e) u = 1 2 3 4
y = 1 9 25 49 81 v = O 3 8 15
(c) x = 2 4 6 8 10 (f) r = 2 3 7 8
y = 2 3 4 5 6 s = 2 4; 24; 32
4. If you kept some chlorine gas in a cylinder always at the same temperature and
applied different pressures to squeeze the gas into a smaller space, you would notice
a relation like this ; pressure = 1 2 3 4
volum.e = 12 6 4 3 Assuming that the rela-
tion is true when you applied more pressure, what is the relation between pressure
and volume?
5. If you were learning Italian you would find that some nouns are called masculine
and some feminine. An Italian grammar states rules by which you decide to which group
any noun belongs. See if you can find that rule by examining the endings of the
following nouns. The letter m is for masculine nouns and f for feminine.
libre m studio m anima f fine stra f
penna f vento rºl C a Sa. f marito II]
6. Italian verbs are separated into three groups. The nature of the ending of the
infinitive form of the verb tells to what group the verb belongs. Find the rule by
examining the following verbs whose group number is given.
parlare 1 movere 2 ridere 2 partire 3
credere 2 preferiro 3 venire 3 rompero 2
dubit are l preparare l visit are l sentire 3
7. Spanish verbs are also separated into three groups which can be recognized by
the ending of the infinitive. Find the rule by examining the following verbs:
C On er 2 8. In allº l explicar l escribir 3
morir 3 servir 3 tener 2 comprender 2
8. The following verbs are from the Swedish langaage. By examining the endings
find the rule that tells to what group the verb belongs.
kallat 1 bundit 4 dragit 4 fodt, 2 skrivit 4
b8.jt 2 klippt. 2 rens at l smort 2 bott 3
trott 3 brukat 1 gripit 4 drivit, 4 planterat l
9. From an examination of the following Swedish adjectives find the rule for
forming the comparative and superlative forms of adjectives:
(rich) rj. K. rikare rikast,
(beautiful) vacker vackr are vackrast,
(faithful) trogen trognare trognast
10. Explain why ex. l to 9 are examples of inductive reasoning.
48
RELIABILITY OF THE INDUCTIVE METHOD. By the reliability of the method is meant:
Does the inductive method always lead to corroct conclusions? It has been said that
the only 100 % correct generalization is: All men are mortal. iio one has ever found
an exception to this statement.
Fxperience shows that the method is not safe unless it passes two tests:
l. Were a sufficient number of examples studied?
2. Were the examples typical ones?
l. If you wanted to make some generalization about the presidents of the United
States it would be possible to examine all the examples because the number of presi-
dents is definite and small. It would be possible to examine all the books written
by any one novelist, and it would be possible to test all the known gases. But there
are other situations in which it would be impossible to test all the examples. You
could never test all the proportions that exist. In such cases we use our judgement
before we accept the conclusion. If the conclusion is afterwards found to be wrong,
we excuse ourselves by saying that it was based on insufficient evidence.
Insufficient is here defined as follows: The number of examples is insufficient if
the conclusion is altered when more examples are studied. This definition agrees
with the one used in Chapter 10.
2. Suppose you wished to investigate the vi ows of the seniors in your school about
football. You could not interview all the seniors but must select some typical ones.
You would not be considering typical pupils if you chose only girls, or only boys, or
only boys on athletic teams, or only bright pupils, or only those who live in one
part of town. You must include all types -
If you wished to investigate high schools in all towns having a population of 50,000
or more, your conclusions would not be reliable if you chose only towns in the east,
or only towns in the west, or only manufacturing towns, or only college towns. You
must include all types. A group is typical if your conclusions are not altered when
you consider more examples.
When organizations take "straw votes" before an election, they include people from
all political parties, all occupations, all incomes, etc. If any type is omitted
then the results are not good for that group. Naturally, if a question involves only
girls, we do not include boys; if the question involves people over forty, we do not
include folks under forty. The conclusions must be based on the assumption: the
examples we have not studied are like those which we have studied.
Read "These Public Opinion Polls" by Spingarn in Harpers Magazine, Dec. 1938
E x e r c i s e s
1. Suppose your class wishes to get the opinion of the teachers on some question
(Select some specific question). Assume you are in a large school having 100 or more
teachers, and that you cannot interview all of them. How would you select lo teachers
who could be considered typical?
2. Suppose your class wishes to get the opinion of the business men in your commun-
ity on some question. Select a specific question. How would you select a group of
10 or 15 typical men?
3. Zane Grey has written about 70 books. How many would you examine before you
could say that you had examined a sufficient number? How would you choose them?
4. Shakespeare wrote about 37 plays. How many would you read before you could say
that you had a sufficient number on which to base some conclusions?
5. In a collection taken in a schoolroom, 5 pupils each gave a dime; 7 pupils each
gave là é; 9 pupils gave 20 g each; 6 pupils gave 25 # each; 3 gave 50 g each.
In a second room, 6 pupils gave a dime; 8 pupils gave 25 ¢, 15 gave 30 & ; and
2 pupils gave a dollar. What was the "typical" gift in each room?
(In a textbook find the meaning of "median", "mode", and "average".)
49
THE SCIENTIFIC METHOD. Some people use the word scientific to mean carefully
planned or systematically or thoughtfully - the opposite of haphazard or muddling.
For example, when i.r. Jones set out the wickets for his croquet set he first drove
two stakes at the ends of the court, stretched a rope from one to the other, and used
that line to guide him in placing the wickets. With a tape measure he made sure that
the side wickets would be at equal distances from the center line and at the correct
distances from each other. !..r. Smith, on the other hand, just guessed at everything
and estimated where the wickets ought to be. The neighbors said that Jones did a
very "scientific job" compared to Smith.
The above use of the phrase "scientific method" is wrong.
A method should not be called scientific Imerely because it uses some facts from
science. We shall see later that Jones r cally used the deductive method.
The scientific method emphasizes two ideas: observation and experiment.
Before Aristotle's time (384 - 322 B.C.) people believed that reasoning or thinking
or arguing could reveal the secrets of the universe. Aristotle opposed this idea; he
favored observations and experiments. In simple language he would have said, "If you
want to find out something about a bird, don't sit and talk about birds. Catch one
and examine it."
Roger Bacon, who lived about sixteen centuries later, was the next prominent man
to emphasize the importance of observations and experiments. But the idea grew very
slowly. After three more centuries Galileo (1564 - 1642) still had trouble convincing
people that they should experiment. For example, before his time it was thought that
a heavy weight would fall faster than a light weight. Galileo would settle such
questions very quickly by dropping some weights and observing which reached the ground
first . This simple method of solving a problem seems so obvious to us now that we
wonder why people were slow to appreciate it. ©
The scientific method is an inductive method because it finds a general statement
by studying particular instances. It can be stated briefly as follows:
1. Know what problem you are trying to solve.
2. , , experiment test the possible solutions.
3. When selecting what seems to be the correct solution, be guided by the evidence
from the experiments not by your hopes, wishes, prejudices, or other influences.
All problems cannot be studied by the scientific method. It is used when we work
with material objects with which we can experiment. You can hardly use the method
with a problem like "What shall I give aunt Flo for Christmas?" Since the books
on General Science contain many examples of the scientific method, none are presented
here, but the following reading is recommended:
The Seven Seals of Science by Joseph Mayer (1927, $3.50, The Century Co.) This
book has many interesting accounts of how notable discoveries were made.
General Science for To-day by Watkins and Bedell (1935, £1.6C, Macmillan Co.) See
pages 607-614 about the migration of cardinals.
Exploring the World of Science by Lake, Harley, Welton (1934, Silver Burdette)
See page 18 dealing with Pascal.
New Yorld Pictures by George R. Gray (1936, § 3.50, Little Brown & Co.) See page 52.
Kepler made 18 laborious attempts to fit the planetary orbits to circles before he
abandoned the idea. See page 56 about Galileo's experiments with falling bodies.
Science Remaking the World by Caldwell and Slosson (1924, Double Day Page & Co.)
See page 65 about Ehrlich.
5O
THE DEDUCTIVE METHOD. A gardener, using inductive and scientific methods, studies
the growth of tulips under various conditions. Finally he discovers that adding some
bonemeal to the soil will produce bigger and better tulips. This knowledge spreads
through magazines and books. One day lirs. Jones notices that her tulips are not so
large and beautiful as those of her neighbor. Her problem is : How can she improve
her tulips? She does not need to experiment and try the many things that the garden-
er tried. The gardener has discovered the general statement:
Adding bonemeal to the soil improves all tulips.
Mrs. Jones merely thinks: Adding bonemeal to the soil will improve my tulips.
The inductive method studies particular examples and finally reaches
a general statement. The deductive method takes a general statement
and applies it to a particular example.
Geometry, as you study it in school, is mostly a deductive process because you use
generalizations (theorems) to solve particular problems. In your science class you
repeat many experiments mostly to enable you to appreciate how certain discoveries
were made. You are hardly brilliant enough to discover for yourself such important
principles as Ohm's Law or Archimedes Law. They were discovered by inductive and
scientific methods. When you understand them you can use them (deductively) to solve
new problems. If you have studied French you have learned the rule; verbs whose in-
finitive end in ir belong to the second conjugation. This rule was not discovered or
proved; it is an agreement made by those who speak French. When you see the verb
finir you use this rule (deductively) to decide how to conjugate that verb.
In most problems we do not use only one method. The inductive method may suggest
some generalization which may suggest another generalization, which may be tested
scientifically, which may suggest some other generalization, and so forth.
References:
An Introduction to the Teaching of General Science by E. P. Downing (1934, $2.00, The
University of Chicago Press) This book is chiefly for teachers, but pages 83-93 give
an example of deductive readoning which pupils can understand and will enjoy.
The Mystery and The Detective by B. C. Williams (1937, $1.00 Appleton-Century Co.)
A collection of 20 short stories.
The Roman Kid, Paul Galli co, Saturday Evening Post, June ll, 1938.
The Gold Bug, Edgar Allan Poe. (What were the hypotheses which Legrand used?)
The Red Headed League, Conan Doyle .
E x e r c i s e s
l. What three kinds of reasoning are discussed in this chapter?
2. What are the differences?
3. From your own experiences give an example of each kind.
4. Give an example of each kind of reasoning from one of the books mentioned in
this chapter or from some other book.
5. If lºr. Jones, in the example on page 49, had been familiar with the scientific
method could he have used it to lay out his croquet court?
6. Which method would be used most by cooks? garage mechanics? bank managers?
salesmen? street car conductors? bricklayers? chemists? lawyers? doctors?
5 l
C H A P T E R ll
M I S T A K E S I N R E A S O N I N G
When driving a car on a busy street we must know not only the correct way to drive
but we must also be able to protect ourselves against the errors of other drivers.
This chapter will discuss some of the cominon errors in reasoning.
ARGUING IN A CIRCLE. ASSUMING THE CONCLUSION . Suppose you were looking for John,
and someone told you that he was with Henry. You ask, "And where is Henry?", and you
get the answer, "With John". We call such an explanation talking in a circle; it
brings you back to the starting point.

Beginners in geometry sometimes make a similar mistake. C / D
For example, they might say l
AB is parallel to CD because < l = < 2. UX
Why does &1 = 4, 2 3 A B
Because AB is parallel to CD. Z
Of course no one makes such mistakes deliberately or consciously; mistakes are made
through carelessness or inability to remember what has been said or what, the data is .
In discussions of economic problems we hear arguments such as:
l. Many people are unemployed because factories are not busy.
2. Factories are not busy because people cannot buy the manufactured goods.
3. The unemployed cannot buy the goods because they have n't the money.
4. They have n't the money because they are unemployed.
Each statement above is true, but the arguments gets nowhere. In such arguments
we can be sure that not all the truth has been told. There is more than one reason
why the factories cannot hire men. When all the reasons for each step are known,
then such arguments will not move in a circle.
DENYING THE CONCLUSION. If someone said, "Yes, I know that Chicago is in Illinois,
and that Mr. Jones lives in Chicago, but I don't believe he lives in Illinois" you
would think either that the speaker was mentally unbalanced or that there was some
"catch" to the statement. We could not argue with someone unless he agreed that
If you accept (believe, admit, grant, assume, agree to) a general
statement, you must accept the conclusion that follows from it.
It would be ridiculous to admit that all the angles of any square are right angles,
and then say that in a certain square the angles are not right angles, or to admit
that all dogs are animals and then say that your dog is not an animal. If that is
the way you feel about your dog, you should not admit the general statement.
Such difficulties do not arise when we argue about geometric figures, laws of science,
and many other fields, but they do arise when we discuss politics, religion, social
questions, personal matters, and all subjects about which we are prejudiced.
I may admit that l. If a man is poor we should help him
and 2. i.r. Smith, my neighbor, is a poor man
and still feel that I do not care to help Lir. Smith. I do not like i.r. Smith; I think
he is lazy; and once he walked through my flower bed and broke some tulips.
I such cases I should withdraw my agreement to the general statement and offer a
new generalization like
If a man is poor, not lazy, and does not injure my flowers, we should help him.
If I accept a general statement I must accept the conclusion that follows from it.
There must be no hesitating, no quibbling, no dodging. must accept . Doctors say
that children should drink a quart of milk every day, but the doctor's children often
do not. Teachers tell us to look both ways before crossing a street, but do the
teachers look both ways? You believe that pupils should do their homework; do you
do the homework?
52
LOGIC VERSUS TRUTH. Are the following syllogisms logically correct?
1. All odd numbers end in the figure 5. l. All 6anadians live in Michigan.
2. 68 is an odd number. 2. All who live in hetroit are Canadians.
3. 68 ends in the figure 5. 3. All who live in Detroit live in Michigan.
Before reading any farther, decide if the above syllogisms are correct.
Most people say that example l is wrong and example 2 is right. Both examples are
correct logically." Most people say that example l is wrong because 68 ends in 8
not in 5; and they say that example 2 is right because Detroit is in Michigan.
But the question at the top of the page is not : Are the conclusions true?
The question is : Is the reasoning correct?
If you knew nothing about numbers, if you had never heard of such things as even
and odd numbers, then you would say that example l was correct. If you had been
born and raised where you had never heard of Canadians, Detroit, and Michigan, then
you would say that example 2 was correct.
The argument in the examples is exactly the same as:
1. The small circle is inside the larger circle.
2. The point A is inside the small circle.
3. Hence, the point A is inside the larger circle.

Examples l and 2 show that
Two False Premises may lead logically either to True or False Conclusions.
The examples should make you realize that an argument involves 3 questions:
1. Are the premises true? 2. Is the logic correct? 3. Is the conclusion true?
E x e r c i s e s
1. What is meant by saying that a syllogism is wrong? (See page 15.)
2. What is wrong with these statements:
a) If the opposite angles of a parallelogram are equal, the figure is a parallelogram.
b) If the medians to the equal sides of a triangle are equal, the triangle is isosceles.
3. What is wrong with the following argument?
Mr. Jones would like to buy a new gas heater for his home because his present one
is inefficient. It costs about $8 a month to use it while a new one with an insulated
tank would cost only $4 a month. But he can't afford a new one because he hasn't the
money because his old heater costs so much to operate.
4. How would you describe the reasoning in this example:
England cannot pay her debts to the United States because we will not buy her merchan-
dise. We do not buy her merchandise because we do not have the money to pay for the
goods. We do not have the money because England does not pay us what she owes us.
5. Discuss the following syllogisms:
a) All even numbers end in 3 c) Sparta was a democracy
72 is an even number A democracy does not have kings
72 ends in the figure 3. Sparta did not have kings.
b) Fish live in water d) Plexicans are citizens of the United States
Horses are not fish New Yorkers are liexi cans
Horses do not live in water New Yorkers are citizens of the U.S.
6. Bring to class an example of
a) Arguing in a circle.
b) A syllogism which is logically correct, the major premise being true, the minor
premise false, and the conclusion true .
c) A syllogism which is logically correct, the major premise being false, the minor
true, and the conclusion true.
d) A syllogism which is logically correct, both premises being false, and the conclu-
sion true.
53
HASTT GENERALIZATIOliS. (See also page 9.) Almost every day you hear remarks like
"I rode on Bus 18 to-day. It was terribly slow. T'll never ride it again."
"... is 5 Hughes always gives low marks. Che marked me 60 on Jester day's test."
"The police are worthless. They haven't caught the thief."
These speakers are making general statements based on insufficient evidence. Police
canſ,ot be worthle.js because they fail to catch one thief. A bus is not slow every
day because it is slow one iai/.
A generalization should be based on a large number of typical examples. Remember
t” at we are always incline i to notice the exceptional cases and overlook the typical
cases. `hen hot water comes from a not water faucet for hundreds of days you think
nothing of it; but if just once the water happens to be cold, you notice it immedi-
ately. Tf a grocery clerk gives you the correct change from a purchase every day for
a year, you think nothing of it; but if just once you get the wrong change, that ex-
ception is noticeable. The fact that you notice the exception and did not make any
romark on the other occasions is a very good reas on for thinking that it is not typ-
ical and that the clerk is honest and careful. One rainy day that makes you talk
about the weather should make you say "most days are sunny" not "most days are rainy".
tri- )
Avoid hasty generalizations based on insufficient evidence. When in doubt, ask
yourself: is there enough data to decide the question?
HASTY COI:CLGSIONS. For many years the railroads have been earning less and less
because they have less freight to haul because trucks are hauling more freight. The
railroads claid that the trucks use highways which are constructed at the expense of
the tax-payers; the trucks did not pay for those concrete concrete roads whereas the
railroads had to pay for their own ºr a cKs.
That sounds like a good arºuſ lent. But the trucking companies answer by saying
that when the railroads were built the . (the railroads) received large "land grants"
from the government; that is , the railroads did not pay for the land over which the
rails were lai j, and in a dition the railroads received from the government certain
other lan'i which was sold to pay for the construction. Hence, say the trucking com-
panies, the railroa is jid not put up their own ILoney to pay for the roads.
That, too, sounds like a good argument . put the railroad companies answer by
saying: first, only a few, not all, railroads received land grants. Second, the
land given to those railroads was worthless until the roads were built. Third, in
return for the land the government was given reduced freight rates. Hence the rail-
roads have now paid for the land. This , too, is a good argument.
The above example shows how our opinions may change as we get more and more evidence.
: any times we cannot be sure that we have all the necessary facts.
INSUFFICIENT DATA OR KluoWLEDGE. We might suppose that if an object will float on
the Atlantic Ocean then it, will float on Lake ...ichigan. Of course the water in the
ocean is salty and the water in the lake is not, but why should the salt that we put
in food have any effect on a floating body? If we do not knowſ Archimedes Law for
floating bodies we do not know that the salt does m.ake a difference.
We may read an advertisement saying "To as your dentist does. Brush your teeth with
a powder." Hence we decide to use a powder instead of a paste. How could we be ex-
pected to know that the dentist makes a paste from the powder before he uses it!
Naturally the only remedy for this kind of trouble is to learn everything about
everything, or else to hesitate continually, fearing that something we do not know may
alter the conclusion. ... ovie wer, there are many situations in which a little bit of
thinking and the use of such education as we have acquired may suggest certain inves-
tigations before deciding on a conclusion.
Hasty generalizations arise from lack of sufficient evidence when reasoning induc-
tively or from lack of sufficient knowledge when reasoning deductively. Then we do
not have enough information to act wisely, we should try to get that information
before reaching a conclusion.
54
MISLEABING ANALOGIES. An analogy is a comparison. In your English classes you
have heard of similes and metaphors; they are also corr parisons , , r. Smith may say to
Kr. Jones, "In England laen with very low incomes pay an income tax; hardly anyone, no
matter how poor, escapes that tax. He, too, ought to make everyone pay an income tax."
The argument is : This thing works well in England; hence it will work well here.
The argument is a good one if the conditions in England regarding taxes are the same
as in this country.
To mend some wooden toy you happen to remember the cement that you used last week
to mend some china. You are thinking: If cement will make pieces of china stick to-
gether, it will make pieces of wood stick together The reasoning is good unless
china and wood differ in their action toward cement.
"Mayor Hoke will make a good governor because he is a good imayor" is a good argu-
ment in so far as the duties of a mayor and a governor are alike.
A certain method, X, will succeed when another mothol, Y, succeeds provi led that
X and Y are alike in whatever causes the success.
%hen you hear an argument base 1 on analogy, ask: Are the conditions the same in
both situations? Of course no conditions are exactly alike; but they must be alike
in so far as they have anything to do with the question being discussed. Gasoline
and alcohol are not alike in all respects; but in a question about evaporation they
are alike. Tf we were discussing weight we could not say they were alike.
We might argue that all the planets are like the earth because they all revolve
around the sun, all have moons, and all obey the same law of gravitation. Hence there
must be life on the other planets just as there is on earth. But life does not depend
on revolving around the sun, nor on the moons, nor of the law of gravitation. Life
depends on other things ; and in respect to these other things the other planets are
not like the earth. iience the analogy is faulty.
We should study the situation in this way:
l. ...hat are we trying to prove? Ans. Life exits on other planets.
2. What objects are compared? Ans. The earth and other planets.
3. Are the objects alike in so far as the necessities of life are concerned?
POOR AUTHORITIES. i. one of us would consult a dertist when the furnace needs re-
pairing. Heither would we consult a garage mec namic when we break a bone. We know
that for certain kinds of work only certain people are able to act with skill or to
speak with authority. Ceing an authority on one kind of work does not make a man an
authority on everything.
But if a problem does not involve special skill or if we incorrectly assume that
the problem does not require special knowledge, then we are inclined to think that
one man's opinion is as good as any other's. And if he is a prominent person we may
think that his opinion is better than that of a less prominent person. If Mr. Hardy
has been very successful in making autolaobiles we may think his opinion of Lincoln
or of the latest movie or of the new public library is more valuable than that of
Bill Smith who works in ' Mr. Hardy's factory.
Before accepting any authority we should a sk: Is special knowledge needed?
Ts special knowledge needed to manage a library, to manage a city, to judge pictures?
Only the ignorant believe that honesty, a kind heart, and good intentions are the only
things needed to settle any question.
In our reading matter we must also be careful of the authorities we consult. An
article on vitamins in a woman's magazine devoted mostly to fashions is hardly good
authority; it may be interested in keeping you thin rather than healthy. An article
on taxes in a magazine devoted to the building trades may not be good authority; it
may minimize taxes so as to encourage the building of hones. An article in a popular
science magazine on detecting crime Lay mention only cases in which scientific methods
were successful and say nothing about any failures .
E x e r c i s e s
1. Mention five well known men or women whose names often appear in newspapers and
state some questions on which they would be good authorities.
2. A. ention some subjects on which the following would be good authorities:
Emily Post Charles Lindbergh The King of England ...us solini
3. State five questions that might appear so simple that anyone could be taken as
an authority. (For example, what is the best way to start a fire when camping?) Then
explain why some special knowledge would be useful in settling the question.
4. Sugar can be compared to sand if we are speaking of a granular substance, but
not if we are discussing sweetness. In the same way give sone examples in Thich the
following would and would not make good analogies.
a) Water and milk. e) A school club and a college fraternity.
b) The United States and Cngland. f) The entrance requirements for college
c) A labor union and a corporation. and the requirements for joining the
d) A rectangle and a parallelogram. plumbers union.
In the next exercises tell what type of reasoning is illustrated and discuss why
the reasoning is good or poor.
5. "Our football team is a bunch of cowards", said ill. "It lost Saturday's game
60 to nothing. Think of that!"
6. If you have been eating lunch at the Goody Shoppe for four years, and one day a
poor meal is served, should you be likely to say, "This places serves good meals" or
"This place is getting worse every day"?
7. If textbooks occasionally make mistakes, then teachers are likely to make sini-
lar mistakes.
8. y neighbor, Air. "'anser, likes the kind of hot-water heater that he has . The
salesman for the manufacturer called on me last week and said that it costs only "2
a month to run it. I think I shall buy that kind of heater.
9. I.iss Amour Lamour, on her way to fiew York from IIollywood where she has just
finished "Love is Love", was asked by our society reporter what she thought of the
design for the new public library. "It's just too too darling." she said.
10. Turing one of his political campaigns Lincoln was advised by some people to
change one of his generals. He answered, "I do not believe in swapping horses when
in the middle of the stream".
ll. Riding in busses is dangerous for school children. Last week a bus overturned
and thirty children were injured.
12. I was thinking of buying a Co-ºell Flatiron. The clerk says it uses only 3 cents
worth of electricity an hour while you iron. But then I went to the Bristo Store and
the clerk there showed me a Just-Fine iron. She said that the cost of the electricity
with it is only 2 cents an hour. So naturally I bought it. Of course a penny is only
a penny, but think of how many hours you iron clothes in a year.
13. John heard i.rs. Burns tell ... rs. Snow how bright and courteous Bill Burns was .
Later John met Bill and found that Bill was not handsome at all. John thought at
once "That shows how little mothers know about their own sons".
14. I.rs. Quick is a talkative soul but very kind hearted. When she met irs. Jones
one morning she said, "I had a terrible cold yesterday, but I, ox-Ache is n't worth a
cent. It didn't help my cold at all. I tried that new cake flour which you recom-
mended last week. It's just terrible. The cake was so lumpy that I could n't serve
it for dinner. We had l.r. Putnam for dinner. Iie's a ſine dentist. IIe lives on
Seventh Avenue in that big house. It's a beautiful house. I think I should consult
him about the new house that Ir. Quick and I are planning."
15. Alfred was arguing with his father about what time small boys should go to bed.
Alfred is only eight years old. "Look at Edison", he said. "He never slept very
much. And look at Edward Bok. He never went to school very much. Why should I go
to bed early, get lots of sleep, and go to school?"
16. Jimmy left school at 2 o'clock. School closes at 3:30. While going from school
Ned said to his friend Dick, "I saw Jimmy leaving school to-day at two o'clock. I did
not know that he was sick."
56
NON-CAUSAL RELATIONS. (Do not confuse the words causal and casual.) No one is
foolish enough to say Since the sun is shining, to-day is blonday but that
sentence illustrates a non-causal relation. The day of the week is not decided by
the shining of the sun. ”e do, however , hear many statements in which the wrong
cause is stated. When i.r. Hill was asked to help in moving a piano he said, "This
must be made of iron; it is ao heavy." It is true that iron is heavy, but there are
many other reasons why a piano is heavy.
Before we say that "A" is the cause of "B" we should make sure that "A" by
itself is sufficient to cause "B"
If two events take place at the same time, or if one event happens soon after
another, an ignorant pers on assumes that one was the cause of the other.
A sentence like "LaFayette was a great French soldier; during his life the United
States became a democracy" suggests that LaFayette was the cause of the democracy.
Tt is true that he helped, but he was not the sole cause.
The night follows the day, and the day follows the night; but neither is the cause
of the other. If Bill fell from a tree and broke a leg on the day that a neighbor's
dog died, the dog was not the cause of either the falling or the breaking. These ex-
amples seem ridiculous but all superstitions are just as foolish. Walking under a
ladder cannot cause ill fortune; collecting 13 people in one room cannot cause death.
If the seeds sown in the garden begin sprouting when the birds fly north, neither
event is the cause of the other; in fact, both are the result of another cause. The
days get shorter when the farmers in hiaine begin digging for potatoes; both events
are the result of another cause.
From Latin we get the phrase non-sequitur meaning "it does not follow". A non-
sequitur argument is one in which certain statements are not causally related.
Events that are closely related in time are not necessarily causal.
SOME OTHER ERRORS IN REASONING. In other chapters we have studied such errors as
l. Assuming that the converses or inverses are true . Pages 52, 38.
2. Using a wrong minor premise in a syllogism. Page 15.
3. Failure to see the difference between necessary and sufficient. Page 44.
4. Failure to define significant words. Page 18.
5. Failure to consider all possibilities in an indirect argument. Page 35.
6. Failure to distinguish between facts and assumptions. Page 20.
There are also other kinds of errors, discussed in books on logic, such as
7. What is true of the members of a group may not be true of the group.
For example, when the corridors are crowded with pupils passing from one class
to another, it may be wise for me to wait a few minutes before trying to get
to my next class. But it would be impossible for everyone to follow that plan.
Every member of congress owns a hat, but congress does not own a hat. Congress
is not the same thing as the members of congress.
8. What is true for a group may not be true for each one in that group.
For example, congress as a body of men may be considered wise, but we cannot
say that every congressman is wise. A mob of people may be persuaded to do a
foolish act although each individual, if alone, would not do it.
EMOTIONAL ARGUMENTS. We try, or should try, to convince people by correct reasoning
and clear thinking. Unfortunately, many people are influenced more easily by appeal-
ing to their emotions than by appealing to their intelligence. This has been explained
by saying that we all have emotions but not all of us have intelligence. We shall
discuss next some of these emotional appeals. They are not only incorrect logically
but are really dishonest since they try to accomplish their purposes by unfair methods.
It is important to recognize these dishonest methods since many people use them, not
always deliberately, but often carelessly.
57
AD HOkiLNEM ARGUMENTS. The Hazeltop Golf Club was discussing whether or not an
a 3dition should be built to the club house. Mr. Cheery favored it; Mr. Grouch did
not.. At a meeting, l, r. Crouch, instead of discussing the reasons for and against the
a dition, said, "Mr. Cheery is one of our poorest players ; he can't do the nine-hole
course in less than 60. He is a terrible dancer, and his bridge playing is worse
than his golf. What does he need a bigger club house for?"
This is an ad hopinem argument - from a Latin phrase meaning regarding the man.
Tn such an argument the speaker does not discuss the question but tries to find fault
with the man, and adversely criticizes his opponent. No one would do this if he had
any good arguments. Hence, attacking the character of your opponent is equivalent to
admitting that you are wrong.
PCOH-POOHIHG THE QUESTICſ, . An ad hominem argument tries to ridicule the oppo-
nents. When a speaker tries to ridicule the question itself, he is said to be pooh-
pc.ohing the question. In the above example, Mr. Grouch might have said, "Why argue
about a new club house? Iv'e heard this question debated so long that I'm tired of
hearing about it. This town is full of poor starving people, and we argue about a
new club house."
"Pooh-poohing" belittles the question, and tries to make it seem ridiculous or un-
important or impossible of solution. We could answer Mr. Grouch by saying, "Your
remarks, Mr. Grouch, have nothing to do with the question. What you say is irrelevant."
(Find the meaning of irrelevant in the diction.ary.)
HiANGIlić, THE QUESTION. The Main Street Business Mens Club had agrred to raise
$1OOO for a Christmas festival, and were discussing the best way to collect the money.
Mr. Hale said, "I don't think we should have a festival like last year's . We spent
too much money on decorations and too little for the band. When you are listening to
a good band, you don't need a fancy stage setting."
We could answer lºr. Hale by saying, "The question is: How shall we collect the
money? not That shall we do with it after we get it?"
When a speaker avoids the question by substituting a new question, he is either not
very intelligent or he has some secret motive for dodging the question.
APPEALS TO PREJUDICE. Let us be honest and admit that we are all biased according
to our training, our education, and our financial star ding. But if these facts have
nothing to do with the questicn then a speaker should not refer to them or take advan-
tage of them. For example, the ouestion about the new Hazeltop golf house has nothing
to do with the fact that Mr. Cheery voted for a republican mayor, attends the Baptist
Church and has three children. These things are irrelevant unless the cuestion happens
to be "Shall a new club house be built for those who belong to the Baptist Church,
who have three children, and who voted for a republican mayor?" In that case Mr. Cheery's
personal affairs are part of the data.
When addressing laborers in a factory a speaker (a poor speaker, of course) may
assume that his audience dislikes farmers ; when speaking to farriers, the speaker may
assume that the interests of the farmers conflict with the interests of laborers. But
a good speaker does not assume prejudices or try to appeal to them.
Many emotional appeals are based on the reverence we all have for old customs. Many
appeals are based on the mental pictures that are aroused by the mere mention of such
words as "Liberty" "a rose covered cottage" "the poor" "tyrants" "Born in a cabin"
"a self made man". Such appeals are an insult to the audience for they imply that the
audience has only slight intelligence and is narrow minded. It may be difficult to
overcome our prejudices but we should try to conquer them.
58
E x e r c i s e S
First make a list of the types of ur rors mentioned in this chapter (there are 21).
Find which errors are illustrated in each exercise. Some of these exercises contain
only a descripti or of a situation, or a possible conversation, or a quotation. Even
if no specific question is asked, you should discuss each exercise as fully as possible.
Are there errors? What kinds? How would you solve the problem? That methods would you
use? Do you agree with the statements'. Why or why not? Make use of everything you
have learned from this or any other chapter of this book or any other book.
l. Archibald DeVry is a graduate of a law school; he was president of his frater-
nity while in college; he lives on Mohawk Boulevard and is a member of the Saddle Club.
Bill Smith... is an at Lorney, a member of the South Side C or munity Club; he lives at
25OC west låth Street with his widowed mother.
frt
DeVry and Smith are candidates for County Judge. The only information you have
about them is that given above. How do you decide for whorl to vote?
2. Mr. *liners must own a great deal of personal property because he gets such a
large salary as president of the Coverall Paint. Cor:pany.
3. To you get low marks in some of your classes because you do not study, or have
you cuit studying because you get low marks?
4. You ought not to prosecute lºr. Adams for embezzling a few dollars. In any large
organization you must expect to find a few dish onest people.
5. A mayor, when called on to explain why there was so much crime in the city,
gave a speech over the radio telling what a beautiful park system the city had, how
it furnished activities for sports, and gave the children an outlet for their youth-
ful energy. I n time these would reduce the petty theft by the children.
6. Turing the depression many statistics were printed showing that it is mostly
unskilled laborers who are unerployed. If all labor would become skilled, there would
not be any unemployment.
7. Mr. Hart made a fortune by buying stock in the l’eepwell Oil Company. How the
stock of that company is so valuable that you cannot buy it. So I am going to buy
some stock in the Fargo Oil Company which doesn't cost so much.
8. In to-night's tiews I found articles on "Hitler's Ceiſlands", "French Cabinet Cver-
thrown" and "Europe and Our Secretary of State". I guess this paper is interested
only in foreign affairs.
9. School children seldom have their lessons on Mondays, and are generally not pre-
pared for their work after holidays. The school ooard might just as well close
schools entirely on Mondays and have school on Saturdays instead.
10. All great statesmen are intellectually keen. Daniel Webster was intellectually
keen . He was a great statesman.
ll. Two of the three bridges across the East, River from lanhattan to Brooklyn and
Queens were built under the regulations of the various labor unions, and one was built
without union interference. In the building of the two union bridges only 6 men lost
their lives, while in the building of the other bridge 55 men were killed.
12. People who believe such statements are fools. You are not fools for you do not
believe such statements.
13. In a debate Jones said that some of Smith's facts were not correct. Smith said,
"Since Smith has called these statements "facts" it is absurd to doubt them. If they
are facts, then they are true."
59
14. The law prohibiting citizens from coining money is an unjust one. If citizens
are not allowed to coin money then the nation should not be allowed to do so either,
for a nation is merely a collection of citizens. But the nation is allowed to coin
money and so the citizens should be allowed to do so.
15. When Mr. Slone of the Bigger Biotors Corporation visited Detroit he as asked
by news reporters what was the secret of success. "Perseverance" was his reply.
When asked if that alone was sufficient, Slone said, "Well, you've got to have it."
16. "Johnny, don't leave your hat on the chair. Hang it up."
"But why should I hang it up, mother?"
"Because it's wrong to leave it on the chair."
"But why is it wrong to leave it on the chair, mother?"
"Because you should hang it up on the hook daddy fixed for you."
17. When we buy goods from foreign countries we get the articles but some foreigner
gets our money. When we sell goods to the foreign countries they get the goods but
we get their money. The sensible thing to do is to buy and sell only in our own
country and not to let foreigners either buy or sell in this country. Then we will
get the manufactured goods and they will not get our money.
l8. The coach said that because of lack of experience we could not have a strong
football team this season. -
19. The Womans Fashion Shoppe began advertising in April in the daily papers. It
had never previously advertised in the daily papers. Its sales in May were five times
as great as in March. This proves that advertising in the daily papers will increase
sales tremenduously.
20. "Isn't it strange" said Alfred, looking at his geography. "All the big rivers
flow past big cities. Why do the cities make the rivers so big?"
21. As they sat down to dinner lºr. Paine said to his wife, "I just came past the
new house being built for Mr. Snyder. The contractor was using second hand bricks
in some places. I really ought to phone Snyder and tell him how he is being cheated."
22. Henry: "I went to high school for four years, 180 days a year, and studied
3 hours a day." John: "I went four years too, but I studied only two hours a day.
They say that a high school education is worth $20,000. So I earned that $20,000 in
less time than you did. I earned more per hour than you did. In other words, the
less you study, the more you earn per hour."
23. By using the following argument a small child reasons that it can touch the
moon: "I can touch bright objects. The moon is a bright object. Therefore I can
touch the moon."
24. Overheard in a streetcar: "Last night I was reading Jones' 'History of Europe"
and I noticed that page 68 was word for word exactly the same as page 82 in Smith's
History of Civilization. This fellow Jones must have copied from Smith. I thought
there was a law against such things."
25. Mr. Adams: "I have kept records for more than 20 years and I always pay for
more coal during January than during February. Of course that's only natural because
January is a colder month than February."
Mr. Williams : "Cold weather reminds me that Christmas is coming. I like Christmas.
But think of the bills you pay in January, especially if you let all your December
expenses go unpaid until January."
60
26. Mrs. Sorr, was not satisfied with the way her car was running, and so she stopped
at a garage and had a mechanic "tune up the engine". She also decided to use ethyl
gasoline instead of the ordinary kind. Thereafter her car ran very smoothly. "That
new gas is certainly the berries" she told her friends.
27. Goldies Department Store has a program on the radio which lasts half an hour.
It ought to do twice as much business as the Hardy Store whose broadcast lasts only
fifteen minutes.
28. More people now have cancer than ever cefore. Last year 639 cases were reported
by doctors. The preceding year there were only 456 cases, and the year before that
only 345 cases were reported.
29. We had airplane service across the Pacific Ocean before we had similar service
a cross the Atlantic. I suppose the airplanes crossed the Pacific first because the
big airplane factories are in California. Otherwise the Atlentic would have been
crossed first because it is smaller than the Pacific.
30. There have been people who knew very little about mathematics and who have still
been successful in life. There have been people who were poor readers and who have
been successful. Many people who could never learn to write legibly have been success-
ful. You can succeed without knowing teading, 'riting, and 'rithmetic. In fact there
is no one thing necessary for success. Select any one thing you wish, and it can be
proved unnecessary for success. You don't need anything to succeed.
31. Every good law ought to be obeyed. The law of gravitation is certainly a good
law. We ought to obey it. .
32. The ruler of a state has often been compared to the captain of a ship. During
a storm it is doubtful if the captain could bring his ship safely into port, if it were
necessary to call the crew together ror a vote when he wished to change the course of
the ship. Likewise the ruler of a state cannot stop to let the people vote during
an emergency. He should be the sole boss. (This analogy is frequently heard when
national policies are discussed.)
33. When a servant girl applies for employment in a family, we demand a letter of
recommendation from her former employer. If a clerk applies for a job, he is asked
for letters from his previous employer so that we may be sure of his honesty. Why
should not every immigrant to this country be required to present similar proof of
his good conduct in the past?
34. Kentucky has many coal mines. It has no coast on any large lake or on an
ocean. States without coast lines usually have coal mines. *
35. Attempts have been made to explain why fewer children die in infancy. The ex-
planation is simple. The old ox carts have gone out of existence and folks now ride
in automobiles. Riding in automobiles has decreased infant mortality.
36. If we wish to prove that "enterprises carried on by the state are not likely
to succeed" why may we not assume that "business enterprises are most successful
when managed by those who have a idrect interest in them"?
37. At a meeting of the city council Alderman Hogan complained about how the city
wasted money. "You spent", he said "$210 putting up that big Danger sign at the foot
of the Ninth Street hill. Think of itſ $210 for a sign that says just 'Danger'. And
there hasn't been an accident there in the last year."
61
C H A P T E R l 2
P R O B L E M S O L W I N G
There are few people who study a subject merely for the pleasure of learning. Most
of us expect to make some use of what we learn. We use our knowledge to solve
problems. The word problems does not mean merely problems of arithmetic or mathe-
matics, but any situation in which we must decide what to do. If the weather is
cloudy my problem may be: Shall I take an umbrella with me? If a friend asks me to
go to a football game with him, my problem may be: Shall I go? If I am offered a
position with Jones & Co. my problem may be : Shall I accept the offer?
We solve our problems by reasoning and by experiment. In previous chapters we
have discussed some elementary ideas about reasoning and about the scientific method
of experimenting. We should therefore next see if we can apply what we have learned
to the solving of problems.
DEFINING THE PROBLEM, . You will be more interested in and do better work on a problem
that has arisen in your own experiences than on one which someone else has asked you
to sclve. Hence you should start, now collecting some problems to which you can apply
the ideas of this chapter. Problems will occur to you if you observe the life around
you. You may have noticed that all the movies in your town are showing two leading
pictures on every program. You begin to wonder why they present two pictures instead
of only one. Or, perhaps you run errands for your mother and have noticed that eggs
cost more in winter than in spring. Your problem then is to find out why this is so.
Or, you may have read that when tulips are planted, the bulbs are set down below the
surface while seeds for flowers are almost all planted near the surface. . .
The ability to observe, compare, and wonder about your surroundings is characteris–
tic of people who are scientifically minded.
Try to state the problem as definitely as possible. After some study of tulips you
may find that conditions vary so much in different parts of the country that you may
decide to limit your problem to the tulips raised in your own state. You may find
that movies in other cities do not present two feature pictures, and so your problem
will refer only to your own community. If your problem is that of comparing the cli-
mate in northern cities with the climate in southern cities, you must decide how you
are going to define and measure differences in climate. All doubtful words will need
defining. All assumptions must be exactly stated.
STUDYING THE PROBLEli. By experiments and by reasoning we next study the problem,
collect data, and try to find possible solutions. If A, B, and C together cause a
certain result, we must find if the result is due to A, or to B, or to C, or to A and
B, and so on. By noting the similarities and differences of various results we can
single out the one cause. We search through our memory for information that may help
us, and recall the general principles that we can apply. Here the diligent student
has the advantage because he can remember the general principles.
If this and this and this is true, then we can understand why tulips are planted
deeply. If this and this is true, then we can understand why theaters have two feat-
ures instead of one. If this and this is true, then we have an explanation of why
eggs are cheaper in spring than in winter. These sentences are purposely stated with
many if's so that you can see why the possible answers are called hypotheses. You
cannot be sure that your first guess, or hypothesis, is the correct one. In ex. 3 (a)
page 47 the rirst guess might be that y = 6x since y = 6 when x = 1; but when we see
that y = 11 when x = 2 we see that y does not equal 6x. We must make another guess
or assumption or hypothesis. We might try y = 6x – l since y = 11 when x = 2. But
this assumption is wrong for the next pair of values. We must make another guess
or assumption or hypothesis.
62
USING COMPARISONS. One of the frequently used ways of getting inspirations or ideas
is that of comparing the problem with others that resemble it. Harvey, who discovered
the circulation of the blood, got this idea by thinking of similar mechanical devices
such as animal traps and water gates. Franklin decided that lightning is electricity
passing from cloud to cloud by noticing that both lightning and electric sparks make
crackling sounds, both killed animals, and both had certain similar features.
You have used this idea many times in algebraic gork. If, for a moment, you have
forgotten how to solve an equation like 2OOO = # r you can get an inspiration
by thinking of a simpler equation like 2O = 3 x*.
If, when working with an expression like x(x + 1)(x + 2) you have forgotten
whether you should first find x times (x + 1) or x times (x + 2) or x times both
(x + 1) and (x + 2) you can decide by thirking of the problem 6 × 5 x 4. Does this
mean 6 times 5 or 6 times 4 or 6 times both 5 and 4?
In geometric problems, if you cannot prove what you wish to prove, you can ask:
What can be proved? or What similar, or only slightly different problem does this
one resemble? In botany, soology or chemistry, you can eec all some experiment like
the one on which you are working.
If investigating the price of eggs, you can compare the prices of other articles.
If theaters are forced to give people more entertainment for their money, perhaps
other establishments are doing likewise. Tulips may not be the only things that are
planted deeply. If the problem is merely that of answering a question like
Can you say that some liberals are democrats because some democrats are liberals?
you can think of a similar geometric question like
Can you say some squares are congruent because some congruent figures are squares?
Your answer to the geometric question will give you an idea for the other question.
In other words, Compare The Problem To Other Similar Problems.
THE SOLUTION. If you have a scientific mind your final judgement will be unpreju-
diced and impersonal. If must be based on the evidence you have collected and
studied, not on any personal opinion that you might have had before you began search-
ing for an answer. It must not be based on your likes and dislikes but on the evi-
dence. It must not be based on something you have heard someone say or merely some
thing you have read somewhere. It must be based on evidence. If you have no exact
proof that your solution is right, you must be willing to admit it, and say that the
problem is still undecided until more evidence can be collected. You must not be
stubborn or think that your answer is right because you have found it.
SUIMARY. State the problem as exactly as possible. Define any necessary terms.
State any assumptions that are to be used. Collect facts, figures, principles, and
other data. Study the collected information. Test the possible solutions. Be
sure that you have evidence to support your conclusion.
EXAMPLE OF A PROBLEli AND ITS SOLUTION. When Jim and Bill went to the zoo they
began guessing the weights of some of the animals. From the zoo-keeper they learned
the correct weights, and then they began keeping a score to determine who was the
better guesser. Jim was closer to the correct weights more times than Bill, but when
Bill was closer his errors were also smaller ; that is, Bill's guesses were either un-
usually good or decidedly poor. Soon the boys began arguing about how to decide who
was the better guesser. After some discussion they stated their problem thus:
How can we define "the best estimator"? -
After more arguing, this question was stated as:
Find an exact formula for measuring the accuracy of an estimate.
63
Some of the data which the boys collected was:
Jim Bill Jim Bill Jim Bill Jim Bill Jim Bill
Correct weight 60 6O 8O 80 210 2lQ 500 500 102 102
Estimate 62 55 85 81 2OO 240 515 495 l2O 135
Error 2 5 5 l lO 3O l 5 5 18 33
Their first plan was to total all the estimates and all the correct weights. This
idea was discarded because a guess of ten pounds too little followed by a guess of ten
pounds too much on another animal would make a perfect score; and this did not seem
right. It would be no better than making a guess of twenty too little followed by a
guess of twenty too much.
Their next decision was to add all the errors, and say that the winner was the one
who had the smaller total. This idea was discarded because a close guess on a large
animal should be considered better judging than a close guess on a small animal. This
suggested the idea that the size of the animal should be consi is red in deciding
between a good and a poor estimate.
Their third decision was to compare the error with the guess since this would take
into consideration the size of the animal. At the
right are shown the figures for one of the animals. Jim Bill
In this case Bill's ratio is smaller than Jim's but Correct weight 100 100
both had made the same error. Estimate 80 l2O
Then Jim thought of comparing the error with the Frror 20 20
true weight. In the above case, this would produce Ratio 1/4 1/6
a tie, both ratios being lſö; and this seemed fair.
They then decided to add the ratios, and the boy with the smaller total would be the
winner. At this point Bill objected to adding the fractions, and Jim suggested that
each ratio be expressed in per cents. Their final decision was the adoption of this
definition: err
Per Cent of Error Garº- 9
expressed in per cents.
and :
"Best Estimator" is the one who has the smallest total of per cents of error.
E x e r c i s e s
1. Using the data for the six animals which boy was the better estimator?
2. Point out the steps in the above example which illustrate the steps mentioned in
the summary on page 62.
3. For practice in getting information in a library, try this problem:
Why do we no longer mine iron from the iron deposits in New Jersey which were used
during the Revolutionary War?
4. The number of feet in which an automobile with good brakes can stop on a concrete
highway when going at various speeds is :
However, it is known that the distance Speed in mi . per hour 2O 3O 40 5O
should vary as the square of the speed. Stopping distance in ft. 43 8O 128 l86
This does not agree with the figures
given here. Explain the contradiction. If there is anything in this statement which
you do not understand, then that itself is a good problem for you to investigate.
5. You are a salesman for a company whose office is in Chicago. You are asked to
plan a trip through all the towns of Illinois with a population of 40,000 or more.
You are to drive your own car and will be paid 5 ¢ a mile, plus $8 a day for expenses.
You are not to drive more than 200 mi. on any one day thus allowing time to call on
the customers. Before you go, the manager wants you to submit a plan showing where
you will be each day, and an estimate of the cost of the trip.
64
6. Harry Jones plays in the backfield on the football team of his high school. His
father is very enthusiastic about football having been a star on his college team .
Harry's father has offered to buy him an expensive camera if Harry's record for yards
gained during the season is good. Harry played in 6 of the 8 games; he carried the
ball 20 times and gained a total of 143 yards. How would you decide whether Harry is
just an ordinary player or an outstanding star on his team?
7. Find out how much money you save by riding a bicycle to school instead of using
the street cars. Consider the cost of the bicycle, the income from that money if put
in a bank, the possible sale of the bicycle several years later, the cost of a pad-
lock, and all possible items of expense.
8. You wish to find how much it costs to heat your home with coal during the winter.
What data must you collect? (Are you sure you have all the necessary data? Have you
considered the cost of removing the ashes, and occasional repairs to the furnace? The
wall paper and the window drapes need cleaning because of the soot from the coal;
should you consider that item? )
9. In a store window I see a glass full of beans with a sign "Guess how many beans
are in the jar, $5 for the nearest guess." What experiments can I perform at home
and what data should I collect to make a good guess?
10. In almost every book of puzzles there are some like these:
If 4 men can build 4 boats in 4 days, how many boats can 6 men build in 6 days?
If 3 cats catch 3 mice in 3 days, how long will it take 12 cats to catch l? mice?
If 3 hens lay 3 eggs in 3 days, how many hens will lay 6 eggs in 12 days?
The problem here is not merely to solve these three problems but to find some gen-
oral method that will solve all such problems. By using comparisons, as suggested
on page 62, these problems ought to suggest a relation like
A times B times C will equal a certain result.
ll. Mr. Jones drove a car for ten years and kept a careful record of his expenses.
They were: 300 qt. of oil, $75.18. 4310 gal. of gas, $76.29. Repairs, $208. 72
Licenses, $80. Insurance premiums, $341.75 Original cost of car, $804.25
New tires, $48. He had driven the car 67, 724 mi . when he traded it for a new one
getting $75 for the old on e.
The problem that Mr. Jones is trying to solve is: Does it pay to drive a car lo
years, or is it better to trade it in for a new one after 5 years? The repairs on
the car were needed during the last 5 years, but the consumption of oil and gas did
not change during the ten years -
If Mr. Jones needs more data, tell what it should be.
12. Imagine that you are in some foreign city where you cannot speak the native lan-
guage. You cannot find an interpreter. You wish to reach a certain address in that
city. How would you learn how to get to it?
13. Suppose the assemblies in your school do not hold the attention of the audience
as they should. The pupils become restless; some open their books and study; others
whisper; at times the pupils have suggested that assemblies be abolished. You are a
reporter for the high school paper, and aim to write a series of articles about the
situation. What data should you collect, and how? How can you find out what the
trouble is? Whom would you interview? What remedies have you to offer? How can you
decide if the remedies will work?
14. You may have heard the phrase "Proportional Representation". Find out what it
is . Even after you have learned what it is , you will find it difficult to explain how
it works. Write an explanation and try it on someone who does not know what it is .
65
E x e r c i s e s in A r r a n g i n g D a t a
When solving problems, skill is needed in arranging the data so that the relations
between the facts can be grasped easily. In the following exercises, collect and
arrange the significant facts, omitting all irrelevant details.
1. Joe, Bill, Edna, and Ruth are the four children of John Green and his wife Mary.
Edna married Tom Carter and they have three children, Alfred, Anna, and Edith. Tom
Carter has a brother Walter who has two children, Kate and Sam. Henry Smith married
Ruth Green and they have two children, Dick and Mary. Joe Green is a bachelor but his
brother Bill married Mary Hinkle. They have no children, but Mary Hinkle's brother
Andrew has a son called Russell.
Are Alfred and Sam cousins? Is John Green the grandfather of Russell Hinkle? Is
Joe Green Dick's uncle? Is Kate's last name Carter? Are Andrew and Henry brothers?
Are Anna and Mary sisters?
2. Rewrite the following paragraph keeping only the facts that would be needed if
you were asked to arrange the articles in order beginning with the most expensive.
The book costs more than the pen but less than the comb. The book has green cover
which matches that of the pen. The watch costs less than the comb. The pencil has
black lead at one end and red lead at the other. The watch costs more than the book.
The pencil costs less than the pen.
3. Rewrite the following six sentences in a single column so that it will be easy
to see which car is the fastest and which the slowest :
Car A is faster than car B Car I is slower than car G
Car H is slower than car F Car B is slower than car H
Car A is fast er than car F Car B is faster than car G
4. Make an exercise like ex. 3 using weights or ages and try it on your classmates.
If n objects are compared how many statements about the objects are necessary?
5. Mrs. Somers had bid 4 spades but irs. Jade got the contract for 5 diamonds.
Mrs. Kelly led the King of Spades and then began talking about some boys who had
ruined the marigolds in her garden. "Harold is l? and ought to know better. John, I
know, is 6 years older than Harold".
Mrs. Jade lead the Ace of trumps and then a ided, "Harold is five years older than
'William. William was 7 last September when he started school."
Mrs. Merry, who was Mrs. Jade's partner, added, "Robert is 3 years younger than
John, but I don't know how old John is."
(You might wish to ask: What is the problem? What do you think it is?)
6. The object is not to answer the question at the end of this exercise (if you
could answer it, you would soon be president of the Okomo National Bank) but to
arrange the information so that the question can be studied.
Okomo has a population of 8,000. Newton, a town six miles away, has a population
of 20,000. In 1935 a new concrete road was laid connecting the towns. In August 1936
the new manager of the DeLux M.ovie began advertising the theater along the country
roads. The average daily attendance at the DeLux Movie, in Okomo, was 250, 280, and
260 respectively for the years lº 32, 1933, and l334. In October 1933 the Okomo Axle
Company closed its shops dues to lack of orders from automobile manufacturers. The
average daily attendance in the years lº 34 to 1938 was respectively, 260, 220, 200,
280, and 310.
What were the reasons for the decline and the later increase in the attendance?
66
INTERPRETING THE DATA. After collecting some data for a problem and arranging it
so that it can be studied, we try to interpret the data; that is, we try to draw some
conclusion from it. We ask "What does this prove? What can we learn from the data?"
Drawing correct conclusions from data requires a great deal of intelligence and
experience. The number of errors that can be made is so great that we could spend a
lifetime studying them. For example, the data at
the right shows the distances of certain planets
from the sun and their approximate diameters.
From this data it is correct to conclude that Mercury
the farther the planet is from the sun, the Wenus
greater is its diameter. 8arth
But the data (not given here) for the other Jupiter
planets shows that the conclusion is wrong.
To be safe, our conclusions should begin with the words
E x e r c i s e s
l. In one state the number of deaths of
children under 10 years for certain years is Pneumonia
shown in the data at the right. - Diptheria
What conclusions can you draw? Measles
distance diameter
from sun
36,000,000 2765 mi.
67,200,000 78.26
92,900,000 7.913
483, 300,000 9 Ol.90
1910
6156
1940
643
That advantage would there be in stating the Scarlet Fever 849
figures for 1920 and 1930 as per cents of the
FROM THIS DATA.
Read again page 48, the reliability of the inductive method.
1920
277O
920
434
388
figure for 1910? Do this, and see what further conclusions you can draw.
2. The data shows the number of Speed A
miles per gallon that certain cars, 10 l9.8
A, B, C, . . . . will go at speeds of 20 26.2
10, 20, 30, . . . . miles per hour. 3O 27.2
Thus Car. A goes 19.8 mi . on a gallon 40 26, 7
of gasoline when driven at a speed of 50 24.9
10 mi. an hour. 60 22. l
B
17 - 5
20.4
20.6
19 - 7
l8. 3
16.8
C D
15.8 l'7.5
20.1 l9.3
20.4 19.6
l9 , 6 l8.8
18, 2 17.7
l6. l lé.3
E
17, 6
19.8
20. 2
l9.5
l8.0
16, 7
1930
1065
90
l62
147
F
l'7.4
21.6
22.l
21.4
19.3
16.6
3. In 1909 there were 51,000 workers in automobile plants and each worker made an
average of 2# cars per year. In 1929, due to labor saving machinery, there were
226,000 workers and each man made an average of 23 cars a year.
In 1860 the number of workers in the steel and iron industry was 22,000 and each
man made 23.1 tons a year. In 1925 there were 400,000 workers and each man made
286 tons a year.
In 1849 there were 957,000 workers in industry and each man produced goods valued
at $1,066. In 1927 there were 8,350,000 workers, and each produced goods
at $7,500.
valued
First rearrange the above data so that it can be easily examined, and then see what
conclusions can be drawn from the data.
4. A manufacturer of candy tested three kinds of
candy bars, called A, B, and C as follows :
loC pupils were each given samples of the bars
and asked to state their first, second, and third
choice. 25 pupils chose A first, B second, and
C third. 6 pupils chose A first, C second, and
B third, etc.
How would you decide which is the most popular candy
bar? Statisticians have certain ways of answering
No. of
pupils lst
25
6
2O
14
22
13
A
A
B
B
C
C
:
3
i
such questions, which you cannot be expected to know, but how would you decide?
67
C H A P T E R 1 3
T H E E U L E R C I R C L E S
If we collected and examined a great many statements we should find that we could
separate them into groups of which the following are examples:
1. Good men are charitable 3. Some metals are soft
2. Dictators are not popular 4. Some flowers are not blue
Notice the word not in the second and fourth statements, and the word some in
the third and fourth statements.
Our problem in this chapter is to learn what conclusions we can draw from two such
statements. For example, what conclusion can we draw from the statements: smºs
Athletes are popular in school. Lazy boys are not athletes.
l. To recognize the hypothesis and the conclusion of Good Men are Charitable
write it as If a man is good, then he is charitable.
Imagine that all the good men of some community are Not
collected in a group. Imagine also that all the char- Chari-
itable men are collected in a group. We can represent table
this situation by drawing a circle within which all the Men
charitable men are standing, and a second circle which
contains all the good men.
The statement "If a man is good then he is charitable" tells us that the circle
containing the good men is inside the circle representing the charitable men.
The space between the circles represents the men who are charitable but not good.
The space outside the large circle represents men who are not charitable.
2. The second example, Dictators are not Popular can be stated
If a man is a dictator, then he is not popular.
As in the other example, we draw a circle to
represent the dictators, and a second circle to
represent men who are popular. Dictators Popular
One circle is entirely outside the other circle Men
because our statement says that no man can be both
a dictator and a popular man.
3. The third statement. Some Metals are Soft can be stated
If it is a metal, then it may be soft.
(The words may be need explaining. When a mother says to a child "you may go" or
"you may not go" the word may denotes permission. In our work, may means that
there are two possibilities, either of which could be correct. If a metal may be soft
then the metal is either soft or not soft.)
One circle represents all metals, and one circle
represents all objects that are soft. A part of
the circle which represents metals is inside or
overlaps the other circle. The part A represents
the metals that are soft; part B represents the
metals that are not soft.
When we use a general statement of the third type
we need additional information telling whether the metal is in part A or in part B.
A Swiss Mathematician, Euler (pronounced oil-er) was the first to use this scheme.


68
4. The fourth statement. Some Flowers are not Blue can be state d
If it is a flower, then it may not be blue.
One circle represents those things which are blue ,
and another circle represents flowers . The circles
overlap because some flowers are in the blue circle
and some are outside that circle. As in the third
example, additional information is needed to decide
whether a flower is in part A or in part B.
Flowev's
B A
Blue
Objects
In conversation, a statement like example 4 tends to emphasize part B, while a
statement like example 3 calls attention to part A. Logically, the statements merely
say that the two parts exist and attention should be given to both parts.
E x e r c i s e s
Draw the circles that represent these statements:
l. Some parallelograms are rectangles . 6. Brave boys deserve honor.
2. All business men are self-confident. 7. All corporations are not monopolies.
3. All triangles are not isosceles. 8. Only democrats voted for Jones.
4. Some teachers are not sarcastic . 9. Those who attend regularly deserve a
5. Some subjects are very dull. high mark.
USING THE CIRCLES. The Euler circles will give us a better understanding of the
four propositions on page 41. The contrapositive of Good Men are Charitable (l)
is If a man is not charitable, then he is not good.
The circles (Draw them!) show quickly that this proposition is true because if a
man is outside the larger circle then he cannot be in the inner circle.
The converse of (l) is : If a man is charitable, then he is good. The circles
show that this converse, like many others, is not true because a man could be inside
the larger circle and not be inside the smaller circle. He might be in the part
between the two circles .
The inverse of (l) is : If a man is not good, then he is not charitable. If a
man is not good, then he is outside the inner circle, but he may be in either of two
places: (a) between the circles, or (b) outside the larger circle. Since his loca-
tion is not decided, we do not know whether he is charitable or not.
Exercise. In the same way as shown above, study the statements 2, 3, and 4 on
page 67, writing their converses, inverses, and contrapositives.
CONCLUSIONS FROM TWO STATEMENTS. Example l. From the two statements
a) Some rules regarding tardiness are unjust
b) The rules made at the faculty meeting to-day dealt with questions of tardiness
Tardi
lies entirely inside circle l, but
can we conclude
c) Some of the rules made to-day are unjust.
l 2 l 2
Jr. D
SBS ºſe SS
we do not know its exact location.
If it lies outside circle 2, the conclusion is: To-day's rules are not unjust.
If it lies inside circle 2, the conclusion is a To-day's rules are unjust.
If only part of circle 3 in in circle 2 (draw your own figure for this situation)
the conclusion is : Some of to-day's rules are unjust.
Circle 1 represents rules about
tardiness. Circle 2 is for unjust
rules. Part of circle 1 is in
circle 2, according to (a).
Circle 3 represents the rules
made to-day. According to (b), it










69
Example 2. What conclusion can be drawn from the statements
a) All seniors are clever
b) No one who fails in mathematics is clever.
Circle 1, representing seniors, is inside circle 2
which represents the clever people.
Circle 3, those who fail in mathematics, is outside
circle 2, according to (b).
Since circles l and 3 cannot overlap, the conclusion
is : Seniors do not fail in mathematics.
Seniors
l
'ailures
in Math.
Example 3. What conclusion can be drawn from the statements
a) Some movies are not long
b) No movies are silent.
According to (a), circle 1 representing movies
overleps circle 2 representing long objects.
According to (b), circle 1 is outside circle 3,
representing silent. But we do not know if
circle 3 overleps circle 2 or is outside circle 2.
What would be the conclusion in each case?
E x e r c i s e s
l. Draw figures showing all the possible ways in which 3 circles may or may not
intersect, overlap, lie inside or outside each other, and so forth.
Make some suitable statements as in the examples above to illustrate what the
circles represent, and then investigate the possible conclusions.
(The circles may be drawn on the blackboard, and the work of making examples divided
among the pupils in the class.)
2. In the second example on page 67, the popular men are supposed to be collected
in a circle and all unpopular men are supposed to be outside that circle. How would
the circles be drawn if all the unpopular men were collected in a circle, and the un-
popular men were outside? Where would the circle for dictators be drawn?
3. At the beginning of this chapter are four typical statements. Find what conclu-
sions can be drawn if you combine a statement like l with a statement like 2, a state-
ment like l with a statement like 3, statement l with statement 4, and so forth,
using all possible combinations. For example, when you combine statements like 2 and
4, you could use
Greedy people are not popular; Some freshmen are not greedy
OI" Greedy people are not popular; Some seniors are not popular.
By drawing the Euler circles find the conclusions from:
Mice are greedy animals. Mice cannot fly.
Some geraniums are red. These flowers are red.
All bankers are rich. Some bankers are handsome.
A planet is not a star. The earth is a planet.
Coal gas is harmful. Coal gas kills people.
Sugar is fattening. Candy is fattening.
:









7O
C H A P T E R 1 4
P R O P A G A N D A
We often hear the statement that our country would never have fought in the World
War if we had not been mislead by propaganda, and that we would not have been mis-
lead by propaganda if people would learn to think clearly and reas on correctly.
POSSIBLE DEFINITIONS. Mr. Jones had read in his daily paper that England was
building enormously large airships capable of making non-stop flights of 6000 miles
and carrying as much as five tons of bombs. He mentioned these facts to his friend
Mr. Smith. "Ah" said Smith, "That's just a lot of propaganda to make you think
that England can beat anybody." If asked for an explanation of propaganda, he
would answer, "It's a lot of stuff you see in the papers put out by somebody to make
you believe something they want you to believe."
If Mr. Smith believes that McCarthy should be reelected mayor, then any article in
the paper which tells how well McCarthy is doing his job is, according to Smith,
nothing but the truth and good publicity. But Jones, who does not like McCarthy,
calls that same article "propaganda". In other words, Smith and Jones call the
articles propaganda if they do not believe it, and good publicity when they do.
Let us consider two possible definitions :
l. Propaganda is an attempt to influence others toward some special end by appeal-
ing to their intelligence or emotions. The organization or persons who make the
attempt may be sincere or insincere; their motives may be selfish or not.
2. Propaganda is any kind of promotion (publicity, advertising, or attempts to
influence others) which is veiled (secret, hidden, camouflaged) in any, ºn several
or in all of the following respects:
(a) its origin (b) its methods (c) the people doing it
(d) the contents (e) the results on the victim.
E x e r c i s e s
l. Would you have a different opinion of the second definition if the last word in
it were reader instead of victim? Does the word "victim" suggest that propaganda is
something dishonest? Is it desirable to use such words in a definition? After fin-
ishing this chapter, try to suggest a different word.
Discuss the following situations according to each definition. Note such matters
as truth, sincerity, selfish or unselfish motives , origin, methods, contents, and
results. If you answer depends on how the article is written, suggest possible ways
of writing the article. You may wish to repeat the exercises after finishing the
chapter.
2. A mother warns her children not to play in the streets because of the danger of
automobiles. (Isn't this an attempt to influence the behavior of children? Is it
veiled? Would the mother's advice be veiled if she said, "Good children do not play
in the streets." Is it propaganda if the mother tells the children not to play in
the streets but does not tell why they should not? If the mother said nothing, but
put a high fence around the play-yard so that the children could not get into the
street, does not the fence satisfy one of the definitions of propaganda? Can a fence
be propaganda?)
3. A road sign which says "Speed Limit 40". Is this not an attempt to influence
other people toward a special action by appealing to their intelligence?
4. A sign which says "No Hunting Allowed". Do you know who put up the sign? Does
it not aim to influence people?
5. An article in a magazine on popular science telling about Newton, the law of
gravitation, and how much less you weigh on some other planet.
71
6. An advertisement in a magazine of a recent popular novel telling the price of the
book, its author, and a brief sketch of the story.
7. A report in a daily paper of a speech by a United States senator telling why he
believes that higher income taxes will be needed to promote certain social benefits.
(Isn't this an attempt to influence other people. Even though you know the speaker,
do you know anything about the possible results on the victim?)
8. According to the second definition of propaganda could not some speech over the
radio be propaganda to one person and not be propaganda to another person?
9. A news item in the daily paper telling about the new course of study introduced
in the high schools of that city. (Do you know who writes the articles in your daily
paper? Do you know what reasons there are for having a new course of study? Is not
the article an attempt to make the tax payers believe that their schools are up-to-
date and efficient?)
10. An item in your high school paper telling about the new requirements for grad-
uation in your high school. (Do you know what was the motive of the people who
changed the requirements? Does it make a difference whether the item tells what the
new requirements are, or whether it states that the new requirements are better than
the former ones, or whether it finds fault with the new requirements and argues for
a return to the old ones?)
ll. A letter from the Parkside Women's Legislative Club telling about a bill now
being discussed in the state legislature, and urging that you write your assemblyman
to vote against the bill. (You know the origin of the letter, since it is signed,
but do you know the motive behind it, and the possible results, and who will profit
by the passing or the failure of the bill?)
12. A large advertisement, covering an entire page in the daily paper, by the local
Milk Drivers Union explaining why the drivers are on a strike. The advertisement
tells what the salaries are, the working hours, and the profits of the company. Is
there anything veiled about this attempt to influence the public? Does the truth of
the statements in the advertisement decide whether it is propaganda or not?
13. An organization of automobile drivers favors a law prohibiting children from
riding bicycles on streets. There is nothing veiled about the organization (it is a
well known club in your town), or about its members, or its methods, its advertising,
or what the results of the law will be . The organization places many advertisements
in the daily papers urging that this law be passed. Is this propaganda?
14. During the civil war in Spain, 1937-1939, many magazines contained advertise-
ments asking that money be collected to help fight the fascists. There was nothing
veiled about the organization (the names of its officers and directors were printed)
the methods they intended to use, and the public could be expected to know what the
money would be used for. But the people who believed that the fascists would im-
prove conditions in Spain said that these advertisements were propaganda, that we
ought to let the Spanish people decide their own form of government, and that we should
mind our own business.
15. An organization is formed to encourage the study of Russian literature. Its
sponsors are well known Russian musicians, writers, and teachers of the Russian lan-
guage in colleges. There is nothing veiled about the organization, its methods, aims,
or what it attempts to accomplish. Its funds are collected by private contributions.
Some people say that the organization should not be allowed to exist because it is
merely propaganda, trying to make people believe that communism is a good form of
government. The sponsors of the organization admit that they do believe that commun-
ism is a good form of government, but they also insist that they are not trying to
win converts to cummunism; they are merely hoping to encourage the study of Russian
literature •
Suppose a similar organization were formed to encourage the study of French litera-
ture. It uses the same methods as that described above. In deciding whether or not
something is propaganda are you influenced by your prejudices?
72
THE DEFINITION. In the rest of our discussion we shall use the second definition
given on page 70. The definition states that the promotion (publicity, advertising,
reading matter, speeches) is veiled. This means that we may not know from where the
material is coming, and who are the people who are trying to influence us, and what
the results will be .
In an advertisement we know that it is the manufacturer who is interested in selling
us some goods. When listening to a speech in a political campaign we know that the
candidate will gain or lose by the election. But if we see an item in a paper about
the great increase in the cotton crop in Brazil, we wonder who wants us to believe
this news. The cotton growers in Brazil? (Why do they want us to know that the crop
has increased?) The cotton growers in the south? (Are they worried, fearing that
they may not be able to sell their cotton?) The business men who may want higher tar-
iffs on cotton? (Should we pass a law prohibiting the importation of Brazilian cotton?)
The manufacturers who use cotton? (Do they want us to believe that cotton is cheap
because Brazil is growing a great deal of cotton?) Who is it that is so anxious to
inform us about the cotton crop in Brazil? Who is going to profit by telling me about
the cotton crop in Brazil? Who? Why?
The definition says nothing about the truth or falsity of the material which is used
to influence us. Even if the statements are true, we may still be mislead unless we
have all the truth, not just part of it. In the argument on page 5 about football,
we can present some good and true reasons why football should be abolished. And we
can also present some equally good and true reasons for not abolishing football. When
the University of Chicago abolished football some people said, "Why can't the univer-
sity have good teams? It has twice as many men students now as it had when its teams
were champions thirty years ago." That statement, by itself, was true; but it was
not all the truth.
If all the facts on both sides of a question were presented, we should be able to
draw the correct conclusion. In propaganda all the truth is seldom presented.
The methods used by propagandists are often veiled. An advertisement may suggest
that you spend your summer at Lake Tohala. It does not say that there is only one
hotel at Tohala and that the advertisement was inserted by that hotel.
EMOTIONAL APPEALS. We all have a desire to be liked by other people. At times
this desire may overcome our judgement. Many advertisements take advantage of this
weakness. They say "Buy this car and people will flock around you to admire your car
and the intelligence you showed in buying it!! We have a natural inclination to do
as everybody else does, as if the crowd always does the right thing. If we cannot be
millionaires actually, we can at least resemble them in some respects; we can drive
over the same roads and use the same brand of toothpaste that they use.
We all desire comfort, easy living, and want to get the most for our money with the
least effort. We all want protection against the imaginary or real dangers. No one
wants to be old or neglected or poor. We all feel that, granted the same opportunities
and a little luck, we could do anything as well as anyone else. Propaganda has always
taken advantage of these weaknesses in human nature. The shrewd politician always
promises to give the people what they want.
No matter how much we try to be liberal minded, we still have certain prejudices
due to our family, our religion, our place in the community, and our finances. I may
have been warned that my friend Mr Jones will try, by flattery, to get me to do what
I do not want to do. In spite of this warning, when Mr. Jones tells me that I am a
leading member of the community and that everyone thinks highly of me, I am inclined
to believe what he says. I believe he is a very keen judge of people and that he
knows a good man when he sees one (his remarks about me proved this). Further, he
belongs to the same golf club that I do, and this proves he knows good golf clubs.
Did the warnings I received do any good? Why not?
73
SOME COMiiON APPEALS. Psychologists (they study human nature) have listed many of
the common devices used by propaganda. Every year new ones will be invented as fast
as the public "catches on" to the old ones, just as a footoall coach invents new
plays when the opponents develop a defense against old plays.
Name Calling. The propagandist tries to associate his ideas with such names as
"liberty" "patriotism" "mother love" "security" or, to be little his opponents, he
associates them with "traitor" "scab" and "plutocrat". Since these words are hard
to define, the propagan dist hopes that the public will associate the good or bad qual-
ities suggested by these names with the good or bad qualities of his proposition.
For example, "Every patriotic citizen will approve of this measure."
"Unless you approve of this measure you have no regard for personal liberty."
"Every man who loves his m.other will favor the élection of a man who has always
been known for his interest in the aged and poor."
"Only a scab would try to take away from a man his chance to make an honest living."
"The plutocrats don't care who starves. They have caviar and steaks."
Glittering Generalities. When a good argument for or against a proposition cannot
be presented, the ideas may be stated in such vague terms that it is impossible to say
that they are false. Thus, if a police commissioner cannot explair, why the police
are inefficient, he can discuss how concrete roads have influenced crime detection,
the use of the two-way radio communication, and the great cost of crime.
The Transfer Device. The propagandist shifts the attention of the reader from the
topic being discussed to some topic about which there can be no disagreement. Thus,
if we favor putting all law enforcers in one county under the control of a single man,
our opponent (who has no good argument against us) may begin talking about the necess-
ity of a better system of catching criminals (to which all of us agree).
Testimonials. "If all the leading actresses of Hollywood use this nail polish, why
shouldn't you use it!"
Plain Folks. "We are just plain ordinary folks like you and everybody else." This
suggests that we are therefore honest, truthful, sincere, generous, and so forth. We
have all the good qualities that plain folks have. There is nothing "ritzy" about us.
We do not pretend to be better than other people.
Card Stacking. A dishonest card player "stacks" the card; that is, he deals himself
the best cards so that you cannot win. Likewise, the propagandist may state the argu-
ment in such ways that the reader cannot win. He says "Either you are a crook and
should be in jail, or you are an honest man, in which case no argument is needed to
convince you of this proposition."
The Band Wagon. "Follow the crowd. Do as your leader does. Don't sulk in a corner.
Get out and enjoy yourself. Even if you do know more than the crowd, even if you are
wiser than the crowd, don't be a wet blanket." The propagandist suggests that you
should not use your intelligence since that implies that you are different from other
people; and people who are "different" are never popular.
Repetition. A clever propagandist has said, "Even if what you say is a lie, if you
just repeat it often enough you will finally get some people to believe it." Likewise,
when the truth does not convinue people (because of their prejudices) the repetition
of the truth may eventually wear away the prejudices.
The devices mentioned above are the tricks which propaganda uses when it does not
wish to tell the truth, all the truth, and nothing but the truth. The presence of
these tricks enable us to detect propaganda. When they are used by any speaker or
writer we are justified in becoming suspicious of the man's intentions.
74
E x e r c i s e s
l. Each of the devices mentioned on page 73 is based on some type of poor reasoning
mentioned in Chapter ll. Thus, name calling may involve poor analogies and lack of
exact definitions. Examine the other devices for similar errors.
Assuming that the following descriptions illustrate propaganda, tell which devices
they illustrate :
2. When Mr. Wills was candidate for sheriff, his publicity manager sent out material
telling how Wills was born in a log cabin, attended a country school for only a few
years as he was compelled to get a job early in life, and he now supports his widowed
mother. He never had much time to study or to acquire the polished social manners of
his opponent because he had to work so hard.
3. While Bill Smith was studying his geometry one evening Henry Fine came in. "Come
on out, Bill" he said. "The bunch is all going to the movies. Why worry about you
geometry. You always get it. Come on out and have some fun. After the movies we're
all going to Dukes for hamburgers. Be human. Don't be such a grind."
4. Going to school is either like going on a picnic or going to the penitentiary. At
a picnic you have a good time and enjoy yourself. If you are not having a good time
at school, then evidently your school is no better than a penitentiary.
5. I don't think Dick Haines should be elected class president. To my mind he is not
worthy of the honor. He should not be elected class president. The presidency of the
class is one of those jobs which is a great honor to the boy who gets it. I don't
believe we should give that honor to Dick Haines. The class presidency is a respons-
ible job, and I don't think we should elect Dick to that job.
6. You ought to buy a Caxton Car. Mr. Wilson, the president of the Electric Company
drives a Caxton. All the smart people along Richton Boulevard drive Caxton Cars.
Paul Monroe, the chairman of the board of directors of the Subway Corporation has said
that he would be ashamed to be seen in any other kind of car.
7. The only people who oppose the extension of the car line to Cicero Avenus are the
plutocrats who drive Cadillacs and Lincolns. Of course those rich guys don't need
street cars . In that district around Cicero Avenue only poor people live. They are
the ones who have no security in their old age. Their rose covered cottages have no
two car garages. Few have ever worked steadily. They raise their own vegetables.
Their homes are far apart. On cold winter days they have to trudge miles in deep
snow to visit a sick neighbor.
8. The new school gymnasium should be built on the east side of the present build-
ing rather than on the west side. Basketball is getting more popular every year and
we may need to increase the gym in a few years. The young folks in our community
need a large gymnasium . We should build now, and remember that no matter how large
we make the gym, in a few years it will be too small. Whenever you build anything
you should consider ; Have you enough space for future additions?
9. We are trying to decide to-night a very important question: Shall America stay
neutral? This country has always been a home for the liberty loving people of the
world. It was settled by people who came here to be free to worship as they chose.
We have created a prosperity which is the envy of the world. We should keep it pros-
perous. Nowe here else in the entire world are there so many automobiles, so many
telephones, so many radios, and so many electrical gadjets for the home. America
will never do anything to destroy that prosperity.
75
10. An advertisement for an automobile selling for $600 pictures a small cottage,
the wife is feeding some chickens, a boy is chopping wood, and a little girl is play-
ing with a rag doll. Everything looks very commonplace and ordinary. This is not
the "band wagon" type of testimonial for a car. On what idea is the advertise:ent
based?
ll. It would be a serious error to assume that all advertising is propaganda. But
advertising may also use some of the devices mentioned on page 73. Select six adver-
tisements from magazines. To what emotions or feelings do they appeal?
Collect the answers from the class and find which appeals are used oftenest.
12. Select some advertisement whose object is to make you buy a certain article.
Make a list of the additional information you would need in order to decide if the
claims in the advertisement are true or false. Tiscuss in class how this additional
information could be obtained. "Would it require much work?
13. Many advertisements are educational or informative in their nature; that is,
they do not chiefly intend to sell any article but to give you information about some
business. See, for example, some of the advertisements of railroads, insurance compan-
ies, and steel corporations. Collect some advertisements of this kind. Examine them
to see if they use any of the devices of the propagandist.
14. When asked to explain the difference between propaganda and education, a speaker
said: Propaganda aims at getting people to do or not to do a certain thing, while
education develops and changes a person, and aims at some ideal.
Explain more fully what the speaker meant.
15. In a newspaper select some item about a social event. Do not change the facts,
but rewrite the account from various points of view such as
(a) A newspaper in a small town.
(b) A paper in a coal mining district.
(c) a paper in a fashionable summer resort.
The object of this exercise is to see how facts can be written in different ways,
creating different impressions on the reader.
16. As in ex. 15, select some item of news dealing with a labor Guestion. Do not
change the facts, but rewrite it from the point of view of
(a) The Business Manager of the Carpenter's Union.
(b) The manager of a chain of grocery stores.
(c) A contractor who hires many plumbers, masons, carpenters, etc.
(d) A candidate for the office of sheriff in a coming election.
17. Would it be possible for an article in a magazine to be propaganda to one reader
and not propaganda to another reader? Is any article propaganda to the man who wrote
the article?
18. Suppose that you are the publicity director for some company or organization.
To be as specific as possible, decide on what the company manufactures or what ideas
you wish to popularize. Write a speech or an advertisement or an article for a maga-
gine which will promote your ideas basing it on one of the appeals mentioned on page
73. For some ideas one of these devices might be more suitable than another but state
which device you are using.
76
M i s c e l l a n e o u s E x e r c i s e S
The following problems are not classified. In real life our problems are never
classified for us; one moment we must deal with one kind of problem, and a few minutes
later we may need to solve an entirely different kind of problem.
These exercises involve ideas from any or all of the previous chapters; some may
even involve ideas that you have never studied previously. In that case your problem
is: What should be done to solve this problem?
In many of the exercises a certain situation is described and no definite question
is asked. You are then expected to make your own questions.
l. During a summer vacation John Jones planned to earn some money by organizing a
business called "The Chore Boys". They worked in the neighborhood cutting lawns,
running errands, washing cars, taking younger boys on trips, and so forth. Their
charges were 25 ¢ an hour. John used his mother's telephone as a place where orders
could be placed. Since he could not do all the work himself, he hired other boys to
do some of the work. The other boys collected 25 & an hour, and paid Jim a commis-
sion of 10 % on all their earnings.
Some of the workers were not honest; they told John they had worked 3 hours and
paid him 8 g when they had worked 4 hours and should have paid him 10 4. John thought
of phoning each person for whom the boys had worked, but a telephone call cost 5 A.
John's problem was ; How could he check on the amount the boys collected so that he
would be sure to get all his commission?
2. A city decided to build an Art Museum. The mayor was asked to appoint a comite
of three citizens to study the question. Among those suggested for the comite were:
Mrs. Haynes, a successful manager of a chain of candy stores.
Fordyce Chomley, a dealer in Art Supplies.
William O'Brien, a builder of homes and a real estate agent.
Charles Spragg, an owner of a wholesale grocery company.
Montcey Boyer, who had once played in the movies in Hollywood.
a) Select three of the above for your comite and give the reasons for your choice.
b) When the comite met, it was decided that one of them should study the question of
how much it might cost to build the museum, another member should decide on a location
for the building, and the third member should find the cost of an art collection. State
who on your comite should perform each of these tasks.
c) The mayor chose a different comite, saying, "I hesitate to appoint a real estate
agent (he would select some building site that would help his own business), or a
building contractor (who would try to make a profit on this job), or an artist (who
would sell us some second rate paintings that no one else wants). We must have dis-
interested parties on this comite." Discuss these reasons.
3. A speaker at a Women's Club said, "There is something peculiar about the figures
which the chairman of the relief comite has presented to us this afternoon. According
to her figures, 17 % of those receiving relief in this town are between the ages of 45
and 54. She also said that lé 7% of the population of this town is between the ages of
45 and 54. Apparently then there are more people, between 45 and 54, who are receiving
relief than there are people between those ages. Evidently there must be something
wrong with her figures.
4. Mr. Hill solicits advertising for a magazine. When he called on Mr. Hart, the
latter said that he did not think this was a good time for advertising furniture since
no one had any money to spend; when times were better he would begin advertising.
Mr. Hill always answers such objections by saying, "When business is good, you must
advertise to get your share of the business. And when business is poor, you must ad-
vertise to create a demand for your goods."
77
5. "I had a lucky escape yesterday", said Mr. Hill to his neighbor. "I had just
left Miami, Florida and was driving north. When I had gone about fifteen miles a
fellow driving about eighty miles an hour passed me. He drove so close to me that I
had to drive off the road but, luckily, I just brushed against some bushes. As I
leaned out of the window to see what damage had been done T could see from the posi-
tion of the sun that I ought to hurry home. But that fellow's speeding was a lesson
to me, and I took my time about getting home."
"sorry, Hill", answered the neighbor. "I don't believe your story."
6. Mr. Jones: I just inherited about $5000. Congratulate me! I'm going to invest
it some good way. At 3 } I'll get an income of $150 a year.
Mr. Smith: You ought to get more than 3 } on an investment.
Mr. Jones: ... ore than 3 } : Mot likely! Hot in a safe investment. The big
life-insurance companies can't do better than that. And I don't pretend to be any
smarter than the big life-insurance companies. If they can't do it, nobody can.
I. r. Smith: Of course they are smarter in such matters. But you don't have to
obey the same laws.
!r. Jones: What do you mean "obey the laws"? Do you want me to start some
crooked business an i land in jail!
7. Sai i lºr. Hall to .rs. Hall, "I guess the strike in Boston has been settled.
Last week the newspapers were full of news about the plumber's strike. To-day all
that you can find in the papers is about the new war in Spain."
8. (a) A barber in a certain village shaves all the men who do not shave themselves.
Tocs the barber shave himself?
(b) All generalizations are false, including this one.
(c) The kind of error in reasoning illustrated by (a) and (b) is not mentioned in
Chapter ll. Tescribe it, and in vent a suitable name for it.
9. Lincoln once made the famous statement
"You can fool some of the people all the time, and all the people some of the
time; but you can't fool all the people all the time."
How would you express just the opposite of this idea?
1C. You are trying to decide which brand of oil to use in your car, and learn that
one of the biggest automobile companies in the country recommends brand A. Is that
a good reason for using brand A3
Would your opinion be changed if you later learned that the president of this auto-
mobile company is also a large stockholder of the oil company which makes brand A 2
Would you opinion be changed if you later learned that the president of the automo-
bile company recommended this oil many years ago when he was not a stockholder of the
oil company?
But suppose the son-in-law of the president of the automobile company was the pres-
ident of the oil company. Now what do you think?
ll. A man rented some land from a company with the understanding that he would not
grow fruit on the land. The land was suitable for growing tomatoes, and a nearby
cannery offered high prices for the tomatoes. When the man lost his job he began
raising tomatoes rather than accept charity. But when he was ready to sell his torna-
toes, the owner of the land objected because the man had agreed not to raise fruit.
The man claimed that, tomatoes were vegetables not fruit.
12. The best cleansing powder consists of small grains of silica. From this state-
ment should you say that "consisting of small grains of silica" is a necessary or a
sufficient ingredient of a good cleansing powder?
78
13. John Slone washed dishes to pay his expenses in college, and later made a huge
fortune as an engineer for the World Export Company. I intend to wash dishes when I
go to college even though my father has enough money to that I don't need to. The
perseverance that such work develops will help me to succeed after T graduate.
14. In the following pairs of statements, is the second the cause of the first, the
first the cause of the second, or is there no causal relation?
(a) Piano playing is poorly taught. Few learn to play well.
(b) A cat has a bone. The dog chases the cat.
(c) John's car has poor brakes. John was in an accident.
(d) A country has a large army and navy. The country is involved in many wars.
15. Many people believe that the government should so manage affairs that every
man who is willing to work will be provided with work; in other words, the government
owes every man a living. But other people say that the government is merely a servant
of the people. We hire the government to do certain things for us such as providing
police protection, water, good roads, etc. And if the government is only our servant
then the government does not owe us a living. Our servants certainly do not make
jobs for us; it is we who make jobs for our servants.
16. Prospective lawyers will be interested in the story of Protagoras and Euathlus,
who lived in Greece several thousand years ago. Protagoras was a lawyer who taught
other people how to become lawyers, and one of his pupils was Fuathlus. They made
this agreement: Euathlus was to pay half his tuiton when he had completed his course
of study, and half when he had won his first case in court.
When Fuathlus had finished his course of study he postponed practicing law, making
no effort to secure clients. Protagoras therefore sued him in court for the unpaid
half of the tuition. Protagoras argues thus: If the judge decides in my favor, then
Euathlus must pay me . But iſ the judge decides against me, then Euathlus will have
won his first case, and so, according to the contract, he must pay me. Hence, win
or lose in court, I will collect from Euathlus.
Euathlus, on his side, argued thus:
If the judge decided in my favor, I need not pay Protagoras.
If the judge decides against me, then I have not won the case, and so, according to
the contract, I need not pay Protagoras .
Hence, whether I win or lose, I need not pay Protagoras.
17. Harry Smith was preparing a debate on the proposition: Cuba should be annexed
to the United States. Cn different cards he had written the following statements.
Now he must rearrange the cards so that they will make a proof of his proposition :
a) The annexation would pay economically.
b) The island has been without a stable government.
c) Annexation would increase our trade with Cuba.
d) When the government is not stable, trade and business decreases.
e) The government was not stable because of the tyranny of the rulers.
f) Annexation would make the government stable.
g) Annexation would increase our trade.
h) Annexation would induce many Americans to invest their money in Cuba.
i) Insurrections of the people against the tyranny of the rulers hindered business.
j) If Americans would invest their money in Cuba, trade would increase.
18. Our work in school is supposed to teach us to adjust ourselves to whatever may
happen in life. In life we are all bound to fail at some time or other in some task.
Hence in school we should learn how to adjust ourselves to failure. The best way to
learn anything is through experience and practice. Therefore in school we all ought
to fail in something in order to learn how to adjust ourselves to the failures which
we will have after we leave school.
79
What type of reasoning (inductive, deductive, indirect method, poor analogy, non-
causal relation, and so forth) do ex. 19 to 31 illustrate?
19. One summer while driving across Rew York we came to a fork in the poad. One
road was marked "To Buffalo"; another was marked "To Cwens"; the third was not marked
at all. Hence we took the third road since we wished to go to Hillsboro.
2O. A fruit is defined as "the ripened seed and adjacent tissue of a plant." The
potato is therefore a fruit because, when plante :, it produces new plants. Sweet
potatoes are not fruit but merely part of the root of the plant.
21. When trains first ran across the country they were called impractical. People
said that it would cost too much money to lay the tracks. Television is now regarded
as impractical. The me chines cost so much money that few can afford to buy them. But
trains proved their value, and some day everyone will have television sets.
22. "I am not going to the picnic of the Ladies Aid next Thursday", said Amantha.
"Haven't you noticed that it always rains on the day of our picnicº Last year I
ruined my dress by that sudden shower."
23. When John Tavis became president of his senior class, his family were proud of
him. "It was due to his home training", said his father.
24. When an income tax was first proposed in the United States some people objected
to it saying that America had alwa, s been the home of the oppressed who wanted liberty,
and we should not place a tax on the people coming into this country.
25. The United States must be expecting another ºorld War since it is increasing the
size of its navy.
26. Alfred claimed that Leethoven wrote the lyinuet in G in 1833. But in a book in
the school library Jim found that Beethoven had died in l827.
27. I.r. Aaron's secretary could not attend to all her duties. A friend urged him to
hire an additional secretary. But i.r. Aaron answered, "Too many cooks spoil the broth."
28. Crime has increased a great deal since automobiles were invented and concrete
roads laid between large cities. The good roads make it easy for a man to com:::it a
crime and then get away before he can be caught. We would have less crime to-day if
there were fewer automobiles and concrete roads.
29. If the value of a fraction is l, then the numerator equals the denominator. But
in the fraction (n + 3)/(n + 4) the numerator does not equal the denominator, and so
this fraction cannot equal l.
30. A chemist for a paint company was trying to find something that he could add to
paint to make it dry faster. He knew that any substance that would evaporate quickly
would make paint dry faster. Hience he poured into the paint the various substances
that evaporate quickly. After many experiments he found that any substance containing
hypoacétate (he invented that name to keep his discovery a secret) would do the work,
and that when hypoacetate was not present the paint dried slowly.
31. Mrs. Heinly thinks that the P.T.A. should be abolished as it is not the parent's
business to advise teachers how to run schools. We do not have a Parent-Coctor Asso-
ciation to advise doctors. We do not have a Parent-Lawyer Association to advise law-
yers how to run their business. He do not have a Parent-Fmgineer Association to tell
engineers how to build bridges. We should not have a Parent-Teacher Association to
advise teachers how to run schools.
80
32. In a debate on the question "The government should encourage the discovery of
synthetic fuels (substitutes for gasoline)" which of the following statements should
you say were facts (needing no proof) and which would need proof?
a) Some day our gasoline supply will be exhausted.
b) A driver gets as many miles per gallong with a synthetic fuel as with gasoline.
c) Synthetic fuel wears out a motor faster than gasoline.
d) In produced on a large scale synthetic fuel would be as cheap as gasoline.
e) The synthetic fuel would not need any special type of tank on a car.
33. In a debate on the question "This state should have a 2 % sales tax" which of
the following statements would need to be proved and which would be assumed as true
by both parties to the debate?
a) The sales tax is a heavier burden on poor people than on rich people.
b) Additional taxes are needed to take care of the growing costs of government.
c) A sales tax is easily collected because no one can postpone paying it.
d) A sales tax brings in a steady flow of money to the treasury whereas other taxes
are paid semi-annually, or annually, or at longer intervals.
34. In a debate on the question "All manufacturing of munitions should be prohibited
by law" which of the following statements are facts and which need proof?
a) Without munitions wars could not be carried on.
b) When war is suddenly declared the country without extensive facilities for manu-
facturing munitions is at a disadvantage.
c) If all nations would agree to the prohibition there would be no sudden wars.
d) If no nation could manufacture munitions, they would settle their disputes in
some other way than by going to war.
35. Our daily conversation, our public speeches, our newspapers and magazines
bristle with words full of ominous meaning. We use these words to blast our foes
and to favor our aims. If we dislike a man's views we call them communistic, and
that settles it. If we want to paint an idea favorably we call it "American" and
that makes it perfect.
36. National defense can be taken to mean anything that the champion of a large
navy wishes it to mean. It may mean the protection of the shores of our country from
attack. It may mean the protection of the shores of the continent plus Hawaii. Or
it may mean the protection of the Phillipine Islands, or it may mean that the navy
must be strong enough to protect American business firms wherever they are located.
37. Television sets will not be purchased by the public till there are more programs
for the public to see. There will not be more programs until advertisers can be found
who will sponsor the programs. And advertisers can not be found until more sets have
been sold to the public.
38. J f our rulers could be trusted to have in mind the best interests of the
people then a monarchial form of government would be the best form of government.
But rulers can not be trusted; they all start well, and then they lose sight of the
common people. Hence a monarchy is not the best form of government.
39. "We don't go to the movies very often", Mrs. Tone told Mrs. Hart when they met
on the bus. "We usually go Saturday nights to give the children a treat, and then on
Wednesday or Thursday nights. That gives us something to do in the middle of the week.
There's always a big Feature on Monday nights, and of course we can't miss that. On
Sundays, after a heavy dinner, ſir. Tone likes to go to the afternoon show, but I
think he sleeps at the show. He wont go alone and he always insists on taking the
children with him. Friday nights we let the children go because there is no school
the next day."
81
40. In the novel "And Tell of Time" by Laura Krey, page 268, Povey when riding a
horse along a road notices that the road is bordered by tall bushes bearing many black
pods. "Coffee beans", thought Povey, leaning over to strip the bushes as he rode past,
"and black jaundice". The two ideas - coffee beans and jaundice - followed one another
in rapid succession, for it was common knowledge that in the years when coffee beans
were thickest, black jaundice was most common.
41. In the book "Thrice a Stranger" Vera Britain mentions that in a cemetery in
Ohio she noticed that a monument mentioned the 60 local boys who had died in the Civil
'''ar and the 2 boys who had died in the World War. She concluded that, in that town
at least, the Civil War had aroused a great deal more patriotism than the World War.
42. One of the two most famous songs by Stephen Foster was written in his youth out
of sheer inspiration. The other was a commercial product written for money. One of
the two songs was "Old Folks at Home"; the other was "ily Old Kentucky Home". Read,
or better still, sing the two songs, and then see if you can decide which song was
written for money and which for the mere joy of somposing.
43. In an argument about chain stores, one speaker mentioned that every year about
one third of the independent retailers became bankrupt. His opponent answered by
saying that even before any chain stores existed it was true that one third of the
independent retailers became bankrupt every year.
44. Some pupils visited the shops of an automobile manufacturer in Detroit. In
their report of the visit the pupils said, "The working conditions in the foundry
must have been very bad since our request to visit the foundry was refused. The com—
pany evidently does not want the public to see the terrible conditions under which
the men work."
45. One day when ºrs. Petty was standing on a pier looking down at the water, she
said, "The water isn't very deep here. I can see the bottom."
46. To test your understanding of the phrase "denies that none but" try this exer-
cise. Do not consult a textbook on Zoology before answering the questions.
John denies that none but prawls are shrimps.
Bill denies that none but shrimps are prawls.
To which of the following statements would Bill agree? To which would. John agree?
a) All prawls are shrimps e) Some prawls are shrimps.
b) All shrimps are prawls f) Some shrimps are prawls.
c) All prawls are not shrimps g) No shrimps are prawls.
d) All shrimps are not prawls h) No prawls are shrimps.
i) If an animal is a prawl then it is a shrimp.
j) If an animal is a shrimp then it is a prawl.
k) To be a prawl it is necessary that the animal be a shrimp.
1) To be a shrimp it is necessary that the animal be a prawl.
Notice in the above work that nothing is said about whether John's or Bill's state-
ments is the correct one. The question is merely: Which statements from a) to l) are
in agreement with his first statement?
If the prawls and shrimps bother you, substitute some letter like A for prawls and
some letter like B throughout the exercise, making the statements read
John denies that none but A are B.
To make the exercise simpler, suppose the statement is John says that all A are B,
and Bill says that all B are A.
47. On a map is the statement that the scale of the drawing is l to 21, 120. The
mystery is: Why did the maker of the map choose such a peculiar number as 21, 120?
82
48. John has just graduated from college as a chemical engineer. In his senior year
he made a special study of the use of steel in skyscrapers. He is a quiet, industrious
young man and never tries to impress others with his brilliance. He is offered his
choice of the following jobs. If his choice depends only on the data given here,
which job should be accept?
a) A job with a chemical coiſ pany that specializes in various kinds of steel. The
first few years he must work in the sales department to become familiar with the
firm's customers. What he does after that depends on his success as a salesman.
b) Pesearch work in the laboratory of a chemical company. The company will let
him do such experiments as he is interested in, but if he does not produce something
valuable in a few years he will be dismissed.
c) A job with a firm that specializes in steel for large buildings and bridges. For
the first two years he must work with men who erect the structures, and then he will
be put in charge of the planning of the buildings.
d) A job with a firm of architects who build only ultra-modern homes. This firm
wants a man who is familiar with the chemistry of air conditioning, the use of steel
in homes, and the ability to sell new ideas to customers.
49. A corporation is said to have a monopoly when it is the only company that can
manufacture certain goods or furnish some service. For example, the telephone company
in your town is very likely a monopoly. There is free competition when several com-
panies manufacture some article. This tends to make the article cheaper because all
the companies are eager to serve the customers.
...hen a monopoly exists, the government says that trade is restrained or hindered,
and so the government has passed many anti-inonopoly laws. But compeitition also
restrains trade; for example, if many competing automobile companies advertise their
goods, then I am tempted to buy an automobile instead of an electric refrigerator.
Hence competition also restrains trade.
The conclusion we reach is that if the government has laws against monopolies, it
should also make laws against free competition. Then eventually no company would
have any monopoly of any business, and no company would have any competition in its
line of business .
50. In many advertisements we see pictures of men or women before and after taking
some certain me decine or treatment. On what general principles are such arguments
based? "...hat assumptions are made? What errors in reasoning may be involved?
5l. hen discussing Heron's formula for the area of a triangle, Jim checked part
of his work by adding s—a, S -b, and 8 -c. The sum should equal s, which represents
half the perimeter. Jim was then asked to prove that
l. (s – a) + (s – b) + (s – c) = s
He wrote the above equation, calling it step l in his proof. He then transposed the
terms – a , – b , and - c, and wrote 3s for 8 + s + 3. Hence his step 2 was
2. 3s = S + a + b + c
Then he substituted 2s for a + b + c in the right member, and had for step 3:
3. 3s = s + 2s which is certainly true.
Henry, who was watching this work, claimed that Jim had made a serious error. He said,
"In step 1, Jim assumed the very thing that he is trying to prove. And in the last
step he tells us that 3s = 8 + 2s. Of course 3s = S + 2s but that is not what
he was asked to prove. Further, '3s equals 3s even if step 1 is not true. 3s equals
3s at any time, any place, anywhere. That proves nothing."
52. A small business company cannot spend money on advertising because it does not
have the money to spend. A large business company should not spend money on adver-
tising because that gives it an unfair advantage over the small companies that do not
have the money. Hence neither a large nor a small company should spend money for ad-
vertising.
8 3
53. Jim was taking an examination. Some of the questions were:
(a) Some A are B. Can we say that some B are A?
(b) No A are B. Can we say that no B are 3.7
(c) All A are B. Can we say that all B are A'.
(d) Some A are not B. Can we say that some B are not A?
Since these questions bothered Jim, he thought he would make use of the geometry
that he had studied, and he changed question (a) to:
Some rectangles are squares. Can we say that some squares are rectangles?
He thought the answer to this ought to be ino; but when he drew the Euler circles
he thought the answer should be Yes.
For question (b) he tried the statements:
No squares are triangles. Can we say that no triangles are squares?
Here the answer was Yes, and this agreed with the answer that he got by drawing
the Euler circles.
Study questions (c) and (d) in the same way by using geometric analogies.
54. Crossing off the sixes in the fraction #——É must be wrong because if
we could cross off the sixes, then we could also cross off the fives ; in that case
the value of the fraction would be l. Since the fraction does not equal l, it is
wrong to cross off the sixes.
55. When asked for the converse of the statement "If a number ends in 5, its
square ends in 25" Dick answered, "If a number ends in 25, then its souare root
ends in 5". Henry claimed that this was wrong, and that the correct converse is:
If the square of a number ends in 25, then the numbers ends in 5.
56. While Mrs. Jones was driving her car, she remarked to her friend, "The driver
in the car behind us is very erratic in his driving. Cometimes he drives very fast
and is close behind me. Cther times he drives slowly and gets far behind me. I pre-
fer to drive at a steady speed all the time."
57. What conclusion can you draw from the fact that the prices of the goods at the
Good-Trade Stores are the same as at the Buy-Low Stores but more people trade at the
first one than at the second one?
58. I had three cards face down before me on the table. I knew that one of them
was the Ace pf Spades. I turned up the first one; it was the Queen of Clubs. Then
I turned up the second one; it was the nine of diamonds. So, without looking at the
third one, I knew it was the Ace of Spades.
59. "Chicago is located on the shores of Lake Michigan and is the third largest
city in the United States. Cities located on shores of large lakes or other large
bodies of water always outgrow other cities, and soon become the largest in the
country" said Tom in his geography class.
"But", answered Jim "What about New Orleans and Pittsburg?"
60. Mr. Dodd was addressing an audience which consisted mostly of portrait paint-
ers, sculptors, and interior decoraters. He had heard that such people disliked
mathematics, and someone had mentioned to him once that painting and mathematical
ability seldom went together. Hence in one part of his speech he said, "To those
who are mathematically minded we shall leave the work of constructing our ugly sky-
scrapers, our soul-killing factories, our man-killing battàeships, and destroyers of
civilization. We shall paint the beauties of nature, the colors of the rainbow, and
express ideals in marble and stone."
6l. When a manufacturer puts on a package of coffee the date when the package left
the factory, what assumptions is he making?
62. In what kind of reasoning (indirect argument, deductive reasoning, and so
forth, .....) do we use the assumption: Because A and B are alike in certain ways,
they are alike in other ways.
63. Granted that "Mr. Winkler would have bought a Linco Car if he had $1200" does
it follow that "Mr. Winkler did not have $1200, if he did not buy a Linco Car" 2
64. At an afternoon tea Amanda Turner said that she always preferred a silver tea
pot to a china one because a silver tea pot keeps the tea hot longer. "As soon as
you have poured tea into a silver pot", she said, "the pot is so hot that you can't
touch it."
65. In the last twelve months the profits of the company have increased 30 % but
the workmen have gained only a 3 } increase in wages. (This statement was taken from
a newspaper in 1936.) What conclusion would one naturally make from this statement?
How would you find out if your conclusion was correct or not?
Page Page
Ad hominem argument 57 Indirect proofs 34
Analogies o 54 Inductive method © 51
Appeals to prejudice 57 Insufficient data 53
Arguing in a circle tº 51 Interpreting data © 66
Assumptions 3, 20, 26 Inverses 38
Authorities 54
Major premise 14
Comparisons 62 Minor premise Q 14
Conclusion ll, 14
Contrapositives 40 Necessary conditions 44
Converses 31, 39 Non-causal relations 56
Deductive method 50 Pooh-poohing the question 57
Defining the problem 6l Postulational systems 25, 30
Definitions 18, 25 Proof 17, 22, 29
Denying the conclusion 51 Propaganda 73
Euler circles 67 Relations between proposi-
tions 42
Hasty conclusions 53
Hasty generalizations 9, 53 Scientific method 49
Hypothesis ll, 61 Sufficient conditions 45
Syllogism 17
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