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BYHAM, F red erick Charles, 1935INDIRECT PROOF IN GEOMETRY FROM
EUCLID TO THE PRESENT.
The Ohio State University, P h.D ., 1969
Education, theory and practice
University Microfilms, Inc., Ann Arbor, Michigan
INDIRECT PROOF IN GEOMETRY
FROM EUCLID TO THE PRESENT
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of the Ohio State University
<•
By
Frederick Charles Byham, B.S. in Ed., M.S.
The Ohio State University
1969
Approved by
Adviser
Department of ]
PLEASE NOTE:
Not original copy. Several pages
have indistinct print. Filmed as
received.
UNIVERSITY MICROFILMS
•ACKNOWLEDGMENT
The author gratefully acknowledges the assistance and
counsel given so generously by Professor Nathan Lazar.
He
is also indebted to Harold P. Fawcett, Professor Emeritus,
for his guidance during the writer's earlier graduate work
in mathematics education.
To his wife, Patricia, the author expresses sincere
appreciation for her encouragement and patience.
11
VITA
April 2, 1935............
Born - Meadville, Pennsylvania
1957•••••••••••••••••••••# B.S. in Ed« , Edinboro St^t6
College, Edinboro, Pennsylvania
1957-1962................
Teacher, Dunkirk High School
Dunkirk, New York
1963«#•••••••••••••♦•••••• M.S., Clarkson College of
Technology, Potsdam, New York
1963-1964................
Instructor, Department of
Mathematics, The Ohio State
University, Columbus, Ohio
1964-1969...... *.......... Assistant Professor, Department
of Mathematics, State
University College at Fredonia,
Fredonia, New York
FIELDS OF STUDY
Major Field:
Education
Studies in Mathematics Education.
P. Fawcett and Nathan Lazar
Professors Harold
Studies in Higher Education. Professors Earl W.
Anderson and Everett Kircher
Studies in Mathematics. Professors Robert J. Bumcrot,
Norman Levine, and Earl J. Mickle
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS..................................... ii
VITA............... '.................................iii
Chapter
I. INTRODUCTION..............................
1
Statement of the Problem
Importance of the Problem
Scope and Limitations of the Study
Outline of the Remainder of the Study
II. THE NATURE OF IMPLICATION................
10
Introduction
Material Implication
Strict Implication
Entailment
Summary and Recommendations
III. ON THE CONTRADICTORY OF AN
IMPLICATIVE STATEMENT......................41
Introduction
Contradictory Statements
Contraries
Contradictory of an Implicative
Statement
Summary
IV. DEVELOPMENT OF INDIRECT PROOF
IN GEOMETRY TEXTBOOKS....................
Introduction
Giving a Definition or. Analysis of
Direct Proof
Showing Examples of Indirect Proofs
Stating Schemes for Indirect Proofs
Contrasting Indirect Proof with
Direct Proof
Giving a Precise Definition of
Indirect Proof
Summary
.
51
V. LITERATURE EXCLUSIVE OF GEOMETRY
TEXTBOOKS......................
109
Introduction
Definitions of Indirect Proof
Importance of Indirect Proof
Dissatisfaction with Indirect Proof
Suggestions on Teaching Indirect Proof
Summary
VI. PROPOSALS FOR INDIRECT PROOF..................
138
Introduction
Inconsistent Statements
Method of Inconsistency
Method of Contradiction
Method of Contraposition
Summary
VII. INDIRECT PROOF IN EUCLID'S ELEMENTS...........
184
Introduction
Illustrations of the Method of
Inconsistency
Illustrations of the Method of
Contraposition
Summary
VIII. SUMMARY AND RECOMMENDATIONS...................
194
Summary of the Study
Recommendations for Teaching Indirect Proof
Recommendations for Further Study
APPENDIX
A ..........
202
B..........................
204
C..........................................
207
D .........................
208
E ...........
209
F ........................
210
v
6
211
BIBLIOGRAPHY...................
vi
213
CHAPTER I
INTRODUCTION
Statement of the Problem
In recent years the mathematics curriculum has
undergone such rapid changes that the situation has been
described as "The Revolution In School Mathematics."
Modern mathematics has placed a new emphasis on the postulational structure of mathematical subject matter which
"lends emphasis to the need for clearer understanding of
the basic techniques used in valid patterns of deductive
thinking ."1
It is to this need as it pertains to Indirect
proof that this study is directed.
The usual manner in which students learn how to
prove a theorem is by imitation, they study model proofs
and then try to arrange their proofs to follow the patterns
exhibited by the model proofs .2
The following statement
from the Report of the Cambridge Conference points out the
inadequacy of'this approach:
"it is hardly possible to do
^Charles H. Butler and P. Lynwood Wren, The Teaching
of Secondary Mathematics (*J-th ed.; New York: McGraw-Hill
Book Co., 1965), p. 70-.
2Myron F. Rosskopf and Robert M. Exner, "Modern
Emphasis in the Teaching of Geometry," The Mathematics
Teacher, L (April 1957) t P» 27^.
1
anything in the direction of mathematical proofs without
the vocabulary of logic and explicit recognition of the
3
inference schemes."^
A survey of the literature indioates that indireot
proofs frequently present difficulty to the student and the
teacher.
This difficulty with indireot proof by students
and teachers has resulted in a continuous decrease in the
number of indirect proofs used in plane geometry.
The
indirect method was used by Euclid to prove eleven of the
ip
forty-eight theorems in Book 1.
In current geometry
programs teachers, too often, use indirect proof for only
one or two theorems.^
The authors of a recently published calculus text
list four points which have proved troublesome to their
students:
In our experience they often have difficulty with
(a) proof by contradiction
(b) implication
(c) finding convenient equivalent forms for the
denial of complicated statements, and
(d) reasoning from false statements.°
^Cambridge Conference on School Mathematics, Goals
for School Mathematics (Boston: Houghton Mifflin Co..
t& tttpryr.-----------
Clifford B. Upton, "The Use of Indirect Proof in
Geometry and in Life," Fifth Yearbook (Washington, D. C.:
National Council of Teachers of Mathematics, 1930), p. 10^.
^Donovan A. Johnson and Gerald R. Rising, Guidelines
for Teaching Mathematics (Belmont, Calif.: Wadsworth
Publishing Co., 1967), p. 79.
^Stoughton Bell et al., Modern University Calculus
(San Francisco: Holden-Day, Inc., 196^), p. 1.
3
It is important to note that all four of these arise in the
normal course of constructing indireot proofs of the
reductio ad absurdum type.
The problem of this study is as follows:
1.
To establish the logic required in the methods
of indirect proof.
2.
To seek methods of teaching indirect proof
which are logically correct, yet simple
enough that they may be readily understood
by students of high school plane geometry.
Importance of the Problem
The development of logical thinking has been an
objective of the teaching of mathematics from the
Pythagorean School (500 B.C.) to the present.?
The Com­
mission on Mathematics in its report published in 1959>
stated that one of the principal objectives of mathematics
in general education is to develop "an understanding of
the deductive method as a method of thought.
This includes
the ideas of axioms, rules of inference, and methods of
proof."®
i
?Myron P. Rosskopf, "Nongeometric Exercises in
Geometry", Twenty-second Yearbook (Washington, D.C.: Nation­
al Council of teachers of Mathematics, 195*0» P. 283.
^Report of the Commission on Mathematics (New York:
College Entrance 'Examination Board, 1959)* P* H •
Today, as in years past, a major objective of the
high school geometry oourse is to develop an understanding
of the nature of proof.
The National Committee on
Mathematical Requirements in its 1923 report stated that a
principal objeotive of instruction in demonstrative geome­
try is "to develop understanding and appreciation of a
deductive proof, and the ability to use this method where
it is applicable."9
Upton, in 1930, stated that "our great
aim in the tenth year is to teach the nature of deductive
proof and to furnish pupils with a model for all their
life thinking ."10
Butler and Wren in their current text­
book on the teaching of mathematics state that "the prime
objective in the geometry of the senior high school is to
build a feeling for a deductive system in which the valid­
ity of conclusions is accepted only when these have been
established by formal reasoning from bases already accepted
or established ."11
Can a complete understanding of the nature of
deductive proof be achieved without an understanding of the
nature of indirect proof?
There are two forms of deductive
^National Committee on Mathematical Requirements,
The Reorganization of Mathematics In Secondary Education
(Boston: Houghton toifflin Co., 1923)* p. 3^.
10Upton, op. clt.t p. 132.
11Butler and Wren, op. clt.. p. ^58.
5
reasoning, direct reasoning and indirect reasoning.
Indirect reasoning is a form of deductive reasoning which
is extensively used by nearly everyone.
In fact, Jevons
has estimated that about one half of all our reasoned
conclusions are arrived at through indirect reasoning.12
It would then seem to necessarily follow that if students
are to gain an understanding of the nature of deductive
proof, they must achieve a basic understanding of the
nature of indirect proof.
H. C. Christofferson expressed
this need for emphasis to be placed on indirect proof as
follows:
If geometry is to be taught largely
because of its inherent possibilities to pro­
vide experiences in the science of reasoning,
...then surely it is a mistake to omit or
neglect to emphasize the method of the indirect
proof. Much of the reasoning which we do in
life is indirect; therefore much of the value of
geometry must be in its treatment of Indirect
proof,13
The need for a better understanding of the nature of
an indirect proof has been underscored by both the
12W. Stanley Jevons, The Principles of Science
(London: Macmillan and Co., 1920), p. 32.
13h. C. Christofferson, Geometry Professionalized
for Teachers (Oxford, Ohio: By the author, 1933)* P# i38.
6
Secondary School Curriculum Committee1^ and the Commission
i5
on Mathematics. ^
Scope and Limitations of the Study
This study will be limited to an analysis of in­
direct proof as it pertains to the high school geometry
course.
The reason for this limitation is that geometry
is the first mathematics course in which students will be
expected to understand and to use methods of indirect proof.
The Report of the Commission on Mathematics states that:
Although the Commission believes that
algebra also should be taught so as to give sound
training in deductive methods, it is likely that
geometry will continue to be the subject in
which the student will receive the most satis­
factory introduction to thinking in terms of
postulates and proofs, undefined terms and
definitions. °
Indirect proofs in Euclid's "Elements" will be
thoroughly analyzed since the "Elements" served as the text­
book for plane geometry for over 2000 years and has had a
tremendous influence on all textbooks in plane geometry.
^Report of the Secondary School Curriculum Committee
of the National Council of Teachers of Mathematics, "The
Secondary Mathematics Curriculum." The Mathematics Teacher.
LII (May, 1959), p. 407.
”
15
^Report of the Commission on Mathematics: Appendices.
op. cit., pp. 116-119.
l 6Ibid., p. 112.
Textbooks on plane geometry will be examined with emphasis
being placed on those published from 1955 to the present.
The year 1955 is chosen since studies by Nathan Lazar*^
18
and Gordon Glabe
have already analyzed the discussions
of indirect proof contained in plane geometry textbooks
published prior to 1955.
Books on the methods of teaching mathematics will
be investigated for suggestions and criticisms on the
teaching of the various types of indirect proof.
Logic
books will be searched for ways of clarifying the logic
used in indirect proofs.
Other sources such as mathema­
tics books, yearbooks, committee reports, periodicals, and
dissertations will be searched for information pertinent
to the problem.
It is hoped that through these sources there may
be evolved pedagogically sound methods of explaining the
logic involved in indirect proof so that indirect proofs
may be understood and appreciated by students in the high
school geometry course.
Outline of the Remainder of the Study
In Chapter II, the nature of implication will be
17
rNathan Lazar, "The Logic of the Indirect Proof in
Geometry," The Mathematics Teacher. XL (May, 19V7),
pp. 225 - 2W .
18
Gordon R. Glabe, "Indirect Proof in College
Mathematics" (unpublished Ph.D. dissertation, Graduate
School, The Ohio State University, 1955).
8
examined.
Since all mathematical theorems are implications
to prove a theorem involves establishing an implication.
This means that the nature of implication must be clearly
understood if the nature of indirect proof -is to be under­
stood and appreciated.
In Chapter II, it will be shown
that there is considerable disagreement among logicians as
to the meaning of implication.
Several different defini­
tions of implication, as proposed by logicians, will be
considered and analyzed.
Then a definition of implication
suitable for mathematics will be proposed.
Chapter III will be devoted to a consideration of
certain logical concepts which are used in various methods
of indirect proof.
tigated —
These logical notions will be inves­
(a) contrary statements, (b) contradictory
statements, and (c) the contradictory of an implicative
statement.
Chapter IV will be devoted to a critical study of
the discussions and explanations of indirect proof given
by authors of secondary school geometry texts.
Chapter V will be taken up with an investigation of
the pertinent literature on indirect proof, exclusive of
high school geometry texts.
In Chapter VI, methods of indirect proof which are
logically correct, yet simple to understand, will be
proposed.
Chapter VII will be directed to an analysis of
indirect proofs in "Euclid's Elements" in light of the
recommended methods of indirect proof proposed in Chapter
VI.
Illustrations of the proposed methods of indirect
proof will be selected from the "Elements."
Chapter VIII will contain a summary of the study.
It will also contain recommendations for the teaching of
the various methods of indirect proof and recommendations
for further related studies.
CHAPTER II
THE NATURE OF IMPLICATION
Introduction
This study is concerned with an analysis of the
nature of indirect proof.
Before an analysis of indirect
proof can be undertaken, the nature of implication must
be investigated.
An implication is expressed in either
the form "If p, then q" or "p implies q."
The connection
between "implication" and "indirect proof" may be ex­
plained as follows:
The relation of implication which binds
sentences together will play a central role in
our development of elementary mathematics. This
relation, as well as the notion of proof - the
means by which we establish an implication - con­
stitutes the principal subject matter of the
study of logic .1
In an indirect proof, then, it is an implication that is
being established.
Since every indirect proof involves
establishing an implication, it is necessary that a stu­
dent understand the nature of implication in order for
him to understand the nature of indirect proof.
Unfortunately the meaning of "p implies q" has been
steeped in controversy among logicians for over 2000 years.
1Leon Henkin et al., Retracing Elementary Mathema­
tics (New York: The Macmillan Co., 1962), p. k.
10
11
The definition of implication was a matter debated at such
lengths among the Megarians and Stoics that Callimachus,
librarian at Alexandria in the 2nd century B.C., said:
•'The very crows on the roofs croak about what implications
2
are sound."
There are two major interpretations of "p implies q"
1 ) "p implies q" is true in all cases except when
p is true and q is false.
2 ) "p implies q" is true only when q is deducible
from p.^
The term "material implication" is usually associated with
the first interpretation while the terms "strict implica­
tion" and "entailment" are usually associated with the
second interpretation.
In this chapter the notions of material implication,
strict implication, and entailment will be critically
analyzed as to their appropriateness as an explanation of
implication for secondary school mathematics.
Material Implication
The view taken by
books
the authors of most
college text­
in mathematics is that the truth of "If p then q" is
2
I. M. Bochenski,
Ivo Thomas (Notre Dame,
Press, 1961), p. 116.
A History of Formal
Ind.: University of
Logic, tr.
Notre Dame
^Boruch A. Brody, "Logic," The Encyclopedia of
Philosophy, ed. Paul Edwards (New York: The Macmillan
Company and the Free Press, 1967), Vol. V, p. 66.
12
completely determined by the truth values of p and q.^
"If p then q" is to be considered true in every case except
when p (the hypothesis) is true and q (the conclusion) is
false.
According to this approach:
"If 2 + 2
tt 4 then the moon is not made of green
cheese" is true since both the hypothesis and
conclusion are true.
"If 2 + 2
/£ 4 then the moon is not made of green
cheese" is true since in this example the hypothesis
is false and the conclusion true.
"If 2 + 2 ^ 4
then the moon is made of green cheese"
is true since in this example both the hypothesis
and conclusion are false.
But "If 2 + 2 = 4
then the moon is made of green
cheese" is false since in this case the conclusion
is false while the hypothesis is true.
The view that the truth of "If p then q" is deter­
mined solely on the basis of the truth or falsity of
p and q originated with Philo of Megara in the 4th century
B.C.^
In modern times this view was adopted by C. S. Peirce
(who was aware of the work of Philo) in his development of
list of college texts which contain a truth value
approach to implication is given in Appendix A.
^Czeslaw Lejawski, "History of Logic," The Encyclo­
pedia of Philosophy, on. cit.. Vol. IV, p. 519.
13
£
logic (1885)
and by Frege (1879) who was unaware that
7
this view had previously been proposed.
This was the
view of implication used by Whitehead and Russell (they
O
were aware of the work of Frege) in their "Principia
Matheraatica" in 1919 and since that time has become widely
accepted by authors of mathematics and logic texts.
When
"If p then q" is considered true for every possible com­
bination of truth values for p and q except when p is true
and q is false, it has become standard practice by logic
books to say that "p materially implies q," denoted by
P > q» and to refer to the implication as "material im­
plication" to distinguish it from other views of impli­
cation.
In 1921 E. Post and L. Wittgenstein independently
introduced a scheme called a "truth table" to display
systematically the possible combinations of truth values
for p and q (or whatever propositions are being consid­
ered).^
For example, the truth table for "p materially
^Collected Papers of Charles Sanders Peirce, ed.
Charles Hartshorne and Paul Weiss (Cambridge, Mass.: Belk­
nap Press of Harvard University Press, 1960),Vol.II,p.336.
^1. M. Bochenski, op. cit.. p. 311.
Q
Alfred North Whitehead and Bertrand Russell,
Principia Mathematica (2d ed., Vol. 1 ; Cambridge:
Cambridge University Press, 1963), p. viii.
•^E. L. Post, "Introduction to a General Theory of
Elementary Propositions," American Journal of Mathematics.
XLIII (1921), pp. 163 - 185, L. Wittgenstein, "Logishphilosophische Abhandlung, Einleitung v. Betrand Russell,"
Annalen der Nat -philosophie. XIV (1921), pp. 185 - 262.
implies q" is:
TRUTH TABLE FOR MATERIAL IMPLICATION
p
q
T
T
F
T
F
F
T
If
id
then q
T
F
T
F____________ T
An examination of the table shows that material
implication results in the following so-called "paradoxes
of material implication":
(1 ) A false proposition materially implies any
proposition.
(2) A true proposition is materially implied by any
proposition.
These paradoxes make it difficult for many people
to accept material implication as a definition for impli­
cation.
The authors of a recent mathematics text point
out the unwillingness of students to accept the paradoxes:
We sometimes express this by saying that a false
statement implies any statement. Students fre­
quently have difficulty at this point. 10
Another college level text expresses the difficulty in
this way:
If you object that it doesn't quite capture the
sense of "implies" or "if...then...," we agree
^George C. Bush and Phillip E. Obreanu, Basic
Concents of Mathematics (New York: Holt, Rinehart and
Winston, 1965), p. 7.
15
with you. The principal difficulty is that it
may connect unrelated propositions. You may very
well take the position that "2 = 1 implies that I
am the Pope" is neither true nor false but simply
meaningless. If so, then "p - » q" fails to be a
proposition at all for certain pairs of proposi­
tions p and q. The difficulties involved in this
position are enormous. If the application of
is restricted to certain related pairs of
propositions, we must state unambiguously which
pairs, and that is quite a task.1'
In mathematics "p implies q" is treated as being
synonomous with:
1 ) P is a sufficient condition for q
2 ) q is a necessary condition for p
3) q is deducible from p
*t) q is a consequence of p
5) q necessarily follows from p
6 ) p only if q .12
It may be noted that all six of these forms convey the
impression that there is necessarily a logical connection
between p and q.
Since a true material implication does
not require a logical connection between the hypothesis
and the conclusion, a true material implication may seem
queer, perhaps even ridiculous, when rephrased in one of
11
Nathan J. Fine, An Introduction to Modern Mathema­
tics (Chicago: Rand McNally and Co., 1965), p. 19.
12
See, for instance: J. Richard Byrne, Modern Ele­
mentary Mathematics (New York: McGraw-Hill Book Co.. 1966).
p. 58.
these six equivalent forms.
16
For example, consider the true
material implication "whales are mammals" implies
"2 + 2 s If,»
In mathematics this material implication may
be restated as:
1) "Whales are mammals" is a sufficient condition
for "2 + 2 a
k"
2 ) "2 + 2 s V 1 is a necessary condition for "whales
are mammals"
3 ) "2 + 2 a if" is deducible from "whales are
mammals"
"2 + 2 = if" is a consequence of "whales are
mammals"
5 ) "2 + 2 = if" necessarily follows from "whales
are mammals"
6 ) "Whales are mammals" only if "2 + 2 = if."
The following quotations support the view that the
sense in which "implies" is used in mathematics is not
captured in material implication:'
fThe paradoxes of material implication]] ^ bring out
the fact that the strictly truth-functional charac­
ter " ^ " prevents it from being a genuine implication
or entailment. It is not the case that (i) if a
statement is true then any statement implies it or
1^
■'For this paper, any phrase in brackets is the
present writer’s.
17
(ii) if a statement is false, then it implies
any statement.
What is absurd is to define materially implies as
we have done and then to forget the definition,
drop the qualification indicated by "materially,"
and thus think of implies as equivalent to entails.
These so-called 'paradoxical* consequences, as
Professor G. E. Moore has pointed out 'appear
to be paradoxical, solely because, if we use
"implies" in any ordinary sense, they are certainly
false. '15
The weak
material
cates it
with the
implication symbolized by 'D * is called
implication, and its special name indi­
is a special notion, not to be_confused
more usual kinds of implication. 1°
Material implication must not be confused with
inferential implication. The former, for example,
holds among all true sentences, which is not the
case, however, for the latter. The "if" of
ordinary English is closer to inferential than
to material implication.1?
The paradox Qof material implication} disappears,
however, if the reader dismisses from his mind
the prejudice in favor of the usual meaning of
the word "implication" and recognizes that, by
definition, we have made it denote something else
in the propositional calculus. The distinction
is recognized by calling the first formal and the
^Nicholas Rescher. Introduction to Logic (New York:
St. Martin's Press, 196*t), p. 192.
15
■'L. Susan Stebbing, A Modern Elementary Logic,
rev. C. W. K. Mundle (5th ed.; New York: Barnes and Noble,
m 3 ) , P. 139.
1
Irving M. Copi, Symbolic Logic (New York: The
Macmillan Co., 1959), P. '7.
17
fJ. M. Bochenski, A Precis of Mathematical Logic,
trans. Otto Bird (Dordrecht, Holland: D. Reidel Publishing
Co., 1959), P. 27.
latter material implication.
18
^®
The so-called paradoxical propositions both of
material and of strict implication are in terms
of the respected systems not paradoxes at all.
It is only when we are told that the symbols 3
and — * represent what is commonly understood by
the word "implication," that these propositions
appear paradoxical.19
Conditional sentence forms (and sentences) are
often called "implications";1 an unfortunate usage
which has led all too many to mistake 'o ', a
conjunction in the grammarians' sense of the
word "conjunction1!,' for the verb 'implies. '20
It might well have its roots in the propositional calculus of "Principia Mathematica" and
in the assumptions, now generally discarded,
that all necessary propositions are truth
functional tautologies, and that the horseshoe
of "Principia Mathematica" designates the
converse of the relation of logical deducibil­
ity.^
Nobody now accepts - if indeed anyone ever
made - the identification of the relation
symbolized by "r>" with the relation which
Moore called "entailment."22
The above quotations point out that many logicians
18
Morris R. Cohen and Ernest Nagel, An Introduction
to Logic and Scientific Method (London: Routledga and
Kegan Paul Ltd., 1934J, p. 127.
19
■'Everett J. Nelson, "Intensional Relations,"
Mind. XXXIX (Oct. 1930), p. 448.
20
Hugues Leblanc, Techniques of Deductive Inference
(Englewood Cliffs, N. J., Prentice Hall, 'inc., 19667,"p.' 4.
21
S. Korner, "On Entailment," Proceedings of the
-------Aristotelian Society. XLVII (1947), P . T 48 .
22
P. F. Strawson, "Necessary Propositions and
Entailment-Statements," Mind. LVII (April, 1948), p. 186.
19
believe that "p materially implies q" does not have the
same meaning as "q is deducible from p."
It might also
be pointed out that Professor Nathan Lazar, Ohio State
University, for more than thirty years in his course
"Philosophy and Logic for Teachers of Mathematics" has
refused to accept material implication on the grounds that
it does violence to the process of deductive reasoning
that is used in mathematics. ^
The defenders of material implication are quick to
point out that the paradoxes of material implication cause
no logical difficulties because they are inferentially
useless.
For example, Bertrand Russell states that:
Whenever p is false, "not-p or q" is true, but is
useless for inference, which requires that p
should be true. Whenever q is already known to
be true, "not-p or q" is of course also known to
be true, but is again useless for inference, since
q is already known, and therefore does not need
to be inf erred. $>•
In mathematics, it is inference which we are concerned
with and if the paradoxes are inferentially useless, why
challenge the intuitions of our students by adopting a
definition of implication that results in these so-called
^Related to the writer by Professor Lazar in a
conversation on December 13, 1968.
^Bertrand Russell, Introduction to Mathematical
Philosophy (London: George”Alien and Unwin, Ltd., 1919),
P. 153.
20
paradoxes?
Max Black answers this question In this manner:
It is also necessary to symbolize the relation
which holds between two propositions p and q when
the second can be deduced from the firstj this is
expressed by saying p implies q, or, in symbols,
pjq,
If however the word implies is used with
this meaning it is found very difficult to develop
a proposltlonal calculus; therefore a modified
definition is adopted and p o q is understood
to mean "either p is false or q is true" which is
equivalent to "it is false that cp is true and q
false.
Thus, the idea of having "p implies q" mean "q is deducible from p" was given up and material implication adopted
so that it would be easier to develop a propositional
calculus.
J. Barkley Rosser in his text "Logic for Mathemati­
cians" gives the following advice to mathematicians:
He should not forget that his intuition is the
final authority, so that, in case of an irrecon­
cilable conflict between his intuition and some
system of symbolic logic, he should abandon the
symbolic logic. He can try other systems of sym­
bolic logic, and perhaps find one more to his liking,
but it would be difficult to change his intuition.26
Thus, why insist that students accept (or at least use)
a definition of implication which is contrary to their
^ M a x Black, The Nature of Mathematics (New York:
Harcourt, Brace and Co., 1934), P.
J. Barkley Rosser, Logic for Mathematicians (New
York: McGraw-Hill Book Co., 19^3), p. 1'l•
21
intuitions when it is not absolutely necessary?
The truth value of a material implication "If p,
then q" is defined to be completely determined by the
truth values of p and q,
But consider the mathematical
2
statement - "If x is an even integer, then x is an even
integer."
Is the hypothesis, "x is an even integer,"
2
true or false? Is the conclusion, "x is an even integer,"
true or false?
The statement "x is an even integer" is neither true
nor false, but any instance of this statement, such as
"2 is an even integer" or "3 is an even integer" is either
true or false.
Whitehead and Russell used the term
"propositional function" to refer to a statement containing
a variable and such that it becomes a proposition (a
statement which is true or false) when a value is substi27
tuted for the variable. '
A material implication expresses a relationship
between propositions but implications in mathematics are
concerned with relations between propositional functions
not propositions.
The following two quotations support
this claim:
Relations
may be true or
class a, and a
together imply
between propositional functions
false. Thus x is a member of the
is contained in the class b,
that x is ab, is true. Here the
27
'Whitehead and Russell, op. cit.. p. 1if.
-
implication is true, and we do not say that the
functions are. The kind of implication we use in
mathematics is of the form: "If 0x is true, then
fx is true"; that is, any particular value of x
which makes 0x true aleo makes Y x true.^®
Pure mathematics is the class of all proposi­
tions of the form "p implies q," where p and q are
[propositional functiondJ29 containing one or more
variables, the same in the two propositional
functions, and neither p nor q contains any
constants except logical constants.
Authors of textbooks commit an obvious logical error
when they define implication as a relation between pro­
positions, say nothing about the corresponding relation
between propositional functions, and then violate their
definition of implication by giving statements such as 2
"If x is even then x is even" - as examples of implica­
tions.
The undesirability of glossing over the distinction
between propositions and propositional functions has been
pointed out by Bertrand Russell:
For clear thinking, in many diverse directions,
the habit of keeping propositional functions
sharply separated from propositions is of the
28
Philip E. B. Jourdain, "The Nature of Mathematics,"
The World of Mathematics (Vol. I; New York: Simon and
Schuster, 19%), p. 70.
^Russell used "propositions" instead of "proposi­
tional functions." It was not until later that he
distinguished between propositions and propositional
functions. See the previous paragraph.
•^Bertrand Russell, Principles of Mathematics (2nd
ed.; New York: W. W. Norton and Co., 1937), P« 3.
23
utmost importance, and the failure to do so in
the past has been a disgrace to philosophy.-5'
Authors of geometry t.exts who have committed this
xp
error are Clarkson, Douglas, Eade, Olson, and Glass^ ;
Henderson,
xx
Pingry, and Robinson^; Jurgensen, Donnelly,
and Dolciani^; and Rosskopf, Sitomer, and Lenchner.^
For the reasons cited, this writer rejects material
implication as constituting a suitable definition of im­
plication for mathematics.
In fact, this writer agrees
with Anderson's and Belnap's statement that material
implication "is no more a kind of implication than a
blunderbuss is a kind of buss."-^
Strict Implication
It was the belief of C. I. Lewis that material
implication with its resulting paradoxes does not capture
^Russell, Introduction to Mathematical Philosophy,
p. 1 61.
^2Donald R. Clarkson et al.. Geometry (Englewood
Cliffs, N. J . : Prentice-Hall, Inc., 1965), P* *+9.
^Kenneth B. Henderson et al., Modern Geometry: Its
Structure and Function (New York: McGraw-Hill Book Co..
1962), p. 67.
^ R a y C. Jurgensen et al.. Modern Geometry (Boston:
Houghton Mifflin Co., 196$), p. 5W .
■^Myron F. Rosskopf et al., Modern Mathematics:
Geometry (Morristown, N. J.: Silver Bur&ett Co., 1966),
p. 12.
^ A l a n Ross Anderson and Nuel 0. Belnap, Jr.,
"The Pure Calculus of Entailment", Journal of Symbolic
Logic. XXVII (March 1962), p. 21.
the "real11 meaning of "implies.11 In 1918, Lewis proposed
a view of implication which he called "strict implication"
which (he claimed) made "p implies q" synonymous with
V? The definition he gave for
"q is deducible from p."^f
strict implication is "»p strictly implies q» is to mean
'It is false that it is possible that p should be true and
q false.'"-'
Restating this:
"p strictly implies q"
means "It is impossible for q to be false while p is true."
An alternate definition of strict implication may
be formulated using the notion of consistent propositions.
For two propositions p and q, "p and q are consistent"
means "it is possible that p and q are both true."-"
The
idea of "possibility" was taken as an undefined term so
it is important to note that it was intended that two
propositions may be considered consistent even though one
or both of the propositions are false.
Thus "Grass is
red" and "Grass has color" are consistent even though
"Grass is red" is false.^
The propositions "Grass is red"
and"Grass is not green" illustrate that two propositions
Clarence Irving Lewis and Cooper Harold Langford,
Symbolic Logic (2d ed.; New York: Dover Publications, 1959),
p. 122.
3 8 I b i d . . p. 12if.
3^Daniel J. Bronstein, "The Meaning of Implication",
Mind. XLV (April, 1936), p. 162.
^Lewis and Langford, op. cit.. p. 156.
25
may both be false, yet still be consistent.
Two propositions which are not consistent are said
to be inconsistent.
as:
Now strict implication may be defined
"p strictly implies q means that p is inconsistent
with the denial of q."4^
The following example shows that material implication
holds in situations where strict implication does not.
"Roses are green" materially implies "Sugar is sweet"
since "Roses are green" is false; but "Roses are green"
does not strictly imply that "Sugar is sweet" since "Roses
are green" and "Sugar is not sweet" are not inconsistent
with one another.42
But whenever p strictly implies q,
it is impossible to have p true and q false, so it is
not the case that p is true and q is false which means that
p materially implies q.
Thus, if p strictly implies q then
p materially implies q but if p materially implies q then
p does not necessarily strictly imply q.
The concept of strict implication may have been
first formulated by Chrysippus of Soli in the 3rd century
B. C.
The writings of Sextus Empiricus tell us that the
Stoic-Megarian logicians had as one of their views of impli­
cation the following:
And those who introduce the notion of connexion
41 Ibid., p. 1
42Ibid., p. 15k*
26
say that a conditional Is sound when the
contradictory of its consequent is incom­
patible with its antecedent.^3
It may well be that their use of incompatible is
synonomous with Lewis' use of inconsistent.
Just as material implication had two so-called
"paradoxes" so does strict implication.
The paradoxes
of strict implication are:
(1) An impossible proposition strictly implies any
proposition.
(2) A necessary proposition is strictly implied by
any proposition.
Lewis considered a proposition, p, to be impossible if
"p is logically inconceivable" and p is necessary if
"It is not logically conceivable that p should be false.
For example, the statement "On a single roll of a pair of
dice the sum of the spots is 13" is impossible since one
can deduce that the sum of the spots connot exceed 12.
The
statement "On a single roll of a pair of dice the sum of
the spots is less than 13" is necessary since one can
deduce that it could not be otherwise.
Necessary proposi­
tions and impossible propositions are statements whose truth
or falsity can be known a priori.
Intuitively it may be seen how these paradoxes result
^William Kneale and Martha Kneale, The Development
of Logic (London: Oxford University Press, 1962), p. 1.29.
^TLewis and Langford, on. cit.. p. 161.
from the definition of strict implication as follows:
If a proposition is impossible, such as
"p and not-p", it is false on purely logical grounds
and hence it will not be logically possible for it
and any other proposition to be both true, i.e.,
an impossible proposition is inconsistent with any
other proposition; so an impossible proposition
strictly implies any proposition.
On the other hand, a necessary proposition like
"p or not-p" is true on solely logical grounds and
so its contradictory is false on logical grounds
and hence inconsistent with any other proposition;
so a necessary proposition is strictly implied by
any proposition.
The question of whether an impossible proposition
entails any proposition is important to mathematics bechuse
in the reductio ad absurdum type of indirect proof, the
argument starts from premises which are later shown to be
impossible.
Von Wright emphasizes the importance in re­
jecting this "paradox" of strict implication as follows:
It should be observed, however, that although the
'paradoxes1 are inferentially useless, strict
Implications with impossible antecedents are not.
Such implications, on the contrary, have a char­
acteristic and important function for purposes of
inference. This is in so-called inverse proof or
proof by reductio ad absurdum. In such proof we
28
demonstrate a proposition p by showing that its
denial ~ p entails (and thus also strictly implies)
the denial of some proposition, the necessity of
which is already established or taken for granted,
from which we then modo tollendo tollens conclude
that p itself is necessary ( and *yp impossible ).
If one could not discriminate between propositions,
which are entailed, and propositions, which are not
entailed, by impossible proposiitions, inverse
proof could not be validly conducted. For, it is
essential to such proof that an impossible propos­
ition should entail exactly such and such conse­
quences, and not anything whatever.
Since, on the view that entailment is the same as
strict implication, one has to say that impossible
propositions entail anything and everything, one
cannot discriminate between what is and v/hat is
not entailed by impossible propositions. Thus this
view of entailment cannot account for inverse proof.
And this, it would seem, is enough to wreck it.^5
Entailment
One of the logicians who was convinced that strict
implication did not give the "real,,meaning of implication
was Everett J. Nelson. In a 1930 article in Mind. Nelson
stated that:
These paradoxes, corresponding to those of material
implication, convince me that Strict implication is
not what is ordinarily meant by "implication". It
is too broad, for it includes cases which are not
implications in any ordinary sense; which fact means
that its "definition" is not convertible. In other
words, /-though we may say that if p implies q then
p - q^ is impossible, we must not say that if p - q
is impossible, p implies q.^7
^Nelson uses p - q to represent p and not-q.
^Nelson, op. cit., p.
29
The paradoxes that Nelson refers to are that (1 ) an
impossible proposition strictly implies any proposition
and, (2) a necessary proposition is strictly implied by any
proposition.
It is Nelson's view that "p implies q" is an intensional relation, a relation holding between the meanings
of the propositions p and q and not their truth-values.^®
Nelson defines a relation which he calls entailment as
"'p entails q' means that p is inconsistent with the
propositional function that is the proper contradictory
of q"**9 ; i.e., "p entails q" means that p is inconsistent
with not-q.
The terra "consistent" is taken as an undefined
term in Nelson's intensional logic.
- -
It is interesting to note that material implication
may be defined as:
"p materially implies q" means that
p is inconsistent with not-q; that strict implication may
be defined as "p strictly implies q" means that p is
inconsistent with not-q; and that Nelson's notion of entailraent is defined as:
"p entails q" means that p is
inconsistent with ftot-q.
The differences then between
material implication, strict implication, and Nelson's
^Ibid., p. 450.
^9Ibid.. p. 445.
30
entailment lie in their different interpretations of the
meaning of the term "inconsistent”,
In material implication propositions p and q are
considered inconsistent if p and q are not both true, i.e.,
if either p or q is false.
Thus a false proposition is
inconsistent with any proposition, including itself.
Since a false proposition is considered inconsistent with
any proposition, a false proposition materially implies any
proposition.
Also when q is a true proposition then not-q
is a false proposition so not-q is inconsistent with any
proposition p, hence a true proposition q is materially
implied by any proposition p.
Some examples of this
interpretation of inconsistent are:
(1 ) "All triangles are isosceles" is inconsistent
with "all triangles are not equilateral" since
"all triangles are isosceles" is false, so "all
triangles are isosceles" materially implies that
"all triangles are equilateral".
(2) "2 + 2 ^ 4" is inconsistent with "the moon is
made of green cheese" since both "2 + 2
4 4"
and "the moon is made of green cheese" are false,
so "2 + 2
4 4" materially implies that "the moon
is not made of green cheese".
(3) "1 + 1 s 2" is inconsistent with "2 + 2 ^ 4"
since "2 + 2 ^ 4" is false, so "1 + 1 =2 materi-
31
ally implies ”2 + 2 = if."
In this approach, inconsistency is determined solely on
the basis of the truth-values of the propositions.
In strict implication, propositions p and q are
considered inconsistent if it is logically impossible that
p and q are both true.
In this interpretation either p or
q or both may be false yet still be consistent.
For exam­
ple, "all mammals are three-legged" and "all horses are
three-legged" are both false but still consistent.
While
"all mammals are three-legged" and "all horses are not
three-legged" are considered inconsistent since it is
logically impossible for both to be true.
The above
approach, as discussed earlier, does result in the two
paradoxes:
(1) An impossible proposition strictly implies
any proposition since it is inconsistent with every
proposition including itself and (2) A necessary proposition
is strictly implied by any proposition since its contradic­
tory is inconsistent with every proposition.
For example:
"squares are triangles" strictly implies "elephants are
fish" since the impossible proposition "squares are tri­
angles" is inconsistent with every proposition; "elephants
are fish" strictly implies "2 + 2 a if" since the contradic­
tory of the necessary proposition "2 + 2 = if" is inconsistent
with every proposition.
Nelson argued it was false that an impossible
32
proposition was inconsistent with any proposition whatsoever.
He considered two propositions as inconsistent only if
there is something in their meanings such that the truth
of one necessitates the falsity of the other.
nothing in the meaning of *2 + 2
V
"There is
which is, or whose
consequences are, inconsistent with the meaning of every
proposition.
In general terms, p may be impossible but
nevertheless consistent with some proposition q . " ^
"The
4' and '3 + 3 / 6 1 are false and
51
impossible yet consistent,"^
These propositions would be
propositions '2 + 2
considered inconsistent under material and strict impli­
cation interpretations.
Thus "2 + 2
implies "3 + 3 = 6" and "2 + 2
"3 + 3 = 6".
4 k" materially
/ 4" strictly implies
The propositions "2 + 2 = 5” and "2 + 2 = 6"
are false, impossible, and inconsistent in Nelson’s view.
Thus "2 + 2 = 5"materially implies, strictly implies, and
entails "2 + 2 ?£ 6".
The following is a summary of the above discussion:
(a) In material implication a false proposition is
considered to be inconsistent with every
proposition.
(b) In strict implication a false proposition is
not considered to be inconsistent with every
50Ibid., p. MfO.
51 Ibid., p.
kk7.
33
proposition but an impossible proposition is
considered to be inconsistent with every
proposition.
(c) In Nelson's entailment neither a false nor an
impossible proposition is considered to be
inconsistent with every proposition.
A second approach to entailment, even more restric­
tive than Nelson's approach, is based upon the observation
that p strictly implies q is of two types.
In Type I, p
strictly implies q although q is not a necessary proposi­
tion and p is not an impossible proposition.
In Type II,
p strictly implies q because either p is an impossible
proposition or q is a necessary proposition.
It is, of
course, the Type II implications which lead to the paradox­
es, thus one can avoid the paradoxes by defining entailment
so as to exclude Type II strict implications.
P. F.
Strawson has followed this approach and has defined "p
entails qM to mean "p and not-q is impossible and neither
p nor q, nor neither of their contradictories are necessary^
Thus, in effect, he has restricted entailment to contin­
gent propositions, propositions which are neither necessary
nor impossible.
52
^ Strawson, loc. cit.
34
A third approach to entailment is to treat entails
as an undefined term although it is intended to convey the
same meaning as when the term "entails” was first used
(in logic) by G. E. Moore in 1920.
Moore explained entails
as:
We require, first of all, some term to express
the converse of that relation which we assert to
hold between a particular proposition q and a
particular p, when we assert that q follows from
or is deducible from x>. Let us use the term
"entails" to express the converse of this relation.
We shall then be able to say truly that "p entails
q", when and only when we are able to say truly „
that "q follows from p" or is deducible from p...
Alan Ross Anderson and Nuel D. Belnap, Jr., pro­
fessors at Yale University, have developed a logical system
of entailment in which entails is taken as an undefined
term.
Their system is a modification of a system of
rigorous implication proposed by Wilhelm Ackermann.
Ackermann asserts that in an implication "There must be
a logical connection between what implies and what is
logically implied."^
In both Ackermann’s system of rig­
orous implication and in Anderson and Belnap*s system of
entailment the paradoxes of strict implication are rejected.
Anderson and Belnap argue that entailments should be
necessarily true and that in an entailment, "p entails q",
^ G . E. Moore, Philosophical Studies (New York:
Harcourt, Brace and Co., 1922), p. 291.
^Wilhelm Ackermann, "Begrundung Einer Strengen
Implikation", Journal of Symbolic Logic. XXI (June, 1956),
P. 113.
35
the hypothesis, p, should be relevant to the conclusion,
q.
Both of these properties are possessed by their system
of entailment.
The need for relevance is explained as follows:
But saying that A is true on the irrelevant assump­
tion that B, is not to deduce A from B, nor to
establish that B implies A, in any sensible sense
of implies.^5
Thus they reject the paradoxes of material implication
and strict-implication on the basis that in these paradoxes,
the hypothesis is irrelevant to the conclusion.
Anderson
and Belnap offer two conditions for relevance:
1 . For A to be relevant to B it must be possible
to use A in a deduction of B from A. It need
not be necessary to use A in the deduction of
B from A - and indeed this is a familiar sit­
uation in mathematics and .logic. It not infre­
quently happens that the hypotheses of a theorem,
though all relevant to a conclusion, are subse­
quently found to be unnecessarily strong.5»
2.
Informal discussions of implication or entail­
ment have frequently demanded "relevance" of
A to B as a necessary condition for the truth
of A — *B, where relevance is construed as
involving some "meaning content" common to both
A and B. A formal condition for "common
meaning content" becomes almost obvious once
we note that commonality of meaning in propos­
itional logic is carried by commonality of
propositional variables. So we propose as a
necessary (but not sufficient) condition for
the relevance of A to B in the pure theory of
55Anderson and Belnap, op. cit., p. 31 •
5^lbid.. p. /+6.
36
entailment, that A and B must share a variable. 57
For instance, "If x is an isosceles triangle, then
the base angles of x are equal" satisfies both conditions
for relevance.
The statement "If x is an even integer,
then y is an odd integer" satisfies neither of the condi­
tions for relevance.
According to Anderson and Belnap, the system which
arises when we recognize that valid inference requires
both necessity and relevance is their system of entailment
which is defined by a set of fourteen axioms and two rules?®
Entailment is defined by Richard B. Angell in a way
which seems to be equivalent to the definition given by
E. J. Nelson
3k years earlier.
Angell states that "S
logically entails S' if and only if S would be inconsistent
with ~ S f by virtue of their connections."^
to represent not-S1.)
(He used ■~SI
He explains what he means by "S
v/ould be inconsistent with ~ S ' by virtue of their con­
nections" as follows ((S*S!) means S and S'):
The problem of distinguishing logical inconsistency
by virtue of connections is Instructive, and relates
directly to the problem of entailment. The state­
ment "S and S 1 is inconsistent" is ambiguous. It
may mean that the conjunction (S • S') is inconsis­
tent, or it may mean that S is inconsistent with S ’,
^ Ibid., p. 48
See, for instance: Alan Ross Anderson and Nuel D.
Belnap, Jr. "Tautological Entallments", Philosophical
Studies. XIII (1961), p. H , for a listing of their
axiomsand rules.
^Richard B. Angell, Reasoning and Logic (New York:
Appleton-Century-Crofts, 1964), p. 199.
37
which is different. The conjunction (S • S') will
be inconsistent if either S or S', taken alone, are
inconsistent. Thus if S s (p • *wp) and S' = q,
then the conjunction (S • S'), i.e., ((p • ~p) • q)
is inconsistent, but only by virtue of the incon­
sistency of (p • ~p). When we say S is inconsis­
tent with S', we are asserting something else;
(p « ~p) cannot be inconsistent with q, since there
are no connections between them. On the other hand,
p is certainly inconsistent with ~p, since they
have the same variable in common, once affirmed and
once denied. The truth-table method does not
distinguish these two notions. And this is exactly
the same problem as the problem of entailment, ,-q
viewed from the point of view of inconsistency.
This explanation of inconsistency corresponds very
closely to the requirement formulated by Anderson and
Belnap that the hypothesis of an entailment must be rele­
vant to the conclusion.
Thus, we might require that for
S to be considered inconsistent with S ’, S must be relevant
to S'.
Implication;
Summary and Recommendations
We have considered three essentially different defi­
nitions of implication - material implication, strict
implication, and entailment.
Material implication is by
far the most widely accepted notion of implication by
authors of mathematics and logic texts today.
Anderson
and Belnap refer to material implication as the current
"Official view" of implication.
be defined as:
Material implication may
"p materially implies q" means "not-p or q".
60Ibid., p. 198.
38
When material implication is defined in mathematics,
since it is the only type of implication considered, the
term "material” is omitted.
Thus, to the wonderment of
students, we have statements such as "2 + 2 = 5 implies
the earth is flat" accepted as true implications.
We have discussed the "paradoxes" of material
implication:
(1) A false proposition materially implies
any proposition and (2) A true proposition is implied by
any proposition.
It has been argued that "material impli­
cation" is not the same as the notion of implication which
is used in deductive reasoning.
For this reason, material
implication was rejected by the writer as a suitable
definition of implication for mathematics.
The strict implication of C. I. Lewis has been
discussed and it has been pointed out that strict implica­
tion fails to hold in some of the paradoxical instances
in which material implication does hold.
Strict implica­
tion tries to capture the concept that in an implication
"p implies q", q is a necessary consequence of p; if you
accept p, you must accept q.
Hence, the definition of
strict implication is "p strictly implies q" means "it is
impossible that p is true and q false."
Although strict
implication is closer to our intuitive untutored notion of
implication than material implication, it also results in
paradoxical implications.
Because of the "paradoxes" of
39
strict implication this writer rejects strict implication,
also, as a suitable definition of implication for mathe­
matics.
The third notion of implication which has been
discussed is called ’'entailment’' to distinguish it from
strict implication and material implication.
The Moore-
Nelson conception of entailment is free from the paradoxes
of strict implication and material implication. Two very
important properties of entailment are:
1 . "p entails q" when and only when ”q is deducible
from p”.
2. It satisfies the intuitive requirement that "p
implies q" only when thereis some logical connec­
tion between p and q, i.e., to accept "All ele­
phants fly implies that 2 is a prime" seems to
"fly in the face" of our intuitions.
This is a
true implication for both material and strict
implication but it is rejected by entailment.
This writer also accepts entailment as being closer
to the way "implication" is used by laymen and professional
mathematicians than either material implication or strict
implication.
The writer recommends that entailment be used
as the definition of implication by mathematics teachers
and authors of mathematics texts.
ko
Entailment may be defined sis:
(A)
p entails q means that q is deducible from p; or
(B)
p entails q means that p is inconsistent with
not-q.
Definition (A) is the manner in which 6. E. Moore intro­
duced the term "entails."
given by
E. J. Nelson.
Definition (B)
is the definition
The writer elects to take definition
(A) as the definition to be used for implication.
tion(B) will be accepted as an axiom.
which may
Defini­
A list of steps
be used to explain the meaning of implication is:
1 . Accept deducible as an undefined term.
examples of simple deductions such as:
Give
If x is
human, then x is mortal.
_________ Socrates is human
Therefore, Socrates is mortal
2. Define implication as:
p implies qmeans that
q is deducible from p.
3. Define inconsistent as:
The statements A and B
are inconsistent if and only if it is possible
to deduce a contradiction from the joint assertion
of A and B.
k. Accept as a logical axiom:
p implies q if and
only if p and not-q axe inconsistent.
The logical equivalence of "p implies q" and "p and
not-q are inconsistent" is of major importance in the anal­
ysis of indirect proof which will be undertaken in the
following chapters.
CHAPTER III
ON THE CONTRADICTORY OF AN IMPLICATIVE STATEMENT
Introduction
The purpose of this chapter is to provide the
necessary preparation for a critical analysis of the
various types of indirect proof proposed by authors of
plane geometry textbooks in the next chapter.
This chapter
will be devoted to a consideration of:
a) the nature of contradictory statements
b) the nature of contrary statements
c) the problem of forming the contradictory of an •
implicative statement.
Contradictory Statements
A typical definition of "contradictories" is:
Two sentences are contradictories (or negations)
of each other if and only if they are necessarily
opposite as regards truth or falsity. '
Thus, two statements, A and B, are called contradictories
if the following two conditions are satisfied:
1. If A is true, then B must be false.
2. If A is false, then B must be true.
2
It also follows from the definition that two contradictory
Stephen F. Barker, The Elements of Logic (New York:
McGraw-Hill Book Co., 1965), p. 317.
^Wesley C. Salmon, Logic (Englewood Cliffs, N.J.:
Prentice-Hall, 1963), p.101.
41
42
statements cannot both be true nor can they both be false.
To summarize, the important properties of contradictory
statements are:
1. Two contradictory statements cannot be true
together.
2. Two contradictory statements cannot be false
together.
3. Of two contradictory statements, one must be
true and the other false.^
This writer rejects the above approach to contradic­
tory statements.
It is too broad.
For instance, consider
the two statements:
a. p is a prime number and p is composite;
b. Every equilateral triangle is isosceles.
In what sense are these statements contradictories?
do satisfy the usual definition of contradictories ~
They
the
first statement is necessarily false while the second is
necessarily true.
But this writer does not believe that
anyone would seriously consider the two statements as
contradictories.
The problem is that the usual definition
of ’'contradictory statements" is a truth-functional defi­
nition, i.e. whether two statements are contradictories
^Clifford B. Upton, "The Use of Indirect Proof in
Geometry and in Life," Fifth Yearbook (Washington. D. C. :
National Council of Teachers of Mathematics, 1930), p. 107.
Wb
depends solely on the truth values of the statements.
This
is similar to the definition of "material implication"
discussed in Chapter II.
An alternative definition of "contradictories" is
given by P. F. Strawson.
Strawson states:
To say of two statements that they are contra­
dictories is to say that they are inconsistent with
each other and that no statement is inconsistent
with both of them.^tRecall from Chapter II that just because two statements
cannot be simultaneously true is not a sufficient condition
to consider them inconsistent.
In order for two statements,
A and B, to be inconsistent, A must be relevant to B.
Hence
a. p is a prime number and p is composite
b. Every equilateral triangle is isosceles
are not inconsistent, they lack relevance, and therefore do
not satisfy Strawson's definition of contradictories, al­
though they do satisfy the usual definition.
Hence, his
definition of "contradictories" is narrower than the usual
definition.
Strawson has defined "contradictories" in terms of
inconsistency, but in Chapter II this writer proposed a
definition of "inconsistency" which utilized the concept of
contradictories.
Thus, although the writer has no logical
objection to Strawson's definition, to avoid having a
circular process exist in the proposed definitions, the
^P. F. Strawson, Introduction to Logical Theory
(London: Methuen and Co., 1952), p. 16.
kk
writer will suggest another alternative method of defining
"contradictories."
When one is asked to form the contradictory of
"p is a prime number" the anticipated response is "p is
not a prime number."
The writer proposes that "p is not
a prime number" be called the proper contradictory of
"p is a prime number."^
The
term "proper contradictory"
is to be defined as follows:
Definition. The proper contradictory of a statement,
p, is the statement "it is not the case that p."
For example, the proper contradictory of "a is an even
integer" is "it is not the case that a is an even integer."
The statement "it is not the
case that a is an even
integer" may be rephrased as
"a is not an even integer."
In general "it is not the case that p: may be rephrased
by placing "not" with the verb of the statement, p.
Some
examples of statements which are proper contradictories
are:
Example 1.
angle A is a right angle
angle A is not a right angle
Example 2.
2 is a prime number
It is not the case that 2 is a prime number
Example 3.
line 1 is perpendicular to line m.
line 1 is not perpendicular to line m
Example
a is less than b
it is not the case that a is less than b
,---------------------------------------------------------------
^The term "proper contradictory" is not original,
see: Everett J. Nelson, Mind. XXXIX, (Oct., 1930), p. W +
w?
It may be observed that in each of the four examples
cited above whenever the first statement of the example
is true, the second statement must be false and whenever
the first statement of the example is false, the second
statement must be true.
This means that whenever two
statements are proper contradictories they are necessarily
opposite as regards truth or falsity.
But just because two
statements have opposite truth values is not a sufficient
condition to assure that they will be proper contradictories.
The statements "2 is less than V' and "the moon is made
of green cheese" have opposite truth values, but they are
not proper contradictories.
It may happen that two statements are relevant to
one another, have opposite truth values, yet still fail to
be proper contradictories.
Consider the example:
a) x is an even number
p
b) x is not an even number
c) x is not an even number.
Clearly (a) and (c) are proper contradictories.
It may
not be obvious, but (a) and (b) have opposite truth values.
In any instance in which (a) is true, (b) is false, and in
any instance in which (a) is false, (b) is true.
The
reason is that (b) and (c) are equivalent, i.e. (c) is
deducible from (b) and (b) is deducible from (c).
Hence
46
(b) is true when and only when (c) is true.
The writer
proposes that (a) and (b) be considered contradictories.
It is suggested that contradictories be defined as follows:.
Definition. A statement, A, is a contradictory of a
statement, B, if and only if A is equivalent to the
proper contradictory of B.
Contraries
Wesley C. Salmon points out that "failure to under­
stand the difference between contraries and contradictories
c
has led to much confusion."
The following two statements
are contraries:
angle A is equal to angle B
angle A is less than angle B.
If either of these statements is true, the other is false.
But if either is false, the other need not be true since
it is possible for angle A to be greater than angle B.
The important distinction between contradictories and
contraries is that one of two contradictory statements must
be true, while two contrary statements may be such that
neither is true.
This is important in certain mathematical
proofs since one can prove a theorem by disproving its
contradictory but disproving a contrary of the theorem does
£
Salmon, op. cit.. p. 101.
kl
not establish the theorem.
7
'
A second distinction is that contradictories must
occur in pairs while there is no limit to the number of
statements that may be contraries of each other.
For
instance, the following statements are contraries of
each other:
Today
Today
Today
Today
is
is
is
is
Monday
Tuesday
Wednesday
Thursday.
Contradictory of an Implicative Statement
The underlying idea of one indirect method of proving
that "p implies q" is to show that the contradictory of
np implies q" is false.
The proper contradictory of "p
implies q" is "p does not imply q."
Unfortunately the only
logical way of proving that "p does not imply qM is false
is by showing that "p implies q" is true which completes
a vicious circle.
The problem, then, is to find a more
useful contradictory of Mp implies q."
Any contradictory
of "p implies q" must be equivalent to its proper contra­
dictory —
"p does not imply q.M
The problem is reduced,
then, to finding a statement equivalent to "p does not
imply q."
Carl B. Allendoerfer states that "the greatest
*^For a misuse of the term "contrary", see:
Richard Courant and Herbert Robbins, What is Mathematics?
(London: Oxford University Press, 1
), p. 8b.
48
single pitfall in handling an indirect proof is the first
step of taking the negation of the proposition to be
O
proved."
An incorrect method of contradicting an impli­
cative statement is given by Howard Eves.
Eves states that
"the contradictory of ?If A then B ’ is ’If A then not-B’."^
A simple example will show these are not contradictories.
Consider the following statements:
a. If x is a composite number then x is an even
number.
b. If x is a composite number then x is an odd
number.
It should be observed that (a) is of the form "If A then B"
while (b) is of the form "If A then not-B."
Statement (a)
is false since 15 is a composite number but 15 is not even,
similarly statement (b) is false since 6 is a composite
number but 6 is not an odd number.
Thus, both (a) and (b)
are false but this violates the necessary condition of
contradictories that they cannot both be false nor can
they both be true.
Hence, (a) and (b) are not contradictor­
ies.
^Carl B. Allendoerfer, "Deductive Methods in Mathe­
matics," Insights into Modern Mathematics: Twenty-Third
Yearbook ('Washington. D.C.: 'Me "National Council of’ Teachers
of Mathematics, 1957), p. 91.
^Howard Eves, An Introduction to the History of
Mathematics (rev. ed.; New York: Holt, Rinehart and
Winston, 1964), p. 399.
h9
The above example points out that Mp implies not-q"
is not a contradictory of "p implies a."
It is, in fact,
the case that a contradictory of an implicative statement
cannot be an implicative statement.1®
In Chapter II the equivalence of "p implies qn and
"p is inconsistent with not-q" was discussed.
This equiv­
alence affords an easy way of forming the contradictory of
"p implies q."
The contradictory of "p is inconsistent
with not-q" must also be a contradictory of "p implies q".
The contradictory of "p is inconsistent with not-q" is
"p is not inconsistent with not-q", i;e., "p is consistent
with not-q".
So, "p implies q" and "p is consistent with
not-q" are contradictories.11
Thus the contradictory of the statement
2
If x is an even number then x is an even::number
would be
the statement "x is an even number" is consistent
with the statement "x is not an even number."
A method of indirect proof based upon the realization
that "p implies q" and "p is consistent with not-q" are
contradictories will be proposed in Chapter VI.
I®Henry W. Johnstone. Jr., Elementary Deductive Loffic
(New York: Thomas Y. Crowell Co.
p. i+y.
II See: Nathan Lazar, "The Logic of the Indirect Proof
in Geometry: Analysis, Criticism and Recommendations,"
The Mathematics Teacher. XL (Hay, 194-7), p. 228.
50
Summary
In this chapter it has been pointed out that a
necessary condition for two statements to be contradic­
tories is that they cannot both be true nor can they both
be false, i.e. the truth of either statement necessitates
the falsity of the other.
It has been argued that although
having opposite truth values is a necessary condition for
contradictories, it is not a sufficient condition, i.e.,
two unrelated statements which merely have opposite truth
values should not be considered contradictories.
With
this view in mind, the following two definitions were
proposed:
Definition. The proper contradictory of a statement,
p, is the statement "it is not the case that p."
Definition. A statement, A, is a contradictory of a
statement, B, if and only if A is equivalent to the
proper contradictory of B.
The distinction between "contradictories" and
"contraries"was also discussed.
Two contradictory statements
cannot both be true.nor can they both be false but although
two contrary statements cannot both be true, they can both
be false.
The problem of forming the contradictory of an impli­
cative statement has been considered.
It was pointed out
that a useful contradictory of "p implies q" is "p is
consistent with not-q."
CHAPTER IV
DEVELOPMENT OF INDIRECT PROOF IN GEOMETRY TEXTBOOKS
Introduction
One characteristic of a "modern mathematics" program
often cited in current literature is an insistence upon
precise language.
The study by Glabe^ indicates that the
plane geometry textbooks published prior to 1955 contained^
considerable confusion as to the meaning of indirect
proof.
This point is clearly made in the following state­
ments taken from the summary of his study:
A survey of plane geometry textbooks reveals
a striking lack of agreement in all the phases of
indirect proof. Authors do not agree on a defini­
tion of indirect proof, and almost none is precise.
Largely because of the lack of precision in defini­
tion, certain variations of proof are classified by
some as direct and by others as indirect. Two such
examples are the "forced coincidence" proofs and
proofs by the method of contraposition.2
The years since 1955 have been turbulent years for
the field of mathematics education.
The "revolution in
mathematics" has stimulated, the publication of many
"modern" plane geometry textbooks.
indirect proof?
How do they treat
The purpose of this chapter shall be to
Gordon R. Glabe, "Indirect Proof in College Mathe­
matics" (unpublished Ph.D. dissertation, Graduate School,
The Ohio State University).
2
Glabe, op. cit.. p. 205.
51
52
make a critical analysis of the discussions of indirect
proof in the plane geometry textbooks published since 1955^
A textbook may convey the meaning of indirect proof
in the following ways:
1. by giving a definition or an analysis of direct
proof
2. by pointing at examples of indirect proofs
3. by giving a scheme that indirect proofs are to
follow
l+, by contrasting indirect proof with direct proof
5. by giving a precise definition of indirect proof.
It is the claim of this writer that all five of these
stages are pedagogically desirable and should be considered
in the textbooks discussion of indirect proof.
In the
following pages we shall consider each of these five stages
and see how they are treated in the geometry texts.
Introducting the Meaning of Indirect Proof by First
Giving a Definition or Analysis of Direct Proof
When a student looks at the word "indirect" there
is an immediate thought process which transforms indirect
into not-direct.
This is the result of past experiences
with words such as "incorrect," "inactive," "incurable,"
and "inexact" where the prefix "in-" is used to mean "not."
Since indirect proof is thought of as not-direct proof, an
^A list of the geometry texts examined in this study
is given in Appendix B.
53
understanding of direct proof is essential to an under­
standing of indirect proof.
In this section the various
definitions and explanations of direct proof that are
given by authors of geometry textbooks will be examined and
criticized.
Schact, McLennan, and Griswold explain direct proof
by exhibiting a pattern that direct proof is to follow.
Most of the proofs in geometry arecalled direct
proofs because they consist of a sequence of syllo­
gisms arranged according to the following pattern:
(1) p is true.
(Something is given)
(2) If p is true, then q is true.
(3) If q is true, then r is true.
(4) Therefore, r is true, (Conclusion)
Statements (2) and (3) are axioms, postulates,
definitions, or previously proved theorems.
Of all the geometry books examined this is the most expli­
cit explanation of the process of direct proof.
In most
books steps (2) and (3) are explained by the vague phrase
"by logical reasoning."
It is doubtful that the phrase
"by logical reasoning" will have meaning to the student
early in the geometry course.
To understand the nature of
logical reasoning is a major goal of the geometry course.
Weeks and Adkins also explain direct proof by
exhibiting a pattern to be followed.
They state:
A direct proof of the assertion ’If A, then B ’ is
^John F. Schact, Roderick C. McLennan, and Alice L.
Griswold, Contemporary Geometry (New York: Holt, Rinehart
and Winston^ 19o2), p. 57.
5k
of the following pattern:
If A, then X
If X, then Y
If Y, then B?
Unfortunately in actual practice direct proofs given in
their text frequently do not conform to this pattern.
Consider the following direct proof from Weeks and Adkins:1
Given:
AB, CD, PR are straight
lines. PR intersects AB
at X and CD at Y. / RYD
= I YXB, that is, £ 1 = / 2.
Prove:
A
C
p
-R
/YXB = /CYX* that is,
/2 = Z3.
Proof
REASONS
STATEMENTS
.
CD and PR are straight
lines intersecting at Y.
1.
Given
2.
Z 1=Z 3
2.
If two straight
lines intersect,
then the opposite
angles formed are
equal in pairs.
3.
L
2
3.
Given
k.
Z 2 =Z 3
4.
Deduced from state­
ments 2 and 3 and
the transitive
property of equality.u
1
1
=Z
To see that the above proof does not follow the pattern of
direct proof suggested by the authors observe that:
a), statement (3) does not imply statement (if), but
^Arthur W. Weeks and Jackson B. Adkins, A Course in
Geometry: Plane and Solid (Boston: Ginn and Co., 19^1 ),
P. "11$.
6Ibid., p. 30.
55
it is statement (2) and (3) that imply statement
(4).
b). it is not the entire hypothesis, the given, that
implies statement (2), but only the conditions
of the hypothesis listed in statement (1).
c). the proof is not explicitly stated as a sequence
of if-then statements although the authors’
pattern for direct proof is a sequence of if-then
statements.
Since the authors make no attempt to explain this apparent
difference to the student,*the student is likely to be
confused.
There are only three geometry texts in the group of
thirty-seven studied that explicitly explain direct proof
by exhibiting a pattern to be followed.
The third text,
by Clarkson, Douglas, Eade, Olson, and Glass, contains the
following:
Given a statement in the form of an implication,
a direct proof looks like the following:
Consider the two statements,
If it is raining, then the ground is wet.
It is raining.
We put these two statements together and conclude,
The ground is wet.7
It is interesting to note that the direct proofs which they
give of geometry theorems are not of this form.
Much of
the reasoning in the above pattern is left tacit and not
"^Donald R. Clarkson et al¥. Geometry (Englewood
Cliffs, N.J.: Prentice-Hall, 1965), p. 54.
56
written in their geometric proofs.
Welchons, Krickenberger, and Pearson state that
’’the direct method consists in putting together known
truths (definitions, assumptions, theorems, the hypothesis),
step by step, to form a proof."
8
They do not discuss what
they mean by "step by step," which is
a crucial part of
their definition.
that they did not
It should be noted
mention the conclusion, perhaps this is because earlier in
the text they said that "the proof consists of a series of
deductions beginning with the facts given in the hypothesis
Q
and ending with the conclusion."7 This explanation of
what constitutes a proof evidently excludes the indirect
method of proof, yet they introduce the indirect method as
a way of proof on page 130.
The step by step idea is also used in the definition
of direct proof given by Mallory, Meserve, and Skeen.
They
explain:
Proofs such as those that we have used thus far are
often called direct proofs. Each consists of direct
step-by-step progress from the known truths (hypo­
thesis, definitions, axioms, postulates, and
,Q
previously proved theorems) to the desired conclusion.'
Authors frequently do not discuss direct proof until
the section in which they introduce indirect proof.
Fehr
Q
A. M. Welchons, W. R. Krickenberger, and Helen R.
Pearson, Plane Geometry. (Boston: Ginn and Co., 1961), p.130.
^Ibid.,
p.
68.
10Virgil S. Mallory, Bruce E. Meserve, and Kenneth C.
Skeen, A First Course in Geometry. (Syracuse, N.Y.: L.W.
Singer Co., 1959), p. 138.
57
and Carnahan give a brief explanation of direct reasoning
as they are about to explain indirect proof.
They say:
In the theorems you have proved thus far you
have reasoned from the given hypothesis until you
reached the conclusion. This is called direct
reasoning.1
This definition obviously lacks preciseness.
Another short explanation is given by Fischer and
Hayden:
a direct proof is one in which f,the conclusion
follows directly from the hypothesis and other accepted
12
statements."
The student must realize what the author
intends by "follows directly" in order to understand this
definition.
Shute, Shirk, and Porter have an explanation which
is essentially that of Fischer and Hayden.^
Price, Peak, and Jones give this description of direct
proof:
In a direct proof we prove a proposition by estab­
lishing the conclusion as a result of previously , .
proved propositions, postulates, and definitions. ^
They neglect to mention the hypothesis, in all proofs the
^Howard F. Fehr and Walter H. Carnahan, Geometry
(Boston: D. C. Heath and Co., 1961), p. 97.
ip
Irene Fischer and Dunstan Hayden, Geometry (Boston:
Allyn and Bacon, 1965), p. 118.
^William G. Shute, William W. Shirk, and George F.
Porter, Geometry-Plane and Solid (New York: American Book
Co., 1960), p. 67.
^ H . Vernon Price, Philip Peak, and Philip S. Jones,
Mathematics - An Integrated Series: Book Two (New York:
Harcourt, Brace and World, 1965), p. 102.
58
hypothesis is used in addition to previously proved propos­
itions, postulates, and definitions.
Keniston and Tully, too, do not mention direct proof
until the section in which they introduce indirect proof.
At that time they state:
All geometric proofs presented so far have been
direct proofs, in which we have proceeded logically
from known facts to the facts to be proved.15
The use of the expression "known facts" may be criticized.
This is contrary to the very nature and spirit of modern
geometry.
The discovery of non-Euclidean geometry has
made this use of "known facts" both improper and incorrect.
Instead of saying they are known facts, it would be more
accurate to say that they were statements assumed to be true.
Edwards, also, discussed direct proof just prior to
a unit on indirect proof.
In a fashion similar to that of
Keniston and Tully, Edwards states:
Nearly all proofs are by direct demonstration, in
which we proceed logically fF9® known facts to the
facts that are to be proved.
To state that nearly all proofs are direct just prior to
introducing indirect proofs may diminish a student's
interest in indirect proofs.
There is but one text left of the thirty-seven geome­
try texts that were examined that contains a discussion of
^Rachael P. Keniston and Jean Tully, High School
Geometry (Boston: Ginn and Co., 1960), p. 267.
1L
Myrtle Edwards. First Course in Geometry (New York:
Exposition Press, 1965;, p. 78'.
59
direct proof and has not been mentioned.
text is by Seymour, Smith, and Douglas.
The remaining
They state:
The direct method of reasoning starts with
definitions and with certain statements which we
assume to be true. From these unproved statements
we derive other statements which, if our logic is
sound, v/e know to be true, and from these, in turn,
we derive still other statements.
In this manner
we build up step by step a chain of ideas every one
of which is true provided the unproved statements
upon which they depend are true.*?
This appears to be a discussion of the axiomatic method
rather than a description of the direct method of proof.
In three books the expression "direct proof" is used
only to indicate that a particular proof is direct without
either analyzing or explaining "direct proof."
18
The majority of the geometry texts which do expli­
citly discuss direct proof wait until they are ready to
introduce indirect proof.
"proof."
Prior to this they often define
Since indirect proofs do not occur until later,
some of them aim their definition of proof towards direct
proof and to the exclusion of indirect proof.
Thus, by
their definition of proof the only mathematical proofs are
direct proofs.
Although when they get to their unit on
indirect proof they will say that an indirect proof
1 «
'
—
.
rF. Eugene Seymour, Paul James Smith, and Edwin C.
Douglas. Geometry for Wigh Schools (New York: The Macmillan
Co., 1958), p. 94.
1A
Irving Allen Dodes, Geometry (New York: Harcourt,
Brace and World, 1965), p.10; Max Beberman and Herbert E.
Vaughan, High School Mathematics: Course 2 (Boston:D.C. Heath
and Co.,1965), p.5&» and University of Illinois Committee on
School Mathematics, High School Mathematics; Unit 6 - Geo­
metry 'Urbana; University of Illinois P r e s s , I960), p.^0.
60
is a proof.
The following four quotations indicate exam­
ples of this observation.
A proof of a theorem is a sequence of true state­
ments. The first statement in the sequence is the
hypothesis of the theorem and the last statement in
the sequence is the conclusion of the theorem.
A mathematical proof is a logical argument. That
is, there is a logical chain of statement leading from
the hypothesis, which is a set of initial conditions
assumed to be true, to a c o n c l u s i o n . 20
To develop a geometric proof, you first determine
the relationships needed to carry the argument from
the given hypothesis to the desired conclusion.21
Reasoning by deduction, the method of formal
geometry starts with the hypothesis and proceeds
logically through a sequence of statements until
the conclusion is obtained. Such a sequence of
statements is a mathematical p r o o f . 22
Summary
In only fifteen of the thirty-seven books examined
was the expression "direct proof" used.
The discussions
of direct proof in these fifteen geometry books have been
investigated in the preceding pages.
It may be observed
that authors describe direct proof in one or more of three
ways:
(1 ) by exhibiting a pattern that direct proof is to
follow, (2) by stating in essence that in a direct proof
IQ
■'Lewis D. Eigen et al., Advancing in Mathematics:
Geometry (Chicago: Science Research Associates, 1966),p.108
2 Price, et al.. pp. eit.. p. 30.
^ F r a n k M. Morgan and ^ane Zartman, Geometry; Plane Solid - Coordinate (New York: Houghton Mifflin Co., 19&3),
P. 67.
22
School Mathematics Study Group, Geometry with Coordinates (New Haven: Yale University Press, 1963), p. 1-7
61
you move logically, step by step, from the hypothesis,
definitions, axioms, and previously proven theorems to the
conclusion, and (3) by merely stating that a particular proof
is an example of a direct proof.
None of the geometry
books examined contained a precise definition of "direct
proof" that conformed to the manner in which direct proofs
were actually written in the text.
It is interesting to note that over one-half of the
geometry texts examined do not mention "direct proof"
although they do mention "indirect proof."
Showing Examples of Indirect Proofs
Real-Life Examples
The examples of indirect proof given in geometry
texts are of either real-life situations or they are geo­
metric examples.
The teacher's manual of a widely-used
geometry book indicated the value of considering real-life
examples as follows:
Indirect proofs, not easily comprehended by some
students, can be made more meaningful if the teacher
proposes simple real-life situations...^3
Some examples chosen as being representative of the reallife examples contained in geometry texts will now be given
along with an analysis of the logic used in the examples.
^ R a y C. Jurgensen, Alfred J. Donnelly, and Mary P.
Dolciani, Modern Geometry (New York: Houghton Mifflin Co.,
1965), teachers manual, p. 23.
62
This interesting example is from the School Mathe­
matics Study Group material:
In a certain small community consisting exclu­
sively of young married couples and their small
children, the following facts are known to be true.
(a) Every boy has a sister.
(b) There are more boys than girls.
(c) There are more adults than children.
Prove that there must be at least one childless
couple.
We begin, then, by assuming that there is no
childless couple. From this, we conclude that every
family must have at least one girl, since by the
first fact there can be no family having only boys.
Thus, there are at least as many girls as there are
families. Moreover, by the second fact, there are
actually more boys than there are families. Hence
the number of boys and girls together is more than
twice the number of families. But this means there
are more children than adults, which contradicts
the third fact. Hence it is false that every family
has a child. In other words, at least one family
must be childless, which is what we were asked to
prove.
Is the above example too difficult as a first introduction
to indirect argument?
This writer believes that it is too
difficult, fortunately the examples given by the remaining
geometry texts are easier.
In this example the proof con­
sists in showing that from statements (a), (b) and the
contradictory of the conclusion (there is no childless
couple) it is possible to deduce a contrary of statement
(c).
The general pattern of proof exhibited in this exam­
ple, then, is that one can prove that "h^ l^h^-*c” by
^School Mathematics Study Group, Geometry with
Coordinates, op. cit., pp. 12 - 13.
showing that h-jh2and not-c — » not-hy.
method of partial contraposition.
63
This is called the
This method of indirect
proof will be discussed later in this chapter and in Chap­
ter VI.
Anderson, Garon, and Gremillion give this simple
example:
Imagine that you arrived in the middle of a
basketball game and observed that the teams were
lined up for a free throw. You would surely con­
clude that a foul had just been called and your
reasoning would be as follows. Suppose that a foul
had not been called. Then, according to the rules
of basketball, the teams would not be lined up
for a free throw. But this contradicts a known
fact, namely that the teams were lined up for a
free throw. Therefore the supposition that a foul
had not been called is false. Hence a foul must
have been called.^5
In this example the statement "A foul has been called" is
proved by showing that its contradictory "A fould has not
been called" is false.
The general pattern of this method
of indirect proof, then, is that one can prove a statement,
A, by showing that its contradictory, not-A, is false.
The statement, not-A, is false if it is possible to deduce
a statement known to be false in the system from not-A.
This method of indirect proof which the writer will call
"the method of contradiction" will be discussed in greater
detail later in this chapter and in Chapter VI.
Welchons, Krickenberger, and Pearson provide the
25
^Richard D. Anderson, Jack W. Garon, and Joseph G.
Gremillion, School Mathematics Geometry (Boston: Houghton
Mifflin Co., 1966), p. 190.
64
last example that will be given here.
Detective Jones used indirect reasoning when he in­
vestigated the death of an elderly woman on Linden
Avenue. He learned from the coroner that the woman’s
death was caused by a bullet which entered her body
below the right shoulder blade and emerged below
the right collar bone.
Mr, Jones knew this was a murder, a suicide, or an
accidental death. After a thorough examination of
the body and surroundings, he knew that the death
did not result from suicide or an accident. There­
fore he knew this was a murder.^6
In this example the detective determined that it was murder
by eliminating the only other possibilities.
In general a
statement, A, may be established by showing that each of
the other possibilities leads to a contradiction.
This
j**'
method of indirect proof is frequently called "the method
of elimination".
The "method of elimination" as a method
of indirect proof of implicative statements will be crit­
icized later in this chapter.
Such varied topics as:
sports, law, ages, court,
detective story, driver's license date, electric motor,
cookies, and rain can be found as subjects of examples of
indirect reasoning.
Some authors use the pattern of the
argument given in their examples of indirect reasoning in
real life as the motivation for the methods of indirect
^Welchons. Krickenberger. and Pearson, op. cit..
pp. 130 - 131.
proof proposed by their text.
65
27
'
Of the thirty-seven geometry texts that were examined
there were sixteen books (see Appendix C) which used reallife examples of indirect reasoning to form the pattern(s)
of indirect proof to be used in indirect proofs of geome­
tric propositions.
In addition, there were eleven texts (see Appendix
D for a listing) that contain either examples or exercises
involving the use of indirect reasoning in real-life sit­
uations, but in these eleven texts the real-life examples
are not explicitly used in the formulation of the methods
of indirect proof given by the authors.
This means that twenty-seven of the thirty-seven
books researched contain some examples of indirect rea­
soning in non-mathematical situations.
It also means that
ten of the thirty-seven texts contained no examples of
indirect reasoning in real-life situations.
Many quotations could be given supporting the desir­
ability of including non-mathematical examples of indirect
reasoning in the geometry course, but the following two
quotations should suffice.
The student may not always be able to see the
similarity between the geometric and the non-geometric situations. These relations, these simil­
arities, and these methods of thinking must be
prp
'For a list of texts which use this approach,
see Appendix C.
66
brought out by the teacher so that the student
is aware of them and so that he may use them in
various situations where non-geometric material
is involved. There is no doubt that if we want
the student to do better thinking in life situ­
ations, we must teach him to do better reasoning
in those situations while he is in the geometry
class.
Much work has been done in recent years to improve
the teaching of indirect proof, but much remains
to be done to make the logical principles involved
so meaningful to a student that he will understand
how to use them to make an indirect proof. Hereto
again, the nongeometry exercise seems essential. y
The above quotations indicate the value of non­
geometric examples of indirect reasoning, yet over
one-fourth of the texts examined failed to contain any
nongeometric examples of indirect reasoning.
Geometric Examples
It has been mentioned that the introductory examples
of indirect reasoning given by geometry texts may be from
either real-life or geometry.
Just as with nongeometric
examples, some texts, eleven of the thirty-seven examined,
(see Appendix E for a listing) use a geometric example of
indirect reasoning to motivate the formulation of a
technique(s) of indirect proof that the student is to
follow in his proofs.
28
William David Reeve, Mathematics For the Secondary
School. (New York: Holt, Rinehart and Winston,! 954)> p.3^5.
29
^Myron F. Rosskopf, "Nongeometric Exercises in
Geometry", Twenty-second Yearbook (Washington, D.C.:
National Council of Teachers of Mathematics, 1954),p.283.
67
Summary
There were five books (see Appendix F for a listing)
that used both real-life and geometric examples of indirect
reasoning as patterns in designing their scheme(s) of
indirect proof.
Twenty-two of the thirty-seven geometry
texts based the pattern of their technique(s) of indirect
proof on illustrations from either everyday life or geome­
try.
To state this another way, fifteen out of thirty-
seven, over 40 per cent, of the texts failed to make use
of any examples of indirect reasoning in originating their
techniques of indirect proof.
The Meaning of Indirect Proof is Given
By Stating Schemes for Indirect Proofs
Most of the geometry books examined gave but one
method of indirect proof which was usually called the
indirect method of proof.
But some books which gave only
one method of indirect proof called the method by a special
name which was descriptive of the particular way in which
the proof was to be done.
Also, special names were used
by those books which gave more than one method of indirect
proof so that they could distinguish between the different
methods of indirect proof.
The following names for methods of indirect proof
are given by authors of geometry textbooks:
"method of
68
elimination", "demolishing the cases", "proof by contra­
diction", "reductio ad absurdum", "method of contraposition"
"proof by coincidence", and "the indirect method of proof."
Unfortunately, as will become evident, different authors
use different names for the same method of indirect proof,
i.e. one author may refer to a particular pattern of indir­
ect proof as "the method of exclusion" while another author
may refer to the same pattern of indirect proof as "the
method of elimination".
Further, to add to the confusion,
different authors use the same name for vastly different
patterns of indirect proof, i.e. one author may use the
term "proof by contradiction" to refer to a particular
pattern of indirect proof while another author uses the
same term, "proof by contradiction", to refer to a differ­
ent pattern of indirect proof.
To clarify this confusing situation, the different
procedures of indirect proof will be categorized as Type I,
Type II, Type III, Type IV, Type V, Type VI, and Type VII.
Each type will be a distinct scheme of indirect proof
proposed by authors of geometry texts.
Under Type I, for
instance, textbook discussions of a particular scheme of
indirect proof will be critically analyzed regardless of
the particular names used by the authors for this distinct
scheme of indirect proof.
69
gyjpeJ
One pattern of indirect proof given by authors of
geometry texts consists of the following steps:
(1 ) State all the possibilities.
(2) Assume a possibility that you expect to be false.
(3) Show that a conclusion based on the assumption
in Step 2, through logical reasoning, contra­
dicts a given fact, another assumption, a
definition, or a theorem previously proved.
Since the conclusion has been proved false the
assumption upon which it was based must be false.
(k) Repeat this process with the other possibilities
except the one which you are trying to establish.
(5) Conclude that, after all other possibilities
have been proved false, the remaining possibility
must be true.30
This pattern of indirect reasoning, which the writer will
call Type I, is called - "method of exclusion", "method of
elimination", "demolishing the cases", "reductio ad absurdum",
"proof by contradiction", or "indirect proof" - by authors
of geometry texts.
a.
Method of Exclusion
Edwards calls this scheme, Type I, of indirect
proof "the method of exclusion".
He explains:
The exclusion method consists in examining all the
different possible conclusions and showing that
none but the one we are to prove can be true.3*
^ Kenneth E. Brown and Gaylord C. Montgomery,
Geometry: Plane and Solid (River Forest, 111.: Laidlaw
Brothers, 1962), p. 297.
^Edwards, op. cit.. p. 78.
70
Welchons, Krickenberger, and Pearson use this same
statement, word for word, in explaining the exclusion
method.^2
Avery and Stone in discussing indirect proof
state that:
Another name given to this form of proof is
the method of exclusion, or proof by contradiction.
Its success depends upon being able to enumerate
all the possible cases and then to exclude all but
one of them by showing that each in turn leads to
some contradiction either of the hypothesis or of
a previously proved statement.33
This implies that they use the terms "proof by contra­
diction" and "method of exclusion" as different names for
the same method of indirect proof.
Shute, Shirk, and Porter give more than one plan of
indirect proof.
They require that for the method of ex­
clusion there must be more than two possibilities to be
considered.
When there are just two possibilities for the
conclusion to be considered, they call it the "method of
reductio ad absurdum."^
Goodwin, Vannatta, and Fawcett, also, give a method
of indirect proof which follows the pattern of a Type I
indirect proof.
They say that this form of proof "is
---------------------------------------------------------
Welchons, Krickenberger, and Pearson.op.cit..p.131 .
•^Royal A. Avery and William C. Stone, Plane Geo­
metry (Boston: Allyn and Bacon, 196*f)» P» H O *
•^Shute, Shirk, and Porter, op.cit.. p. 68.
71
sometimes called the method of exclusion.
b.
Method of Elimination
The method of elimination is described by Schact,
McLennan, and Griswold as follows:
A frequently used variation of proof by contradic­
tion might be called indirect proof by elimination.
Here the procedure is to list all possible conclu­
sions - the desired conclusion and all of those
arising from its negation - and then to eliminate
all except the one to be established by deducing
a statement that contradicts a known fact.36
Fischer and Hayden state that:
An indirect proof may also be called a proof by
elimination of other possibilities. This name
suggests the procedure of the proof...One might
think that besides the conclusion q there could
be other possibilities, q f, q " , ......... The proof
consists in showing that none of these other possi­
bilities is compatible with our geometric system,
and therefore only q can be true.37
c.
Demolishing the Cases
The only text in which the name "demolishing the
cases" was found is the book by Clarkson, Douglas, Eade,
Olson, and Glass.
indirect proof.
These authors give three methods of
They called the method of indirect proof
70
which follows the Type I pattern "demolishing the cases.
^ A Wilson Goodwin, Glen D. Vannatta, and Harold
P. Fawcett, Geometry. A Unified Course (Columbus, Ohio:
Charles E. Merrill Books, 1965), p.' 2j)6.
•^Schacht, McLennan, and Griswold, op. cit.. p. 322.
•3 7
-''Fischer and Hayden, op. cit.. p. 170.
-^Clarkson et al., op. cit., p. 156.
72
d.
Reductio Ad Absurdum
Birkhoff and Beatley state that the Type I scheme
of indirect proof is often called the method of reductio
ad absurdum.
They say:
It is often called the method of reductio ad
absurdum because by it every possibility except one
is shown to be absurd and contrary to previously
established propositions.39
e.
Indirect Proof.
The name given most frequently by authors of geo­
metry texts for their method of indirect proof which
followed the pattern of Type I was simply nthe method of
indirect proof."
Weeks and Adkins explain the method of indirect
proof as follows:
In many situations it is very difficult, or
even impossible, to establish the direct form of
argument. The method of indirect proof is then
used. The basis of the method is that if there are
a number of possible results, one of which must be
true, then if all but one of the possibilities are
shown to be false, the remaining one must be the
true one.^O
The Type I pattern is also called "indirect proof"
in the texts authored by Brown and M o n t g o m e r y D o d e s ^ ;
^ G e orge David Birkhoff and Ralph Beatley, Basic
Geometry (3rd ed.; New York: Chelsea, 1959), P. 35.
^W e e k s and Adkins, op. cit.. p. 113.
^ B r o w n and Montgomery, op. cit.. p. 297.
^Dodes, op. cit.. p. 10.
73
Keniston and Tully^; Leary and Shuster^; Morgan and
Zartman^; Price, Peak, and Jones^; Seymour, Smith and
Douglas^; and the School Mathematics Study Group^®.
All of the texts which contain the Type I pattern of
indirect proof have now been mentioned.
There were eighteen
of the thirty-seven texts that were examined, almost onehalf, which contained a method of indirect proof which
followed the Type I scheme.
It has been substantiated
that authors refer to the Type I scheme of indirect by these
names:
"method of exclusion", "method of elimination",
"demolishing the cases", "proof by contradiction", "reductio
ad absurdum", and "indirect proof."
Criticism of Type I Indirect Proof
A Type I indirect proof is a method of indirect
proof which essentially follows this pattern:
1 . State or list all the possibilities (possible
conclusions).
2. Show that each possibility other than the desired
conclusion leads to a contradiction.
^Keniston and Tully, op. cit.. p. 269.
^Arthur F. Leary and Carl N. Shuster, Plane Geo­
metry (New York: Charles Schribner’s Sons, 1955), p. ^88.
^Morgan and Zartman, op. cit.. p. 138.
^Price, Peak, and Jones,
op.
cit.. p. 102.
^Seymour, Smith, and Douglas, op. cit.. p. 96.
^School Mathematics Study Group, Geometry with
Coordinates, op. cit.. p. 326.
7k
3. The desired conclusion is then proved.
Notice that in a Type I indirect proof the attention
is focused on the conclusion, while the hypothesis of the
theorem is ignored.
But consider the example on the next
page of a Type I indirect proof taken from the text by
hq
Avery and Stone^7.
Avery and Stone, op. cit., p. 278.
75
Proposition IV. Theorem
366. If two triangles have two sides o f one respectively equal to
two sides o f the other and the third sides unequal, the triangle which
has the greater third side has the greater included angle.
C
F
0
G iven: The two &ABC and DEF with AC - DF, CB - FE,
and AB > DE.
To prove: AC > AF'.
Plan oj Attack: method to be used — proof by exclusion or in­
direct proof.
Proof:
1. Either AC ~ AF, or A C
1. In this case only three sup­
< AF, or AC > AF.
positions are possible.
2. If AC - AF, the AABC
2. s.a.s.
S £J)EF.
3.
AB - DE.
3. Why?
4. This is impossible.
4. By hy£., AB > DE.
5. If AC < AF, then AB
5. 365.
< DE.
6. This is impossible.
7.
AC > AF.
6. Why?
7. By argument it is impossi­
ble for A C to be equal to A F or
to be less than A F.
76
The hypotheses do play an important role in the
previous example.
Statement (2) of theproof is a result
of the hypotheses AC = DF and CB = FE while statement (6 )
is a result of the condition in the hypothesis that AB > DE.
Thus, the conditions in the hypothesis are assumed in a
Type I indirect proof even if they are not mentioned in the
statement of the method.
In fact one would be very sus­
picious of a proof in which the conditions in the hypothesis
were not utilized.
Another fault of the Type I scheme is that it leads
the student to believe that it is the conclusion that he
is proving but what he really is proving is "If the hypo-
50
thesis then the conclusion."^
It is the theorem that is
being proven, not the conclusion.
An additional weakness of the Type I pattern is the
vagueness of the command to state all possible conclusions.
Consider the theorem:
"In a circle, if two arcs are equal
then their chords are equal."
conclusions?
What are the possible
The following are possible conclusions for
the condition that two arcs of a circle are equal:
1 . the chords are equidistant from the center
2 . the central angles of the arcs are equal
^ S e e : Stanley A. Smith, "What Does a Proof Really
Prove?", The Mathematics Teacher, LXI (May, 1968),pp.483 484.'
77
3 . the chords are equal
k. the chords are not equal.
Obviously authors only expect students to list statements
(3) and (Zf) but statements (1 ) and (2) are possible con­
clusions for the condition that two arcs of a circle are
equal.
Further, the student would find it impossible to
eliminate (1 ) and (2) since these statements are deducible
from the hypothesis.
What authors really intend is for
the student to list the conclusion and all its contraries
but this is not explicitly stated in any of the geometry
texts.
Type II
Type II shall denote a method of indirect proof
which essentially follows this pattern:
1 . State the contradictory (negative) of the
conclusion.
2. Reason from the contradictory of the conclusion
until a contradiction is reached.
3. The desired conclusion is proved.
Type II has been called - "reductio ad absurdum", "method
of elimination", and "indirect proof" - by authors of
geometry texts.
Note that all three of these terms were
also applied to a Type I indirect proof.
Type II differs
from Type I in the number of possibilities that may need
to be considered.
In Type II only the contradictory of
the conclusion needs to be considered whereas in Type I
more than one possibility other than the desired conclusion
may be considered.
a. Reductio Ad Absurdum
Shute, Shirk, and Porter discuss the reductio ad
absurdum method as one of the methods of indirect proof
(they give three methods).
They explain reductio ad
absurdum as follows:
Reductio ad absurdum method consists in reducing
an assumption to an absurdity. The negative state­
ment of the conclusion of the proposition is assumed
to be true and then a new conclusion is_obtained
which is contrary to established facts. '
This is the only text which calls a Type II pattern of
indirect proof in mathematics the "reductio ad absurdum"
method.
b. Method of Elimination
i
Harry Lewis is the only author which calls a Type
II pattern of indirect proof the "method of elimination".
Lewis states:
Basically, the mechanics of the proof by
elimination are as follows:
(1 ) Examine the conclusion you are asked to prove.
(2) Set up the statement that is contradictory to
^ Shute, Shirk, and Porter, on. cit.. p. 68
79
this conclusion.
(3) Show that the acceptance of this possibility
leads to a logical inconsistency; that is, it
leads to a statement that is contradictory to
one of the following:
(a) The given data
(b) An assumption
(c) A definition
(d) A theorem
(/f) The conclusion then follows, for having elim­
inated one of the two possibilities that
existgd, the remaining one will have to be
true.
c. Indirect Proof
In the eight remaining texts which contain the Type
II pattern of indirect proof the method is called "indirect
proof".
Jurgensen, Donnelly, and Dolciani give the following
directions to the studetn for writing an indirect proof:
1 . Suppose that the negative of the conclusion is
true.
2. Reason from your assumed statement until you
reach a contradiction of a known fact.
3. Point out that the assumed statement must be
incorrect and that the desired conclusion is
This is essentially the pattern of indirect proof
^2Lewis, op. cit.. p. 227.
^Jurgensen, Donnelly, and Dolciani, op. cit..
p. 165.
5Zi
which is also given by Smith and Ulrichr^ and by Keedy,
55
Jameson, Smith, and Mould.
80
Henderson, Pingry, and Robinson explain indirect
proof as follows:
(1) When the theorem is a conditional p-*q, take
the antecedent p [hypothesis} as the given and
the consequent q [conclusiorfj as the prove.
(2) List the consequent q and its contradictory
q.
(3) Choose /%-q to start your chain of reasoning.
That is, assume ^ q to be true.
(k) Reason from ~ q until you reach a contradiction
of a postulate, of a theorem, or of the ante­
cedent of the theorem you are proving.
(5) Since you are led to a contradiction, then the
statement ~ q you made in step 3 must be false.
Hence, the other statement q must be true because
This proves the theorem.^
The explanations of indirect proof given by Ander­
son, Garon, and Gremillion^; Mallory, Meserve, and Skeen^®;
^ S m i t h and Ulrich, op. cit.. p. 330.
^ K e e d y et al.. p. 219 .
^Henderson, Pingry, and Robinson, op. cit.. p. 179.
^Anderson, Garon, and Gremillion, op. cit.. p. 192.
■^Mallory, Meserve, and Skeen, op. cit.. p. 138.
81
Moise and Downs'^; and the School Mathematics Study Group^0
is similar to the scheme of Henderson, Pingry, and Robinson.
All of the textbooks which contained the Type II
pattern of indirect proof have now been mentioned.
There
were ten of the thirty-seven geometry texts examined which
contained this pattern of indirect proof.
Criticism of Type II Indirect Proof
A Type II indirect proof is a method of indirect
proof which essentially follows this pattern:
1 . State the contradictory of the conclusion.
2. Reason from the contradictory of the conclusion
until a contradiction is reached.
3. The desired conclusion is then proved.
The impression given is that~ the goal of this method of
proof is to show that the conclusion is true by establish­
ing that the contradictory of the conclusion is false.
But consider the example on the next page of a Type II
indirect proof taken from the text by Shute, Shirk, and
Porter.^
^ M o i s e and Downs, on. cit.. p. 1
60
School Mathematics Study Group, Geometry.
op. cit.. p. 160.
61
Shute, Shirk, and Porter, op. cit.. p. 69.
134. Corollary: Two lines In the tame plane parallel to the same line are
parallel to each other.
a
ll
II ?
e
It
ll
«t
1
------------------6
II
Given: a and b le.
To prove a !! b.
Analysis: Apply the indirect method of proof reductio ad absurdum.
Proof:
1. a and b are cither || or
1. 2 lines in the same plane must be
either |[ or they must intersect.
2. Suppose they are H.
3. Then they will meet in some 3. Non |I lines in the sairie plane in­
tersect.
pt. P.
4. Then there will be 2 lines through
a pt. || to a 3d line.
5. But this is impossible.
5. Through a given pt. only one line
can be drawn [J to another line.
6.
a and b cannot meet and a |J b. 6. Lines in the same plane which
cannot meet however far they
are produced are J.
In this example, does one really show that "a is
parallel to b" is true by demonstrating that "a is not
parallel to b" is false?
Is the isolated statement "a is
parallel to b" a true statement?
The reader can undoubt­
edly visualize instances in which two lines, a and b,
intersect as well as instances in which two lines, a and
b, are parallel.
Hence, the statement "a is parallel to
b" cannot be classified as either true or false.
Type
II indirect proof, as with Type I, focuses attention on
the conclusion of the theorem and fails to explicitly
recognize the important role of the hypotheses of the
theorem.
A careful analysis of the previous example will
indicate that the authors really show that it is the
conditions of the hypothesis, a and b are parallel to c,
along with the contradictory of the conclusion, a is not
parallel to b, which leads to the deduction of a statement
which is the contradictory of an axiom (step l+).
Thus, it
is the joint assertion of the hypothesis and the contra­
dictory of the conclusion which is really shown to be
false, false in the sense that this joint assertion is
inconsistent with the axiom system.
Another fault of the Type II scheme is that it, as
84
does Type I, leads the student to believe that it is the
conclusion that he is proving but the purpose of the proof
is to prove the theorem, not the conclusion.
Type III
Type III shall denote a method of indirect proof
which follows this pattern:
1. State the contradictory of the conclusion.
2. Reason from the contradictory of the conclusion
and the hypothesis until a contradiction is
reached.
3. The desired conclusion is then proved.
The distinction between Type III and Type II is that in
Type III it is explicitly stated that the hypothesis is
used along with the contradictory of the conclusion in
deducing a contradiction, while in Type II the use of the
hypothesis is not mentioned.
Only two of the geometry texts examined contained
a Type III scheme for indirect proofs.
Both texts referred
to the method as simply "indirect proof."
Hart, Schult, and Swain list the following steps
for an indirect proof:
1 . The proof starts by assuming the negative of
the desired conclusion.
2. Logical consequences of the hypothesis and the
assumption lead to a contradiction.
3. Since there must be an error somewhere, we con­
clude that the assumption is false. Therefore the
85
negative of it, the desired conclusion, is
true.^2
Eigen, Kaplan, Krouse, and Rosenfeld also give a
list of steps that an indirect proof is to follow:
1. We state the negation of the conclusion as an
additional hypothesis.
2. From the additional hypothesis we deduce
statements, using postulates, definitions,
theorems already proved true, the original
hypothesis, and rules of logic.
(If necessary,
we may also use true statements from algebra.)
3. From the statements deduced from the additional
hypothesis we obtain one or more statements that
contradict a statement known to be true in our
geometry.
if. We conclude that the additional hypothesis is
false and hence that the conclusion of the
conditional statement is true.°*
Criticism of Tyne III Indirect Proof
In a Type III indirect proof the impression is
given, as in Type II, that the goal of this scheme of proof
is to show that the conclusion is true by establishing that
the contradictory of the conclusion is false.
The inad­
equacy of this approach was discussed in the criticism of
Type II indirect proof.
Another fault of the Type III scheme is that it, as
does Type I and Type II, leade the student to believe that
it is the conclusion he is proving.
-
—
The purpose of the
......
Hart, Schult, and Swain, op. cit.. p. 91*
^Eigen. Kaplan. Krouse. and Rosenfeld. op. cit..
pp. 235 - 236.
86
proof, as has been pointed out, is to prove the theorem,
not the conclusion.
Type IV
Type IV shall denote a method of indirect proof which
follows this pattern:
1. State the contradictory of the conclusion.
2. Reason from the contradictory of the conclusion
and the hypothesis until a contradiction is
reached.
3. The theorem is then proved.
Authors of geometry texts call the Type IV scheme of
indirect proof either "proof by contradiction" or "indirect
proof".
a. Proof by Contradiction
Schacht, McLennan, and Griswold give more than one
method of indirect proof.
They explain "proof by contra­
diction" as follows:
The most common form of an indirect proof is called
proof by contradiction. We make a proof by contra­
diction when we establish the validity of a statement
by showing that from its negation we can deduce
contradictory statements. In this form of proof
it is usually the case that a statement of the form
"p implies q" is negated. The negation of the
statement "p implies q" is the statement "It is
false that p implies q". Since this is not a
convenient statement form to use in proofs, we use
its logical equivalent, which is "p and not-q". We
then proceed to deduce, from "not-q" a conclusion
that contradicts some known fact.64
^Schacht. McLennan, and Griswold, op.cit.. p. 320.
87
These authors should state that they proceed to deduce
from "p and not-q" a conclusion that contradicts some known
fact for in their method it is "p and not-q" they wish to
show is false.
The previous explanation is essentially the same
as the explanation of "proof by contradiction" which is
given by Clarkson, Douglas, Eade, Olson, and Glass.
Al­
though, one notable difference is that Clarkson, Eade,
Olson, and Glass do explicitly state that in order to prove
"p implies q" it is necessary to show that "p and not-q"
implies a contradiction.^
b. Indirect Proof
The explanation of "indirect proof" given by Rosskopf,
Sitomer, and Lechner is essentially the same as the explan­
ation of proof by contradiction" given by Clarkson,
Douglas, Eade, Olson, and Glass.
66
Fehr and Carnahan state the following procedure for
proving a theorem indirectly:
1 . Assume the contradictory of the conclusion in
the theorem.
2. Reason from this assumption (and any of the
hypotheses of the theorem) to a statement known
to be false.
3. Then the theorem consisting of the hypotheses
and the conclusion,is a true statement and the
theorem is proved.®'
^Clarkson et al.. on. cit.. p. 155.
Rosskopf, Sitomer, and Lenchner, on.cit.. pp.46-47.
^ F e h r and Carnahan, op. cit.. p. 103.
88
There were only five of the thirty-seven texts
examined which contained an explanation of the Type IV
scheme of indirect proof.
and Ladd.
The remaining text is by Kelly
Prior to explaining indirect proof Kelly and
Ladd explain the term "principle of contradiction".
Principle of Contradiction. If statements A and B
together imply both the statement T and the state­
ment '•'T, then A implies
and B implies ~A.°°
They then explain "indirect proof" as follows:
From the principle of contradiction, we can
obtain a very useful method of proving theorems in
an indirect way. Suppose that H stands for certain
given conditions or hypotheses, that C stands for a
conclusion, and that we want to prove the theorem
"H implies C". To prove this theorem, we can
consider H and
together. If together theyimply
some statement T, and if T is the denial of a
proven statement or axiom, then H and C together
give us T and ^T.
By the principle of contradiction,
it follows that H — > *'('*•0), that is, H implies
not-not C. But not-not C is just C, so H -*C
as we wished to prove.°9
This explanation of the Type IV scheme of indirect proof
is esentially the "method of inconsistency" which will be
discussed in detail in the next chapter.
It will be argued
in the next chapter that the "method of inconsistency"
is both a suitable and a desirable method of indirect
proof for high school students of geometry.
^®Kelly and Ladd, on. cit., p. 35.
69Ibid.. pp. 35 - 36.
89
Criticism of Type IV Indirect Proof
A Type IV indirect proof is a method of indirect
proof which follows this pattern:
1. State the contradictory of the conclusion.
2. Reason from the contradictory of the conclusion
and the hypothesis until a contradiction is
reached.
3. The theorem is then proved.
In the Type IV scheme of indirect proof it is explicitly
asserted that one reasons from both the hypothesis and the
contradictory of the conclusion until a contradiction has
been deduced.
In the Type IV scheme it is also pointed
out that it is the theorem which is being proven, not the
conclusion.
Thus the Type IV scheme is free of the faults
contained in the Type I, Type II, and Type III schemes.
An example is given on the next page to show the reader
how the steps in the Type III scheme are actually followed
in one form of indirect proof.
The example is taken from
the text by Clarkson, Douglas, Eade, Olson, and Glass.
--------------------------------------
'Clarkson et al.. op. cit.. p. 155
70
T w o distinct linos intersect in at m ost one point.
For convenience, we rewrite this in the form of an im plication.
If tw o distinct lines intersect, then th ey intersect in at m ost one point.
We now wish to show that p A <+-q im plies a contradiction. In other
words, we must show that the statem en t “ tw o d istinct lint ; intersect
and they intersect in more than one p o in t” is false. It is more idio­
m atic English to say, “ T w o distinct lines intersect in more titan cue
p o in t.” We can prove this last statem ent false by show ing that it con­
tradicts a postulate or a previot: ly proved theorem .
PR OO F 8 Y C O N T R A D IC T IO N Suppose tw o distinct ’’r.e>, ’ :.,.r.d m, inter­
sect in more than one point, say points /' nu Q. T ..e a . by P o stu la te - ,
points P and Q are contained in one and o.uy on !i
T ..is c o n ; .
■'..diets
the statem ent in the hypothesis that I and m ..re uistinct lines. I; nee, it
is false that I and m intersect in more d'.aa one p oint; and, therefore,
they intersect in at most one point, which was to he proved.
91
Although all five texts containing the Type IV
scheme use the same steps in constructing their Type IV
indirect proofs, they do not all us 3 the same logic in
justifying this scheme of indirect proof.
The texts by Clarkson, Douglas, Eada and Gloss;
Schacht, McLennan, and Griswold; and Hosskopf, Sitomer, and
Lechner use the same logic in justifying the Type IV
scheme.
The logic used is:
1 . We can show that "p implies q" is true by showing
that its contradictory is false.
2. The contradictory of "p implies q" is "p and
not-q".
3. If a contradiction is deducible. from "p and not-q"
then "p ana not-q" is false.
/f. Therefore a sufficient condition to show that
"p implies q" is true is to deduce a contradiction
from "p and not-q".
The inadequacy of this approach lies in step 2.
Consider
the two statements:
a.
If x is a prime number then x is an
odd number.
b.
x is a prime number and x is not an
odd number.
Are they really contradictories?
to realize that statement
integers,
The student is expected
(a) isintended to mean "For all
if x is a prime number then x is an
odd number'1
92
and that since 2 is a prime number which is not odd,
statement (a) is false.
If the student interprets state­
ment (b) in the same way, which seems only natural, he
will have "For all integers, x is a prime number and x is
not an even number" which is false since 3 is a prime
number which is odd.
Under this interpretation (a) and
(b) are not contradictories since they are both false.
What
is intended by the authors, although it is not explained
to the student, is that "x is a prime number and x is not
an odd number" is to be interpreted as "For some integer,
x is a prime number and x is not an odd number" which is
true since 2 is prime and 2 is not an odd number.
A method of indirect proof based on the logical
notion that "p implies q" is true if its contradictory is
false will be proposed in the next chapter.
will be called the "method of contradiction".
The method
The lack of
clarity concerning the contradictory of an implicative
statement, discussed in the previous paragraph, wall be
avoided by using "p is consistent with not-q” as the
contradictory of "p implies q".
This was suggested in
Chapter III.
Kelly and Ladd use the logical equivalence of "p
implies q" and "p and not-q imply a contradiction" in
justifying the Type IV scheme.
This method will be dis­
cussed in the next chapter where it will be called the
93
"method of inconsistency".
Fehr and Carnahan do not attempt to give a logical
analysis of the Type IV scheme.
But they do use several
examples in the attempt to make the scheme plausible to
the student.
Tyne V
Another method of proof which is called an indirect
method by some authors of geometry texts is the method of
proof by coincidence.
Shute, Shirk, and Porter describe
this method in the following manner:
Coincidence method consists in constructing a
geometric figure which possesses all she properties
in question and then showing that it coincides with
the given geometric figure. Y.'e assume that if two
geometric figures coincide, either possesses all the
properties of the other. That is a line assumes
all the properties of the other.
An illustration of the method of proof by coincidence
is given on the next page.
geometry text by Edwards.
71
The example is taken from the
72
Shute, Shirk, and Porter, on. clt., p. 68.
"^Edwards, on. cit.. p. 78.
9k
THEOREM
■>6. If two linos are parallel, a line perpendicular to one is perpendicular to
the other.
E
Gwen: A B j| C D , E F 1 AB a t G c a d
in te rse c tin '' C D a t il.
Prove: E F L C D .
Plan: In d irect proof. Shov,- th a t E F
is perpendicular to a hoe w hich
coincides w ith C D .
--M
nCASc:.’;>
STATEM! NTS
AB !! C D
E F J. AB a t G and inter.''-cting C D a t H .
Given.
Suppose LM d. E F a t p o in t IL
§-Jt3.
T hen L.M |J AB.
T v -j lines in the f . . . o plane pcrpcad icular to th e «ar.-.e line are parallel
LM coincides w ith C D .
T h ro u g h a "iv ca pc'. •' ■ '
a g iv ...
line <ai!y one line c;.r.
n:..vv,i - .■r.hh ! to the ;::Ven line (.
r).
.". E F ± C D .
I*. ,* .s il jji" i pe,, i *. u ,.ir to .j .'.a
coir . '.icc v !:h C r .
r.
95
It should be noted that the contradictory of the
conclusion, EF is not perpendicular to CD, is not tenta­
tively assumed at any stage of this proof.
proof be classified as an indirect proof?
Should tho
Upton has
suggested that it would be more correct to classify proofs
by coincidence as direct proofs rather than indirect
ox
proofs. ^
In Chapter V a definition of indirect proof
will be proposed which excludes proof by coincidence as
a method of indirect proof.
The method of coincidence is discussed in only six
of the thirty-seven geometry books examined.
All six of
these books classify the method as an indirect method of
proof.^
Type VI
The authors of geometry texts commonly define the
contrapositive of an implication in this manner:
DEFINITION: The contrapositive of a statement of
the form “If P, then Q'th-s a statement of the form
"If not Q, then not
The contrapositive of an implication, then, is formed by
73
^Clifford Brewster Upton, “The use of Indirect
Proof in Geometry and in Life", Fifth Yearbook (V/ashington
D.C.: The National Council of Teachers ox mathematics,
1930 ), p. 121 .
^ T h e six texts were: Avery and Stone, on.cit.,
p.111; Keniston and Tully, on.cit., p.269; Edwards, op.cit
p.78; Shute, Shirk, and Porter, on.cit., p. 60 ; Smith and
Ulrich, op.cit., p.3^1; and Welchons, Krickenberger, and
Pearson, op.cit., p.159.
^ R o s s k o p f , Sitomer, and Lenchner, o p . cit.. p.238.
96
interchanging the contradictory of the conclusion and the
contradictory of the hypothesis.
Since an implication and its contrapositive are
equivalent statements, proof by contraposition consists of
proving the contrapositive of the given implication instead
of the given implication.
An example of a proof by contra­
position is given on the next page.
from Henderson, Pingry, and Robinson.
P.
195.
The example is taken
76
76
'Henderson, Pingry, and Robinson, op. cit..
THEOREM 5 .3 (Alternate Statement): If t w o lines in tho t a m o p la n e aro inlorscctod b y a tr a n s v o r s a l so th a t tho a l te r n a t e interior a n g l o s a r e not o q u a l , th o n
tho linos a r e not p a r a lle l.
n
g iv en :
Lines I and k are in t h e s a m e
plane and are intersected by n
i n A and B respectively.
m(Zx)?* m (Z y )
pro v e:
I X k {I and k intersect)
p p .o o k
1.
2.
3.
4.
5.
6.
7.
:
S ta te m e n ts
1 and k are in the same plane and
are intersected by n in points A and
B respectively.
m ( Z x) t* m ( Z y )
There exists line V through A such
that m(X.x') = m ( Z y )
II k
A £ / and A £ /'
.‘. 1 intersects I'
1 intersects k
R e a son s
1.
Why?
2.
3.
Why?
Problem 5, page 141
4.
5.
6.
Why?
Statements 1, 3
Definition of intersecting lines,
Statements _L_
Theorem 5.4, Statements J L
7.
B y proving the above theorem, you have also proved its contrapositive
because o f logical equivalence. T his contrapositive is Theorem 5.3: I f two
parallel lines in the same plane are intersected b y a transversal, th en the
alternate interior angles are equal.
98
The contrapositive of an implication is discussed
in twenty-nine of the thirty-seven books examined, i.e.,
there were only eight books in which no mention of the
contrapositive could be found in either the index or the
text.^
Only two of the twenty-nine texts which mention the
contrapositive state that proof by contraposition or the
method of contraposition is an indirect method of proof.
nO
The text by Clarkson, Douglas, Eade, Olson, and Glass'
70
and the book, Geometry With Coordinates'-7, by the School
Mathematics Study Group are the two texts which assert
that the method of contraposition is an indirect method of
proof.
Schacht, McLennan, and Griswold state that "proof
by contraposition is frequently classified as a form of
oq
indirect proof."
But they do not commit themselves as
to whether they believe it is an indirect method of proof
or not.
77
''The eight texts are: Anderson, Garon, and Gremillion, op. cit. ; Edwards, op.cit.; Eigen et al.. op.cit.;
Keniston and Tully, op.cit: Seymour, Smith, and Douglas,
op.cit. ; School Mathematics Study Group, Geometry, op.cit.
and Lee R. Spiller, Franklin Frey, and David Reichgott,
Today's Geometry (Englewood Cliffs. N.J.: Prentice-Iiall.
1956).
^Clarkson et al., op. cit., p. 154.
79
'^School Mathematics Study Group, Geometry V/ith
Coordinates, op. cit.. p. 328.
UA
Schacht, McLennan, and Griswold, op. cit., p.323.
99
In Chapter V "proof by contraposition" will be
proposed as a method of indirect proof.
Tyne VII
The concept of partial contraposition is an exten­
sion of the concept of contraposition.
The directions
given by Schacht, McLennan, and Griswold for forming the
partial contrapositives of given conditional statements
are:
The contrapositives of conditional
having more than one condition and only
clusion are formed by interchanging the
of one given condition and the negation
conclusion.
statements
one con­
negation
of the
Thus, for the given implicative statement "If p and q,
then r" the partial contrapositives are "If p and not r,
than not q" and "If q and not r, then not p".
Although
this illustration contains only two conditions in the
hypothesis, this procedure may be used with any finite
number of conditions in the hypothesis.
Each of the partial contrapositives is logically
equivalent to the given implicative statement.
The method
of partial contraposition consists in proving a partial
contrapositive of the given implicative statement which
then establishes the given implicative statement because
of their logical equivalence.
The question of whether
O 1
Schacht, McLennan, and Griswold, on. cit.. p. 176.
restrictions need to be placed on partial contraposition
in order for it to be a valid method of proof will be
discussed in Chapter VI.
The authors of geometry texts
who discuss partial contraposition do not place any
restrictions upon the method of partial contraposition.
The method of partial contraposition is discussed
in the geometry texts by Schacht, McLennan, and Griswold
Dodes^; Brown and Montgomery^; and Welchons, Kricken-.
berger, and Pearson^.
An example of the method of
partial contraposition, taken from the text by Dodes
RA
,
is given on the following page.
^2 Ibid.
®^Dodes, op. cit.. pp. 402 - 403.
^ B r o w n and Montgomery, op. cit.. p. 319.
^Welchons, Krickenberger, and Pearson, op. cit..
p. 151
►
^Dodes, op. cit.. pp. 402 - 403.
82
101
or r
THEOREM
ARTIAl CCNi'-APOSlTlvcS, F o r a n y c o n d itio n a l <i
th e form p A q ~» r, if tin- <<uid itiu n al ininn-, th e n »•»p a rtia l *■(.:>
trupositivcR a re tru e ; « n d if 'h e c o n d itio n al in fah.e, t l n u its p . . i : J
c o n tr a p o s itiv e s arc fa lse . (Alt ernat e form: A c o n d itio n a l o f th e
fo r m p A q —*r h a s th e sam e t r u th v a lu e as its p a rtia l contra;x>s;.
fig. M . ( i l l = ^ 3 ) A
(SO >* DC) — (At
AC)
tivcs.)
It is u n d e rs to o d t h a t the th e o re m is in t h e fo rm /> * q — r. The p r o o f o f
th is th e o re m is d iscu ssed in ih e appendix to C h a p te r 3, w h e re t r u t h ts h ie *
•re explained.
ILLUSTRATIVE J’Kom.F.M. l a eq u ila te ral tria n g le ADC, m ed ian B M
is prolonged to point
and A D is draw n. A D does n o t e q u al AD.
Prove th a t DM and M D a re not equaL
Solution: T h e given conditions and conclusion are:
lllwk fro k .
Given: A D — DC — A C
A M = MC
ABj-A D
Prove:
DM y= M D
In stead o f proving th is, we shall prove the folio contrapositive.
Given: A B AM =
DM =
Prove: A D =
Plan: D raw
:
a d
-
DC = A C
MC
MD
AD
DC. Now A B C D is a parallelogram .
CK0£f»m:
A D = DC.
ad.
It is an easy exercise to p ro v e th a t A H = A D. W h en this is
accom plished, we have proved th a t: I f A D = DC = AC, and
A M ® MC, and D M — MD, then A D = A D . In o th e r words, the
following is a true conditional:
(AD = DC s A C ) A ( A M = M C ) A ( B M = MD ) — ( A D = AD)
Since this is tru e , th e p artial co n tra p o sitiv e m ust also b e tru e bv
the Theorem of P artial C ontrapositives:
( A B = DC = A C ) A ( A M = M C ) A ( A D
A D ) — ( D M /-■ MD)
i
p a rt iai
102
None of the four geometry texts which mention the
method of partial contraposition state that it is an indir­
ect method of proof.
But then, neither do they state that
it is a direct method of proof.
The method of partial
contraposition, with necessary restrictions, will be
proposed as a method of indirect proof in Chapter VI.
Summary
There are five books of the thirty-seven which are
§7
'
.
Of the remaining thirty-two texts, twenty-eight contain
procedures of indirect proof, Type I or Type II, which
contain no mention of the use of the hypothesis.
It is
tacitly assumed that the student will realize from examples
of indirect proof the manner in which the hypothesis is to
be used.
This approach seems to be pedagogically unsound
and inconsistent with the goal of precise expression
claimed by modern mathematics programs.
Many of the textbooks give the impression that the
ultimate objective in a proof is to establish the truth
of the conclusion.
In fact, only eleven textbooks, Type
'These are: Beberman and Vaughan, on. cit.; Charles
F. Brumfiel, Robert E. Eicholz, and Merrill E. Shanks,
Geometry (Reading, Mass.: Addison-Wesley, I960); Kenner,
Small, and Williams, op. cit.; Spiller, Frey, and Reidgott,
op. cit.; and University of Illinois Committee on School
Mathematics, op. cit.
IV or those indicated in Type II, indicate that it is the
theorem, the hypothesis implies the conclusion, that is
proven.
This focus of attention on the conclusion and
ignoring the hypothesis may be detrimental to the student's
understanding of indirect proof.
The Meaning of Indirect Proof is Given By
Contrasting Indirect Proof with Direct Proof
Many of the authors of geometry texts implicitly
contrast direct proof with indirect proof.
An author who
explains "direct proof" and also explains "indirect proof"
is implicitly contrasting the two even if he does not
explicitly point out their differences.
Similarly authors
who give examples of direct proof and also give examples
of indirect proof are implicitly contrasting the two
methods of proof even if their differences are not ex­
plained.
Very few authors explicitly contrast indirect
proof with direct proof.
Authors who do explicitly contrast direct proof and
indirect proof focus their attention on the manner in which
the conclusion of the theorem is used.
Dodes states that:
Unlike the indirect method, which proves a
proposition true by proving other propositions
false, there are many direct methods which deal
onlggWith the proposition we are really interested
OO
Dodes, on. cit.. p. 10.
104
In a similar vein Price, Peak, and Jones state:
The proof is indirect because you do not
examine the correct conclusion, but systematically^
eliminate all of the others until this one remains?
Lewis contrasts indirect proof with direct proof
as follows:
This method of proof is called the indirect
proof, for we do not attempt to move from the given
data directly to the conclusion, but rather we
start with a statement that is contradictory to the
conclusion and somehow justify that this statement
leads but to a logical inconsistency.90
The above three quotations assert that the differ­
ence between a direct proof and an indirect proof is the
manner in which the conclusion is established.
In a
direct proof the conclusion of the theorem is deduced from
the hypothesis whereas in an indirect proof the conclusion
is established by showing that other possibilities are
false.
Summary
Only three textbooks of the thirty-seven geometry
books examined contained statements contrasting indirect
proof with direct proof.
The authors of all three texts
appear to be saying that the difference between indirect
proof
and direct proof is the manner in which the conclu­
sion is established. It has already been pointed out that
------- go--------------------------------------------Price, Peak, and Jones, op. cit.. p. 102.
^°Lewis, op. cit.. p. 228.
105
the purpose of a proof is not to establish the conclusion
but to establish the theorem.
The Meaning of Indirect. Proof is Given
by a Precise Definition
Not one book of the thirty-seven geometry books
reviewed contained a statement on indirect proof which the
author(s) labelled as a definition of indirect proof.
However, several of the texts do have statements on in­
direct proof which apparently are intended as definitions
of indirect proof.
Price, Peak, and Jones state that "A proof in which
91
we use indirect reasoning is called an indirect proof .1'7
Since they do not define "indirect reasoning" this obvious­
ly is not a satisfactory definition of "indirect proof".
Brown and Montgomery also give a definition which
lacks preciseness.
They state:
The conclusion was not deduced directly but
indirectly. This method of reasoning is called an
indirect proof.
Mallory, Meserve, and Skeen say that an indirect
proof of a statement "is a proof that the statement cannot
be f a l s e . U n f o r t u n a t e l y this does not distinguish a
^ Price, Peak, and Jones, on. cit.. p. 102.
^2Brown and Montgomery, on. cit.. p. 295.
^Mallory, Meserve, and Skeen, on. cit.. p. 138 .
106
direct proof from an indirect proof.
A direct proof of a
statement will establish that the statement is true and
hence cannot be false.
Several authors present a scheme of proof which they
call "indirect proof".
It has been pointed out in section
three of this chapter that the Type I, Type II, Type III,
and Type IV schemes of proof are called simply "indirect
proof" by various authors.
But these schemes are really
directions for constructing indirect proofs rather than
definitions of indirect proof so the schemes will not be
repeated again in this section.
In this section the "definitions" of indirect
proof given by authors of geometry texts have been examined.
It has been argued that none of the authors of the geometry
texts examined give an adequate definition of indirect
proof.
Summary
A textbook may convey the meaning;\-of indirect proof
in the following ways:
1 . by giving a definition or an analysis of direct
proof
2 . by an analysis of examples of indirect proofs
3 . by giving a scheme that indirect proofs are to
follow
if. by contrasting indirect proof with direct proof
107
5 . by giving a precise definition of indirect proof.
The thirty-seven geometry books included in this study
have been examined for their treatment of each of these
five stages.
The findings may be summarized as follows:
(a) Over one-half of the geometry texts examined do
not mention "direct proof".
(b) None of the texts give a precise definition of
direct proof that conforms to the manner in v/hich direct
proofs are actually written.
(c) Over 40 per cent of the texts failed to make use
of examples of indirect reasoning in originating their
techniques of indirect proof.
(d) Twenty-seven of the thirty-seven texts, over
70 per cent, contain schemes of indirect proof of a
theorem, Type I and Type II, in which the use of the
hypotheses of the theorem are not mentioned.
(e) Twenty-four of the thirty-seven texts, over 60
per cent, contain schemes of indirect proof of a theorem,
Type I, Type II, and Type III, in which it is claimed
that the conclusion is proven, not the theorem.
(f) A single pattern of indirect proof may be
referred to by several different terms.
Also, a single
term may be applied to more than one pattern of indirect
proof by different authors.
108
(g) The method of coincidence is discussed in six
of the thirty-seven geometry texts.
All six classify it
as an indirect method of proof.
(h) The contrapositive of an implication is dis­
cussed in twenty-nine texts.
In only two texts is it
claimed to be an indirect method of proof.
(i) The partial contrapositive of an implication is
discussed in only four texts.
None of the authors state
that the method of partial contraposition is an indirect
method of proof.
(j) Very few of the authors explicitly contrast
direct proof with indirect proof.
(k) None of the geometry texts contain an adequate
definition of "direct proof".
CHAPTER V
L I T E R A T U R E E X C L U S I V E OP CT30M.3TRY TEXTBOOKS
I ntroduction
W h y teach "indirect proo f " ?
If Indirect proof
is difficult for students to understand,
one possible
s o l u t i o n to the p r o b l e m is simply to exclude methods
of indirect p r o o f from the school curriculum.
In this
chapter the l i terature exclusive of g e o m e t r y texts will
be searched for discussions
on the
importance of indirect
p r o o f and for d iscussions w hich express d i s s a t i s f a c t i o n
w ith Indirect proof.
In the last chapter it was argued that none of
the g e o metry texts e x a m i n e d contained an adequate
d e f i n i t i o n of "indirect proof".
In this chapter the
literature e x clusive of plane g e o m e t r y .texts will be
searched for precise defini t i o n s of "indirect proof".
A d e f i n i t i o n of indirect proof, b a s e d on the definitions
found in the literature,
The analysis
will then be proposed.
of g e o m e t r y texts
in the last chapter
also indicated several, other shortcomings w h i c h are p r e ­
valent
in the usual treatment
of indirect proof.
In this
chapter the literature will be e x a mined for suggestions
109
110
on the teaching of Indirect proof.
In the present chapter,
matics books
then,
logic books,
(other than g e o m e t r y texts),
mathe­
and pertinent
literature in mathematics e d u c a t i o n will be searched
for:
1. Definitions of indirect proof
2. Statements
on the
importance of indirect proof
3. Statements w h i c h express d i s s a t i s f a c t i o n with
indirect proof.
i)-. Suggestions
on the t e a c h i n g of indirect proof.
Definitions
of Indirect P r o o f
In a search for definitions
of "indirect proof"
one
can not help but observe that altho u g h many authors give
d i rections on ho w to construct an indirect proof very few
authors define what they mean by "indirect proof".
Perhaps
this
is beca u s e there appears to be little a g r e e ­
ment on how indirect p r o o f should be defined.
A p a rticular
point of conte n t i o n is whether contrapositive proofs
(Type VI and Type VII of the last chapter)
are indirect or
direct proofs.
One of the ways
in w hich indirect p r o o f is defined
is to characterize an indirect proof as a proof
a c o n t radiction is deduced.
illustrate this approach:
in w h i c h
The f o l lowing three quotations
Ill
Indirect proof (r e d uctio ad- a b s u r d u m ) . An
argument which proves n proposition A by showing
that the denial of A, together with the accepted
propositions
, B_, . .
B , leads to a co n t r a ­
diction.1
1
'
n
A c o nclusion C follows indirectly from an h y p o t h ­
esis H if tho conjunction of the hypothesis and a
denial of the conclusion, (H and ~ C ) , leads to a
contradiction.2
An indirect proof is one in w hich a logical contra­
d i c t i o n is obtained at some stage of the work, such
as an a s s e r t i o n that some p a r t i c u l a r statement is
b o t h true and f a l s e . 3
A proof using contrap o s i t i o n is not an indirect proof by
these three definitions since a c o n t r a d i c t i o n is not
deduced
in a contrapositive proof.
Another method
by Zehna and Johnson.
of d e f i n i n g
Indirect proof is given
The y define "indirect proof" as:
We say that an Indirect proof has b e e n given when
the n e g a t i o n of the c o n c l u s i o n has b e e n used as
one of the hypotheses in the proof.
1
Boruch A Brody, "Logic," The Encyclopedia of
P h i l o s o p h y , ed. Paul Edwards (New York: Ma c m i l l a n Co.
and The Free Press, 1967), Vol. V, p. 6 6 .
2
'Resort of the C o m m i s i o n on Mathematics, App endices
(New York: College Entrance E x a m i n a t i o n Board, 1955T»
p. 119.
3
^Robert L. Swain, "Logic: For Teacher, For Pupil",
T w e n t y - S e v e n t h Y e a r b o o k (Washington, D. C.: National
Council of Teachers of Mathematics, 1963) > P» 296 .
*
^
Peter W. Zehna and Robert L. Johnson, Elements of
Set Theory (Boston: A l l y n and Bacon, Inc., 1962), p. 21.
112
By this definition not only are those proofs in which
a contradiction is deduced considered as indirect proofs
but also contrapositive proofs would be indirect proofs.
For in a contrapositive proof the contradictory of the
conclusion is used as one of the hypotheses.
The definition of indirect proof in the mathematics
dictionary by James and James is interesting.
Indirect Proof. (1) Same as Reductio Ad Absurdum
Proof. (2)" Proving a proposition by first proving
another theorem from which the given proposition
follows.5
The first meaning given in the definition would mean that
in all indirect proofs a contradiction is deduced and
hence the first meaning excludes contrapositive proofs
as indirect proofs.
But the second meaning would include
contrapositive proofs since by proving the contrapositive
of a theorem the given theorem follows.
In deciding upon a definition of "indirect proof"
the crucial question, then, is —
"Should a proof in which
the complete or partial contrapositive of a given theorem
is proven, instead of the theorem itself, be considered as
an indirect proof or as a direct proof of the theorem?"
% l e n n James and Robert C. James (eds.), Mathe­
matics Dictionary: Student*s Edition (Princeton, N. J.:
D. Van No'strand Co., 1959), p. "2 02 . '
The q u e s t i o n is enhanced b y the man y theorems which are
p r o v e n by the method
of contraposition.
In fact
L o w e n h e i m asserts that part i a l c o n t rapositive proofs
"constitute fully
95% of all so-oalled indirect proofs."
This a s s e r t i o n receives support from De Morgan.
stated,
"It is an easily ascert a i n e d fact,
c
De Morgan
that really
7
indirect d e m o n s t r a t i o n is u n c o m m o n in g e o m e t r y
. . . ."
Neither De M o r g a n nor L o w e n h e i m considers contrapositive
proofs as "really" b e i n g indirect proofs.
Tarski considers
a c o n t rapositive p r o o f as b e i n g an Indirect proof but he
does agree with L o w e n h e i m and De M o r g a n on the frequency
of contrap o s i t i v e proofs.
Tarski states that con t r a ­
po s itive proofs " consti tute the most usual type of in-
O
direct proof".
The wr i t e r considers a c o n trapositive proof to be
an Indirect proof for the f o llowing two reasons:
f.
Direct",
Leopold Lowenheim, "On M a k i n g Indirect Proofs
Scripts M a t h e m a t l c a , XI I (June, 1946), p. 133*
7
Augustus De Morgan,
"On Indirect D e monstration,"
L. E. D. Philosophical Magazine, IV (Dec., 1852), p. 436.
Q
Alfred Tarski, Introduction to Logic and to the
Methodology of Deductive Sciences (New York: Oxford ”
University Press, 1946)', p. l8o. '
114
1. It is desirable to be able to classify a proof
as "direct" or "not-direct", i.e. "indirect".
A direct proof starts with the assumptions of
the theorem, and of course, the axioms, which are
always understood to be part of the assumptions of
every theorem, and demonstrates directly that the
conclusion of the theorem must be true.9
That is, in a direct proof the conclusion of the theorem
is deduced from the hypotheses of the theorems, axioms,
definitions, and previously proven theorems.
A contra­
positive proof does not follow this pattern, since the
contradictory of the conclusion is used as one of the
hypotheses in a contrapositive proof.
Therefore a
contrapositive method of proof is not a method of direct
proof.
Since a contrapositive proof is not a direct proof
it should be considered an Indirect proof.
2. Frequently reductio ad absurdum proofs, proofs
in which a contradiction is deduced, are merely disguised
contrapositive proofs.
For instance, consider the
following proof:
Theorem: If xy is an odd. number, then x and y are
both odd.
Proof:
Suppose xy is an odd number and x and y
were not both odd. Then one of them
would be even, say x = 2k. Then xy =
2ky would be an even number. Hence xy
would be both an even number and an odd
number, a contradiction.
^R. B. Kershner and L. R. Wilcox, The Ana t^
Mathematics (New York: The Ronald Press, T 950T 7
115
A careful analysis of the p roof will show that from "x
and y are not both o d d ” it was deduced that "xy is an
even number," but this
g i v e n theorem.
is the contrapositive of the
Thus a l t hough the proof is w r i t t e n in
the form of a reduc t i o ad absur d u m proof,
the same reasoning
will prove the contrapositive of the theorem.
Because
of the similarity b e t w e e n c o n trapositive proofs and many
r e d uctio ad a b s u r d u m proofs,
the methods of contrap o s i t i o n
as w ell as the method of reductio ad absurdum should be
considered as
indirect methods of proof.
The wri t e r will propose a d e f i n i t i o n of "indirect
proof" which is a slight m o d i f i c a t i o n of a d e f i n i t i o n
g i v e n by G o r d o n Glabe.
In his d i s s e r t a t i o n G lobe suggested
this d e f i n i t i o n for "Indirect proof":
An Indirect Method of P roof is any method in
which, instead of p r o v i n g a g i v e n proposition,
(1) an e quivalen t p r o p o s i t i o n is proved, or (2)
the contr a d i c t o r y of the g i v e n p r o p o s i t i o n is
disproved, or (3) a p r o p o s i t i o n equivalent to the
contradictory of the g i v e n p r o p o s i t i o n is disproved.
In Chapter III
it was argued that a p r o p o s i t i o n e quivalent
to a contr a d i c t o r y of a g i v e n p r o p o s i t i o n is itself a
contradictory of the g i v e n proposition.
Hence
(3)
in
G l a b e ' s d e f i n i t i o n is unnecessary.
^ G o r d o n B. Glabe, "Indirect P roof in College M a t h e ­
matics" (unpublished Ph.D. dissertation, G r a d u a t e School,
The Ohi o State U n i v e r s i t y ) .
116
In v i e w of the d i s c u s s i o n p r esented
in this
s ect i o n it is p r o posed that "indirect proof" be defined
as follows:
D e f i n i t i o n . A n Indirect Met h o d of P r o o f is any
method’ in which,' Instead of proving a g i v e n
statement, (1) an e q uivalent statement is proved,
or (2) a c o n t radictory of the g iven statement is
disproved.
By this d e f i n i t i o n c o n trapositive proofs are
proofs.
indirect
This d e f i n i t i o n would classify Type I, Type II,
Type III, Type IV, T ype VI,
chapter as indirect proofs.
and Type VII of the previous
But Type V,
the method of
coincidence,
would not be classified as an indirect
proof.
is not a disa d v a n t a g e of the proposed
This
d e f i n i t i o n of indirect proof since the method of coi n ­
cidence would not be classified as a n indirect proof by
any of the definition::; of indirect p roof m e n t i o n e d in this
section.
I mportance of Indirect Proof
One r e a s o n g i v e n for the importance of teaching
Indirect p roof in the mathematics
class
is the value of
this me t h o d of r e a s o n i n g in n o n - m a t h e m a t i c a l situations.
This
is the p o s i t i o n taken b y H. C. Christofferson.
C h r i s t o f f e r s o n states:
If g e o m e t r y is to be taught largely because
of its inherent possi b i l i t i e s to provide
experiences in the science of reasoning, . . .
11?
then surely it is a mistake to omit or neglect
to emphasize the method of the indirect proof.
Much of the re a s o n i n g which we do in life is
Indirect; therefore m u c h of the value of geometry
must be in its treatment of indirect p r o o f . 11
Alfred Milnqs argues the importance of indirect
pr o o f in a similar manner.
Milnes states:
The process of reductio ad absurdum is of the
greatest importance.
It is the most prominent of
all the methods by which m e n learn those truths
of Mature that are u n i tedly k n o w n b y the name
of s c i e n c e . 12
Another rea s o n g iven by writers for the importance
of indirect proof
is that some theorems
can only be p r o v e n indirectly.
four quotations
in mathematics
In each of the following
the ass e r t i o n is made that some theorems
can only be proved by the
indirect method.
In fact, a few of our basic theorems, and
certain exercises^ can be proved only by the
indirect m e t h o d . 1-'
11
K. C. Christofferson, G e o m e t r y Professionalized
for Teachers (Oxford, Ohio: Pri v a t e l y published, 1953)V
p. 138.
12
(London:
Alfred Milnes, E lementary Motions of Logic
W. Swa n S o n n e n s c h e i n and Co., 18"i?/0 ,"p. 93 .
■^Clifford 3 r e wster Upton, l!The Use of Indirect
P roof in G e o m e t r y and in Life", The F ifth Yearbook
(Washington, D. C.: National Coun c i l of~Teac'ners of
Mathematics, 1930), p. 102.
118 F r e q u e n t l y it is Impossible to p rove a statement
except i n d i r e c t l y .1^
There are some theorems which by their very
.nature preclude the p o s s i b i l i t y of a direct proof.
Often e direct p roof of a cert a i n theorem
presents great difficulty, it sometimes eve n
proves to be i m p o s s i b l e 7 °
A l t hough the previous quotations all assert that
there are theorems which can only be p r o v e n indirectly
whether this is really the case or not
is debatable.
In speaking of r e d u c t i o ad a b s u r d u m proofs,
asserts that,
"But
C. S. Peirce
it is very easy to convert any such
p r o o f into a direct p r o o f T h u s ,
Pe i r c e claims that
any theorem provable at all is p r o vable directly.
L e o p o l d L o w e n h e i m supports the claim made by
P eirc e that
indirect proofs can be converted
into direct
IhRo b e r t
Katz, Axi o m a t i c Analysis: An Introduction
to Logic and the Rea l Num'be'r Sy'st'em (Boston:- D. C. Heath
and“ C o . 7 W , 1 r r ~ 5 5 7
------ -----
16
-'Claire Fi s h e r Adler, M o d e r n Geometry, A n
Integrated First Course, (New York: M c G r a w - H i l T -Boo k Co.,
19^87,■ p.T*:-------16
I.
S. Grsdshtein, Direct and Converse T h e o r e m s ,
trans. T. B o d d i n g t o n (New Y o r k : M a c m i l l a n Co., 1963 )”,
p. ^6.
17
C. S. Peirce, D i c t i o n a r y of Phil o s o p h y and
P s y c h o l o g y , ed. James M ark B a l d w i n (Gloucester, Mass.:
P e t e r Smith, 1957), Vol. II, p. kjk.
119
proofs by actually exhibiting schemes that when followed
18
will convert any indirect proof into a direct proof.
Some support for the claim by Lowenheim and Peirce
that indirect proofs may be converted into direct proofs
is furnished by Aristotle.
Aristotle states:
Everything which is concluded ostensively can
be proved per impossible and that which is proved
per impossible can be proved ostensively.'9
The writer will argue that regardless of which
group is correct, those who claim that some theorems must
be proved indirectly or those who claim that every theorem
can be proven directly, the importance of indirect proof
is assured.
If it is the case that some theorems can only
be proven by an indirect method then the importance of
indirect proof is obvious.
On the other
hand if every
theorem can be proved directly then it will be argued that
the way one would discover direct proofs for some of the
theorems usually proven indirectly is by examining the in­
direct proofs.
Hence, indirect proof would still be an
essential method of mathematical reasoning.
18
19
Lowenheim, op. cit.. pp. 125-139.
7The Works of Aristotle, Vol. I: Great Books of
the Western World (Chicago: Encyclopedia of Dritannica.
1952)7 P. #3.
120
Two examples will be.given to show how an indirect
20
proof may be converted into a direct proof.
The first
example was chosen because of its simplicity, the second
example was selected because it is a classic example of an
indirect proof.
Example 1. If x is a positive integer, then
x + - > 2.
Indirect Proof
Direct Proof
Suppose x is a positive
(x - 1 )2 > 0
integer and
Then x2 - 2x
x + 1 ^2.
Then x2 + 1
So
+ 1 > 0
2x
So x2 + 1 ^ 2x
x2 - 2x + 1 < 0
Thus (x - 1 )2
Thus x + — ^ 2
A *
0 which
contradicts the theorem
that the square of any
number is non-negative.
The key to the direct proof is that the student must
p
realize that he should start with (x - 1 )
0. But there
is nothing in the statement of the theorem
to indicate
to the student that he should start with (x - 1 ) ^ O .
2
The student will not doubt that (x - 1 ) > 0 is true but
20
For methods of converting indirect proofs into
direct proofs, see: Lowenheim, on. cit... pp. 125-139.
121
he will probably be curious as to how anyone would know
to use it to start the proof.
In comparing the direct
proof and the indirect proof the reader may confirm that
the direct proof was obtained by taking the contradictory
of each step in the indirect proof and reversing the steps.
That is, the first step in the direct proof is the
contradictory of the last step in the indirect proof, the
second step in the direct proof is the contradictory of
the next to last step in the indirect proof, etc.
Example 2.
J Z is irrational, i.e. if a and b
are integers which are relatively
prime then f; ^
b
Since the indirect proof of this theorem is so well known
only the direct proof which the writer obtained by con­
sidering the usual indirect proof will be given.
A
preliminary theorem will be proven from which the desired
theorem will quickly follow.
The preliminary theorem can
be incorporated into the body of the proof of the desired
theorem.
It is given separately here in the hope that it
increases the clarity of the proof.
Lemma 1 .
If a and b are integers and a and b are not
2
2
both even then a 4 2b .
Proof Case 1.
Suppose a is not even
2
2
2
Since a is odd this implies a is odd which implies a 4 2k
for any integer k.
122
2
2
Thus, in particular we have a ^ 2b
for the integer b.
Case 2.
Suppose a is even
Since a is even this means by hypothesis that b is odd
2
2
2
which implies b is odd which implies b ^ 2k for any
integer k, now since a is even (in this case) we know
2
that ^ is an integer and hence b2 = 2(^)^ so b2 ^ 2(^-),
so b^ / ^
which implies a2 / 2b2 .
Since these are the only 2 possibilities the proof
of the lemma is established.
'fz is irrational proceeds as
Nov/ a direct proof that
follows:
Theorem
If a and b are integers which are relatively
prime, then £
£
.
Proof
!
1 . a and b are
relatively
prime
2. a and b are
not both even
3. a2 / 2b2
2
4.
{ 2
b
5. § / J T
1. hypothesis
2. Definition of relatively
prime
3. Lemma 1
= *s + ='s are =
5. If two numbers are not
equal, then their square
roots are not equal.
123
Although the writer believes that this direct proof that
[2 is irrational is logically correct, he does not believe
that students will find it as intuitively acceptable as
the usual indirect proofs of the theorem.
In particular
the student may wonder how it was known to make the
inference from "a and b are relatively prime" to "a and b
are not both even, i.e. a and b are not both divisible by
2".
An answer such as "It makes the proof work out" may
not be satisfying to the student.
Another reason given by some writers for the
importance of indirect proof is that sometimes the contra dictory of the conclusion is better known than the
conditions of the hypothesis of the theorem.
In these
instances an indirect proof of the theorem is often easier
than a direct proof.
The following quotations support
this position:
Due to the fact that the definition is a
"negative" one, in the sense that it states what an
infinite set is not, many proofs of the "infinite­
ness" of certain sets proceed by contradiction.
That is, we assume the set is finite and try to show
that a contradiction ensues.^1
When the falsity of the conclusion is the better
known we use reductio ad impossible; when the major
premise of the syllogism is the more obvious, we
use direct demonstration,22
21
Anthony R. Lovaglia and Gerald C. Preston,
Foundations of Algebra and Analysis: An Elementary
Approach (hew York: Harper and Row. '19^6). p. 390.
22
The Basic Works of Aristotle, op. clt.. p. 152.
12Zf
It seems that the contrapositive proposition
is often more accessible than the positive one,
because we know more about the negative terms
than about the positive ones: and we have to
proceed from the more known to the less known.
Some authors mention that indirect proof is
important because of its use in establishing converse
theorems.
Davis makes this point as follows:
The indirect method is often employed in
geometry, especially to establish converse theorems.
Its application is thus quite important and
deserves the earnest attention of the student. ^
Finally, it should be emphasized that indirect
reasoning is a valuable tool of investigation in mathe­
matics.
As is well known, the discovery of non-Euclidean
geometry was the result of an attempted reductio ad
absurdurn.
Dissatisfaction with Indirect Proof
In a recent article in "Scripta Mathematics" Leigh
S. Cauman states, "Indirect proof has been variously
^Augustus De Morgan, on. cit., p. W b l •
^ D a v i d R. Davis, The Teaching of Mathematics
(Cambridge, Mass. : Addison-Wesley Press, 1951 ), p. 1
125
criticized as inelegant, nonintuitive, uninformative,
25
unnecessary, and inconclusive". ^ Each of these criti­
cisms will be examined in this section.
Inelegance
Lowenheim claims that an indirect proof "is an
26
inelegance".
But Raymond Wilder states that proof by
reductio ad absurdum is "in good repute among many
formalists and logicians both for its elegance and its
27
incisive character".
It would appear, then, that it is
merely a matter of personal taste whether an indirect
proof is elegant or inelegant.
Nonintuitive
It has been observed by teachers that "pupils feel
po
that indirect proof 'beats around the bush1".
This
appears to be the same complaint expressed by writers
who say that an indirect proof of a theorem does show that
25
^Leigh S. Cauman, "On Indirect Proof", Scrinta
Mathematica. XXVIIII, No. 2, p. 101 .
^Lowenheim, op. cit.. p. 126.
^ R . L. Wilder, "The Nature of Mathematical Proof",
American Mathematical Monthly. LI (June-buly, 19^/f),
p. 312 .
28
Upton, op.cit.. p. 103.
126
the theorem is true but the indirect proof does not show
why the conclusion of the theorem follows from the hypoth­
eses.
This complaint is expressed in the "Port-Royal
Logic" as follows:
Our minds are unsatisfied unless they know not
only that a thing is but wh y it is. But the why
is never learned from a demonstration which
employs a reductio ad a b s u r d u m . ^ °
It would seem that the reason some students feel that
indirect proofs "beat around the bush" is that the logic
underlying indirect proof has not been made sufficiently
clear to them.
As an attempt to rectify this situation,
methods of indirect proof, based on logic which students
may be expected to understand, will be proposed in the
next chapter.
Lowenheim says that, "No one who is guided mainly
by geometrical intuition is likely to construct an
indirect proof".^®
If Lowenheim were correct then
it
would seem futile to argue that greater emphasis should
be placed on indirect proof in the plane geometry course.
Antoine Arnauld, The Art of Thinking, Port
Royal Logic (New York: Bobbs-Merrill Co., 1964), p. 330.
^Lowenheim, on. cit.. p. 126.
1?.7
But the many indirect proofs that were given in "Euclid's
Elements" cast doubt upon the correctness of Lowenheim's
statement.
It would seem that the first proofs given for
the theorems of geometry would make greater use of
in tuition than the proofs which have been the result of
refinements,
Therefore,
through the ages,
of these original proofs.
the ver y fact that Euclid used methods of
indirect proof so
frequently is sufficient reason to doubt
Lowenheim's statement.
Some of Euclid's indirect proofs
will be examined in Chapter VII.
Uninformative
Some writers point out that once an indirect proof
of a theorem has b een completed all we knov; is that the
theorem is true.
We cannot assert that the subsidiary
deductions made in the p roof are true since they are
based on premises which are shown by the proof to be i ncon­
sistent
(false).
Lo w e n h e i m expresses the situation as
follows:
Almost always, in a direct proof, something
more is proved than merely B; every single
inference d rawn adds to our knowledge, and, in so
far as it is not merely a special case of the
thesis, constitutes a by-product which (though
actually it be dropped from consideration on
completion of the proof) might sometime have been
of further use.
A direct proof always extends our
"
128
knowledge, even when the proof is incomplete.
But with indirect proof the case is otherwise.
Here the actual thesis B is the only gain in
knowledge which the entire proof affords.31
The fact that the subsidiary deductions of a direct proof
are necessarily true must be granted as an advantage of
direct proof over the reductio ad absurdum type of
indirect proof.
But it should be observed that the real
purpose of a proof is to establish the theorem and this
goal is accomplished by an indirect proof.
Unnecessary
It is the claim of some logicians that every true
statement in mathematics can be proved directly; hence,
there is no need for indirect proofs.
This has already
been discussed in the section on the importance of in­
direct proof in this chapter.
Inconclusive
There is a well-known group referred to as
"intuitionists" who reject certain indirect proofs.
In
particular they consider invalid the indirect proofs of
existence theorems which show that an entity exists by
assuming that the entity does not exist and deducing a
Ibid.. p. 126.
129
contradiction from which it is concluded that the entity
must therefore exist. 52
The intuitionists thus reject
the Law of the Excluded Kiddle which states that "either
S holds or the denial of S holds".
xx
Moot mathematicians
and logicians do accept the Law of the Excluded Middle.
Courant and Robbins point out that to accept the position
of the intuitionists would result in "the partial destruct­
ion of the body of living mathematics."-^
Suggestions on Teaching Indirect Proof
In this section suggestions on how to teach indirect
proof as given by authors of books and articles on the
teaching of mathematics vail be discussed.
Johnson and Rising in their "methods" book suggest
that students should get more practice in the use of the
methods of indirect proof.
They express this suggestion
as :
^Raymond L. Wilder, Introduction to the Foundations
of Mathematics (2d. ed.; New York: John Wiley and Sons,
1965), pp. 2^6-262.
^ Ibid.. p. 26.
■^Richard Courant and Herbert Robbins, What is
Mathematics? (London: Oxford University Press, 1 9 ^ 0 7
p. 87.
130
Be sure to provide plenty of experience with
indirect proof, a very important form of proof
often applied - and misapplied - in modern society.
Too often, teachers use indirect proof for only
one or two theorems and fail to give students any
real understanding of the techniques involved.35
The suggestion that students receive more practice in the
use of indirect proof had previously been made, thirtyseven years earlier, by Upton.
•zC.
Johnson and Rising also suggest that a non-mathematical example of indirect reasoning be used to introduce
students to indirect proof.
After the students have
discussed examples of non-mathematical indirect reasoning
then the students "are encouraged to abstract from their
discussion the technique of indirect reasoning.
xn
In his "methods" book Stephen Willoughby suggests
that "contrapositive proofs and indirect proofs should
probably be reserved until pupils acquire an understanding
- ^ D o n a v a n a. Johnson and Gerald R. Rising,
Guidelines for Teaching Mathematics (Belmont, Calif.:
Wadsworth' Publishing Co., 1967), p. 78.
■^°Upton, op.cit.. p. 121.
Johnson and Rising, on. cit.. p. 1 60.
131
of direct proofs and have had some experience with the
contrapositive form of an i m p l i c a t i o n . T h i r t y - s e v e n
years earlier, Upton had stated, "It seems to be
pedagogical common sense to delay the introduction of
indirect proof until the pupil has become quite familiar
■50
with direct proof."^'
Butler and Wren suggest the use of the Greek
letters 0 and 0 to represent the conclusion,of the
theorem to be proven, and the contradictory of the
conclusion.
Thus, if the conclusion of a theorem is
"a is parallel to b" then 0 represents "a is parallel to
b" while 0 represents "a is not parallel to b."
According
to Butler and Wren the advantage in using these Greek
letters is that:
The statement "0 is not true is much more concise
and less confusing than the verbal statement "It
is not true that a is not parallel to b." More­
over, experience has shown beyond doubt that the
use of this symbolic representation of the two
possibilities in setting up the theorem distinctly
^Stephen S. Willoughby, Contemporary Teaching of
Secondary School Mathematics (New York: John Wiley and
Sons, 1967), p.' 209 .
xq
■"Upton, o p . cit., p. 121.
132
increases the students' perception of the essenti­
ally contradictory nature of the two statements ,q
and clarifies for them the mechanics of the proof.
Butler and Wren give the following scheme for
constructing an indirect proof:
1. Set up a pair of contradictory propositions,
one of which you desire to prove true. Select
the latter at the outset.
. 2. Assume, for the time being, that the other of
the two is true and test the consequences by
deductive reasoning to see whether this assump­
tion leads to a contradiction or an inconsis­
tency.
3. If the assumption of step 2 does lead, by correct
reasoning, to an inconsistency or a contradic­
tion, conclude that it was a false hypothesis.
if. Under the conditions of step 3, conclude that
the contradictory proposition selected in step
1, i.e., the one you want to prove true, is
necessarily true, since the only alternative
proposition has been shown to be false.^1
In the examples given by Butler and Wren it is clear that
the pair of contradictory statements to be set up under
step 1 is the conclusion of the theorem and the contra­
dictory of the conclusion.
This is the scheme of indirect
proof called Type II in Chapter IV.
In Chapter IV it was
pointed out that
a) This scheme of indirect proof completely ignores
the use of the conditions in the hypothesis; and
^Charles H. Butler and F. Lynwood Wren, The Teach­
ing of Secondary Mathematics (ifth ed.; New York: McGrawHill Book Co., 1965), p. ^63.
W 1Ibid.. p. 80.
133
b) This scheme of indirect proof leads the student
to believe his goal is to establish that the
conclusion is true, but, of course, the purpose
of the proof is to prove the theorem, not the
conclusion.
The form for an indirect proof given by Johnson
and Rising is:
1 . One of the following (say A, B, C) is true,
2. All but one (say, A and B) are false.
3. The remaining one (in this case C) is necessarily
true.^
This scheme of indirect proof was called Type I in Chapter
IV.
It is subject to the same criticisms as Type II
indirect proof, given in the previous paragraph, but in
addition it also has the disadvantage that the first
step in this scheme is to "list all the possibilities.”
The vagueness of the phrase "list all the possibilities"
was discussed in Chapter IV.
In his "methods" book Willoughby explains the method
of indirect proof as follows:
Indirect proof begins by assuming that the
theorem to be proved is false. Using the usual
inference schemes, we find that assumption leads
to a contradiction (or statement known to be false).
Thus, the negation of our theorem implies a false
statement. By modus tollems or contrapositive
inference, we conclude that the negation of our
theorem is not true and that the theorem itself
^ J o h n s o n and Rising, op. cit.. p. 79.
134
must therefore be true.^
In using the scheme suggested by Willoughby the student
must form the negation (contradictory) of the theorem.
Since the theorem will usually be expressed as an impli­
cative statement, the student must know how to form the
contradictory of an implicative statement which is not an
easy task.
But Willoughby gives no instructions on how
to form the contradictory of an implicative statement;
A
hence his discussion of indirect proof is incomplete.
Summary
In books and articles on the teaching of mathematics
the following suggestions on teaching indirect proof have
been found:
1 . Provide plenty of experience with indirect
proof in order for students to understand this
method of proof.
2. Use non-mathematical examples of indirect
reasoning in the introduction of indirect proof.
3. Delay the introduction of indirect proof until
the pupil has become quite familiar with direct
proof.
4. Use letters to represent the conclusion and the
contradictory of the conclusion.
The suggested techniques of indirect proof given by
^Willoughby, on. cit.. p. 195.
135
authors of recent, published since 1955, "methods" books
were also analyzed.
All of the "methods" books examined,
there were only three, give techniques of indirect proof
which are either incomplete or inaccurate.
An article by Nathan Lazar which does contain
suggestions on the teaching of indirect proof has not
been mentioned in this section.^
This article will be
discussed in detail in Chapter VI, where methods of
indirect proof based on the suggestions in this article
will be proposed.
Summary
In this chapter the literature exclusive of plane
geometry texts has been searched for discussions on
indirect proof.
It has been observed that some authors consider a
contrapositive proof to be a direct proof while other
authors consider a contrapositive proof to be an indirect
proof.
The following definition of "indirect proof" has
been proposed:
An Indirect Method of Proof is any method in
which, instead of proving a given statement,
(1) an equivalent statement is proved, or (2) a
contradictory of the given statement is disproved.
According to the proposed definition a contrapositive proof
^ N a t h a n Lazar, "The Logic of the Indirect Proof
in Geometry: Analysis, Criticism, and Recommendations,"
The Mathematics Teacher. XL (Kay, 1947), pp. 225 - 240.
136
is an indirect proof.
The importance of indirect proof has also been
investigated.
The following reasons for the importance
of indirect proof are given by writers:
1. Indirect reasoning is of considerable value in
non-mathematical situations.
2. Some theorems of mathematics can only be
proven indirectly.
3. Sometimes the contradictory of the conclusion
is better known than the conditions of the
hypothesis.
k. Indirect proof is important because of its use
in establishing converse theorems.
The criticisms against indirect proof were also
investigated in this chapter.
Indirect proof has been
variously criticized as inelegant, nonintuitive, uninfor­
mative, unnecessary, and inconclusive.
The criticism
that indirect proof is uninformative must be granted but
the other criticisms were found to be a matter of personal
opinion rather than fact.
Suggestions by writers on the teaching of indirect
proof were also examined in this chapter.
The following
suggestions have been made for the teaching of indirect
proof:
1. Give the student plenty of experience with
indirect proof.
2. Use non-mathematical examples of indirect
reasoning to introduce indirect proof.
3. Delay the introduction of indirect proof until
the student has become quite familiar with
direct proof.
k. Use letters to represent the conclusion and
the contradictory of the conclusion.
CHAP T E R VI
PROPOSALS FOR INDIRECT PROOF
Introduction
In current mathematics programs the practice of
manipulation without understanding is generally considered
to be undesirable.
In this chapter methods of indirect
proof will be proposed which will make clear to the student
not only how to prove a theorem indirectly but why the
steps taken in the indirect proof do constitute a proof of
the theorem.
In an article in "The Mathematics Teacher" in 1947
Nathan Lazar suggested three methods of indirect proof:
1.
The Method
ofInconsistency
2.
The Method ofContradiction
3.
The Method ofContraposition^.
A discussion of these three methods of indirect proof may
also be found in the dissertation of Gordon Glabe, a
2
former student of Lazar .
The logical basis for the methods
of indirect proof suggested by Lazar and Glabe relies
on the concept of inconsistent statements.
^Nathan Lazar, "The Logic of the Indirect Proof in
Geometry: Analysis, Criticism and Recommendations," The
Mathematics Teacher. XL (May, 1947) pp. 223 - 240.
2
Gordon R. Glabe, "Indirect Proof in College Mathe­
matics" (unpublished Ph.D. dissertation, Graduate School,
The Ohio State University, 1955), PP. 155 - 172.
138
139
The three methods of indirect proof proposed in
this chapter are essentially the three methods of in­
direct proof originally proposed by Lazar.
But the
analysis of the concept of inconsistent statements
presented in this chapter does differ from the analysis
of inconsistant statements given by Lazar and Glabe.
Inconsistent Statements
The logical terms "consistent" and "inconsistent"
have been discussed in Chapter II.
The meaning of these
terms will be analyzed in this section because of their
importance to the methods of indirect proof that will be
proposed in this chapter.
The term "consistent" is sometimes defined as:
Propositions which are such that both can be
true are said to be "consistent", or "compatible",
with each other.3
This means that two statements are consistent when and
only when it is possible for both of the statements to be
simultaneously true. This is the manner in which
"consistent" was defined by Lazar^ and Glabe^.
Each of
the following is an illustration of a pair of consistent
3
Alice Ambrose and Morris Lazeroitfitz, "Fundamentals
of Symbolic Logic (2d ed. rev.; New York: Holt, Rinehart
and WinstonV' 1962 ), p . 89
4
Lazar, op. cit., p. 227.
^Glabe, on. cit., p. 132.
1/fO
statements:
1. a. x and y are Integers such that x Is less
than zero.
b. xy is greater than zero.
2. a. ABC is a triangle which has AC equal to BC.
b. AB is not equal to BC.
3 . a. x is an even integer,
b. x is divisible by 3 .
When the above definition is given for "consistent”
the definition for "inconsistent" is - Two statements ..re
inconsistent if and only if they cannot both be true.
That is, inconsistent statements are such that cannot
possibly be true at the same time.
Examples of statements
which are Inconsistent by this definition are:
1 . a. x is a negative integer,
b. 2x is a positive Integer.
2 . a. yis an even integer.
b. y is not divisible by 2 .
2
3 . a. x is an integer such that x is less than
zero.
b. All equilateral triangles are isoceles.
It may be observed that the third example differs from the
other two in that the inconsistency of the third example
is a result of the fact that ”x is aninteger such
x 2 isless than zero" can never
that
be truewhile.in the
other two examples the inconsistency is the result of the
joint assertion of the two statements.
Should the statements "2 + 2 = 5" ana "the number
^Ambrose and Lazerowitz, o p . cit., p. 89.
of prime numbers is infinite" be considered inconsistent?
Because of the falsity of "2 + 2 = 5" both of these
statements cannot be simultaneously true.
Hence, they
are inconsistent according to the definition of "inconsis­
tent" given in the previous paragraph.
But note that
these two statements cannot both be true solely because
of the falsity of "2 + 2 = 5" and not because of a
relationship that exists between the two statements.
Further, the reader may recall that it was pointed out in
Chapter II that "p implies q"~"is equivalent to "p is
inconsistent with not-q".
So to consider "2 + 2 = 5"
inconsistent with "the number of prime numbers is
infinite" would mean that ”2 + 2 = 5 " implies the number
of primes is finite" is true.
In Chapter II it was argued
that implications such as this should be rejected.
For
these reasons the writer believes that "Two statements are
inconsistent if and only if they cannot both be true"
should not be used as the definition of "inconsistent".
A second way of defining "consistent" is
A set of propositions has consistency (or is
consistent) when no contradiction can be derived from the
joint assertion of the propositions of the set.'
That is, two statements are consistent if it is not possible
7
B o r u c h A. Brody, "Logic," Encyclopedia of
P h i l o s o p h y , ed. Paul Edwards (New York: M a c m i l l a n Co.
and the F ree Press, 1967), Vol. V, p. 61.
H2
to deduce a contradiction from the joint assertion of the
two statements.
When this definition is used for
"consistent" the definition of "inconsistent" is - Two
statements are inconsistent if and only if it is possible
to deduce a contradiction from the joint assertion of the
statements.®
This definition of "inconsistent" is more
restrictive than the definition given previously.
For
although two statements cannot be simultaneously trv.
if
it is possible to deduce a contradiction from the joint
assertion of the two statements, it is possible to have
two unrelated statements that cannot both be true and yet
still not be able to deduce a contradiction from the joint
assertion of the two statements.
A contradiction has been derived whenever it is
possible to deduce both a statement and its contradictory.
But it is a well established practice in mathematics to
also use the term "contradiction" for certain other
situations.
For instance, if it is possible to deduce the
contradictory or contrary of an axiom, definition, or
previously proven theorem, it is claimed that a contra­
diction has been deduced.
Also if from a set of premises
?Bor u c h A. Brody, "Logic," Encyclopedia of
P h i l o s o p h y , ed. P a u l Edwards (New York: M a c m i l l a n Co. and
The Free Press, 1967 ), Vol. V, p. 61.
^I b i d . , p.
9
66.
Ibid., p. 61.
9
1^3
it is possible to deduce a contradictory or contrary of
one of the premises it is asserted that a contradiction
has been deduced.
The writer suggests that "inconsistent'' be defined
as - "A set of statements is inconsistent if and only if
it is possible to deduce a contradiction from the joint
assertion of the statements of the set."
A contrad'ction
may be deduced from a set of statements in each of the
following four ways:
a.) By deducing from all of the statements of the
set a contradictory or a contrary of an axiom,
definition, or previously proven theorem.
b.)
By deducing from all
the statements of the set
both a statement and
its contradictory, or
contrary.
c.) By deducing from all the statements of the set
a contradictory or a contrary of one of the
statements of the set.
d.)
By deducing from all
but one of the statements
of the set a contradictory or a contrary of
the remaining statement.
Examples in which statements are shown to be inconsistent
by each of these methods will be given in the next section
of this chapter.
Method of Inconsistency
In Chapter II it was pointed out that "P implies Q"
is equivalent to "P is inconsistent with not-Q."
The
logloal equivalence of "P implies Q" and "P and not-Q
form an inconsistent set” forms the basis for a method of
indirect proof which has been called "the method of
inconsistency."^0
The method of inconsistency consists
in proving that "P implies Q" by establishing the equi­
valent statement "P and not Q form an Inconsistent set."
Prom the previous section any one of the following methods
may be used to show that a set of statements is an
inconsistent set:
(1) Show that it is possible to deduce from all
of the statements of the set a statement which
is a contradictory or a contrary of an axiom,
of a definition, or of a previous theorem.
(2) Show that from all of the statements of the set
it is possible to deduce two statements which
are contraries or contradictories of each
other.
(3) Show that from all of the members of the set
it is possible to deduce a contradictory or a
contrary of one of the statements of the set.
10
Lazar, op. olt.. p. 235.
145
(4) Show that from oil but one of the statements
of the set It is possible to deduce a contra­
dictory or a contrary of the remaining one.
Each of these four methods will be explained in the order
in which they appear above.
The First Method
The rationale behind this method is that if it is
possible to deduce from a set of statements a contra­
dictory of an axiom, definition, or previous theorem
then we have both a statement R (the axiom, definition,
or previous theorem) and its contradictory not-R which
is a contradiction.
Since a contradiction may be obtained
from the set of statements, the set of statements is
inconsistent.
If instead of deducing a contradictory .of
an axiom, definition, or previous theorem only a contrary
is deduced then it must be observed that from the contrary
of an accepted statement a contradictory of the accepted
statement may be deduced by just one more step in the
deduction.
An example will now be given which illustrates
this point.
The following is an example from geometry in which
a set is shown to be inconsistent by deducing from the
set a contrary of a previously proven theorem.
Theorem.
If one angle of a rhombus is bisected, then the
bisector passes through the vertex of the opposite side.
146
H-^s ABCD is a rhombus
H2 : DE bisects angle D
C:
DE passes through B
Plan: Show that the following set of statements is
Inconsistent:
H , : ABCD is a rhombus
1
H2 : DE bisects angle D
not-C: DE does not pass through B
Proof.
I. Draw diagonal DB
2. In triangles ABD and CBD
a. AB = DC
b. AD = BC
C.
DB = DB
3.
4 A B D £ ABCD
4.
^1 =
5.
JL 2
DB bisects angle D
6 . Since DE also bisected angle D, angle D has two
angle bisectors, DB and DE, which contradicts
the theorem that an angle has exactly one
bisector.
The hypothesis and the contradictory of the conclusion of
the theorem have been shown to be inconsistent since it
was possible to deduce a contradiction from them.
The
desired theorem is therefore established.
In the above proof it may be noted that "an angle
has two bisectors1' and "an angle has exactly one bisector"
are contraries, not contradictories.
But from "an angle
has two bisectors" it may be deduced in one step that "an
angle does not have exactly one bisector'" which is the
contradictory of "an angle has exactly one bisector".
This illustrates that whenever it is possible to deduce a
contrary of an accepted statement it is also possible
to deduce in an additional step a contradictory of the
accepted statement.
The joint assertion of astatement
and its contradictory is a contradictionbut if
from the
contrary of a statement it is always possible to deduce
a contradictory of the statement, why not save a step and
also consider the joint assertion of a statement and a
contrary as a contradiction?
This is a well established
practice in mathematics and will also be adopted by this
writer.
The following is an exar. ole in which a set of
statements is shown to be inconsistent by deducing from
the set a statement which is a contrary of an axiom.
Theorem.
In a plane if each of two lines is parallel to
the same line, then they are parallel to each other.
K^: x, y, and z are lines in the same plane
H2 : x is parallel to z
H^: z is parallel to y
C:
x is parallel
to y
148
Plan. Show that the following set of statements is
inconsistent:
x, y, and z are lines in the same plane
Hg? x is parallel to z
Hy
y is parallel to z
not-C: x is not parallel to y
Proof.
1. Since x is not parallel to y, x intersects y
in some point P.
2 . x is parallel to z and y is parallel to z.
3. From steps 1 and 2, x and y both pass through P
and both are parallel to z.
This contradicts
the axiom that there is one and only one line
through P parallel to z.
4. Therefore the statements H 1, H , H , and not C
X
£
J)
form an inconsistent set and the original
theorem is proven.
The following is an illustration of a proof in
which a set of statements is shown to be inconsistent by
deducing a contradictory of a definition.
Theorem.
If two parallel lines are intersected by a
transversal then the alternate interior angles are equal.
149
H :1 is parallel to m
/
1
Hgil and m are intersected by
n in points B and A
m
C:iy
= i x.
n
Plan.
Show that
, H2, and not C (/y ^ i x )
form an
inconsistent set.
Proof.
1. Since i y
£ i x , with B as a vertex and A3 as a
side construct an angle, i z, such that ^z and
ly
are equal alternate interior angles.
2. By a previous theorem p is parallel to m since
lz
and /y are equal alternate interior angles.
3. It is given that iis parallel to m.
4.
p is parallel to 1 .
5. But p and 1 intersect in B which contradicts the
definition of parallel lines.
The Second Method
The second method of showing that a set of statements
is an inconsistent set is to show that from all of the
statements of the set it is possible to deduce two state­
ments which are contraries or contradictories of each
other.
The rationale underlying this method is that if it
is possible to deduce two statements which are contra­
dictories, then a contradiction is deducible from the set
and the set of statements is inconsistent.
Also if it is
150
possible to deduce two contrary statements from the set
then it is possible to deduce two statements which are
contradictories, a contradiction, and hence the set of
statements is inconsistent.
The following unusual proof that V"2 is irrational
is an example in which a set is shown to be inconsistent
by deducing from all of the statements of the set two
statements which are contradictories of each other.
The
proof given is a rephrasement of a proof given by Sherman
K. Stein .11
Theorem.
If a and b are integers then ^ ^ vT.
H: a and b are integers
C: ® t {2
b
Plan
Show that the following form an inconsistent set:
H: a and b are integers
not-C: — = \T2"
b
Proof
1. It is given that a and b are integers and £ = {"2*.
3.
11
a
2
= 2b
2
Sherman K. Stein, Mathematics': The Man-made
Universe (San Francisco: W. H. Freeman and C o .,'T9'63),
p7T7.
151
k.
By the unique factorization theorem the integer
a may be expressed uniquely as product of primes.
5. Among the prime factors of a the prime 2 appears
a certain number of times, say D times.
6 . When a
is written as
product of primes, the
prime 2 appears D + D times.
Thus 2 appears an
even -number of times among the primes whose
2
product is a .
7. Now in the expression of b as the product of
primes, 2 appears, say, E times. Thus in the
2
expression of b as the product of primes, 2
appears E + E times, an even number of times.
8 . When 2b
is written as a product of primes, 2
appears 2E + 1 times, i.e., an odd number of
times.
2
2
9. From steps 6 and 8 , since a = 2b we have that
the number of appearances of 2 in the prime
2
factorization of a is both odd and even, a
contradiction.
10. Therefore the hypothesis and the contradictory
of the conclusion of the theorem form an
inconsistent set and the desired theorem is
established.
An illustration in which a sot of statements is
shown to be inconsistent by deducing statements from the
152
set which are contraries ox" each other is quoted below.
The example is taken from the college geometry text by
Nathan Altshiller-Court.
Theorem. If two internal
’bisectors of a triangle
are equal, the triangle
is isoceles.
Let bisector BV
equal bisector CW. If
the triangle is not
isosceles, the one angle,
say B, is larger than the
other, C, and from the
two triangles BVC and BCW,
in which BV = CW, BC = BC,
and angle B is greater
than angle C, we have CV
greater than BW. Now
through V and W draw
parallels to BA and
BV respectively. From the parallelogram BVGW we have
BV = WG = CW. Hence the triangle GWC is isosceles, and
^(g + g') = ^ (c + c'). But zg = zb. Hence Z(b + g')
= Z(c + c 1) and therefore g* is smaller than c'. Thus
in the triangle GVC, we have CV smaller than GV, but GV =
BW. Hence CV is smaller than BV/. Consequently, the
assumption of the inequality of the angles B, C leads
to two contradictory results. Hence B = C, and the
triangle is isosceles.12
In this example both the statement "CV is greater than
BV/" and its contrary "CV is smaller than BW" were deduced
from the joint assertion of the hypothesis and the
contradictory of the conclusion of the theorem.
This is
sufficient to establish that the hypothesis is
12Nathan Altshiller-Court, College Geometry
(Richmond: Johnson Publishing Co., £ 92 $), pp. 63-c"6 .
153
inconsistent with the contradictory of the conclusion
and hence the theorem is proven.
The Third Method
The third method of showing that a set of statements
is inconsistent is to show that from all of the statements
of the set it is possible to deduce a contradictory or a
contrary of one of the statements of
the set.
The
rationale for this method is that ifa contradictory
of
one of the statements of a set is deducible from all of
the statements of the set then the deduction leads to the
joint assertion of both a statement and its contradictory
which is a contradiction and hence the set of statements
is inconsistent.
When a contrary of one of the statements
is deducible from all of the statements of a set then the
deduction leads to the joint assertion of s. statement and
a contrary of the statement which means that a contra­
diction is deducible from the set; hence the set of
statements is inconsistent.
It is always possible to
deduce contradictories from a pair of contrary statements.
For instance from "Today is Friday" which is contrary of
"Today is Monday" it is possible to deduce "Today is not
Monday" which is a contradictory of "Today is Monday".
In the following example a set is
inconsistent by deducing from all of
shown to be
the statements of the
set the contradictory of one of them.
Theorem.
If p is a prime and p divides a
p
then p divides a.
15k
H^: p is a prime
2
H2 : p divides a
C:
Plan*
p divides a
Show that the following set of statements is
.inconsistent:
: p is .a prime
O
H2 : p divides a
not-C:
p does not divide a.
Proof.
Since p does not divide a ana p is a prime
we have that p and a are relatively prime.
By an
important theorem in number theory since p and a
are relatively prime there exist integers m and n
such that
pm + an = 1
2
apra + a n = a, multiplying by a.
Now p divides apm since p divides p and p divides
o
2
a^n since by hypothesis p divides a therefore p
divides (apn + a 2n).
But this means that p divides a, since a = apm
2
an.
But p divides a contradicts the assumption that p
does not divide a.
Hence the set of statements
H2 , and not-C is
inconsistent which means that H-i
C and
1 Ho
c. implies
*■
the theorem is proven.
155
It is interesting to note that in this example the
conclusion, C, is deduced from the joint assertion of the
hypothesis and the contradictory of the conclusion,
“1» ^2» anc*
In the following example a set is shown to be
inconsistent by deducing from all of the statements of
the set a contrary of one of the statements of the set.
The example is paraphrased from a proof given by J.
Barkley Rosser.^
Theorem.
If two lines in a plane are cut by a
transversal and a pair of alternate interior
angles are equal, then the lines are parallel
lines AC and BD are
cut by the
transversal AB
C:
AC is parallel
?
to BD
Plan.
Show that the following set of statements
is Inconsistent:
H^: lines AC and ED are cut by the transversal AB
H2 : 11 = 1 2
not-C: AC is not parallel to BD.
13
J. Barkley Rosser, Logic For Mathematicians
(New York: McGraw-Hill Book Co., 1953")» P P . 33-39 •
156
Proof.
1. Since AC is not parallel to BD, AC inter­
sects BD at some point, say E.
2. Lay off BF equal to AE.
3. In triangles ABE and BAF
a. AE = BF
b. Z 3 = I k since 11 = 12 and the supplements
of equal angles are equal.
C.
4.
AB = AB
A ABE =
A BAF (two sides and the
included angle equal)
5.
ZBAF - 1 1
6 . But since / B A F < / 2 we have 11 < 1 2 which
is contrary to LI = 1 2 ,
7. Therefore H1 , H2 , and not C form an
inconsistent set and the desired theorem
is established.
The case in which AC meets BD on the other side may be
carried out in the same manner.
It is interesting to note
that in this example from the hypothesis and the contra­
dictory of the conclusion of the theorem a contrary of a
condition in the hypothesis of the theorem was deduced.
The Fourth Method.
The fourth method of showing that a set is incon­
sistent is to show that from all but one of the state­
ments of the set it is possible to deduce a contradictory
157
or a contrary of the remaining statement.
This is closely
connected with the method of indirect proof known as the
"method of contraposition" which will be diecussed later
in this chapter.
Suppose it is desired to prove a theorem of the
form "H^ H 2
to "H1 H 2
^ C".
Since
H2
-»C" is equivalent
and not-C form an inconsistent set" the plan
in this fourth method is to show H 1 H2
and not-C are
inconsistent by deducing from all but one of these state­
ments a contradictory or contrary of the remaining state­
ment, for instance by showing that
H-j.
H2 and not-C— >not-
That is, the set will be shown to bo inconsistent by
establishing a partial contrapositive of the original
implicative statement.
Unfortunately, there are instances in which a
partial contrapositive of a statement appears to be
nonsensical.
A simple example will illustrate this point.
Consider the statement - "If x is an even number and y is
an even number, then xy is an even number."
For this
statement:
x is an even number
H2 : y is an even number
C:
xy is an even number
Now "If x is an even number and xy is an odd number, then
y is an odd number" is a partial contrapositive,
not-C — »not-H2 of the original statement.
and
Although the
original statement is true, should the partial contra­
positive "If x is an even number and xy is an odd number,
then y is an odd number" be considered true?
The reader
may think that it should not be considered true.
This
would mean that it is not always the case that a statement
in implicative form and a partial contrapositive of the
statement are equivalent.
The problem with "If x is an
even number and xy is an odd number, then y is an odd
number" is that the hypothesis is inconsistent.
The
statement "x is an even number" Is inconsistent with "xy
is an odd number".
In the two examples which will be
given in this section each implicative statement and each
of its partial contrapositives have a hypothesis which is
consistent so the difficulty is avoided.
The question of
whether or not a statement in the form of an implication
is always equivalent to each of its partial contrapositives
will be investigated later in this chapter in the section
on "the method of contraposition".
The following Is an example from geometry in which
a set of statements Is shown to be inconsistent by deducing
from all but one of the statements the contradictory of
the remaining one.
Theorem.
If a line is in the plane of two parallel
lines and intersects one of the parallel lines,
then it also intersects the other.
1 59
io m » an(i n are
^*'3 same plane
H2 : k Is parallel to m
H^: n intersects k
C:
n intersects m
Plan.
Show that the following set of statements
is inconsistent.
: k, m, and n are in the same plane
H2 : k is parallel to m
H^: n intersects k
not C: n does not intersect m.
Proof.
1 . n is parallel to m since n does not intersect
m and n and m are in the same plane.
2. But it is given thatk is parallel to m.
3.
k is parallel to n since they are both
parallel to the same line.
4. But this contradicts the given statement
that n intersects k; so the set is
inconsistent and the desired theorem is
proved.
The following is a simple example from algebra in which a
set is shown to be inconsistent by deducing from all but
I
one of its statements a contrary of the remaining one.
Theorem.
: a < b
If a < b and c < d then a + c < b + d.
Hgt o < d
C:
a + c < b
Plan.
+ d
Show that the following set of statements
is inconsistent:
H^: a < b
H2 : c < d
not C: a + C 2- b + d.
Proof.
1. Since a < b ,
a +e < b + c
2. It is given that a + c u b
+ d
3. From steps (1) and (2 ), b + c >■ b + d
c > d which is contrary of the given
h.
statement that c < d.
5. Therefore the set of statements
E^
and not G are inconsistent and the desired
theorem is established.
Advantages of the Method of Inconsistency
Perhaps the most important argument in favor of this
approach to indirect proof is the simplicity of the logical
basis for this method of proof.
about the bush."
a theorem "H^ H 2
There is no "beating
The student knows that he may establish
—> C" by proving an equivalent theorem,
H 2 E j and not-C form an inconsistent set".
The
conclusion
to be
student is not asked to assume that the
of an implication is
true just for the sake of
false when he believesit
the argument only tolater
1 61
show that the concl u s i o n is true.
trying to show wrong,
the
Instead the student
is
joint ass e r t i o n of the hypothesis
and the contradictory of the conclusion,
what he probably
suspects to be wrong.
The m e t h o d of inconsistency also illuminates the
importance of the h ypotheses
the
of an implication.
It is
joint a s s e r t i o n of the h y potheses and the c o n t r a ­
dictory of the c o n c l u s i o n that
The h y p o t h e s e s of the
is shown to be i n c o n s i s t e n t .
implication are
just as important
as the c o n t radictory of the c onclusion in the met h o d of
inconsistency.
M any textbook explanations
of indirect
proof focus a t t e n t i o n on the c o nclusion and neglect the
hypotheses.
Anot h e r advantage of the me t h o d of inconsistency
is that in this me t h o d no m e n t i o n is made
or the falsity of the conc l u s i o n of the
m a t h e m a t i c a l theorems
implication.
In
such as "If x is an even number
then x 2 is a n eve n number" the conclusion,
number",
of the truth
"x2 is an even
is a p r o p o s i t i o n a l function and cannot be
c lassified as true or false.
not w het h e r "x
The important q u e s t i o n is
is a n e ven number"
but w het h e r " x 2 is an eve n number"
is an eve n n u m b e r " .
is true or is false
is d e d u c i b l e from "x
1 62
The Method of Contradiction
In the method of contradiction1^ the truth of
"H^ H 2
— » C" is established by showing that it has a
contradictory statement which is false.
In Chapter III
it was pointed out that ”H 1 H 2 H ^ — >C" and ,,H 1 H 2
is consistent with not-C" are contradictories.
"H^ H 2
Thus
— »C" may be established by showing that
"Hi H 2 H 3 is consistent with not-C" is false.
that "Hi H 2
Is
Observe
consistent with not-C" is false
whenever "Hi H 2 H^ is inconsistent with not C" is true.
The steps in proving
H 2 H^ — >C" by the method
of contradiction are:
(1) Show that Hi H 2 H 3 and not-C form an inconsis­
tent set.
(2) Conclude that "H^ H 2 H^ is consistent with
not-C" is false.
(3) State that "H^ H 2
»C" must be true.
This
follows from the definition of contradictories.
With the intention of clarifying the use of this
method the following example is offered:
Theorem.
If n is a positive integer then
h + — y- 2 .
n H: n is a positive integer
For a discussion of this method see: Lazar,
op. clt., p. 235*
163
c: n -i—
Plan,
1
>2
n -
Show that H ->C by showing: the falsity of
its contradictory, "H is consistent with
not-C", which will be established by
showing that H and not C are inconsistent.
Proof.
1.
We are given H and not C, i.e., n is a pos­
2.
itive integer and n + — < 2 .
n
n^ + 1 < 2n, multiplying n + £ < 2 by the
n
positive integer n.
3.
n 2 - 2n + 1 < 0 .
4.
;.(n - I )2 < 0 .
5.
But this contradicts the theorem that the
square of any real number is non-negative.
6.
H and not-C are inconsistent.
7.
"H is consistent v.’ith not-C" is false.
8 . /#"H —> C" and the theorem is established.
The logic underlying the method of contradiction
is obviously not as simple as the logic used in the
method of inconsistency.
For this reason the writer
suggests that greater emphasis be placed on the method of
Inconsistency than on the method of contradiction in the
high school mathematics program.
1 6/+
The Method of Contraposition
The use of contraposition has been established in
logic since the time of Aristotle.
Aristotle said,
"If the honorable is pleasant, what is not pleasant is
not honorable, while if the latter is untrue so is the
former, likewise also, If what is not pleasant is not
honorable, then what is honorable is pleasant." ^
The general rule that this example illustrates is:
If P then Q is equivalent to if not-Q then not-P.
The statement "If not-Q then not-P" is called the
"complete contrapositive" of the statement "If P then Q".
The complete contrapositive of an implication is
formed by taking the contradictory of the hypothesis and
the contradictory of the conclusion of the original
statement and then interchanging their positions.
For
example, the complete contrapositive of "If x is an
irrational number, then 2x is an irrational number" is the
statement "If 2x is not an irrational number, then x is
not an irrational number."
Also the complete contrc-
positlve of "if two triangles are congruent, then they
^William Kneale and Martha Kneale, The
Development of Logic (London: Oxford University Press, 1962),
P. 4i.
1 65
are similar” is the statement "If tvio triangles are not
similar, then they are not congruent."
In his work with syllogisms another rule of which
Aristotle was conscious was:
Should ’If p and q, then r' be valid then also
'If not-r and q, then not-p' is valid.^
The statement "If not r and q, then not-p" is called a
partial contraposiitive of "If p and q, then r".
In
general:
A partial contrapositive of a theorem containing
more than one hypothesis and only one conclusion
may be obtained by the interchange of the contra­
dictory of one of the hypotheses with the contra­
dictory of the conclusion.
This means that a theorem having n conditions in the
hypothesis and one conclusion will have n partial contra­
positives .
An illustration of a theorem and its contrapositives
may be helpful.
Consider the theorem.
H 2 — *C: If x / 0 and y ^ 0 then xy ^ 0 .
Its partial contrapositives are:
and not-C — >not-H2 : If x ^ 0 and xy = 0 then y = 0.
and
and not-C — ♦ not-H^: If y / 0 and xy = 0 then x = 0.
1
I.
Bochenski, A History of Formal L q t Ic , tr.
Ivo Thomas (Notre Dame: University of Notre Dame Press,
1961 ), p. 78 .
17
Nathan Lazar, "The Importance of Certain Concepts
and Laws of Logic for the Study and Teaching of Geometry",
The Mathematics Teacher, XXXI (1938), p. 107.
166
The complete contrapositive is:
not-C — not (H^ and \\^): If xy = C then not
(x
0 and y / 0) which may he restated as:
If xy = 0 then x = 0 or y - 0.
It is Important to note that the theorem is true and each
of its contrapositives is true.
Is a theorem always equivalent to each of its
contrapositives?
Earlier in this chapter the statement
"If x is an even number and y Is an even number, then xy
is an even number" and its partial contrapositive "If
x
is an even number and xy is an odd number, then y is an
odd number" were briefly discussed.
equivalent?
Are these statements
The original statement is obviously true but
is its partial contrapositive true?
The difficulty in this
situation is that the hypothesis of the original state­
ment, "x is an even number and y is an even number",
contains more information than is essential to deduce the
conclusion, "xy is an even number."
If x is an even
integer it is not necessary that the integer y be even in
order to assure that xy will be even.
Korner calls an
implication "not pure" when it contains conditions in the
hypothesis which are not essential to the deduction of the
1 67
conclusion.
18
hypothesis".
Lazar calls this "ov e r l o a d i n g the
1q
7
In order to d e t ermine the effect of overloading
the hypothesis
some examples
be examined
on the process of partial c o n traposition
in w hich the h y pothesis
is overloaded will
in detail.
As a first example consider the statement:
If a and b are both
integers and a is odd,
then
a2 ^ 2b2 .
In this example:
: a and b are b oth integers
Hg
:
a
is odd
C
:
a 2 / 2b2
The c o n dition
"a is odd", H 2 ,
proof
is irrational establishes
that
isu n n e c e s s a r y since the
The two p a r t i a l contrapositives
that
of this example of
overloading the hypothesis are:
1.
and not-C — > n o t - H 2 , that
b o t h integers and a
2
2
"If a and b are
2
= 2b , then a is an e v e n i n t e g e r " .
2. H g and not-C — y n o t - H ^ , that
integer and a
is,
is,
"If a is an odd
9
= 2b‘c then a and b are not both integers" .
Xorner, "On S n t a i l m e n t " , Proceedings
Ari s t o t e l i a n S o c i e t y , X L V I I (19^7)* p T T 5 9 .
19
of the
Lazar, "Logic of Indirect Proof in Geometry",
op. c i t . , p. 228.
The second partial contraposit ive has a consistent hy­
pothesis and is obviously a true implication.
But the
truth of the first partial contrapositive is questionable.
In the first partial contrapositive the conditions in the
hypothesis are inconsistent, i.e. "a and b are both
integers" is inconsistent with "a
2
2
= 2b ".
Some people
would consider the first partial contrapositive false
since it is not possible to have the conditions of the
hypothesis satisfied.
20
In fact they would consider any
implication with an inconsistent hypothesis false.
Other
individuals, notably C. I. Lewis, would consider as true
21
any implication with an inconsistent hypothesis.
The
writer has adopted neither of these approaches.
The
writer committed himself in Chapter II to considering an
Implication as true if and only if it is possible to deduce
the conclusion from the conditions (using all of them) of
the hypothesis.
The first partial contrapositive may be
proved as follows:
If a and b are both integers and a
p
p
= 2b , then a
is an even integer.
Proof.
Since a and b are integers it follows that
20
21
See: Glabe, op. clt., p. 135*
See the discussion of strict implication in
Chapter II.
1 69
2
2
Now since a = 2b and
2
2
b is an integer we have that a = 2k for some
a
p
and b
2
are integers.
2
2
integer k, namely k = b . a = 2k implies that
O
a is even which in turn implies that a is an even
integer.
The proof establishes that the first partial contrapositive is true and hence the original implication,
although overloaded, is true and each of its contra­
posit ives is true.
A second example of a true implication with an
overloaded hypothesis is:
Ii x is an even integer and x
2
is an even
integer then
p
x + x' is an even integer.
: x is an even integer
I-I2 : x
C
2
is an e v e n integer
: x + x
2
is an even integer.
It is possible to deduce
from I-L so the hypothesis
contains an unnecessary condition.
The two partial
contrapositives are:
1. H-l and not-C — >not-H2, that is, "If x is an
even integer and x + x
2
is an odd Integer, then x
2
is
an odd integer."
2.
c
and not-C — >not-H.,
that is, "If x
i.
even integer and x + x
odd integer".
2
2
is an
is an odd integer, then x is an
1 70
In each of these partial contrapositlves the hypothesis
is Inconsistent.
These partial contrapositlves appear
especially strange since in each case it is possible to
deduce the proper contradictory of the conclusion from
one of the conditions of the hypothesis.
Nevertheless
in each of these partial contrapositlves the conclusion
is deducible from the joint assertion of the hypothesis.
The first partial contrapositive may be established as
follows:
If x is an even integer and x + x
integer, then x
2
2
is an odd
is an odd integer.
Proof.
2
Since x + x is an odd integer this implies
2
that x + x = 2k + 1 for some integer k. 31"ice
x is an even integer, x = 2m for some integer m.
2
o
Now since x + x = 2k + 1 we have rS - = 2k + x 2
x and since x = 2m this means x = 2k + 1 - 2ra
p
which may be rewritten as x = 2(k - m) + 1 .
Since k and m are integers, k - m is an integer,
say j.
Therefore x
2
=
1 where j is an
integer which by definition means that x ‘ is an
odd integer.
A similar argument will establish the other partial
contrapositive.
In this example of an implication with
an overloaded hypothesis it has been argued that the
171
original implication and each of its contrapositlves
should be considered as true.
A hypothesis may be overloaded in essentially
three ways.
One way of overloading is by adding an
unnecessary condition to the hypothesis which is relevant
to the conclusion but not deducible from the other
conditions of the hypothesis or the axioms of the system.
A condition- is to be considered relevant to the conclusion
if it is possible to actually use the condition in a
deduction of the conclusion.
The first example examined
in detail, "If a and b are both, integers and a is odd
p
then a
p
^ 2b ”, v;as overloaded in this manner.
condition "a is ode." may be used in deducing "a
The
2
2
^ 2b "
but "a is odd" is not deducible from the other condition
in the hypothesis, "a and b are both integers", nor is it
deducible from the axioms of the system.
It may be
recalled that this overloaded implication and all its
contrapositlves were true.
A second way of overloading the hypothesis is
by adding an unnecessary condition to the hypothesis
which is relevant to the conclusion but which is deducible
from the other conditions of the hypothesis or the axioms
of the system.
The second example examined in detail,
2
"If x is an even integer and x is an even integer, then
172
x + x
2
is an even integer", was overloaded in this manner.
It is possible to deduce the conclusion using both the
conditions of the hypothesis although "x^ is an even
Integer" is deducible from "x is an even integer". It
may be recalled that it was also the case that this
overloaded implication and all its contrapositlves were
true.
The third way of overloading a hypothesis is by
introducing a condition which is not relevant to the
conclusion.
A simple example is:
If x is an even
integer and y is an odd integer
then x2 is an even integer.
The condition "y is an odd integer" is irrelevant to the
conclusion "x^ is an even integer".
Although "x is an
even integer implies x2 is an even integer" is true, "x
2
is an even integer and y is an odd integer implies x '
is an even integer" is false since it is not possible
to deduce the conclusion from both conditions of the
hypothesis.
The two partial contrapositlves of this
example are:
1. If x is an even integer and x
2
is an odd integer
then y is an even integer.
2. If y is an odd integer and x 2 is an odd integer
then x is an odd integer.
173
3ach of these partial contrapositives is also false since
the hypothesis contains condition(s) which are not
relevant to the conclusion.
For the examples of implications wiuh an overload­
ed hypothesis that have been examined in detail it has
been shown that whenever 3 is deducible from P and Q.
then not-Q is deducible from ? and not-R.
It will now be
argued that acceptance of complete contraposition forces
acceptance of partial contraposition without restrictions.
Suppose it is possible to deduce R from P and Q,.
This means that it is possible to start with ? and Q and
make successive inferences until the conclusion R is
reached.
Imagine that all the statements derived are
written one below the other in the order in which they
are derived, say S, T, U, V, W, ••• .
Let us further
require that each derived statement is deducible from the
preceding statement in the sequence.
Proofs are not
usually written in this manner so an illustration will
be given to clarify what is intended.
The left column
in the following proof is the way the proof would usually
be written while the right column indicates what is
intended in requiring that each statement derived in the
proof be deducible from the preceding statement in the
proof.
174
Theorem.
If x and y are even integers then
x + y
is an eve n integer.
Proof.
1. x and y are even
integers
(given)
2. Since x is even,
P and Q: x and y
are e v e n Integers
S: x = 2k for
x = 2k for some
some integer k
integer k
and y is a n even
integer
3. Since y is an
T: x = 2k for some
e ven integer,
integer k and y
y = 2m for some
= 2m for some
integer m
integer m
4. F r o m (2) and (3)»
x + y = '2k + 2ra
U: x + y = 2k + 2m
for some
integers
k and m
5.
x + y = 2(k + m)
V: x + y = 2(k + m)
for some
integers
k and m
6.
x + y is a n even
integer
E: x + y is a n even
integer
Observe that in the p r o o f g i v e n in the right column each
statement in the sequence is ded u c i b l e from the pre c e d i n g
statement whereas
statement
in the p roof g i v e n in the left column
(4) is not ded u c i b l e from statement (3).
175
Now if It is possible to deduce R from P and 0,
then there exists a sequence of statements, say S, T,
U, V, W, such that S is deducible from P and Q,, T is
deducible from S, U is deducible from T, V is deducible
from U, W is deducible from V, and R, the conclusion,
is deducible from W.
From a proof that R is deducible
from P and Q, it is always possible to construct a proof
that not-Q is deducible from P and not-R by using complete
contraposition and simply drawing inferences in reverse
order.
That is, from not-R (given) it must be possible
to deduce not-W since R was deducible from W.
Similarly
it must be the case that: not-V is deducible from not-W,
not-U is deducible from not-V, not-T is deducible from
not-U, not-S is deducible from not-T, ana not (P and Q)
is deducible from not-S.
Thus from not-R it is possible
to deduce not (P and Q).
Finally from not (P and Q),
since P is given, the desired conclusion not-Q. follows.
22
The writer has attempted to argue the plausibility
of accepting the logical equivalence of an implication
and each of its contrapositlves even when the implication
contains an overloaded hypothesis.
22
The advantage of this
For a discussion of the type of reasoning
employed in this argument, see: Lowenheim, on. cit.
176
approach is that it is not necessary to determine whether
or not an implication has an overloaded hypothesis before
making use of partial contraposition.
Korner and niabe
permit partial contraposition only when the hypothesis is
neither overloaded nor inconsistent.
implication,
Hg H
The hypothesis of an
. . . Hn — >C, will not be over­
loaded or inconsistent whenever each subset containing all
but one of the statements, H^, H^, H_, . . ., H^, not-C,
23
is consistent.
Thus to show Ghat
H2 — >C does not
have an overloaded or inconsistent hypothesis one must show
that
and H
and that
is consistent,
and not-C is consistent,
and not-C is consistent.
Two statements A and B can be shown to be con­
sistent by constructing an example or a model in which
both of the statements are satisfied.
statements "x is even" and "x
2
For instance the
is even" can be shown
consistent by merely observing that when x = 2 both of
the statements are true.
"x
2
< 0" and "x
2
- 1 <0"?
But how about the statements
These two statements
are
consistent but it is not possible to construct an example
or model in which both of the statements are satisfied
2^See: Korner, op. cit. p. 159; and Glabe, on. cit.,
p. 136.
since "x2 < 0" Is false for all values of x.
177
But it
may be observed that "x2 - 1 < 0 " is deducible from
2
"x < 0", hence, two statements may also be shown to
be consistent by demonstratin," that one of the statements
is deducible from the other.
2k■
The above discussion
indicates that it may be a difficult task to determine
whether or not a particular implication has an overloaded
or inconsistent hypothesis.
It has been argued that an implication is equivalent
to each of its contrapositives.
This may be postulated
as:
Law of Contraposition.
An implication is equivalent
to each of its contrapositives.
In the method of contraposition a theorem is established
by proving a contrapositive of the theorem.
The contra­
positive may be either a complete or a partial contra­
positive of the theorem.
The logical rationale of the
method of contraposition is simple.
Instead of directly
proving the theorem a logically equivalent statement is
proven, namely a contrapositive of the theorem.
In Chapter IV a geometric example was given in which
Any two independent statements are also
consistent, for example "2 is a prime number*' and "Today
is Monday" are consistent.
178
a theorem was established by proving its complete contra­
positive, and also a geometric example was given in
which a theorem was established by proving a partial
contrapositive of the theorem.
3inc.o geometric examples
have already been given two algebraic examples will now
be given to illustrate the use of the method of contra­
position.
In the following example a theorem is established
by proving the complete contrapositive of the theorem.
2
Theorem. If x is an even integer then x is an
even integer.
O
H : x is an even integer
C : x is an even integer
Plan.
To establish the theorem, K — »C, by proving
its contrapositive equivalent, not-C — >not-H.
That is:
GIVEN
not-C: x is an odd integer
DEDUCE
not-H: x2 is an odd integer.
Proof.
Since x is an odd integer we have that x = 2k + 1
for some integer k.
x
2
= 4k
2
Upon which squaring yields
+ ^k + 1 which may in turn be factored
to give x^ = 2 (2k2 + 2k) + 1.
Since 2k2 + 2k is
179
an integer, x2 = 2J + 1 for some integer j.
Therefore x2 is an odd integer.
Since i
has been shown that "not-C — >not-Ii" .h: true,
it thereby follows that its equivalent statement, H — »C
(the original theorem), is also true.
*
The following is an example in which a theorem
is established by proving a partial contrapositive of
the theorem.
Theorem.
If x ^ 0 and y
0 then xy
0.
: x ^ 0
H2 ; y ^ 0
C
: xy / 0
Plan.
To establish the theorem,
— »C, by
proving its partial contrapositive equivalent,
and not-C - » not-H .
That is;
GIVEN
H1 : x £ 0
not-C; xy = 0
DEDUCE
not-H2 : y = 0
Proof.
y = 1 • y, since 1 is the multiplicative
identity
*
= (_ . x)y, since ^ • x = 1 when x r 0.
X
X
18o
i
= — (xy) by the associative property of
A
multiplication
1
= — . 0 since it is given that xy = 0
x
“ 0 since a - 0 ® 0 for any real number a.
Since it has been shown that "H
and not-C->not-H" is
1
2
true, it thereby follows that the equivalent statement,
and H2 — »C (the original theorem), is also true.
Advantages of the Method of Contraposition
One advantage of the method of contraposition is
that it is sometimes easier to construct a direct proof of
a contrapositive of a theorem than it is to construct a
direct proof of the theorem itself.
For instance it would
be difficult, Hennessey claimed it was impossible2^, to
construct a direct proof of "If two lines are cut by a
transversal to form equal alternate interior angles, then
the lines are parallel" following the sequence of theorems
given in "Euclld*s Elements".
But it is a relatively
easy task to construct a direct proof of this contrapositive
of the theorem - "If two lines are cut by a transversal
and the lines are not parallel, then the alternate interior
angles are not equal" - -following the sequence of theorems
2*5
John Pope Hennessy, "On Some Demonstrations in
Geometry", L.E.D. Philosophical Magazine, (Series 4),
Vol. IV (Deo.- 1852)’,’ p.
181
given in "Euclid's Elements".
An advantage of the method of contraposition when
compared to the method of inconsistency is that it is
possible to have a figure which conforms to the
information given in the contrapositive while in the
method of inconsistency it is not possible to have a
figure which conforms to the assumed statements, since
the set of statements is inconsistent.
When teachers and students are aware of the Law of
Contraposition time and energy may be saved since it is
unnecessary to construct a proof of a contrapositive of
a theorem that has already been established.
For instance,
if it has already been proven that "If a is an even
integer, then a
2
is an even integer" then there is no neea
2
to provide & proof of "If a is an odd integer, then a
is an odd integer" since the second theorem follows
immediately from the first because it is a contrapositive
equivalent.
Contraposition also provides an easy method of
discovering new theorems.
Once an implication,
H2
H -5 . . . H„-r>C, has been established then there are
J
n + 1 other true implications which the student may dis­
cover by contraposition, n partial contrapositives and
one complete contrapositive.
A l t h o u g h "If a is an even integer,
e ven integer" and its contrapositive,
integer,
then a is an odd integer",
then a
"If a
2
2
is an
is an odd
are logically equiv­
alent they do evoke d i f ferent thought patterns b y the
student.
In the first theorem the student may be
expected to focus his a t t e n t i o n on e ven integers whereas
in the contrapositive his a t t e n t i o n will be focused on a
p r o p e r t y of odd
integers.
It may be argued that a student
w ill b e t t e r see the r a m i fications
of a theorem after he
has also considered all the contrapositives
theorem.
"Indeed,
contrapositives
of the
it is only after studying all the
of a theorem that a student appreciates
its true significance."
2^
Summary
In this chapter three methods
of indirect proof
were p r o p o s e d - the m e t h o d of i n c o n s i s t e n c y , the method
of contradiction,
Examples
and the method of contraposition.
of indirect proofs u sing e a c h of these methods
w ere given.
The advantages
of the m e t h o d of inconsistency and
Frank B. Allen, The L a w of Contrapositives",
Him' -School M a t h e matics Notes, No. 3 (Boston: G i n n and Co.,
196'-7, p« 5.
133
the method, of contraposition were enumerated.
The method
of contradiction, although logically correct, was not
considered to be as desirable a method of indirect proof
for high school mathematics as the methods of inconsistency
and contraposition.
The logical concepts of "inconsistent statements"
and "overloaded hypothesis" were also analyzed in this
chapter because of their close connection with the
proposed methods of indirect proof.
CHAPTER VII
INDIRECT PROOF IN EUCLID'S ELEMENTS
Introduction
In this chapter examples of indirect proofs in
Euclid's Elements" will be examined in light of the
proposed methods of indirect proof given in the previous
chapter.
The "Elements" has been selected for examination
for two reasons.
The first reason is the historical
importance of the "Elements" and its influence as a
prototype of modern mathematics.
Sir Thomas Heath states,
"Euclid's work will live long after all the text-books of
27
the present day are superseded and forgotten."
A second
reason for selecting the Elements is that the "Elements"
28
contains many indirect proofs.
Illustrations of the Method of Inconsistency
The following proof is the first indirect proof that
27
Sir Thomas L. Heath, The Thirteen Books of
Euclid's Elements (2d ed. rev.; New York: Dover Publication,
1956), Vol. f Tpp. vi-vii.
28
Of the 465 propositions given in Heath's edition
of "Euclid's Elements" indirect proofs are given for 101
propositions. The propositions which are proven
indirectly are listed in Appendix G.
184
185
appears in "Euclid's Elements" ,?'9
If in a triangle two angles be equal to one
another, the sides which subtend the equal
angles will also be equal to o n e another.
Let ABC be a triangle having the angle ABC equal
to the angle ACB;
I say that the side AB
is also equal to the
side AC.
A.
For, if A3 is unequal to
AC, one of them is
greater.
Let AB
.AB the
be cut
A.C the
be greater; and from
greater let DB
off equal to
less;
C
let DC be joined.
Then, since D3 is equal
to AC, and 3C is common,
the two sides DB, BC are equal to the two
sides AC, C3 respectively; and the angle DBC
is equal oo the angle ACB;
therefore the base DC is equal to the bane
AB, and the triangle DBC will be equal to the
triangle ACB
the less to the greater:
which is absurd.
Therefore A3 is not unequal to AC;
it is therefore equal to it.
Therefore etc.
r\
7 7
vv • — J • iJ
•
In this proof Euclid seeks to establish the
implication "If ABC is a triangle and L ABC = *AC3 then
29
The proof given here is taken from: Heath,
op. cit., Vol. I, pp. 255-258.
186
AB = AC".
To establish this implication, as with every
implication, it must be shown that the conclusion is
deducible from the hypothesis.
What has been done by
Euclid is to show that from the hypothesis and. the
contradictory of the conclusion it is possible to deduce
a statement which contradicts the axiom that "The whole
is greater than the part."
From which Euclid tacitly
assumes that it has been established that the conclusion
is deducible from the hypothesis.
Hence, although he
does show that the hypothesis and the contradictory of
the conclusion form an inconsistent set by deducing a
contradiction from them, the proof of the original impli­
cation is logically incomplete until the logical equiv­
alence between the inconsistent set and the implication
has been formally asserted.
In the next example Euclid shows the hypothesis of
the theorem to be inconsistent with the contradictory of
the conclusion by deducing from them a statement which
contradicts a previously established theorem.
\
If a straight line falling on two straight lines
make the alternate angles equal to one another,
the straight lines will be parallel to one another.
For let the straight line EF falling on the two
straight lines AB, CD make the alternate angles
A.EF, EPD equal to one another;
30
The proof given here is taken from: Heath,
on. cit., Vol. I, pp. 308-309.
137
I say that A3 is
p a r allel to CD.
For, if not, AB,
CD when produced
viill meet either
in the d i r e c t i o n
of 3, D or
towards A, C.
Let them be
produced and meet,
in the d i r e c t i o n
of B, D, at C-.
Then, in the triangle GEF, the e x t e r i o r angle
AEF is eoual to the interior ana opposite angle
EFG;
1.16
w h i c h is impossible.
Therefore, A3, CD w h e n produced will not meet
in the d i r e c t i o n of B, D.
S i milarly it can be pro v e d that n e i t h e r will
they meet towards A, C.
But straight lines w h i c h do not meet in either
d i r e c t i o n are parallel; therefore A3 is p a r allel
to CD.
Def. 23
T h erefore etc.
1..3.D.
In this proof,
from the h ypothesis and the contradictory
of the c o n c l u s i o n E u c l i d deduces that an e x t e r i o r angle
of a triangle
is equal to an opposite
interior angle
w hich contradicts a p r e vious theorem to the effect
a n exterior angle of a triangle
that
is greater than either
of the interior and opposite angles.
E u c l i d then tacitly
assumes
this e stablishes that the conclusion of the
theorem
is d e d ucible from the hypothesis
Once a g a i n the p r o o f is l o gically
of the theorem.
incomplete u ntil the
188
equivalence of an inconsistent set and an implication has
b e e n established.
In the d i a g r a m for this p roof it may be observed
that A 3 G and CDG are assumed to be straight lines but
they certainly do not appear to be straight lines.
in the m e t h o d of inconsistency the idea
the diagram
is wrong,
But
is to show that
i.e. to show there is no diagram
w h i c h satisfies b o t h the hypot h e s i s and the contradictory
of the conc l u s i o n b e c a u s e they are inconsistent statements.
Illustrations
of the Method
A l t h o u g h there
of Co n t r a p o s i t i o n
is no m e n t i o n of c o n traposition
"Euclid's El e m e n t s " there are
indirect proofs
in
in the
"Elements" w h i c h c o n t a i n a d i s guised use of the me t h o d of
contraposition.
proofs
In this section two of the
in the "Elements"
indirect
involving a di s g u i s e d use of the
m e t h o d of c o n t r a p o s i t i o n will b e examined.
In the following example the theorem
"H^ H 2
is of the form
—* Ci C2 Cc" wh i c h means that three distinct
implications must be established,
namely "K^ H 0 — >
C±ts>
H g — > C 2 ", and "11^ H g — ► C y , in order to prove the
t h e o r e m . 31
A straight line falling on p a r a l l e l lines makes
31
T h e proof g i v e n here is taken from:
on. c i t . , Vol. I, pp. 3 H - 3 1 2 .
Heath,
139
the alternate angles equal to one another, the
exterior angle equal to the interior avid opposite
angle, and the interior angles on the same side
equal to two right angles.
For let the straight line SF fall on the
parallel straight lines AB, CD;
I say that it makes the alternate angles' AGH,
GHD equal, the exterior angle SGB equal to the
interior and opposite angle GHD, and the interior
angles on the same side, namely BGH, GHD, equal to
two right angles.
For, if the angle
AGH is unequal to
the angle GHD, one of
them is greater.
Let the angle AGH
be greater,
Let the angle BGH
be added to each;
therefore the angles
AGH, BGH are greater
than the angles BGH,
GHD.
C
T-r
F
But the -ngles AGH, BGH are equal to two right
angles.
1.13
therefore the angles BGH, GHD are less than
two right angles.
But straight lines produced indefinitely from
angles less than tv.ro right angles meet; Post. 5
therefore AB, CD, if produced indefinitely,
will meet; but they do not meet, because they are
by hypothesis parallel.
Therefore the angle AGH is not unequal to the
angle GHD, and is therefore equal to it.
Again, the angle AGH is equal to the angle SGB;
1.15 therefore the angle SGB Is also equal to
the angle G H D .
C .H .1
Let the angle BGH be added to each; therefore
the angles SGB, BGH are ecual to the angles BGH,
GHD.
*
C.N.2
190
But the angles EGB, BGH are equal to two right
angles; 1.13 therefore the angles BC-H, GHD are
also equal to two right-angles.
Therefore etc.
Q.E.D.
One of f'he Implications to be established in this proof
is that the hypothesis implies that the alternate interior
angles are equal, i.e. "If the straight line EF intersects
lines AB and CD and if A3 is parallel to CD, then /AGH =
/GHD."
What is done in the proof is to establish the
partial contrapositive, "If the straight line EF intersects
lines AB and CD and if /AGH ^ /GHD then AB and CD will
intersect."
The proof of the partial contrapositive
establishes the original implication assuming that it has
been established that an implication is equivalent to each
of its contrapositives.
Since Euclid did not assert the
equivalence of an implication with each of its contrapositives his proof is logically incomplete.
As a second example of the method of contraposition
in "Euclid’s Elements" consider the followin': demon­
stration:
If two magnitudes be commensurable, and the one
of them be incommensurable with any magnitude, the
remaining one will also be incommensurable with
the same.
Let A, B be two commensurable magnitudes, and
32
-' The proof given here is taken from: Heath,
op. cit., Vol. Ill, p. 36.
191
-
let one of then, A, be incommensurable with any
other magnitude C;
r
i-.___
I say that the
remaining c w , B,
will also be
incommensurable with C.
_
C___________
For, if B is
B_____ ____________
commensurable with C,
~
while A is also commensurable
with 3, A is also commensurable
with C.
x. 12
But it is also incommensurable with it:
which is impossible.
Therefore B is not commensurable with C;
therefore it is incommensurable with it.
Therefore etc.
In this proof Euclid is attempting to establish the
implication, "If A and B are commensurable magnitudes
and A is Incommensurable with C, then B is incommensurable
with C."
In the proof, from "A and B are commensurable
magnitudes and 3 is commensurable with C" Euclid deduces
that "A is commensurable with C", i.e. the implication,
"If A and 3 are commensurable magnitudes and B is
commensurable with C, then A is commensurable 'with C", is
established.
Since a partial contrapositive of the desired
implication has been proven the original implication has
also been established providing it has been previously
accepted that an implication is equivalent to each of its
partial contrapositives.
Since Euclid did not postulate
(or prove) that an Implication is equivalent to each of its
-
192
-
contrapositives his proof is logically incomplete.
In the preceding analysis of examples from the
"Elements" it has been pointed out that the indirect
proofs in the "Elements" are deficient in that Euclid
tells us nothing about the logical principles that he
assumes in his proofs.
It should be added that Euclid
not only fails to mention the logical principles he
uses in his indirect proofs but he also fails to mention
the logical principles used in his direct proofs.
In
describing this deficiency, R. L. Goodstein states:
One of the respects in which the Elements
falls short of the requirement of a modern
presentation of mathematics is in Euclid's
failure to give an account of the logical
machinery by which his deductive system operated.
We are told about the mathematical presuppositions
but not about the logical ones, and are left to
extract Euclid's notion of a valid argument from
the examples he gives us.33
The reader may have observed that every one of
Euclid's proofs which have been given end with "Therefore
etc."
In fact, all of Euclid's proofs end with "Therefore
etc." where "etc." represents a complete restatement of the
theorem.
Current textbooks have not adopted this practice
but it does emphasize to the student that it is the theorem
which has been proved, not the conclusion.
The writer
■^R. L. Goodstein, Mathematical Logic (Leicester,
England: Leicester University Press, 1961), p. 11.
193
recommends that in the early stages of the geometry
course every p r o o f be concluded with a r estatement of
the theorem.
w h e n he
The teacher may discon t i n u e this practice
is convinced that the students kno w that the
p r o o f establishes the theorem and not the conclusion.
Summary
In this chapter indirect proofs
Elements" hav e b e e n examined.
in "Euclid's
A l t h o u g h E u c l i d fails
to est a b l i s h a logical b asis for his methods
of indirect
p r o o f the "Elements" does contain indirect proofs w hich
appear to intuitively be fo l l o w i n g the p a t t e r n of indirect
p r o o f called the "method of inconsistency"
chapter.
There are also indirect proofs
in the p r e c e d i n g
in the "Elements"
w h i c h are really proofs by the method of contraposition.
A c o n s i d e r a t i o n of E u c l i d ’s style of p roof has
r e s u l t e d in the r e c o m m e n d a t i o n that
in the early stages
of the g e o m e t r y course e very proof,
direct
conclude wit h a r e s t a tement of the theorem.
emphasise to the student that
or indirect,
This will
it is the theorem w hich the
p r o o f establishes and not the c o nclusion of the theorem.
CHAPTER VIII
SUMMARY AND RECOMMENDATIONS
Summary oi’ the Study
(1) A survey of plane geometry texts reveals a lack of
unanimity among authors as to the meaning of the crucial
terms "direct proof" and "indirect proof".
Over one-half
of the geometry texts examined do not even bother to men­
tion the term "direct proof".
None of the geometry texts
examined contain an adequate definition ox "indirect proof
(2) A majority of the geometry texts contain schemes of
indirect proof in which the use of the hypotheses of the
theorem is not mentioned.
Also a majority of the texts
contain schemes of indirect proof in which it is claimed
that the conclusion is proven,
not the theorem.
It is
perhaps little wonder that students are bewildered by
indirect proof and fail to understand how a theorem is
proven by following such schemes.
(3) There has been an increase in the use of examples of
indirect reasoning in non-mathematical situations.
Over
one-half of the geometry texts examined contain examples
of indirect reasoning in non-mathematical situations.
The
increase may have been a combined result of Upton's articl
194
195
and the movement toward "teaching f<v transfer".
The use
of non-mathematical examples can help to clarify the logic
used in the early mathematical examples of indirect proof.
(4) The use of contraposition has increased, perhaps
because of the articles by Lazar.
The complete contra-
positive of an implication is discussed in most current
geometry texts, but unfortunately partial contraposition
has been adopted by very few authors.
This is particularly
unfortunate since partial contraposition provides an
excellent opportunity for students to discover new theorems.
(5) Perhaps because of the lack of precise definitions of
"direct proof" and "indirect proof" there is confusion
over whether contrapositive proofs and "forced coincidence"
proofs are direct or indirect proofs.
Some authors treat
them as though they were direct proofs others consider them
to be indirect proofs.
(6) It is recommended that "indirect proof" be defined as:
An Indirect Method of Proof is any method in which,
instead of proving a given statement, (1) an
equivalent statement is proved or (2) a contradictory
of the given statement is disproved.
By this definition contrapositive proofs are indirect
proofs and proofs by "forced coincidence" are direct proofs.
(7) Because implications play a central role in indirect
proof the nature of implication was also examined in this
study.
The three major notions of implication are called
196
"material implication",
"entailment".
"strict implication", ana
The majority of the mathematics texts that
define "implication" use the material implication approach.
Material implication has two major deficiencies when used
as the definition of implication in mathematics:
a) In a true material implication there does not
need to be any connection between the hypothesis
and the conclusion.
This is evidenced by the
"paradoxes of material implication".
b) In a material implication, ?-*Q, P and Q are
propositions but in most mathematical implications
the hypothesis and conclusion a're propositional
functions.
(8) The logical concept of "contradictories"
indirect proofs.
is used in
Contradictories are usually defined as
two statements which cannot both be true and cannot both
be false.
This would strangely mean that "2 + 2 ^ 4"
and "every equilateral triangle is isosceles" are contra­
dictories since both cannot be true and both cannot be
fa l s e .
( 9) Writers give the following reasons for the importance
of indirect proof:
a) Indirect reasoning is of considerable value in
non-mathematical situations.
b) Some theorems can only be proven indirectly.
197
c) Sometimes the contradictory of the conclusion
is better known than the conditions of the
hypothesis.
a) Indirect proof is important because of its use
in establishing converse theorems.
(10) Indirect proof has been variously criticized as
inelegant, nonintuitive, uninformative, unnecessary, and
inconclusive.
(11) Writers give the following suggestions for teaching
indirect proof:
a) Give the student plenty of experience with
indirect proof.
b) Use non-mathematical examples of indirect reason­
ing to introduce indirect proof.
i
c) Delay the introduction of indirect proof until
the student has become quite familiar with direct
proof.
d) To clarify an indirect proof use letters to
represent the conclusion and the contradictory
of the conclusion.
Recommendations for Teaching Indirect Proof
(1) Because of the central role implication plays in
indirect proof it is imperative that the student understands
the nature of implication.
It is recommended that impli­
cation be defined as: P imolies Q, means that Q is deducible
from P.
Examples of simple deductions should be given to
198
help clarify this definition to the student.
(2) The logical terras "contradictories '1 and "inconsistent
statements" must he explained to the students prior to a
formal discussion of methods o ' indirect proof.
The
following definitions are recommended:
a) The proper contradictory of a statement, p, is
the statement "it is not the case that p".
b) A statement, A, is a contradictory of a statement,
B, if and only if A is equivalent to the proper
contradictory of B.
c) A set of statements is inconsistent if and only
if it is possible to deduce a contradiction from
the joint assertion of the statements of the set.
Simple examples should be provided to help clarify for
the students the meaning of each of these terms.
(3) The method of inconsistency is recommended as a method
of indirect proof suitable for piano geometry students.
It is further recommended for mathematics students at all
levels.
The most important argument in favor of the method
of inconsistency is the simplicity of the logical basis for
this method of proof.
The student knows that he may estab­
lish a theorem "H^ I^jH^ — »C" by proving an equivalent
theorem,
and not-C form an inconsistent set".
is recommended that the equivalence'between"H-,
and
It
— >C"
and not-C is an inconsistent set" be clearly
stated as an axiom on a par v/ith the other axioms of the
course.
(Zf) The method of contraposition is also recommended as a
method of indirect proof appropriate for use in mathematics
at all levels.
The Law of Contraposition, which states that
an implication is equivalent to each of its contrapositives,
should be accepted as an axiom in the mathematics course.
Students should consider examples to convince themselves
that it is reasonable to accept the Law of Contraposition.
The logic of the method of contraposition is quite simple.
The student should realize that in the method of contrapos­
ition instead of proving the stated theorem he is going to
prove an equivalent statement, a contrapositive of the stated
theorem.
Once he has proven a contrapositive of the theorem
then the theorem follows immediately by the Lav; of Contra­
position .
(5) Contraposition should also be emphasized as a method of
discovering new theorems.
An implication with n conditions
in the hypothesis has n + 1 contrapositives all of which
will be true if the implication is true.
Students should
be forewarned that if an implication has an overloaded hypo­
thesis then some of the contrapositives may appear to be
"odd”.
(6 ) The method of contradiction is not recommended in gen­
eral for high school mathematics.
But the method of contra­
200
diction is logically correct and some teachers may wish to
use this method with superior students as an optional topic.
The method of contradiction consists of establishing an
implication by showing that its contradictory is false.
It
should bo observed by teachers and students that " P - * not-Q"
is not a contradictory of "P — >Q".
A useful contradictory
of "P-— >Q" is "P is consistent with not-Q".
"P is consis­
tent with not-Q" can be shown to be false by showing that
"P is inconsistent with not-Q" is trve.
Thus in the method
of contradiction it must be shown "P is inconsistent v/ith
\
not-Q" is true from which it follows that "P is consistent
v/ith not-Q" is false from which it follows that its contra­
dictory "P — S>Q", the desired implication, is true.
Recommendations for Further Study
(1 ) In Chapter V the question arose as to whether every true
theorem can be proven directly.
Statements may be found in
the literature to the effect that there are theorems which
can only be proven indirectly.
But it is also possible to
find statements in the literature to the effect that any
indirect proof may be converted into a direct proof.
An
investigation to determine whether indirect proofs are log­
ically necessary would be a welcome addition to the litera­
ture on indirect proof.
(2) A statistical study using the methods of indirect proof
recommended here in some classes and the traditional
methods in control groups would be welcomed.
(3 ) It is possible that there are indirect proofs in
higher mathematics which are not an unconscious use of
either the method of inconsistency or the method of
contraposition.
An investigation to determine whether
the methods recommended here are applicable to all
indirect proofs in higher mathematics might be worthwhile.
APPENDIX
Appendix A
Some College Textbooks on Ms the. .atics Which Use a
Truth Value Approach to Implication
Allendoerfer, Carl B. and Oakley, Cletus 0. P r inciples
of Mathematics. 2d ed. revised.
New York:
McGraw-Hill Book Co., 1963, P» 21 •
Bush, George C., and Obreanu, Philip E.
B a s ic Concepts
of Mathe m a t i c s . New York: Holt, Rinehart and
Winston, 19&57 P» 7Dinklnes, Flora.
Elementary C oncer-:.:? o^ Modern Mathe ­
matics . New York: Applebon-^entury-Crofts, 19&A,
p. 2 oT.
Eves, Howard and Newsome, Carroll V.
An Introduction to
the Foundations and Fundamental Goncep ,s of Mathe­
m a t i c s . 2d ed. revised.
New York:''HoltV"Rinehart
and Winston, 1965, P» 277*
Fine, Nathan J. Introduction to Modern M athematics.
Chicago: Rand McNally and CoV, 1*962, p. 18.
Gemignani, Michael C. Basic Concepts of Mathem atics and
Logic.
Reading! M a s s .: A d d i o n - W e sley Publishing
Co., 1968, p. 33.
Groza, Vivian Shaw.
A Survey of Mathematics: Elementary
Concepts and Their Hi's tori oal Develop l e n t . New
York: Holt*,’ Rinehart ana Wins Yon, 1*9*6*8’, p. 111 .
Haag, Vincent H. and Western, Donald W. Introduction to
C ollege Mathematics. New York: Holt, Rinehart and
Winston, 195B/ p. 13 .
Horner, Donald R. A Survey of College Mr ^hematics. New
York: H o l t ,“TTinehart arid~WinstonT i"967, p. 2A.
Jones, Burton W. Elementary Concepts of Mathematics. 2d
ed. revised.
New York: Macmillan Co., 1963, p . 16.
202
203
Keedy, Mervin L. A Modern Introduction to Baric Mathe­
matics. Reading, H a s s .: iTud*i 3oh-VJesley Publishing
Co., 1963, p. 97.
Laffer, Walter 3. Math e m a tics f-~: G~ -ar a? Ecu;"'t i o n .
Belmont, C a l T f .: Dickens H'Ph;,,nn Y g Co.,
66 .
p.
Lightstone, A . K . Symbolic Logic ana the Leal Number S y s t e m :
An IntroduoVion to oHe "Pound at ions of ?himbe:r
Systems"
New York: Harper'and Row, fc
J 6 * 5 pT-''<■.
Lovaglis, Anthony R. and Preston, Gerald C.
Fcundc, .ions
of Algebra and Analysis: An E lementary Accroach.
New York: Harper and Row, 1 9 o 6 , p . 5.
Me serve, Bruce E. and Sobel, Max A. Introduction to Vis thematics.
Englewood Cliffs, W . J .: "Prentice-Hsll,
Inc., 196^, p. 182.
Moore, Charles S. and Little, Charles E. B asic Concents
of Mathematics.
New York: McGraw-Hill Book Co.,
T 967Y p._ 2T;
Nahikian, Howard M. Tonics in M o d ern Mathematics.
York: Macmillan Co., I900 , p . 73.
Nev;
Sanders, Paul.
Elementary Mathematics: A Logical tnoroach.
Scranton, Pa.: International Textbook Co., 1963 , p . 12.
Stoll, Robert R . Sets, Lonlc and Axiomatic T h eor i e s .
San Francisco: W. H. Freeman and Co., 13*61, n .62.
Witter, G. S. Mathematics: The Study of Axiom Systems.
Nev; York: Blaisdell Publishing Co'., 1 9 6 1 / p. 6 .
Wilierding, Margaret F. Elementary M a thematics: Its
Structure and C o n c e p t s . New York: John Wiley and
Sons, I966 , p. 32.
VJiHerding, Margaret F. and Hayward, Ruth A. M a thematics:
The Alphabet of S c i e n c e , .lev; York: John Wiley ana
S ons^ T f S E T T . l'H
Yandl, Andre L. Introduction to IJniyersity Mathematics.
Belmont, C a T i T 7:^0"ick'ensoh PuoTisning Co., 19*5’?, p. 6 .
Young, Frederick H. The Nature of Mathematics.
John Wiley and Sons, i'9'5'5, p. 119 .
New York:
Append5.x 3
Plane Geometry Textbooks Analyzed in liis Study
Anderson, Richard D., Garon, Jack VJ., and Greraillion,
Joseph G. School Mathema t ics G e o m e t r y . Boston:
Houghton lixTTin" 'Co~T, f9o’63
Avery, Royal A ., and Stone, William C. Plane G e o m e t r y .
Boston: Allyn and Bacon, I 96A.
Beberman, Max, and Vaughan, Herbert E. High School Hathematics: Course 2. Boston, D. C."Ifea'fYT’ahd'Co.,
19^ ------------Birkhoff, George David, and Beat ley, Ralph.
B a s ic
G e o m e t r y . 3rd ed. Hew York: Chelsea, 1959*
Brown, Kenneth E., and Montgomery, Gaylord C. Geometry:
Plane and Solid.
River Forest, 111.: Laidlaw
Brothers, 1962 .
Brumfiel, Charles F., Eicholz, Robert E., and Shanks,
Merrill E. G e o m e t r y . Reading, Mass.: AddisonWesley, i960 .
Clarkson, Donald R., et al. Geometry.
N. J.: Prentice-Hall, W & T .
Dodes,
Irving Allen.
G e o m etry.
Brace, and World, 1’953>7
Englewood Cliffs,
New York: Haroourt,
Edwards, Myrtle. First Course in G e o m e t r y .
Expos it ion~Pres's ,''”1'9£33
Nev; York:
Eigen, Lev;is D., et a l ., A d v a n cing in Mathematics:
G eom e t r y . Chicago: Science Research AxTs'o'ciates,
Fehr, Howard F., and Carnahan, Walter H. Geometry.
D. C. Heath and Co., 196I.
Fischer, Irene, and Hayden, Dunstan.
Allyn and Bacon, I965 .
C-eometry.
3oston
Boston:
205
Goodwin, A. Wilson, Vannetts, Glen D . , and Fawcett,
Harold ?. O e o m e t r y , A 7~nif'ed. Course.
Columbus,
Ohio:
Charles ^ r ’ner'rill "k oil: , l9~oy.
Hart, Walter N ., Schult, V e r y l , e -a Sv." in, Henry.
How
Plane ' eometry sac Suupl
n V . 3 a :t
D. ‘CV*
Hoa to and C^: , i 9 o .
Henderson, Kenneth 3., Pingry, Robert S., and Robinson,
George A.
Modern G e o m etry: Its Structure ana
F u n c t i o n . Nev/ 1'orlc: McGraw-Hill jtToo'k “C o ., 1962 .
Jurgensen, Ray C., Donnell?/, Alfred J., and Dolciani,
Mar?/ P. Modern Geometry: Structure and M e t h o d .
Boston: Houghton Mifflin Co., I'Jop.
Keedy, Mervin L., et a l . Exploring G e ometry.
Holt, Rinehart and Winston, ~i£9o?.
New York:
Kelly, Paul J., and Ladd, Norman S.
F u ndamental Mathe­
m a t ical Structures: G eo m e t r y . Chicago: Scott,
Foresman ana Co., 19*551
Keniston, Rachael P., and Tull?/ Jean.
Hisrh School
Geometr y . Boston: Ginn and Co., f/oOT
Kenner, Morton R . , Small, Dwain 2., and W i l l i a m s , Grace N.
Concents of Modern M athematics, Boo-: 2.
Hew York:
American Book Co., T y S j .
Lear?/, Arthur F., and Shuster, Carl M. P l a n e G e o m e t r y .
New York: Charles Scribner's Sous, 19531
Lev/is, Harry.
G e o m e t r y , A Contemnorary C o urse.
N. J.: DTX/an Nootrend Co., 19 o 4.
Princeton,
Mallory, Virgil S., Reserve, Bruce S., and Skeen, Kenneth
c • A First Course in C-ec.ne tr •. Syr a cus e , H . Y .,:
L. W. Singer Co.', 19593
"
,
Moise, Ed w i n E . , and Downs, Floyd L. G eometry . Palo
Alto: Addis on-Wes ley Publishing Co., 1*967 .
Morgan, Frank M., and Zartman, Jane.
G e omet r y : PIaneSolid-Coordinate.
Boston: Houghton I'iiTlin Co.,
19535
Price, H. Vernon, Peak, Philip, and Jones. 7 nilip S. Mathe­
matics - An Integrated Series: Book f :q. Nev; York:
Harcourt, Brace and World, i9’55l
*~
206
Rosskopf, Myron P., Sitomer, Harry, and Lenchner, George.
Modern M a t hematics, G eoinr or;/. Morristown, N. J.,:
STlver Luraett Co., T'Jbb.
Schacht, John P., McLennan, Roderick G., and Griswold,
Alice L. Contemn or any Ge.-r._eCy;.-. New York: Holt,
Rinehart and Winston, f962.
School Mathematics Study Group.
Geometry.
Yale University Press, 1961.
New Haven:
School Mathematics Study Group.
Geometry with Coordinates.
New Haven: Yale University Press, fsTS^
Seymour, P. Eugene, Smith, Paul J., and Douglas, Edv.un C.
Geometry for High Schools.
New York: The Macmillan
G o . , 1958• ~
'
Shute, William G . , Shirk, William W., and Porter, George
^ • Geometry - Plane and 3o l i d . New York: American
Book Co .",T9 6oC
Smith, Rolland R., Ulrich, James P., and MacDougall,
Alice K. G e ometry: A Mo dern Course.
New York:
Harcourt, Brace and World, ”i9*o£.
Spiller, Lee R., Prey, Franklin, and Reichgott, David.
T o d a y 8s G e ometry. Englewood Cliffs, N. J . :
Prence-ffail',' 19’5 6 .
University of Illinois Committee on School Mathematics:
High School Mathem a t i c s : Unit 6_ - G eometr y .
Uroana: University o f T Y l i n o i s Press, f9o0.
Weeks, Arthur VI., and Adkins, Jackson B. A C o u rse in
Geometry: Plane and Solid.
Boston: Girin and~~Co.,
I 9 S 1 ~ ------------Welchons, A. M., Krickenberger, W. R., and Pearson,
Helen R.
Plane Geometry.
Boston, Ginn and Co.,
1961.
207
Appendix C
Texts which Use Examples of Tr.-ilrecReasoning in Life
Situations in the Motivation of Moth ods of '.fnuirect Proof
(Complete References are given in Appendix B)
Anderson, Garon, and Gremilllon,
At-ery end Stone,
on. c i t . , p. 190.
on. c i t . , pp. 109-110.
Birkhoff and Beatley,
on. c l t . ,
Brown and Montgomery, on. c i t . ,
pp. 33-35pp. 295-296.
Dod.es, on. c i t . , pp. 8-9Goodwin, Vannatta, and Fawcett,
on. c i t . , p. 23^.
Jurgensen, Donnell?/, and Dolciani,
Keniston and Tally,
Lewis,
on. c i t . , pp. lo^i— 165-
on. cit . , pp. 217-268.
on. c l t . , p. 225 .
Moise and Downs,
on. cit . , pp. 153-15!: •
Morgan and Zartmsn,
on. clt . , pp. 138-139•
School 'Mathematics Study Group,
(Geometry, on. c i t . , p. 160.
School Mathematics Study Group,
Geometry with Coord m a t e s ,
on. c i t . , p. 12.
Smith, Ulrich, and MacDougall,
Se?/mour, Smith, and Douglas,
Welchons, Krickenberger,
on. c l t . , pp. 329-330.
on. cit., pp. 95-96.
and Pearson,
o p , c i t .., pp. 13C-131.
208
Appendix D
•Texts which Contain Examples of Indigent Reasoning in
Life Situations, But the Examples are Not Used in the
Formulation of Methods of Indirect Proof
(Complete References may be found in Appendix R)
Clarkson, et al., op. cit., p. 157.
Fohr and Carnahan, on. cit., p. 103.
Hart, Schult, and Swain, on. cit., p. 9 1 .
Keedy, et al., on. clt., p. 221.
Mallory, Meserve, and Skeen, on. clt., p. 139.
Price, Peak, and Jones, on. cit., p. 103.
Weeks, and Adkins, op. cit., p. 127.
Kelly and Ladd, on. clt., p. 36.
Leary and Shuster, op. cit., p. ^89.
Schacht, McLennan, and Griswold, op. cit., p. 322.
Shute, Shirk, and Porter, op. cit., pp. 67-68.
20S
Appendix 3
Texts which Use Geometric Exrmoles of
Indirect Reasoning in tho Hoc ivatior.
of Methods of Indirect Ircwf
(Complete References are given in Appendix B)
Fehr and Carnahan, op. cit., pp. 98-99.
F i r m e r and Hayden, op.
n o . 117-118.
Goodwin, Vannatta, and Fawcett, on. cit.
Hart, Schult, and Swain, on. clt., p. 90.
Henderson, Pingry, and Robinson, on. cit., pp. 178-179.
Keedy, et al., on. cit., p. 219 .
Keniston and Tully, on. cit.
Moise and Downs, on. cit., pp. 153-15^.
Price, Peak, and Jones, on. cit., pp. 101-102.
School Mathematics Study Group, Geometry, on. cit., p. 160.
School Mathematics Study Group, Geometry v.’inh Coordinates,
op. cit., p. 326.
210
*
I
Appendix ?
Texts v:hich Used both Real-Life end Geometric
Examples in the Motivation of Mothoaa of
Indirect Proof
(Complete References are given in Appendix B)
Goodwin, Vannstta, and Fawcett, on. clt., pp. 23^-236.
Keniston and Tully, on. clt., pp. 267-26-3.
Moise and Downs, on. clt., pp. 153-15^*
School Mathematics Study Group, Geometry, op. cit., p. 160.
School Mathematics Study Group, Geometry With Coordinates,
op. cit., p. 12 and p. 326.
211
Appendix G
'Propositions which are Proven Indirect;ly in
Euclid's E l e m o n to°
-
6b
III
-
23
VIII
-
15
X
•t*
X
-
7
III
-
24
VIII
-
17
X - 47
T
-
8
III - 27
IX
-
10
X
-
I
-
14
V
-
9
IX
-
12
X
-
81
I
-
19
V
-
10
IX
-
13
X
-
82
I
-
25
V
-
18
IX
-
14
X
-
83
I
-
26
VI
-
7
IX
-
16
X
-
CO
I
-
2?
VI
-
26
IX
-
17
X
-
111
I
-
29
VII
-
1
IX
-
18
XI
-
1
I
-
39
VII
-
2
IX
-
19
XI
-
2
I
-
40
VII
-
3
IX
-
20
XI
-
3
III
-
1
VII
-
20
IX
-
30
XI
-
5
III
-
2
VII
-
21
IX
-
31
XI
-
7
III
-
4
VII
-
22
IX
-
33
XI
-
13
III
-
5
VII
-
23
IX
-
34
XI
-
14
III
-
6
VII
-
24
IX
-
36
XI
-
16
III
_
7
VII
28
X
_
4
XI
_
19
S
-
46
■3
I
°This list was compiled from: Sir Thomas L. Heath,
The Thirteen Books of Euclicrs Elements (2d ec.. rev.; New
York: Dover Publications, 195o," Vol. I, II, III.
"b
1-6 means Book I, Proposition 5.
212
III - 8
VII - 29
X - 7
III - 10
VII - 33
X - 8
XII - 2
11
VII - 34
X - Q
XII - 5
III - 12
VII - 35
X - 13
XII - 10
III - 13
VII - 36
X - 26
XII - li
III - 16
VII - 39
X - 42
XII - 12
XII - 18
III
VIII - 1
X - 43
III - 19
VIII - 4
X
VIII _
7
-j*
l
III - 18
XI - 23
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Smith, Holland P., Ulrich, Jones ?., c .id MacDougall, Alice
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Altshiller-Court, Nathan. College Geometry.
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*’
21 6
Richmond:
Bell, Stoughton, et al. Modern Urlyerstty Calculus .
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19 ^
Bryne, J. Richard. Modern Elementary Mathematics.
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Nev;
Courant, Richard and Robbins, Herbert. What is Mathe­
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Dinkines, Flora. Elementary Concents of Modern Mathe ­
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21 ?
James, Glenn and James, RoberJ C. (ed.) Mathematics
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Butler, Charles I-I. and Wren, F, Lynwood, The Teaching of
Secondary A '.thematics. tth ed. revised. Rev/ York:
McGraw-Hill Book Co., 1965Cambridge Conference on School Mathematics. GooLo for
School Mathematics. Boston: Houghton MiY-iin Co.,
19to.
Christofferson, II. C. Geometry Professionalized "or
Teachers. Oxford, Ohio: By the author, 1933.
Commission on Mathematics. Program for College Preparatory
Mathematics: Report and Annera-ices. New York:
College Entrance Examination Board, 1959.
Davis, David P. The Teaching of Mathematics.
Mass.: Addison-Wesley Press, 1991.
Cambridge,
Glabe, Gordon R. "Indirect Proof in College Math omatics.11
Unpublished Ph.D. dissertation, Graduate School, The
Ohio State University, 1955.
Johnson, Donovan A. and Rising^ Gerald R. Guidelines for
Teaching Mathematics. Belmont, Calif.: Wadsworth
Publishing Co., 19b7.
National Committee on Mathematical Requirements. The
Reorganization of Matr.ematics in Secondary
Education, Boston: Ho ig'nton Mifflin Co., 1923.
National Counci], of Teachers of Mathematics. The Tea china:
of Geometry. Fifth Yearboo.
National Council of Teaoho
tionel
Council of Teachers of Mathematics, 195^
.• Insights Into Modern Mathematics. Twenty-third
YearUooicT Washington, D.C.: “National Council of
Teachers of Mathematics, 1957.
.• The Growth of Mathematical Ideas. Twenty-fourth
’Yearbook. Washington, D .C.f" National Council of
Teachers of Mathematics, 1959.
.♦ Tnr1chment Mathematics for the Grades. Twenty­
’sevent’
rTYearb ook. Washington, ~D.C.: National
Council of Teachers of Mathematics, 1963.
Reeve, William David. Y
School. Mew York”
211 the Seconiary
,onsr o and. '.Iin
is uon
uon, 193^»
Willoughby, Stephen S. Contemneror- .
rieachirm of Secondary.
School Mathematic
ev; York: Jo m V.iley and Sons,
D. Books on the Foundations of Mathematics
Max. The Nat1
Brace and Co
New York; Kareourt
Gradshtein, S. Direct and Corve
by T. 3oddington. New Y
Rosser, J. Barkley.1 Logic for Mathematicians. Nev; York
McGr av;-Hill Book ’“"Co., if 53 •
Rus se11, Bertrand. Int roduct ion _to Ma thema tical J^hil ~nonhy
London: George Allen and U.-iw’in,“"5Jta. £919 ."
.* Principles of Ms thematic ;.
W. \I~. Norton and Co., 193*?
2d ed. New York:
220
Wilder, Raymond L. Introduction to the Foundations of
Mathematics. 2cL ed. ~New~i ork: John VIiley "ana
S ons^ 1 9 ^
S. 3ooks on logic
Ambrose, Alice and Lazerowitz, M- -rio . Fundamentals _of
■Symbolic Logic. 2d c C , re /.sec . Nov Ybrx: Hoit7
niuehart and Winston, 196 -'.
Angell, Richard B. Reasoning and Ionic,
Appleton-Century-Crofts, -96 d.
n ew
York:
Arnauld, Antoine. The Art of Thinking, Port Royal Logic.
New York: BoBbs-fterriTT Go'., 1§'dl.
Baldwin, James Mark (ed.) Dictionary of Ph ilosophy and
Psychology. 3 vols". Glouddster, Mass.f P’eter
SmitWr^i"95? Barker, Stephen F. The E' err.ants of Logic.
McGraw-Hill Boom Co., 1905 .
New York:
Bochenski, I. M. A History of Formal Lon-j r .. Translated
by Ivo Thomas." lioore Dame, Ind.:~~bliiversity of
Notre Dame Press, 1961.
__________. A Precis of Mathematical Lor i . e .
Otto Bird.
Co., 1959.
Translated by
bordre'cntf, HoTrana;“~D. Re idol Publishing
Cohen, Morris R. and Nagel, Ernest.
logic and Scientific Method.
Kegan Pa’ul Ltd., i9>t.
Copi, Irving M. Symbolic Logic.
Co., 1959.
‘
An Introduction to
London: ~&‘bu'cTSV.ge and
New York: The Macmillan
Edwards. Paul (ed.). The Encyclopedic of Philosophy. 8
vols. New York: The Mscmiilan ”oo. ana tfie*"?ree
Press, 1967.
Great Books of the Western World. Vol. 1: The Works of
Arris'to~STo. ’Chfcago: Encyclopedia Br’itahnice", 1^52.
Goodstein, R. L. Mathematical Logic. Leicester, England:
Leicester University Press’, I96 I.
221
Hartshorns, Charles and Weisr, rani (eds.). Collected
Papers ox" Charles Sanders Peirce. 8 vols".
Canbridge, Mass.": Belknap Press of Harvard
University Press.
Jevons, W. Stanley. The Principe •3 ch Sc tor. oe.
Macmillan and Co., 1^7'5V
.... .
London,
Johnstone, Henry W., Jr. Elements.-v P 'in. hive r .
wgic.
New York: Thomas Yl Orow\7lV"“Co. iyb'h.
Kneale, William and Kneale, Martha. The Development of
Lo.plc . London: Oxford University Press, '£ $ o 7 ,
Leblanc, Hugues. Teohnicn.es of Deductive Inference.
Englewood Cliffs, N.J.: Prentide-HaII, Inc., 1966.
Lewis, Clarence Irving and Langford, Cooper Harold.
Symbolic Logic. 2d ed. ilw Yet:: Lover Publications,
W7 C
J.
Milnes, Alfred. Elementary Notions o" Lcr-jc.
Swan iO^nliCl.s oixii c*no.
,
± bo
Moore, G. E. Philosophic cl Stud!or.
Brace and Co., iy22."
New York: Hare curt
Rescher, Nicholas, j.ntroduotlor to Lo 'o.
St. Martin's Preno, 1961.
Salmon, Wesley C. Logic.
Hall, 1963.
London: w.
New York:
Sngle’
wood Cliffs, N.J.: Prentice-
St ebbing, L. Susan. A Modern Elementary L c-ric. y t h ed.,
revised by C. W. K. Mundle. Mew Y'ork: Barnes and
Noble, I9 I 3 .
Stoll, Robert R. Sets, Logic and Axiomatic Theories.
San Francisco: W. Hi’ Freeman” ;.Id "Co., 1961.
Strawson, P. F. Introduction to Logical Theory.
Methuen and Co., 1$0 2.
London:
Tarski, Alfred. Introduction to_Logic and to the
Methodology of Deducelve~Sciences. New York: Oxford
Univer 3 itfy Press, T9US1
Von Wright, George Henrik. Lorleal Studies.
The Humanities Press, £ 9 5 7 .
New York:
222
Whitehead, Alfred North and Rusr.cll, Bertrand. Principle.
Mathematica. 3 vols. 2d cd. Cambridge: Cambridge
University Press, 1963.
F. Periodicals and Notsr,
Ackcrmann, Wilhelm. "Bc.gruc.un'-; IMnor Strengen
Implikation " Journal of Symuo: io Ir~;lc, XXI
(June, 1936;, pp. 1 f"3 - 123.
Allen, Frank B. "The Lav; of Contrapositives," Eigh-School
Mathematics Notes, No. 3 . Boston: Ginn ana Go.,
l~9oo".
Anderson, Alan Ross and Belnap, Fuel D., Jr. "The Pure
Calculus of Entailment," Journal of Symbolic Lo.ric
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. "Tautological Entailments," Philosophical Studio
XIII (1961 ), pp. 9 - 2.'+.
Bronstein, Daniel J. "The Meaning of Implication," Mind,
XI.V (April, 1936), pp. 137 “ 130.
Cauman, Leigh S. "On Indirect Proof," Scr m t aMat hem c.tica
XXVIII (June, 1963),
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DeMorgan, Augustus. "On Indirect Demonstration," London,
Edinburgh, Dublin Philosophical. Matazlre, 1V
'(’December, 1352), PP- 435 - 433.
Hennessy, John Pope. "On Some Demons orations in Geometry
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Tv '(’December, 1o52)"," pn. 41 7 - Tip.
Korner, S. "On Entailment," Pro: -edlrrs ofthe Aris^otelt
Society, XLVII (1947), pp7 T4T - 132.
Lazar, Nathan. "The Importance of Certain Concepts and
Laws of Logic for the Study and Teaching of Geome­
try." The Mathematic o toa jiiCr ^ xi.j\.^ iicv.e0Ity .A.'.j 1 1 ^
May, 193S7, PP. 9*9-Trj;' pp. '1*56-174; PP. 216-240.
_______ . "The Logic of the Indirect Proof in Geometry:
Analysis, Criticism, and Recommendations," The
Mathematics Teacher, XL (May, 1947), pp. 223-240.
Lov/enheim, Leopold. "On Making Indirect Proofs Direct,"
S crip ta Me v.h ema 11.ca , XII ( J unc, 1946), pp. 125-139
223
Nelson, Everett J. "Intensional Relations,1' Kind, XXXIX
(Oct., 1930), pp. 440-453.
Post, E. L. "Introduction to s O'no -031 Theory of El vnentsry Propositions," Amor 1c r Jcyrnal cf Mi tho­
rnstics, XLIII (1921), pp~.
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Rosskopf, Myron P. and Exner, RoV. rt 1 . "Modern En-hasls
in the Teaching of Geor.ie try ." The Eat hems ■ p..-;
Teacher, L (April, 191/),
2Y-;,*-279’.
Secondary School Curriculum Committee of the National
Council of Teachers of Mathematics. "The Secondary
Mathematics Curriculum," The Mathematics Teacher,
LII (May, 1959), pp. 359-91?.
Smith, Stsnle?/ A. "What Does a Proof Really Prove?",
The Mathematics Teacher, LXI (May, 1968 ), 0 0 .
------------------
Strawson, P. F. "Necessary Propositions and EntoilmentStatements," Mind, LVII (Aoril, 1948), po. 184200 .
Wilder, R. L. "The Nature of Mathematical Proof,"
American Mathematical Monthly, LI (June-July,
1 9 & P t, pp. 309-323.
Wittgenstein, L. "Logisch-philosophische Aohondhrng,
Einleitung v. Betrand Russell ." Anna 1 or. aor Natohiloso-ohie XIV (1921), pp. 185-lo2.