This dissertation has been microfilmed exactly as received 69-22,104
BYHAM, Frederick Charles, 1935- INDIRECT PROOF IN GEOMETRY FROM EUCLID TO THE PRESENT.
The Ohio State University, Ph.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
1963-196 4...... Instructor, Department of Mathematics, The Ohio State University, Columbus, Ohio
1964-196 9...... *...... Assistant Professor, Department of Mathematics, State University College at Fredonia, Fredonia, New York
FIELDS OF STUDY
Major Field: Education
Studies in Mathematics Education. Professors Harold P. Fawcett and Nathan Lazar
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...... 51
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 . 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...... 1 84
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...... 213
vi 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 postu-
lational 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 i 5 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. ” 1 5 ^Report of the Commission on Mathematics: Appendices. op. cit., pp. 116-119.
l6Ibid., 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
1 7 rNathan Lazar, "The Logic of the Indirect Proof in Geometry," The Mathematics Teacher. XL (May, 19V7), pp. 225 - 2W . 1 8 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. These logical notions will be inves tigated — (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.
1 Leon 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 the authors of most college text books in mathematics is that the truth of "If p then q" is
2 I. M. Bochenski, A History of Formal Logic, tr. Ivo Thomas (Notre Dame, Ind.: University of Notre Dame Press, 1961), p. 116. ^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, "Logish- philosophische Abhandlung, Einleitung v. Betrand Russell," Annalen der Nat -philosophie. XIV (1921), pp. 185 - 262. implies q" is: TRUTH TABLE FOR MATERIAL IMPLICATION
p q If id then q T T T T F F F T T F 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. 16 these six equivalent forms. 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 implication symbolized by 'D * is called material implication, and its special name indi cates it is a special notion, not to be_confused with the 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. 1 5 ■'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. 18 ^ ® latter material implication.
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 proposi- tional 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. 1 9 ■'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., 1 9667,"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 deduci- ble 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 substi- 27 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 between propositional functions may be true or false. Thus x is a member of the class a, and a is contained in the class b, together imply 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 Yx 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^ ; xx Henderson, 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? "q is deducible from p."^f The definition he gave for 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 Ibid. . 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. Now strict implication may be defined as: "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 inten- sional 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 en- tailraent 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. "There is nothing in the meaning of *2 + 2 V 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 propositions '2 + 2 4' and '3 + 3 / 6 1 are false and 51 impossible yet consistent,"^ These propositions would be considered inconsistent under material and strict impli cation interpretations. Thus "2 + 2 4 k" materially implies "3 + 3 = 6" and "2 + 2 / 4" strictly implies "3 + 3 = 6". 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."^ (He used ■~SI to represent not-S1.) 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. Material implication may be defined as: "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." Definition (B) is the definition given by E. J. Nelson. The writer elects to take definition
(A) as the definition to be used for implication. Defini tion (B) will be accepted as an axiom. A list of steps which may be used to explain the meaning of implication is:
1 . Accept deducible as an undefined term. Give
examples of simple deductions such as: If x is
human, then x is mortal.
______Socrates is human Therefore, Socrates is mortal
2. Define implication as: p implies q means 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 2. If A is false, then B must be true.
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? They do satisfy the usual definition of contradictories ~ 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.^t-
Recall 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 7 not establish the theorem.'
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 is Monday Today is Tuesday Today is Wednesday Today is 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. How do they treat indirect proof? 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 are called 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 A C lines. PR intersects AB at X and CD at Y. / RYD = I YXB, that is, £ 1 = / 2. p -R Prove: /YXB = /CYX* that is, / 2 = Z 3 . Proof
STATEMENTS REASONS
1 . CD and PR are straight 1. Given lines intersecting at Y. 2. Z 1 = Z 3 2. If two straight lines intersect, then the opposite angles formed are equal in pairs. 3. L 1 = Z 2 3. Given k. Z 2 = Z 3 4. Deduced from state ments 2 and 3 and the transitive property of equality.u
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), 8 step by step, to form a proof." They do not discuss what they mean by "step by step," which is a crucial part of their definition. It should be noted that they did not 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. i p 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. 1 L Myrtle Edwards. First Course in Geometry (New York: Exposition Press, 1965;, p. 78'. 59 direct proof and has not been mentioned. The remaining text is by Seymour, Smith, and Douglas. 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 18 either analyzing or explaining "direct proof."
The majority of the geometry texts which do expli citly discuss direct proof wait until they are ready to introduce indirect proof. Prior to this they often define
"proof." 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. 1 A 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 Press,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 conclusion. 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. ^Frank 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 Coor- dinates (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 real- life 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. 63 showing that h-j h2and not-c —» not-hy. This is called the method of partial contraposition. 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. 65 27 proof proposed by their text. ' Of the thirty-seven geometry texts that were examined there were sixteen books (see Appendix C) which used real- life 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-geo- metric 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.
2 8 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 gyjpe J 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» HO* •^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. These authors give three methods of indirect proof. 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, 1 965), p.' 2j)6. •^Schacht, McLennan, and Griswold, op. cit.. p. 322. •37 -''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 MontgomeryDodes^;
^George David Birkhoff and Ralph Beatley, Basic Geometry (3rd ed.; New York: Chelsea, 1959), P. 35. ^Weeks and Adkins, op. cit.. p. 113. ^Brown 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 one- half, 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, o p . 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 of 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
Given: 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. 6. Why? 7. AC > AF. 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." What are the possible conclusions? 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. 80 5Zi which is also given by Smith and Ulrichr^ and by Keedy, 55 Jameson, Smith, and Mould.
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^®;
^Smith and Ulrich, op. cit.. p. 330. ^Keedy 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.^
^Moise 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 ?
------1 6 It « t
e ll II Given: a and b le. To prove a !! b. Analysis: Apply the indirect method of proofreductio 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 pt. P. tersect. 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, 66 Douglas, Eade, Olson, and Glass.
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. ^Fehr 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. The remaining text is by Kelly
and Ladd. 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 7 0 the text by Clarkson, Douglas, Eade, Olson, and Glass.
------'Clarkson et al.. op. cit.. p. 155 Two distinct linos intersect in at most one point.
For convenience, we rewrite this in the form of an implication.
If two distinct lines intersect, then they intersect in at most one point.
We now wish to show thatp A <+-q implies a contradiction. In other words, we must show that the statement “ two distinct lint ; intersect and they intersect in more than one point” is false. It is more idio matic English to say, “ Two distinct lines intersect in more titancue point.” We can prove this last statement false by showing that it con tradicts a postulate or a previot:ly proved theorem.
PROOF 8Y CONTRADICTION Suppose two distinct ’’r.e>, ’ :.,.r.d m, inter sect in more than one point, say points /' nuQ. T..ea. by Postulate -, points P and Qare contained in one ando.uy on !i T..is con;.■'..diets the statement in the hypothesis thatI and m ..re uistinct lines. I; nee, it is false that I and m intersect in more d'.aa one point; 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 us3 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? The student is expected to realize that statement (a) is intended to mean "For all integers, 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. The method will be called the "method of contradiction". 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. The example is taken from the geometry text by Edwards. 7 2
71 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: AB j| CD, E F 1 AB a t G cad intersectin'' CD at il. Prove: E F L CD. Plan: Indirect proof. Shov,- that EF is perpendicular to a hoe which coincides with CD. --M
STATE M! NTS nCASc:.’;> AB !! CD EF J. AB at G and inter.''-cting CD at H. Given. Suppose LM d. EF at point IL §-Jt3. Then L.M |J AB. T v-j lines in the f...o plane pcrpca- dicular to the «ar.-.e line are parallel r. LM coincides with CD. Through a "ivca pc'. •' ■ ' a g iv ... line 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. ^Rosskopf, 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. The example is taken 76 from Henderson, Pingry, and Robinson. 76 'Henderson, Pingry, and Robinson, op. cit.. P. 195. THEOREM 5.3 (Alternate Statement): If two lines in tho tam o plane aro inlor- scctod by a transvorsal so that tho alternate interior anglos are not oqual, thon tho linos are not parallel. n g iv e n : Lines I and k are in th e s a m e plane and are intersected by n in A and B respectively. m(Zx)?* m(Zy) p r o v e : I X k {I and k intersect) p p . o o k : Statements Reasons 1. 1 and k are in the same plane and 1. Why? are intersected by n in pointsA and B respectively. 2. m(Z x) t* m(Zy) 2. Why? 3. There exists line V through A such 3. Problem 5, page 141 that m(X.x') = m(Zy) 4. II k 4. Why? 5. A £ / and A £ /' 5. Statements 1, 3 6. .‘. 1 intersects I' 6. Definition ofintersecting lines, Statements _L_ 7. 1 intersects k 7. Theorem 5.4, Statements JL By proving the above theorem, you have also proved its contrapositive because of logical equivalence. This contrapositive is Theorem 5.3: If two parallel lines in the same plane are intersected by a transversal, then 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. n O The text by Clarkson, Douglas, Eade, Olson, and Glass' 7 0 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. 7 7 ''The eight texts are: Anderson, Garon, and Gremil- lion, 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. 7' ^School 9 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 statements having more than one condition and only one con clusion are formed by interchanging the negation of one given condition and the negation of the conclusion. 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 82 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 R A partial contraposition, taken from the text by Dodes , is given on the following page. ^2Ibid. ®^Dodes, op. cit.. pp. 402 - 403. ^Brown and Montgomery, op. cit.. p. 319. ^Welchons, Krickenberger, and Pearson, op. cit.. p. 151 ► ^Dodes, op. cit.. pp. 402 - 403. 101 THEOREM or rARTIAl CCNi'-APOSlTlvcS, F o r a n y co n d itio n al trupositivcR are true; «nd if 'he conditional in fah.e, tln u its p ..i:J contrapositives arc false. (Alternate form: A conditional of the fig. M. (ill = ^ 3) A fo rm p A q —* r has the same truth value as its partial contra;x>s;. (SO >* DC) — (At AC) tivcs.) It is understood that the th e o re m is in th e form /> * q — r. The p r o o of f this theorem is discussed in ih e appendix to Chapter 3, where truth tshie* •re explained. ILLUSTRATIVE J’Kom.F.M. la equilateral triangle ADC, m edian BM is prolonged to point and AD is drawn. AD does not equal AD. Prove that DM and MD a re not equaL Solution: The given conditions and conclusion are: lllwk frok. Given: AD — DC — AC AM = MC A B j-A D Prove: DM y= MD Instead of proving this, we shall prove the folio- p artiai contrapositive. Given: AB - DC = AC CK0£f»m: AM = MC DM = MD Prove: AD = AD Plan: Draw DC. Now ABCD is a parallelogram. AD = DC. : a d - ad. It is an easy exercise to prove thatAH = AD. W hen this is accomplished, we have proved that: If AD = DC = AC, and i AM ® MC, and DM — MD, then AD = AD. In other words, the following is a true conditional: (AD = DC s AC) A (AM = MC) A (BM = MD) — (AD = AD) Since this is true, the partial contrapositive must also be true bv the Theorem of Partial Contrapositives: (AB = DC = AC) A (AM = MC) A (AD AD) — (DM /-■ MD) 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 O O 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 false.Unfortunately 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. 13 8 . 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 LITERATURE EXCLUSIVE OP CT30M.3TRY TEXTBOOKS Introduction Why teach "indirect proof"? If Indirect proof is difficult for students to understand, one possible solution to the problem is simply to exclude methods of indirect proof from the school curriculum. In this chapter the literature exclusive of geometry texts will be searched for discussions on the importance of indirect proof and for discussions which express dissatisfaction with Indirect proof. In the last chapter it was argued that none of the geometry texts examined contained an adequate definition of "indirect proof". In this chapter the literature exclusive of plane geometry.texts will be searched for precise definitions of "indirect proof". A definition of indirect proof, based on the definitions found in the literature, will then be proposed. The analysis of geometry texts in the last chapter also indicated several, other shortcomings which are pre valent in the usual treatment of indirect proof. In this chapter the literature will be examined for suggestions 109 110 on the teaching of Indirect proof. In the present chapter, then, logic books, mathe matics books (other than geometry texts), and pertinent literature in mathematics education will be searched for: 1. Definitions of indirect proof 2. Statements on the importance of indirect proof 3. Statements which express dissatisfaction with indirect proof. i)-. Suggestions on the teaching of indirect proof. Definitions of Indirect Proof In a search for definitions of "indirect proof" one can not help but observe that although many authors give directions on how to construct an indirect proof very few authors define what they mean by "indirect proof". Perhaps this is because there appears to be little agree ment on how indirect proof should be defined. A particular point of contention is whether contrapositive proofs (Type VI and Type VII of the last chapter) are indirect or direct proofs. One of the ways in which indirect proof is defined is to characterize an indirect proof as a proof in which a contradiction is deduced. The following three quotations illustrate this approach: Ill Indirect proof (reductio ad- absurdum). An argument which proves n proposition A by showing that the denial of A, together with the accepted propositions , B_, . . B , leads to a contra diction.1 1 ' n A conclusion C follows indirectly from an hypoth 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 which a logical contra diction is obtained at some stage of the work, such as an assertion that some particular statement is both true and f a l s e . 3 A proof using contraposition is not an indirect proof by these three definitions since a contradiction is not deduced in a contrapositive proof. Another method of defining Indirect proof is given by Zehna and Johnson. They define "indirect proof" as: We say that an Indirect proof has been given when the negation of the conclusion has been used as one of the hypotheses in the proof. 1 Boruch A Brody, "Logic," The Encyclopedia of Philosophy, ed. Paul Edwards (New York: Macmillan Co. and The Free Press, 1967), Vol. V, p. 66. 2 'Resort of the Commision on Mathematics, Appendices (New York: College Entrance Examination Board, 1955T» p. 119. 3 ^Robert L. Swain, "Logic: For Teacher, For Pupil", Twenty-Seventh Yearbook (Washington, D. C.: National Council of Teachers of Mathematics, 1963)> P» 296. * ^ Peter W. Zehna and Robert L. Johnson, Elements of Set Theory (Boston: Allyn 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. "202. ' The question is enhanced by the many theorems which are proven by the method of contraposition. In fact Lowenheim asserts that partial contrapositive proofs c "constitute fully 95% of all so-oalled indirect proofs." This assertion receives support from De Morgan. De Morgan stated, "It is an easily ascertained fact, that really 7 indirect demonstration is uncommon in geometry . . . ." Neither De Morgan nor Lowenheim considers contrapositive proofs as "really" being indirect proofs. Tarski considers a contrapositive proof as being an Indirect proof but he does agree with Lowenheim and De Morgan on the frequency of contrapositive proofs. Tarski states that contra positive proofs "constitute the most usual type of in- O direct proof". The writer considers a contrapositive proof to be an Indirect proof for the following two reasons: f. Leopold Lowenheim, "On Making Indirect Proofs Direct", Scripts Mathematlca, XII (June, 1946), p. 133* 7 Augustus De Morgan, "On Indirect Demonstration," 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, T950T 7 115 A careful analysis of the proof will show that from "x and y are not both odd” it was deduced that "xy is an even number," but this is the contrapositive of the given theorem. Thus although the proof is written in the form of a reductio ad absurdum proof, the same reasoning will prove the contrapositive of the theorem. Because of the similarity between contrapositive proofs and many reductio ad absurdum proofs, the methods of contraposition as well as the method of reductio ad absurdum should be considered as indirect methods of proof. The writer will propose a definition of "indirect proof" which is a slight modification of a definition given by Gordon Glabe. In his dissertation Globe suggested this definition for "Indirect proof": An Indirect Method of Proof is any method in which, instead of proving a given proposition, (1) an equivalent proposition is proved, or (2) the contradictory of the given proposition is disproved, or (3) a proposition equivalent to the contradictory of the given proposition is disproved. In Chapter III it was argued that a proposition equivalent to a contradictory of a given proposition is itself a contradictory of the given proposition. Hence (3) in Glabe's definition is unnecessary. ^Gordon B. Glabe, "Indirect Proof in College Mathe matics" (unpublished Ph.D. dissertation, Graduate School, The Ohio State University). 116 In view of the discussion presented in this section it is proposed that "indirect proof" be defined as follows: Definition. 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. This definition would classify Type I, Type II, Type III, Type IV, Type VI, and Type VII of the previous chapter as indirect proofs. But Type V, the method of coincidence, would not be classified as an indirect proof. This is not a disadvantage of the proposed definition of indirect proof since the method of coin cidence would not be classified as an indirect proof by any of the definition::; of indirect proof mentioned in this section. Importance of Indirect Proof One reason given for the importance of teaching Indirect proof in the mathematics class is the value of this method of reasoning in non-mathematical situations. This is the position taken by H. C. Christofferson. Christofferson states: If geometry is to be taught largely because of its inherent possibilities 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 reasoning which we do in life is Indirect; therefore much of the value of geometry must be in its treatment of indirect proof.11 Alfred Milnqs argues the importance of indirect proof 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 men learn those truths of Mature that are unitedly known by the name of science.12 Another reason given by writers for the importance of indirect proof is that some theorems in mathematics can only be proven indirectly. In each of the following four quotations the assertion 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 method.1-' 11 K. C. Christofferson, Geometry Professionalized for Teachers (Oxford, Ohio: Privately published, 1953)V p. 138. 12 Alfred Milnes, Elementary Motions of Logic (London: W. Swan Sonnenschein and Co., 18"i?/0 ,"p. 93. ■^Clifford 3rewster Upton, l!The Use of Indirect Proof in Geometry and in Life", The Fifth Yearbook (Washington, D. C.: National Council of~Teac'ners of Mathematics, 1930), p. 102. 118 - Frequently it is Impossible to prove a statement except indirectly.1^ There are some theorems which by their very .- nature preclude the possibility of a direct proof. Often e direct proof of a certain theorem presents great difficulty, it sometimes even proves to be impossible7 ° Although the previous quotations all assert that there are theorems which can only be proven indirectly whether this is really the case or not is debatable. In speaking of reductio ad absurdum proofs, C. S. Peirce asserts that, "But it is very easy to convert any such proof into a direct proofThus, Peirce claims that any theorem provable at all is provable directly. Leopold Lowenheim supports the claim made by Peirce that indirect proofs can be converted into direct Ih Robert Katz, Axiomatic Analysis: An Introduction to Logic and the Real Num'be'r Sy'st'em (Boston:- D. C. Heath and“ Co.7 W , 1 r r ~ 5 5 7 ------1 6 -'Claire Fisher Adler, Modern Geometry, An Integrated First Course, (New York: McGraw-HilT-Book Co., 19^87,■ p.T*:------16 I. S. Grsdshtein, Direct and Converse Theorems, trans. T. Boddington (New York: Macmillan Co., 1963)”, p. ^6. 17 C. S. Peirce, Dictionary of Philosophy and Psychology, ed. James Mark Baldwin (Gloucester, Mass.: Peter Smith, 1957), Vol. II, p. kjk. 119 proofs by actually exhibiting schemes that when followed 1 8 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. 1 8 Lowenheim, op. cit.. pp. 125-139. 1 9 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. + 1 > 0 Then x2 + 1 2x So x2 + 1 ^ 2x So x2 - 2x + 1 < 0 Thus x + — ^ 2 A * Thus (x - 1)2 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. JZ 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 122 2 2 for any integer k. 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. Nov/ a direct proof that 'fz is irrational proceeds as follows: Theorem If a and b are integers which are relatively prime, then £ £ . Proof ! 1 . a and b are 1. hypothesis relatively 2. Definition of relatively prime prime 2. a and b are 3. Lemma 1 not both even = *s + ='s are = 3. a2 / 2b2 2 5. If two numbers are not 4. { 2 equal, then their square b 5. § / J T 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 bl• ^David 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 why it is. But the why is never learned from a demonstration which employs a reductio ad absurdum.^° 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 intuition than the proofs which have been the result of refinements, through the ages, of these original proofs. Therefore, the very 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 been completed all we knov; is that the theorem is true. We cannot assert that the subsidiary deductions made in the proof are true since they are based on premises which are shown by the proof to be incon sistent (false). Lowenheim expresses the situation as follows: Almost always, in a direct proof, something more is proved than merely B; every single inference drawn 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 xx S holds or the denial of S holds". 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, 19^07 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, thirty- •zC. seven years earlier, by Upton. Johnson and Rising also suggest that a non-mathe- matical 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 xn discussion the technique of indirect reasoning. 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 implication.Thirty-seven 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. x q ■"Upton, op. 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: McGraw- Hill Book Co., 1965), p. ^63. W 1 Ibid.. 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 ^Johnson 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 ^Nathan 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. CHAPTER 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 of Inconsistency 2. The Method of Contradiction 3. The Method of Contraposition^. 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. y is 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 an integer such that x2 is less than zero" can never be true while.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 Boruch A. Brody, "Logic," Encyclopedia of Philosophy, ed. Paul Edwards (New York: Macmillan Co. and the Free Press, 1967), Vol. V, p. 61. H 2 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 9 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 ?Boruch A. Brody, "Logic," Encyclopedia of Philosophy, ed. Paul Edwards (New York: Macmillan Co. and The Free Press, 1967), Vol. V, p. 61. ^Ibid., p. 66. 9 Ibid., p. 61. 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. 4ABD £ ABCD 4. ^1 = JL 2 5. 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 a statement and its contradictory is a contradiction but 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 H y 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 H1, 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 / Hgil and m are intersected by 1 n in points B and A m C:iy = ix. n Plan. Show that , H2, and not C (/y ^ ix) 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 1 50 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*. 2 2 3. a = 2b 11 Sherman K. Stein, Mathematics': The Man-made Universe (San Francisco: W. H. Freeman and C o., ' T 9 ' 6 3 ) , 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 if a 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 shown to be inconsistent by deducing from all of the statements of the set the contradictory of one of them. p Theorem. If p is a prime and p divides a then p divides a. 15k H^: p is a prime 2 H2 : p divides a C: p divides a Plan* 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 + a2n). But this means that p divides a, since a = apm a2 n . 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-i1 Hoc. implies *■ C and 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 =12 not-C: AC is not parallel to BD. 13 J. Barkley Rosser, Logic For Mathematicians (New York: McGraw-Hill Book Co., 1953")» PP. 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 = Ik since 11 = 12 and the supplements of equal angles are equal. C. AB = AB 4. A ABE = A BAF (two sides and the included angle equal) 5. ZBAF - 1 1 6 . But since /BAF is contrary to LI =12, 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^ H2 ^ C". Since H2 -»C" is equivalent to "H1 H2 and not-C form an inconsistent set" the plan in this fourth method is to show H1 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 H2 and not-C —>not- H-j. 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, and not-C — »not-H2 of the original statement. 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. If a < b and c < d then a + c < b + d. : a < b Hgt o < d C: a + c < b + d Plan. 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 2. It is given that a + c u b + d 3. From steps (1) and (2 ), b + c >■ b + d h. c > d which is contrary of the given 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. There is no "beating about the bush." The student knows that he may establish a theorem "H^ H2 —> C" by proving an equivalent theorem, H2 Ej and not-C form an inconsistent set". The student is not asked to assume that the conclusion of an implication is false when he believes it to be true just for the sake of the argument only to later 1 61 show that the conclusion is true. Instead the student is trying to show wrong, the joint assertion of the hypothesis and the contradictory of the conclusion, what he probably suspects to be wrong. The method of inconsistency also illuminates the importance of the hypotheses of an implication. It is the joint assertion of the hypotheses and the contra dictory of the conclusion that is shown to be inconsistent. The hypotheses of the implication are just as important as the contradictory of the conclusion in the method of inconsistency. Many textbook explanations of indirect proof focus attention on the conclusion and neglect the hypotheses. Another advantage of the method of inconsistency is that in this method no mention is made of the truth or the falsity of the conclusion of the implication. In mathematical theorems such as "If x is an even number then x2 is an even number" the conclusion, "x2 is an even number", is a propositional function and cannot be classified as true or false. The important question is not whether "x is an even number" is true or is false but whether "x2 is an even number" is deducible from "x is an even number". 1 62 The Method of Contradiction In the method of contradiction1^ the truth of "H^ H2 —» C" is established by showing that it has a contradictory statement which is false. In Chapter III it was pointed out that ”H1 H2 H ^ — >C" and ,,H1 H2 is consistent with not-C" are contradictories. Thus "H^ H2 — »C" may be established by showing that "Hi H2 H3 is consistent with not-C" is false. Observe that "Hi H2 I s consistent with not-C" is false whenever "Hi H2 H^ is inconsistent with not C" is true. The steps in proving H2 H^ —>C" by the method of contradiction are: (1) Show that Hi H2 H3 and not-C form an inconsis tent set. (2) Conclude that "H^ H2 H^ is consistent with not-C" is false. (3) State that "H^ H2 »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 1 c: n -i— > 2 n - Plan, 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 itive integer and n + — < 2 . n 2 . n^ + 1 < 2 n, multiplying n + £ < 2 by the n positive integer n. 3 . n2 - 2 n + 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.’it h 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 2 x 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. H2 —*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 Lq 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. Are these statements equivalent? 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 1 8 conclusion. Lazar calls this "overloading the 1q hypothesis". 7 In order to determine the effect of overloading the hypothesis on the process of partial contraposition some examples in which the hypothesis is overloaded will be examined 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 both integers Hg : a is odd C : a 2 / 2b2 The condition "a is odd", H2 , is unnecessary since the proof that is irrational establishes that The two partial contrapositives of this example of overloading the hypothesis are: 1. and not-C —>not-H2 , that is, "If a and b are 2 2 both integers and a = 2b , then a is an even integer". 2. Hg and not-C — ynot-H^, that is, "If a is an odd 2 9 integer and a = 2 b ‘c then a and b are not both integers" . Xorner, "On Sntailment", Proceedings of the Aristotelian Society, XLVII (19^7)* p T T 5 9 . 19 Lazar, "Logic of Indirect Proof in Geometry", op. cit., 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 2 2 integers" is inconsistent with "a = 2b ". Some people would consider the first partial contrapositive false since it is not possible to have the conditions of the 20 hypothesis satisfied. 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: p p If a and b are both integers and a = 2b , then a is an even integer. Proof. Since a and b are integers it follows that 20 See: Glabe, op. clt., p. 135* 21 See the discussion of strict implication in Chapter II. 1 69 p 2 2 2 a and b are integers. Now since a = 2b and 2 2 b is an integer we have that a = 2k for some 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 contra- positive 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: 2 Ii x is an even integer and x is an even integer then p x + x' is an even integer. : x is an even integer 2 I-I2 : x is an even integer 2 C : x + x 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 2. and not-C — >not-H., that is, "If x is an c i. 2 even integer and x + x is an odd integer, then x is an odd integer". 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: 2 If x is an even integer and x + x is an odd 2 integer, then x 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 r S - = 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, 2 say j. Therefore x = 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 p then a ^ 2b ”, v;as overloaded in this manner. The 2 2 condition "a is ode." may be used in deducing "a ^ 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 2 x + x 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: 2 1. If x is an even integer and x is an odd integer then y is an even integer. 2. If y is an odd integer and x2 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 even integer. Proof. 1. x and y are even P and Q: x and y integers (given) are even Integers 2. Since x is even, S: x = 2k for x = 2k for some some integer k integer k and y is an even integer 3. Since y is an T: x = 2k for some even integer, integer k and y y = 2m for some = 2m for some integer m integer m 4. From (2) and (3)» U: x + y = 2k + 2m x + y = '2k + 2ra 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 an even E: x + y is an even integer integer Observe that in the proof given in the right column each statement in the sequence is deducible from the preceding statement whereas in the proof given in the left column statement (4) is not deducible 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), 22 since P is given, the desired conclusion not-Q. follows. 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. The advantage of this 22 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. The hypothesis of an implication, Hg H . . . 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 is consistent, and not-C is consistent, and that 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. For instance the 2 statements "x is even" and "x is even" can be shown consistent by merely observing that when x = 2 both of the statements are true. But how about the statements 2 2 "x < 0" and "x - 1 <0"? 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. 177 since "x2 < 0" Is false for all values of x. 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. Upon which squaring yields 2 2 x = 4k + ^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 2 integer, then a 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 HJ-5 . . . H„-r>C, has been established then there are n + 1 other true implications which the student may dis cover by contraposition, n partial contrapositives and one complete contrapositive. 2 Although "If a is an even integer, then a is an 2 even integer" and its contrapositive, "If a is an odd integer, then a is an odd integer", are logically equiv alent they do evoke different thought patterns by the student. In the first theorem the student may be expected to focus his attention on even integers whereas in the contrapositive his attention will be focused on a property of odd integers. It may be argued that a student will better see the ramifications of a theorem after he has also considered all the contrapositives of the theorem. "Indeed, it is only after studying all the contrapositives of a theorem that a student appreciates 2 ^ its true significance." Summary In this chapter three methods of indirect proof were proposed - the method of inconsistency, the method of contradiction, and the method of contraposition. Examples of indirect proofs using each of these methods were given. The advantages of the method of inconsistency and Frank B. Allen, The Law of Contrapositives", Him' -School Mathematics Notes, No. 3 (Boston: Ginn 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. fTpp. 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 A. side AC. For, if A3 is unequal to AC, one of them is greater. Let AB be greater; and from . AB the greater let DB be cut off equal to A.C the 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. rvv \ • —J 7 • iJ 7 • 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 parallel to CD. For, if not, AB, CD when produced viill meet either in the direction of 3, D or towards A, C. Let them be produced and meet, in the direction of B, D, at C-. Then, in the triangle GEF, the exterior angle AEF is eoual to the interior ana opposite angle EFG; 1.16 which is impossible. Therefore, A3, CD when produced will not meet in the direction of B, D. Similarly it can be proved that neither will they meet towards A, C. But straight lines which do not meet in either direction are parallel; therefore A3 is parallel to CD. Def. 23 Therefore etc. 1..3.D. In this proof, from the hypothesis and the contradictory of the conclusion Euclid deduces that an exterior angle of a triangle is equal to an opposite interior angle which contradicts a previous theorem to the effect that an exterior angle of a triangle is greater than either of the interior and opposite angles. Euclid then tacitly assumes this establishes that the conclusion of the theorem is deducible from the hypothesis of the theorem. Once again the proof is logically incomplete until the 188 equivalence of an inconsistent set and an implication has been established. In the diagram for this proof it may be observed that A3G and CDG are assumed to be straight lines but they certainly do not appear to be straight lines. But in the method of inconsistency the idea is to show that the diagram is wrong, i.e. to show there is no diagram which satisfies both the hypothesis and the contradictory of the conclusion because they are inconsistent statements. Illustrations of the Method of Contraposition Although there is no mention of contraposition in "Euclid's Elements" there are indirect proofs in the "Elements" which contain a disguised use of the method of contraposition. In this section two of the indirect proofs in the "Elements" involving a disguised use of the method of contraposition will be examined. In the following example the theorem is of the form "H^ H2 —* Ci C2 Cc" which means that three distinct implications must be established, namely "K^ H0 —> C±ts > H g — > C 2", and "11^ Hg —► C y , in order to prove the theorem.31 A straight line falling on parallel lines makes 31 The proof given here is taken from: Heath, on. cit., Vol. I, pp. 3H-312. 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, T-r C Let the angle BGH be added to each; therefore the angles F AGH, BGH are greater than the angles BGH, GHD. 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 GHD . 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 contra- positives 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 cw, B, will also be C______incommensurable with 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 proof be concluded with a restatement of the theorem. The teacher may discontinue this practice when he is convinced that the students know that the proof establishes the theorem and not the conclusion. Summary In this chapter indirect proofs in "Euclid's Elements" have been examined. Although Euclid fails to establish a logical basis for his methods of indirect proof the "Elements" does contain indirect proofs which appear to intuitively be following the pattern of indirect proof called the "method of inconsistency" in the preceding chapter. There are also indirect proofs in the "Elements" which are really proofs by the method of contraposition. A consideration of Euclid’s style of proof has resulted in the recommendation that in the early stages of the geometry course every proof, direct or indirect, conclude with a restatement of the theorem. This will emphasise to the student that it is the theorem which the proof establishes and not the conclusion 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 (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", "strict implication", ana "entailment". 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" is used in indirect proofs. 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 false. (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 1 98 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". It is recommended that the equivalence'between"H-, — >C" and 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. Principles of Mathematics. 2d ed. revised. New York: McGraw-Hill Book Co., 1963, P» 21 • Bush, George C., and Obreanu, Philip E. Basic Concepts of Mathematics. New York: Holt, Rinehart and Winston, 19&57 P» 7- Dinklnes, Flora. Elementary Concer-:.:? o^ Modern Mathe matics . New York: Applebon-^entury-Crofts, 19&A, p. 2oT. Eves, Howard and Newsome, Carroll V. An Introduction to the Foundations and Fundamental Goncep ,s of Mathe matics. 2d ed. revised. New York:''HoltV"Rinehart and Winston, 1965, P» 277* Fine, Nathan J. Introduction to Modern Mathematics. Chicago: Rand McNally and CoV, 1*962, p. 18. Gemignani, Michael C. Basic Concepts of Mathematics and Logic. Reading! Mass.: Addion-We sley Publishing Co., 1968, p. 33. Groza, Vivian Shaw. A Survey of Mathematics: Elementary Concepts and Their Hi's tori oal Develop lent. New York: Holt*,’ Rinehart ana Wins Yon, 1*9*6*8’, p. 111. Haag, Vincent H. and Western, Donald W. Introduction to College Mathematics. New York: Holt, Rinehart and Winston, 195B/ p. 13. Horner, Donald R. A Survey of College Mr ^hematics. New York: Holt,“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, Hass.: iTud* i3oh-VJesley Publishing Co., 1963, p. 97. Laffer, Walter 3. Mathematics f-~: G~ -ar a? Ecu;"'tion. Belmont, CalTf.: Dickens H'Ph;,,nn Yg Co., 66. p. Lightstone, A . K . Symbolic Logic ana the Leal Number System: An IntroduoVion to oHe "Pound at ions of ?himbe:r Systems" New York: Harper'and Row, fcJ 6 * 5 pT-' '<■. Lovaglis, Anthony R. and Preston, Gerald C. Fcundc, .ions of Algebra and Analysis: An Elementary Accroach. New York: Harper and Row, 19o6, p . 5. Me serve, Bruce E. and Sobel, Max A. Introduction to Vis the- matics. Englewood Cliffs, W . J .: "Prentice-Hsll, Inc., 196^, p. 182. Moore, Charles S. and Little, Charles E. Basic Concents of Mathematics. New York: McGraw-Hill Book Co., T 967Y p._ 2T; Nahikian, Howard M. Tonics in Modern Mathematics. Nev; York: Macmillan Co., I900, p . 73. Sanders, Paul. Elementary Mathematics: A Logical tnoroach. Scranton, Pa.: International Textbook Co., 1963, p. 12. Stoll, Robert R . Sets, Lonlc and Axiomatic Theories. 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'., 1961/ p. 6. Wilierding, Margaret F. Elementary Mathematics: Its Structure and Concepts. New York: John Wiley and Sons, I966, p. 32. VJiHerding, Margaret F. and Hayward, Ruth A. Mathematics: The Alphabet of Science, .lev; York: John Wiley ana S ons^ T f S E T T . l'H Yandl, Andre L. Introduction to IJniyersity Mathematics. Belmont, CaTiT7:^0"ick'ensoh PuoTisning Co., 19*5’?, p. 6 . Young, Frederick H. The Nature of Mathematics. New York: John Wiley and Sons, i'9'5'5, p. 119. Append5.x 3 Plane Geometry Textbooks Analyzed in liis Study Anderson, Richard D., Garon, Jack VJ., and Greraillion, Joseph G. School Mathema t ics Geometry. Boston: Houghton lixTTin" 'Co~T, f9o’63 Avery, Royal A ., and Stone, William C. Plane Geometry. Boston: Allyn and Bacon, I96A. Beberman, Max, and Vaughan, Herbert E. High School Hathe- matics: Course 2. Boston, D. C."Ifea'fYT’ahd'Co., 19^ ------ Birkhoff, George David, and Beat ley, Ralph. Basic Geometry. 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. Geometry. Reading, Mass.: Addison- Wesley, i960. Clarkson, Donald R., et al. Geometry. Englewood Cliffs, N. J.: Prentice-Hall, W&T. Dodes, Irving Allen. Geometry. New York: Haroourt, Brace, and World, 1’953>7 Edwards, Myrtle. First Course in Geometry. Nev; York: Expos it ion~Pres's ,''”1'9£33 Eigen, Lev;is D., et al., Advancing in Mathematics: G eometry. Chicago: Science Research AxTs'o'ciates, Fehr, Howard F., and Carnahan, Walter H. Geometry. 3oston D. C. Heath and Co., 196I. Fischer, Irene, and Hayden, Dunstan. C-eometry. Boston: Allyn and Bacon, I965. 205 Goodwin, A. Wilson, Vannetts, Glen D. , and Fawcett, Harold ?. Oeometry, A 7~nif' ed. Course. Columbus, Ohio: Charles ^ r ’ner'rill "k oil: , l9~oy. Hart, Walter N ., Schult, Veryl, 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 Geometry: Its Structure ana Function. Nev/ 1'orlc: McGraw-Hill jtToo'k “Co ., 1962. Jurgensen, Ray C., Donnell?/, Alfred J., and Dolciani, Mar?/ P. Modern Geometry: Structure and Method. Boston: Houghton Mifflin Co., I'Jop. Keedy, Mervin L., et al. Exploring Geometry. New York: Holt, Rinehart and Winston, ~i£9o?. Kelly, Paul J., and Ladd, Norman S. Fundamental Mathe matical Structures: Geometry. 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 Williams, Grace N. Concents of Modern Mathematics, Boo-: 2. Hew York: American Book Co., T y S j . Lear?/, Arthur F., and Shuster, Carl M. Plane Geometry. New York: Charles Scribner's Sous, 19531 Lev/is, Harry. Geometry, A Contemnorary Course. Princeton, N. J.: DTX/an Nootrend Co., 19o 4. Mallory, Virgil S., Reserve, Bruce S., and Skeen, Kenneth c • A First Course in C-ec.netr •. Syr a cus e, H . Y .,: L. W. Singer Co.', 19593 " , Moise, Edwin E., and Downs, Floyd L. Geometry . Palo Alto: Addis on-Wes ley Publishing Co., 1*9 67. Morgan, Frank M., and Zartman, Jane. Geometry: PIane- Solid-Coordinate. Boston: Houghton I'iiTlin Co., 19535 Price, H. Vernon, Peak, Philip, and Jones. 7nilip 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 Mathematics, Geoinr 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. New Haven: Yale University Press, 1961. 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 Go., 1958• ~ ' Shute, William G., Shirk, William W., and Porter, George ^ • Geometry - Plane and 3olid. New York: American Book Co .",T96oC Smith, Rolland R., Ulrich, James P., and MacDougall, Alice K. Geometry: A Modern Course. New York: Harcourt, Brace and World, ”i9*o£. Spiller, Lee R., Prey, Franklin, and Reichgott, David. Today8s Geometry. Englewood Cliffs, N. J.: Prence-ffail',' 19’5 6 . University of Illinois Committee on School Mathematics: High School Mathematics: Unit 6_ - G eometr y. Uroana: University ofTYlinois Press, f9o0. Weeks, Arthur VI., and Adkins, Jackson B. A Course 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, on. cit., p. 190. At-ery end Stone, on. cit., pp. 109-110. Birkhoff and Beatley, on. clt., pp. 33-35- Brown and Montgomery, on. cit., pp. 295-296. Dod.es, on. cit., pp. 8-9- Goodwin, Vannatta, and Fawcett, on. cit., p. 23^. Jurgensen, Donnell?/, and Dolciani, on. cit., pp. lo^i— 165- Keniston and Tally, on. cit., pp. 217-268. Lewis, on. clt., p. 225. Moise and Downs, on. cit., pp. 153-15!: • Morgan and Zartmsn, on. clt., pp. 138-139• School 'Mathematics Study Group, (Geometry, on. cit., p. 160. School Mathematics Study Group, Geometry with Coord ma t e s , on. cit., p. 12. Smith, Ulrich, and MacDougall, on. clt., pp. 329-330. Se?/mour, Smith, and Douglas, on. cit., pp. 95-96. Welchons, Krickenberger, and Pearson, op, cit.., 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. 91 . 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. Firmer and Hayden, op. no. 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 Elemo n t o ° I - 6b III - 23 VIII - 15 X - 46 •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 ■3 CO I - 26 VI - 7 IX - 16 X - 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 S 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 °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 XI - 23 III - 10 VII - 33 X - 8 XII - 2 III 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 III - 18 VIII - 1 X - 43 XII - 18 -j* X III - 19 VIII - 4 l VIII _ 7 BIBLIOGRAPHY A. Plane Geometry LhextbOoks Anderson, Richard D., Garon, Jack W., and Cremillion, Joseph G. School Mathematics Geometry, Boston: Houghton Mifflin Co., 19oo. Avery, Royal A., and Stone, William C. Plane Geometry. Boston: Allyn and Beacon, 1964. Beberraan, Max, and Vaughan, Herbert 3. H~ rh School Ko-1-he matics : Course 2. Bcsto,.: D. C. neath anu Co., 19o ‘ Birkhoff, George David, and Beatley, Ralph. Bajrfc Geometry. 3rd ed. Nev; York: Chelsea, 1959. Brovm, Kenneth E., and Montgomery, Gaylord C. Geometry: Plane and Solid. River Porest, 111. : Laidlaw Brotners, 1962. Brumfield, Charles F., Eicholz, Robert S., and Shanks, Merrill 2. Geo ire try. Reading, Mass. : Addison- Wesley, 1§60. Clarkson, Donald R., et al. Geometry. Englewood Cliffs, N.J.: Prentice-hail, 19o5. Dodes, Irving Allen. Geometry. Nev; York: Harcourt, Brae e, and W g rl a, 1963. Edwards, Myrtle. First Course jr. Geometry. Nev; York: Exposition Press, 1963. Eigen, Lewis D., et al., Advancing in Mathematics: Geometrv. Chicago: Science nosearch Associates, 1966. Fehr, Howard F., and Carnahan, Walter K. Geometry. Boston: D.C. Heath and Co., 1961. Fischer, Irene, and Hayden, Dunstan. Geometry. Boston: Allyn and Bacon, 1965. 21 k Goodwin, A. Wilson, Vannetts, Glen D., and Fawcett, Harold P. Geometry: A Unified Course. Coliwr.bus, Ohio: Charles E. Herr£TT~Books, i'9^5. Hart, Walter W., Schult, Veryl, and Swain, Ho nr jr. New- Plane Geometry ard Supplements. Boston: D. C. Heath and C'o.“V "1 yS^'T Henderson, Kenneth B., Pinery, Robert E., and Robinson, George A. Modern Geometry: Its Structure and Function. New'" York: McGr a w - Hill Booh Co., 1962. Jurgensen, Ray C., Donnelly, Alfred J., and Dolciani, Mary P. Modern Geometry: Structure and Method. Boston: Houghton Mifflin Co., 1965~* Keedy, Mervin L., et al. Exn 1 orIng Geom'e t r y. New York: Holt, Rinehart ana Winston, 19^7- Kelly, Paul J., and Ladd, Norman E., Fundaments" Ms the- - matleal Structures: Geometry. Chicago: ocott, Forseman and Co., I9S5 . Keniston, Rachael P., and Tully, Jean. High School Geometry. Boston: Ginn ana Co., I 9S0 . Kenner, Morton R., Small, Dwain E., ana Williams, C-race N. Concentsi of Modern Mathematics, Eqok 2. New York: American Book Co.," 1963. Leary, Arthur F., and Shuster, Carl N. Plane Geometry. Now York: Charles Scribner's Sons, i95ih Lewis, Harry. Geometry. A Conter.roorary Course. Princeton, N . ' J d T ’D . Van icstrahd Co., i~964. Mallory, Virgil S., Meserve, Bruce E., and Skeen, Kenneth ^0,» A First Course in Geometry. Syracuse, N. Y.: L. W. Singer Co., T$T59- Moise, Edwin E . , and Downs, Floyd 1.. Geometry;. Palo Alto: Addison-Wesley Publishing Co., Morgan, Frank M., and Zartman, Jane. 0 ometry: Plane-Solid- Coordinate. Boston: Bought on HYT'flin Co.^ 1963. Price, H. Vernon, Peak, Philip, and Jones, Philip S. Mathematics an Integrated Series-. Book Tv;0. New York Harcourt, Brace and World, *19;o5*. 215 Rosskopf, Myron P., Sitomer, Harry, and Lenchner, C-eorge. Modern_Mathema_ti;,s, Geometry. Morristown, N.J.,: Sirve'r"lBurde tt Co., Schacht, John P., McLennan, Roderick C., and Oriswold, Alice L. Contorn-oorary -eorotrv. Nev; York: Holt, Rinehart a no w £neton, 1962. ~ School Ha theme tics Study Group. Geometry ♦ New Haven: Yale University Press, 1961”.” School Mathematics Study Group. Geometry with Coordinates. New Haven: Yale University Press, Seymour, P. Eugene, Smith, Paul J., ana Douglas, Edwin C. Geometry for High Schools. Nev/ York: The Mac...... laa Co., 193d. Shute, William G., Shirk, William. W„, and Porter, George P. Geometry - Plana and So" d. New York: American 3 ook-Co., TySoZ Smith, Holland P., Ulrich, Jones ?., c .id MacDougall, Alice K. Oeomitry: A Moo am Pour I lev; York: Harcourt, Brace anc~Norlc, 19*6^3” Spiller. Lee R., Prey, Franklin ,.x Re iohgott, David. Today’s C-somev.ry. Englev/ood Cliffs, N.J.: ? r e noice-Ha11, 1956. University of Illinois Committee on School Mathematics. Hirrh School Mathematics: Unit 6 - Geometry. Uroena: University of Illinois Press, i960. Weeks, Arthur W ., and Adkins, Jackson B. A Course in Geom etry: Plane avia Solid. Boston: Ginn ana’"‘Co ., 1961. Welchons, A.M., Krickenbcrger, W.E., and Pearson, Helen R. Plane Geometry. Boston, Ginn ana Co., I96I. B. Other Mathematics Books Adler, Claire Fisher. Modern Geometry, An Integrated First Course. New York: KcC-raw-Hill Book Co., Iv’pd. Allendoerfer, Carl B. and Oakley, Cletus 0. "rir.c doles of Mathematics. 2nd ed. revised. New York: Flcgraw- Hill Book Co., 1963. 21 6 Altshiller-Court, Nathan. College Geometry. Richmond: Johnson Publishing Co., 19HJI *’ Bell, Stoughton, et al. Modern Urlyerstty Calculus. San Francisco: Holden-bay 1; d., 1'9'6 6 '. Bush, George C. and Obreanu, Phil" ip E. Basic Concervos of Mathematics. Nev; York: Holt, RindHart an'.; wYriitcn, 19^ Bryne, J. Richard. Modern Elementary Mathematics. Nev; York: McGraw-Hill Book Co., 19SoT Courant, Richard and Robbins, Herbert. What is Mathe matics? London: Oxford University Press', 19'^IT Dinkines, Flora. Elementary Concents of Modern Mathe matics . NevT’York: ifpp v.eton-Century-Crof ts, i ~ 9 . Eves, Howard. An Introductlon to the History of Mathe matics . 2d ed. revised. Nev; Yor'k':"~Holt, Rinehart ana W ins ton, 196A. Fine, Nathan J. An Introduction to Modern Mathematics. Chicago: Hand McNally and Co., 1 9 & 5• Gemignsni, Michael C. Basic Cor.cents of Nations tics and Logic. Reading, Mass.: Idolsoh-Wc;ley"Fuolishing Co., '1968. Groza, Vivian Shaw. A Survey of M athematics: Elementary Concents and their Historical development. New' 1'ork: Ho-t, Rinehart and Winston, 19o8. Haag, Vincent H. and Western, Donald W. Introduction to College Mathematics. Nev; York: Holt, Rinehart and Winston^ T91TB. Heath, Sir Thomas L. The Thirteen Books of Euclid’s Elements. 3 vols. 2d ed. revised. Nev; York: Dover Publications, 195o. Henkin, Leon, et al. Retracing Elementary Mathematics. The Macmillan Co., T9 S2 . Horner, Donald R. A Survey of College Mathematics. Nev; York: Holt, Rinehart and” Winston, 1967. 21 ? James, Glenn and James, RoberJ C. (ed.) Mathematics Diet ionarv ; Student8s YditAon. Princeton, N.J.: D. Van Nostrand Co., iCJ59 •" Jones, Burton VJ. Elementary Concerts of Mathematics. 2d ed. rovisbd. NewTLoYn: c:d!Cit;;i o ... , ipcj- Katz, Robert. Ax i o m tic a-1" is• A_u ~ntreduction to Lotto an.', the HelTr lure A" . p.:/ x*.. £•:>;.cotx: D.C. Ho;.uh and Co., 19C‘1t. Keedy, Kervin L., A Modern Introduction to Basic Kathe- mafcics. Reading, 'Mass.: Adoison-Wesley Puollining Co., 1963. Kershner, R. B. and Wilcox, L. R. The Anatomy of Math^- matics. New York: The Ronald Press, 195C. Laffer, Walter 3. Mathematics for General Education. Belmont, Calif.: Dickenson Publishing Co., 1968. Light stone, A. H. Symbolic Logic c;-.d the Roal Number System: An Introduce lb... bo "Aha foundations of Nnm0e~r Systems. New York: harper and Row, 19*65. Lovaglia, Anthony R. and Preston, Gerald C. Foundatic ,s of .Algebra avid Analysis:_ An 11 omentary Auoroacn. New York: Harper and Rdw, Me serve, Bruce E. and Sobel, Kao: A. Introduction to Mathe- matics. Englewood Cliffs, N.o.: ProntTce-Ka11, Inc., 19*61. Moore, Charles S. and Little, Charles E. Basic Connrots of Mathematics♦ New York: McGraw-Hill book Co., 1967. Nahikian, Howard M. Topics in Modern Mathematics. New York: Macmillan Co., 1961. Newman, James R. (ed.) The World of Mathematics. 4 vols. New York: Simon and Shuster, 19*3*6. Sanders, Paul. Elementary Mathematics: A Logical Approach Scranton, Pa.: International Textbook "Co., 19'o3. Stein, Sherman X • Xs themet ics: Tne Man-mode Universe. San Francisco: V/. 1. Freeman and Co., i960. 218 Witter, G. E. Mathematics:__'nhe Study of Axiom Systems. New York: Blaise.' 11 I blis/.ing Co., 19 b k » V/illerding, Margaret F. Elerr.-ntar-' , 'c.' c-ed ' c>: I us Structure a: a Co.-qco \ Y'.rk: «.onn i.iley and Sons, 1 >06. V/illerding, Margaret F. and Hayward, Ruth A. mat hematics: The Alphabet of Science. New York: John Wiley and Sons, 1 too. Yandl, Andre L. Introduction to University Mathematics. Belmont, Calif.: Dickenson Publishing Co., 19^7. Young, Frederick H. The Nature of Mathematics. New York: John Wiley and Sons, 1 9od. Zehna, Peter W. and Johnson, Robert L. Elements of Set Theory. Boston, Allyn and Bacon, I 9o2. C. Books on the Teaching of Mathematics Butler, Charles I-I. and Wren, F, Lynwood, The Teaching of Secondary A '.thematics. tth ed. revised. 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