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Logic, logic, and logic Harvard University Press George Boolos, John P. Burgess, Richard Jeffrey ปี: 1998 ภาษา: english ไฟล์: DJVU, 7.71 MB แท็กของคุณ: 0 / 0

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Philosophy of mathematics: selected readings Cambridge University Press Paul Benacerraf, Hilary Putnam ปี: 1983 ภาษา: english ไฟล์: DJVU, 5.54 MB แท็กของคุณ: 0 / 0

LOGIC
MADE
EASY

A L S O BY D E B O R A H J .

Randomness

BENNETT

LOGIC
MADE
EASY
How to Know When Language Deceives You
DEBORAH
W • W • NORTON & COMPANY

J.BENNETT
I ^

I N E W YORK

LONDON

Copyright © 2004 by Deborah J. Bennett
All rights reserved
Printed in the United States of America
First Edition
For information about permission to reproduce selections from this book, write to
Permissions, WW Norton & Company, Inc., 500 Fifth Avenue, New York, NY 10110
Manufacturing by The Haddon Craftsmen, Inc.
Book design by Margaret M.Wagner
Production manager: Julia Druskin
Library of Congress Cataloging-in-Publication Data
Bennett, Deborah J., 1950Logic made easy : how to know when language deceives you /
Deborah J. Bennett.— 1st ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-393-05748-8
1. Reasoning. 2. Language and logic. I.Title.
BC177 .B42 2004
160—dc22

2003026910
WW Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110
www. wwnor ton. com
WW Norton & Company Ltd., Castle House, 75/76Wells Street, LondonWlT 3QT

1234567890

CONTENTS

INTRODUCTION: LOGIC IS RARE

I1

The mistakes we make l 3
Logic should be everywhere 1 8
How history can help 19
1 PROOF

29

Consistency is all I ask 29
Proof by contradiction 33
Disproof 3 6
I ALL 40

All S are P 42
Vice Versa 42
Familiarity—help or hindrance? 41
Clarity or brevity? 50

7

8

CONTENTS

3 A NOT TANGLES EVERYTHING UP

53

The trouble with not 54
Scope of the negative 5 8
A and E propositions s 9
When no means yes—the "negative pregnant"
and double negative 61
k SOME Is PART OR ALL OF ALL

64

Some is existential 6 s
Some are; some are not

68

A, E, I, andO JO
5 SYLLOGISMS

73

Sorites, or heap

8s

Atmosphere of the "sillygism" 8 8
Knowledge interferes with logic 89
Truth interferes with logic 90
Terminology made simple 91
6 WHEN THINGS ARE IFFY

96

The converse of the conditional
Causation

10 8

112

The contrapositive conditional

US

7 SYLLOGISMS INVOLVING IF, AND, AND OR

Disjunc; tion, an "or" statement
Conjunction, an "and" statement
Hypothetical syllogisms

124

118

119
121

CONTENTS

9

Common fallacies 130
Diagramming conditional syllogisms
8 SERIES SYLLOGISMS

134

137

9 SYMBOLS THAT EXPRESS OUR THOUGHTS

145

Leibniz's dream comes true:
Boolean logic 15 J
10

LOGIC MACHINES AND TRUTH TABLES

160

Reasoning machines 160
Truth tables 16 s
True, false, and maybe 1 68
11 FUZZY LOGIC, FALLACIES, AND PARADOXES

Shaggy logic 1J3
Fallacies 177
Paradoxes 1 8 J
M COMMON LOGIC AND LANGUAGE

13 THINKING WELL—TOGETHER

Theories of reasoning
NOTES

219

REFERENCES

233

ACKNOWLEDGMENTS

INDEX

24s

243

210

192

202

173

INTRODUCTION: LOGIC IS RARE
Crime is common. Logic is rare.
SHERLOCK

HOLMES

in The Adventure of the Copper Beeches

Logic Made Easy is a book for anyone who believes that logic is
rare. It is a book for those who think they are logical and wonder
why others aren't. It is a book for anyone who is curious about
why logical thinking doesn't come "naturally." It is a book for
anyone who wants to be more logical. There are many fine
books on the rules of logic and the history of logic, but here you
will read the story of the barriers we face in trying to communicate logically with one another.
It may surprise you to learn that logical reasoning is difficult.
How can this be? Aren't we all logical by virtue of being human?
Humans are, after all, reasoning animals, perhaps the only animals capable of reason. From the time we are young children,
we ask Why?, and if the answer doesn't make sense we are rarely
satisfied. What does "make sense" mean anyway? Isn't "makes
sense" another way of saying "is logical"?
Children hold great stock in rules being applied fairly and
rules that make sense. Adults, as well, hold each other to the
standards of consistency required by logic. This book is for any-

i2

I N T R O D U C T I O N : LOGIC I S R A R E

one who thinks being logical is important. It is also for anyone
who needs to be convinced that logic is important.
To be considered illogical or inconsistent in our positions or
behaviors is insulting to us. Most of us think of ourselves as
being logical. Yet the evidence indicates something very different. It turns out that we are often not very logical. Believing
ourselves to be logical is common, but logic itself is rare.
This book is unlike other books on logic. Here you will learn
why logical reasoning isn't so easy after all. If you think you are
fairly logical, try some of the logic puzzles that others find
tricky. Even if you don't fall into the trap of faulty reasoning
yourself, this book will help you understand the ways in which
others encounter trouble.
If you are afraid that you are not as logical as you'd like to be,
this book will help you see why that is. Hopefully, after reading
this book you will be more logical, more aware of your language. There is an excellent chance that your thinking will be
clearer and your ability to make your ideas clearer will be vastly
improved. Perhaps most important, you will improve your
capability to evaluate the thinking and arguments of others—a
tool that is invaluable in almost any walk of life.
We hear logical arguments every day, when colleagues or
friends try to justify their thoughts or behaviors. On television,
we listen to talking heads and government policy-makers argue
to promote their positions. Virtually anyone who is listening to
another argue a point must be able to assess what assumptions
are made, follow the logic of the argument, and judge whether
the argument and its conclusion are valid or fallacious.
Assimilating information and making inferences is a basic
component of the human thought process. We routinely make
logical inferences in the course of ordinary conversation, reading, and listening. The concept that certain statements necessar-

INTRODUCTION: LOGIC I S RARE

13

ily do or do not follow from certain other statements is at the
core of our reasoning abilities. Yet, the rules of language and
logic oftentimes seem at odds with our intuition.
Many of the mistakes we make are caused by the ways we use
language. Certain nuances of language and semantics get in the
way of "correct thinking." This book is not an attempt to delve
deeply into the study of semantics or cognitive psychology.
There are other comprehensive scholarly works in those fields.
Logic Made Easy is a down-to-earth story of logic and language
and how and why we make mistakes in logic.
In Chapter 2 , you will discover that philosophers borrowed
from ideas of mathematical proof as they became concerned
about mistakes in logic in their never-ending search for truth. In
Chapters 3, 4 , and 5, as we begin to explore the language and
vocabulary of logical statements—simple vocabulary like all,
not, and some—you will find out (amazingly enough) that knowledge, familiarity, and truth can interfere with logic. But how can
it be easier to be logical about material you know nothing about?
Interwoven throughout the chapters of this book, we will
learn what history has to offer by way of explanation of our difficulties in reasoning logically. Although rules for evaluating
valid arguments have been around for over two thousand years,
the common logical fallacies identified way back then remain all
too common to this day. Seemingly simple statements continue
to trip most people up.

Hie Mistakes We Make
While filling out important legal papers and income tax forms,
individuals are required to comprehend and adhere to formally
written exacting language—and to digest and understand the

i4

INTRODUCTION: LOGIC I S RARE

fine print, at least a little bit. Getting ready to face your income
tax forms, you encounter the statement "All those who reside in
New Jersey must fill out Form 203." You do not live in New
Jersey. Do you have to fill out Form 203? Many individuals who
consider themselves logical might answer no to this question.
The correct answer is "We don't know—maybe, maybe not.
There is not enough information." If the statement had read
"Only those who reside in New Jersey must fill out Form 203"
and you aren't a New Jersey resident, then you would be correct
in answering no.
Suppose the instructions had read "Only those who reside in
New Jersey should fill out Form 203" and you are from New
Jersey. Do you have to fill out Form 203? Again, the correct
answer is "Not enough information. Maybe, maybe not ."While
only New Jersey residents need to fill out the form, it is not necessarily true that all New Jersey-ites must complete it.
Our interpretations of language are often inconsistent. The
traffic information sign on the expressway reads "Delays until
exit 26." My husband seems to speed up, saying that he can't
wait to see if they are lying. When I inquire, he says that there
should be no delays after exit 26. In other words, he interprets
the sign to say "Delays until exit 26 and no delays thereafter." On
another day, traffic is better. This time the sign reads "Traffic
moving well to exit 26." When I ask him what he thinks will
happen after exit 26, he says that there may be traffic or there
may not. He believes the sign information is only current up to
exit 26. Why does he interpret the language on the sign as a
promise about what will happen beyond exit 26 on the one
hand, and no promise at all on the other?
Cognitive psychologists and teachers of logic have often
observed that mistakes in inference and reasoning are not only
extremely common but also nearly always of a particular kind.

INTRODUCTION: LOGIC IS RARE

IS

Most of us make mistakes in reasoning; we make similar mistakes; and we make them over and over again.
Beginning in the 1960s and continuing to this day, there
began an explosion of research by cognitive psychologists trying
to pin down exactly why these mistakes in reasoning occur so
often. Experts in this area have their own journals and their own
professional societies. Some of the work in this field is revealing
and bears directly on when and why we make certain errors in
logic.
Various logical "tasks" have been devised by psychologists
trying to understand the reasoning process and the source of our
errors in reasoning. Researchers Peter C. Wason and Philip
Johnson-Laird claim that one particular experiment has an
almost hypnotic effect on some people who try it, adding that
this experiment tempts the majority of subjects into an interesting and deceptively fallacious inference. The subject is shown
four colored symbols: a blue diamond, a yellow diamond, a blue
circle, and a yellow circle. (See Figure 1.) In one version of the
problem, the experimenter gives the following instructions:
I am thinking of one of those colors and one of those
shapes. If a symbol has either the color I am thinking
about, or the shape I am thinking about, or both, then I
accept it, but otherwise I reject it. I accept the blue diamond.
Does anything follow about my acceptance, or rejection, of
the other symbols?1

OOoo
Figure 1. "Blue diamond" experiment.

i6

INTRODUCTION: LOGIC IS RAKE

A mistaken inference characteristically made is to conclude
that the yellow circle will be rejected. However, that can't be
right. The blue diamond would be accepted if the experimenter
were thinking of "blue and circle," in which case the yellow circle
would not be rejected. In accepting the blue diamond, the experimenter has told us that he is thinking of (1) blue and diamond,
(2) blue and circle, or (3) yellow and diamond, but we don't
know which. Since he accepts all other symbols that have either
the color or the shape he is thinking about (and otherwise rejects
the symbol), in case 1 he accepts all blue shapes and any color
diamond. (He rejects only the yellow circle.) In case 2 , he
accepts all blue shapes and any color circle. (He rejects only the
yellow diamond.) In case 3, he accepts any yellow shapes and any
color diamonds. (He rejects only the blue circle.) Since we don't
know which of the above three scenarios he is thinking of, we
can't possibly know which of the other symbols will be rejected.
(We do know, however, that one of them will be.) His acceptance
of the blue diamond does not provide enough information for us
to be certain about his acceptance or rejection of any of the other
symbols. All we know is that two of the others will be accepted
and one will be rejected. The only inference that we can make
concerns what the experimenter is thinking—or rather, what he
is not thinking. He is not thinking "yellow and circle."2
As a college professor, I often witness mistakes in logic. Frequently, I know exactly which questions as well as which wrong
answers will tempt students into making errors in logical thinking. Like most teachers, I wonder, Is it me? Is it only my students? The answer is that it is not at all out of the ordinary to find
even intelligent adults making mistakes in simple deductions.
Several national examinations, such as the Praxis I™ (an examination for teaching professionals), the Graduate Records Examination (GRE®) test, the Graduate Management Admissions Test

INTRODUCTION: LOGIC I S R A R E

17

(GMAT®), and the Law School Admissions Test (LSAT®), include
logical reasoning or analytical questions. It is these types of questions that the examinees find the most difficult.
A question from the national teachers' examination, given in
1992 by the Educational Testing Service (ETS®), is shown in Figure 2 . 3 Of the 25 questions on the mathematics portion of this
examination, this question had the lowest percentage of correct
responses. Only 11 percent of over 7,000 examinees could
answer the question correctly, while the vast majority of the
math questions had correct responses ranging from 32 percent to
89 percent.4 Ambiguity may be the source of some error here.
The first two given statements mention education majors and the
third given statement switches to a statement about mathematics
students. But, most probably, those erring on this question were

Given:
1. All education majors student teach.
2. Some education majors have double majors.
3. Some mathematics students are education majors.
Which of the following conclusions necessarily follows
from 1,2, and 3 above?
A. Some mathematics students have double majors.
B. Some of those with double majors student teach.
C. All student teachers are education majors.
D. All of those with double majors student teach.
E. Not all mathematics students are education majors.
Figure 2. A sample test question from the national teachers'
examination, 1992. (Source: The Praxis Series: Professional Assessments
for Beginning Teachers® NTE Core Battery Tests Practice and Review
[1992]. Reprinted by permission of Educational Testing Service, the copyright owner.)

is

INTRODUCTION: LOGIC IS RARE

seduced by the truth of conclusion C. It may be a true conclusion, but it does not necessarily follow from the given statements. The correct answer, B, logically follows from the first two
given statements. Since all education majors student teach and
some of that group of education majors have double majors, it
follows that some with double majors student teach.
For the past twenty-five years, the Graduate Records Examination (GRE) test given by the Educational Testing Service
(ETS)
consisted of three measures—verbal, quantitative, and
analytical. The ETS indicated that the analytical measure tests
our ability to understand relationships, deduce information
from relationships, analyze and evaluate arguments, identify
hypotheses, and draw sound inferences. The ETS stated, "Questions in the analytical section measure reasoning skills developed
in virtually all fields of study."5
Logical and analytical sections comprise about half of the
LSAT, the examination administered to prospective law school
students. Examinees are expected to analyze arguments for hidden assumptions, fallacious reasoning, and appropriate conclusions. Yet, many prospective law students find this section to be
extremely difficult.

Logic Should Be Everywhere
It is hard to imagine that inferences and deductions made in
daily activity aren't based on logical reasoning. A doctor must
reason from the symptoms at hand, as must the car mechanic.
Police detectives and forensic specialists must process clues logically and reason from them. Computer users must be familiar
with the logical rules that machines are designed to follow. Business decisions are based on a logical analysis of actualities and

I N T R O D U C T I O N : LOGIC I S R A R E

19

contingencies. A juror must be able to weigh evidence and follow the logic of an attorney prosecuting or defending a case: If
the defendant was at the movies at the time, then he couldn't
have committed the crime. As a matter of fact, any problemsolving activity, or what educators today call critical thinking,
involves pattern-seeking and conclusions arrived at through a
logical path.
Deductive thinking is vitally important in the sciences, with
the rules of inference integral to forming and testing hypotheses. Whether performed by a human being or a computer, the
procedures of logical steps, following one from another, assure
that the conclusions follow validly from the data. The certainty
that logic provides makes a major contribution to our discovery
of truth. The great mathematician, Leonhard Euler (pronounced
oiler) said that logic "is the foundation of the certainty of all the
knowledge we acquire."6
Much of the history of the development of logic can shed
light on why many of us make mistakes in reasoning. Examining
the roots and evolution of logic helps us to understand why so
many of us get tripped up so often by seemingly simple logical
deductions.

How History (an Help
Douglas Hofstadter, author of Godel, Escher, and Bach, said that
the study of logic began as an attempt to mechanize the thought
processes of reasoning. Hofstadter pointed out that even the
ancient Greeks knew "that reasoning is a patterned process, and
is at least partially governed by statable laws."7 Indeed, the
Greeks believed that deductive thought had patterns and quite
possibly laws that could be articulated.

20

INTRODUCTION: LOGIC I S RARE

Although certain types of discourse such as poetry and storytelling may not lend themselves to logical inquiry, discourse that
requires proof is fertile ground for logical investigation. To prove
a statement is to infer the statement validly from known or
accepted truths, called premises. It is generally acknowledged
that the earliest application of proof was demonstrated by the
Greeks in mathematics—in particular, within the realm of
geometry.
While a system of formal deduction was being developed in
geometry, philosophers began to try to apply similar rules to
metaphysical argument. As the earliest figure associated with the
logical argument, Plato was troubled by the arguments of the
Sophists. The Sophists used deliberate confusion and verbal
tricks in the course of a debate to win an argument. If you were
uroop/iisricated, you might be fooled by their arguments.8 Aristotle, who is considered the inventor of logic, did not resort to
the language tricks and ruses of the Sophists but, rather,
attempted to systematically lay out rules that all might agree
dealt exclusively with the correct usage of certain statements,
called propositions.
The vocabulary we use within the realm of logic is derived
directly from Latin translations of the vocabulary that Aristotle
used when he set down the rules of logical deduction through
propositions. Many of these words have crept into our everyday
language. Words such as universal and particular, premise and conclusion, contradictory and contrary are but a few of the terms first
introduced by Aristotle that have entered into the vocabulary of
all educated persons.
Aristotle demonstrated how sentences could be joined
together properly to form valid arguments. We examine these in
Chapter 5. Other Greek schools, mainly the Stoics, also con-

INTRODUCTION: LOGIC I S R A R E

21

tributed a system of logic and argument, which we discuss in
Chapters 6 and 7.
At one time, logic was considered one of the "seven liberal
arts," along with grammar, rhetoric, music, arithmetic, geometry, and astronomy. Commentators have pointed out that these
subjects represented a course of learning deemed vital in the
"proper preparation for the life of the ideal knight and as a necessary step to winning a fair lady of higher degree than the
suitor."9 A sixteenth-century logician, Thomas Wilson, includes
this verse in his book on logic, Rule of Reason, the first known
English-language book on logic:
Grammar doth teach to utter words.
To speak both apt and plain,
Logic by art sets forth the truth,
And doth tell us what is vain.
%

Rhetoric at large paints well the cause,
And makes that seem right gay,
Which Logic spake but at a word,
And taught as by the way.
Music with tunes, delights the ear,
And makes us think it heaven,
Arithmetic by number can make
Reckonings to be even.
Geometry things thick and broad,
Measures by Line and Square,
Astronomy by stars doth tell,
Of foul and else of fair.10

22

INTRODUCTION: LOGIC I S RARE

Almost two thousand years after Aristotle's formulation of the
rules of logic, Gottfried Leibniz dreamed that logic could
become a universal language whereby controversies could be settled in the same exacting way that an ordinary algebra problem is
worked out. In Chapter 9 you will find that alone among
seventeenth-century philosophers and mathematicians, Leibniz
(the co-inventor with Isaac Newton of what we today call calculus) had a vision of being able to create a universal language of
logic and reasoning from which all truths and knowledge could
be derived. By reducing logic to a symbolic system, he hoped
that errors in thought could be detected as computational errors.
Leibniz conceived of his system as a means of resolving conflicts
among peoples—a tool for world peace. The world took little
notice of Leibniz's vision until George Boole took up the project
some two hundred years later.
Bertrand Russell said that pure mathematics was discovered
by George Boole, and historian E.T. Bell maintained that Boole
was one of the most original mathematicians that England has
produced.11 Born to the tradesman class of British society,
George Boole knew from an early age that class-conscious snobbery would make it practically impossible for him to rise above
his lowly shopkeeper station. Encouraged by his family, he
taught himself Latin, Greek, and eventually moved on to the
most advanced mathematics of his day. Even after he achieved
some reputation in mathematics, he continued to support his
parents by teaching elementary school until age 35 when Boole
was appointed Professor of Mathematics at Queen's College in
Cork, Ireland.
Seven years later in 1854, Boole produced his most famous
work, a book on logic entitled An Investigation of the Laws of
Thought. Many authors have noted that "the laws of thought" is
an extreme exaggeration—perhaps thought involves more than

I N T R O D U C T I O N : LOGIC I S R A R E

23

logic. However, the title reflects the spirit of his intention to
give logic the rigor and inevitability of laws such as those that
algebra enjoyed.12 Boole's work is the origin of what is called
Boolean logic, a system so simple that even a machine can employ
its rules. Indeed, today in the age of the computer, many do. You
will see in Chapter 10 how logicians attempted to create reasoning machines.
Among the nineteenth-century popularizers of Boole's work
in symbolic logic was Rev. Charles Lutwidge Dodgson, who
wrote under the pseudonym of Lewis Carroll. He was fascinated
by Boole's mechanized reasoning methods of symbolic logic and
wrote logic puzzles that could be solved by those very methods.
Carroll wrote a two-volume work called Symbolic Logic (only the
first volume appeared in his lifetime) and dedicated it to the
memory of Aristotle. It is said that Lewis Carroll, the author of
Alice's Adventures in Wonderland, considered his book on logic the
work of which he was most proud. In the Introduction of Symbolic Logic, Carroll describes, in glowing terms, what he sees as
the benefits of studying the subject of logic.
Once master the machinery of Symbolic Logic, and you
have a mental occupation always at hand, of absorbing
interest, and one that will be of real use to you in any subject you take up. It will give you clearness of thought—the
ability to see your way through a puzzle—the habit of
arranging your ideas in an orderly and get-at-able form—
and, more valuable than all, the power to detect fallacies,
and to tear to pieces the flimsy illogical arguments, which
you will so continually encounter in books, in newspapers,
in speeches, and even in sermons, and which so easily
delude those who have never taken the trouble to master
this fascinating Art. Try it. That is all I ask of you!13

24

INTRODUCTION: LOGIC I S RARE

Carroll was clearly intrigued with Boole's symbolic logic and
the facility it brought to bear in solving problems, structuring
thoughts, and preventing the traps of illogic.

The language of logic employs simple everyday words—words
that we use all the time and presumably understand. The rules
for combining these terms into statements that lead to valid
inferences have been around for thousands of years. Are the
rules of logic themselves logical? Why do we need rules? Isn't
our ability to reason what makes us human animals?
Even though we use logic all the time, it appears that we
aren't very logical. Researchers have proposed various reasons
as to the cause of error in deductive thinking. Some have suggested that individuals ignore available information, add information of their own, have trouble keeping track of information,
or are unable to retrieve necessary information.14 Some have
suggested that ordinary language differs from the language used
by logicians, but others hypothesize that errors are due to our
cognitive inability. Some have suggested that familiarity with the
content of an argument enhances our ability to infer correctly,
while others have suggested that it is familiarity that interferes
with that ability.15 If the problem is not faulty reasoning, then
what is it in the material that causes us to focus our attention on
the wrong things?
As we progress through the following chapters, we will examine the ways that we use (or misuse) language and logic in everyday life. What insight can we gain from examining the roots and
evolution of logic? How can the psychologists enlighten us about
the reasoning mistakes we commonly make? What can we do to
avoid the pitfalls of illogic? Can understanding the rules of logic

INTRODUCTION: LOGIC I S RARE

25

foster clear thinking? Perhaps at the journey's end, we will all be
thinking more logically.
But let's not get ahead of ourselves; let us start at the beginning. What is the minimum we expect from each other in terms
of logical thinking? To answer that question, we need to examine
the roots of logic that are to be found in the very first glimmerings of mathematical proof.

LOGIC
MADE
EASY

1

PROOF

No amount of experimentation can ever prove me right;
a single experiment can prove me wrong.
ALBERT

EINSTEIN

Consistency Is All I Ask
There are certain principles of ordinary conversation that we
expect ourselves and others to follow. These principles underlie
all reasoning that occurs in the normal course of the day and we
expect that if a person is honest and reasonable, these principles
will be followed. The guiding principle of rational behavior is
consistency. If you are consistently consistent, I trust that you
are not trying to pull the wool over my eyes or slip one by me.
If yesterday you told me that you loved broccoli and today
you claim to hate it, because I know you to be rational and honest I will probably conclude that something has changed. If nothing has changed then you are holding inconsistent, contradictory
positions. If you claim that you always look both ways before
crossing the street and I see you one day carelessly ignoring the
traffic as you cross, your behavior is contradicting your claim
and you are being inconsistent.
These principles of consistency and noncontradiction were

29

3o

LOGIC MADE EASY

recognized very early on to be at the core of mathematical
proof. In The Topics, one of his treatises on logical argument,
Aristotle expresses his desire to set forth methods whereby we
shall be able "to reason from generally accepted opinions about
any problem set before us and shall ourselves, when sustaining
an argument, avoid saying anything self-contradictory."1 To that
end, let's consider both the law of the excluded middle and the law
of noncontradiction—logical truisms and the most fundamental of
axioms. Aristotle seems to accept them as general principles.
The law of the excluded middle requires that a thing must
either possess a given attribute or must not possess it. A thing
must be one way or the other; there is no middle. In other
words, the middle ground is excluded. A shape either is a circle
or is not a circle. A figure either is a square or is not a square.
Two lines in a plane either intersect or do not intersect. A statement is either true or not true. However, we frequently see this
principle misused.
How many times have you heard an argument (intentionally?)
exclude the middle position when indeed there is a middle
ground? Either you're with me or you're against me. Either you
favor assisted suicide or you favor people suffering a lingering
death. America, love it or leave it. These are not instances of the
excluded middle; in a proper statement of the excluded middle,
there is no in-between. Politicians frequently word their arguments as if the middle is excluded, forcing their opponents into
positions they do not hold.
Interestingly enough, this black-and-white fallacy was common even among the politicians of ancient Greece. The Sophists,
whom Plato and Aristotle dismissed with barely concealed contempt, attempted to use verbal maneuvering that sounded like
the law of the excluded middle. For example, in Plato's Euthydemus, the Sophists convinced a young man to agree that he was

PROOF

31

either "wise or ignorant," offering no middle ground when
indeed there should be.2
Closely related to the law of the excluded middle is the law of
noncontradiction. The law of noncontradiction requires that a
thing cannot both be and not be at the same time. A shape cannot be both a circle and not a circle. A figure cannot be both a
square and not a square. Two lines in a plane cannot both intersect and'not intersect. A statement cannot be both true and not
true. When he developed his rules for logic, Aristotle repeatedly
justified a statement by saying that it is impossible that "the same
thing both is and is not at the same time."3 Should you believe
that a statement is both true and not true at the same time, then
you find yourself mired in self-contradiction. A system of rules
for proof would seek to prevent this. The Stoics, who developed
further rules of logic in the third century B.C., acknowledged
the law of the excluded middle and the law of noncontradiction
in a single rule, "Either the first or not the first"—meaning
always one or the other but never both.
The basic steps in any deductive proof, either mathematical
or metaphysical, are the same. We begin with true (or agreed
upon) statements, called premises, and concede at each step that
the next statement or construction follows legitimately from
the previous statements. When we arrive at the final statement,
called our conclusion, we know it must necessarily be true due to
our logical chain of reasoning.
Mathematics historian William Dunham asserts that although
many other more ancient societies discovered mathematical
properties through observation, the notion of proving a general
mathematical result began with the Greeks. The earliest known
mathematician is considered to be Thaïes who lived around
600 B.C.
A pseudo-mythical figure, Thaïes is described as the father of

32

LOGIC MADE EASY

demonstrative mathematics whose legacy was his insistence that
geometric results should not be accepted by virtue of their intuitive appeal, but rather must be "subjected to rigorous, logical
proof."4 The members of the mystical, philosophical, mathematical order founded in the sixth century B.C. by another semimythical figure, Pythagoras, are credited with the discovery and
systematic proof of a number of geometric properties and are
praised for insisting that geometric reasoning proceed according
to careful deduction from axioms, or postulates. There is little
question that they knew the general ideas of a deductive system,
as did the members of the Platonic Academy.
There are numerous examples of Socrates' use of a deductive
system in his philosophical arguments, as detailed in Plato's dialogues. Here we also bear witness to Socrates' use of the law of
noncontradiction in his refutation of metaphysical arguments.
Socrates accepts his opponent's premise as true, and by logical
deduction, forces his opponent to accept a contradictory or
absurd conclusion. What went wrong? If you concede the validity of the argument, then the initial premise must not have been
true. This technique of refuting a hypothesis by baring its inconsistencies takes the following form: If statement P is true, then
statement Q^is true. But statement Q^ cannot be true. (Q^is
absurd!) Therefore, statement P cannot be true. This form of
argument by refutation is called reductio ad absurdum.
Although his mentor Socrates may have suggested this form
of argument to Plato, Plato attributed it to Zeno of Elea
(495^-35 B.C.). Indeed, Aristotle gave Zeno credit for what is
called reductio ad impossibile—getting the other to admit an
impossibility or contradiction. Zeno established argument by
refutation in philosophy and used this method to confound
everyone when he created several paradoxes of the time, such as
the well-known paradox of Achilles and the tortoise. The form

PROOF

33

of Zeno's argument proceeded like this: If statement P is true
then statement Q^is true. In addition, it can be shown that if
statement P is true then statement Q_is not true. Inasmuch as it
is impossible that statement Q^is both true and not true at the
same time (law of noncontradiction), it is therefore impossible
that statement P is true.5

Proof by Contradiction
Argument by refutation can prove only negative results (i.e., P
is impossible). However, with the help of the double negative,
one can prove all sorts of affirmative statements. Reductio ad
absurdum can be used in proofs by assuming as false the statement to be proven. To prove an affirmative, we adopt as a premise the opposite of what we want to prove—namely, the
contradictory of our conclusion. This way, once we have refuted
the premise by an absurdity, we have proven that the opposite of
what we wanted to prove is impossible. Today this is called an
indirect proof or a proof by contradiction. The Stoics used this
method to validate their rules of logic, and Euclid employed this
technique as well.
While tangible evidence of the proofs of the Pythagoreans has
not survived, the proofs of Euclid have. Long considered the
culmination of all the geometry the Greeks knew at around 300
B.C. (and liberally borrowed from their predecessors), Euclid's
Elements derived geometry in a thorough, organized, and logical
fashion. As such, this system of deriving geometric principles
logically from a few accepted postulates has become a paradigm
for demonstrative proof. Elements set the standard of rigor for all
of the mathematics that followed.6
Euclid used the method of "proof by contradiction" to prove

34

LOCK MADE EASY

that there is an infinite number of prime numbers. To do this, he
assumed as his initial premise that there is not an infinite number of prime numbers, but rather, that there is a finite number.
Proceeding logically, Euclid reached a contradiction in a proof
too involved to explain here. Therefore—what? What went
wrong? If the logic is flawless, only the initial assumption can be
wrong. By the law of the excluded middle, either there is a finite
number of primes or there is not. Euclid, assuming that there
was a finite number, arrived at a contradiction. Therefore, his
initial premise that there was a finite number of primes must be
false. If it is false that "there is a finite number of primes" then it
is true that "there is not a finite number." In other words, there is
an infinite number.
Euclid used this same technique to prove the theorem in
geometry about the congruence of alternate interior angles
formed by a straight line falling on parallel lines (Fig. 3). To
prove this proposition, he began by assuming that the alternate
interior angles formed by a line crossing parallel lines are not
congruent (the same size) and methodically proceeded step by
logical step until he arrived at a contradiction. This contradiction forced Euclid to conclude that the initial premise must be
wrong and therefore alternate interior angles are congruent.
To use the method of proof by contradiction, one assumes as
a premise the opposite of the conclusion. Oftentimes figuring
out the opposite of a conclusion is easy, but sometimes it is not.
Likewise, to refute an opponent's position in a philosophical

Figure 3. One of the geometry propositions that Euclid proved:
Alternate interior angles must be congruent.

PROOF

35

argument, we need to have a clear idea of what it means to contradict his position. Ancient Greek debates were carried out
with two speakers holding opposite positions. So, it became
necessary to understand what contradictory statements were to
know at what point one speaker had successfully refuted his
opponent's position. Aristotle defined statements that contradict
one another, or statements that are in a sense "opposites" of one
another. Statements such as "No individuals are altruistic" and
"Some individual(s) is (are) altruistic" are said to be contradictories. As contradictories, they cannot both be true and cannot
both be false—one must be true and the other false.
Aristotle declared that everv affirmative statement has its
own opposite negative just as every negative statement has an
affirmative opposite. He offered the following pairs of contradictories as illustrations of his definition.
Aristotle's Contradictory Pairs 7
It may be

It cannot be.

It is contingent [uncertain].

It is not contingent.

It is impossible.

It is not impossible.

It is necessary [inevitable].

It is not necessary.

It is true.

It is not true.

Furthermore, a statement such as "Every person has enough
to eat" is universal in nature, that is, it is a statement about all
persons. Its contradictory statement "Not every person has
enough to eat" or "Some persons do not have enough to eat" is
not a universal. It is said to be particular in nature. Universal
affirmations and particular denials are contradictory statements.
Likewise, universal denials and particular affirmations are
contradictories. "No individuals are altruistic" is a universal
denial, but its contradiction, "Some individuals are altruistic," is

36

LOGIC MADE E A S Y

a particular affirmation. As contradictories, they cannot both be
true and cannot both be false—it will always be the case that
one statement is true and the other is false.
Individuals often confuse contradictories with contraries.
Aristotle defined contraries as pairs of statements—one affirmative and the other negative—that are both universal (or both
particular) in nature. For example, "All people are rich" and "No
people are rich" are contraries. Both cannot be true yet it is possible that neither is true (that is, both are false).
"No one in this family helps out .""Some of us help out."
"Don't contradict me."
"Everyone in this family is lazy." "I hate to contradict
you, but some of us are not lazy."
"No one in this family helps out." "We all help out."
"Don't be contrary."
"Everyone in this family is lazy." "To the contrary, none
of us is lazy."
John Stuart Mill noted the frequent error committed when
one is unable to distinguish the contrary from the contradictory.8
He went on to claim that these errors occur more often in our
private thoughts—saying that if the statement were enunciated
aloud, the error would in fact be detected.

Disproof
Disproof is often easier than proof. Any claim that something is
absolute or pertains to all of something needs only one counterexample to bring the claim down. The cynic asserts, "No human

PROOF

37

being is altruistic." If you can think of one human being who has
ever lived who is altruistic, you can defeat the claim. For example, you might get the cynic to admit, "Mother Teresa is altruistic." Therefore, some human being is altruistic and you have
brought down the cynic's claim with one counterexample. As
Albert Einstein suggested, any number of instances will never
prove an "all" statement to be true, but it takes a single example
to prove such a statement false.
In the face of an "all" or "never" statement, one counterexample can disprove the statement. However, in ordinary discourse
we frequently hear the idea of a counterexample being used
incorrectly. The idea of argument by counterexample does not
extend in the reverse direction. Nonetheless, we sometimes
hear the illogic that follows: She: All women are pacifists. He:
I'm not a woman and I'm a pacifist. (This is not a counterexample. To disprove her statement, he must produce a woman who is
not a pacifist.)
Psychologists have found that people can be extremely logical
when they can notice a contradiction but that correct inference
is often hindered when a counterexample is not obvious. For
example, in Guy Politzer's study on differences in interpretation
of the logical concept called the conditional, his subjects were
highly successful in evaluating a rule logically when direct evidence of a contradiction was present. Specifically, Politzer's subjects were given a certain statement such as, "I never wear my
dress without wearing my hat," accompanied by four pictures
similar to those in Figure 4 . Subjects were asked to label each
picture as "compatible" or "incompatible" with the given statement. Inasmuch as the pictures illustrated the only possible
combinations of information, subjects weren't required to
retrieve that information from memory. These visual referents
facilitated the retrieval of a contradiction.9

38

LOGIC MADE EASY

Examine the pictures in the figure for yourself. From left to
right, they illustrate hat/dress, no hat/dress, hat/no dress, and
no hat/no dress. The claim is made, "I never wear my dress without wearing my hat," and we are to judge whether the pictures
are consistent or inconsistent with the claim. Since the claim is
about what I will or will not do when I wear my dress, we judge
that the last two pictures are "compatible" with the claim as they
are not inconsistent with it. The first two pictures must be
examined in more detail since the wearing of a dress is directly
addressed by the claim. "I never wear my dress without wearing
my hat" is clearly consistent with the first picture and is clearly
violated by the second. So the correct answers are that all the
pictures are "compatible" with the claim except the second,
which is "incompatible" with it.
In this experiment, subjects were not obliged to rely on memory or imagine all possible dress/hat scenarios.The subjects were
presented with pictorial reminders of every possibility. With
visual images at hand, subjects could label those pictures that
contradicted the statement as incompatible; otherwise the pictures were compatible.
From very ancient times, scientists have sought to establish

Figure 4. Evaluate each picture as compatible or incompatible
with the statement "I never wear my dress without wearing
my hat."

PROOF

39

universal truths, and under the influence of Thaïes, Pythagoras,
and Euclid, universal truths required proof. Armed with the law
of the excluded middle and the law of noncontradiction, ancient
mathematicians and philosophers were ready to deliver proof. All
that remained was an agreed-upon set of rules for logical deduction. Aristotle and the Stoics provided such a framework for
deductive inference, and the basics of their systems remain virtually unchanged to this day.
As the Greek philosophers attempted to establish universal
truths about humans and the world around them, definitions
were set forth in an effort to find a common ground in language.
Aristotle defined statements of truth or falsity and words like
all. Do they really need any definition? He felt that for one to
articulate a system of correct thinking, nothing should be taken
for granted. As we'll see in the next chapter, he was right.

z
ALL
You mayfoolall the people some of the time;
you can evenfoolsome of the people all the time;
hut you can'tfool all of the people all the time.
ABRAHAM

LINCOLN

Aristotle's works in logic consisted of six treatises: Categories, On
Interpretation, Prior Analytics (or Concerning Syllogism), Posterior
Analytics (or Concerning Demonstration), Topics, and Sophistical
Elenchi (or On Sophistical Refutations). After Aristotle's death in
322 B.C., his followers collected these treatises into a single
work known as the Organon, or instrument of science.
The title, On Interpretation, reflects the notion that logic was
regarded as the interpretation of thought.1 In this treatise, Aristotle set down rules of logic dealing with statements called
propositions. A proposition is any statement that has the property
of truth or falsity. A prayer, Aristotle says, is not a proposition.
"Come here" and "Where are you?" are not propositions. "2 + 2
= 5" is a proposition (it is false). "Socrates was a man" is a proposition (it is true). Propositions can be true or false and nothing
in between (law of the excluded middle), but not both true and
false at the same time (law of noncontradiction).2 "All tornadoes
are destructive" might be a false proposition if it is true that
some tornadoes are not destructive, even if only one is not.

40

ALL

41

"That tornado is destructive" would certainly be either true or
false but not both. We would know whether the proposition is
true or false by checking the facts and agreeing on a definition of
"destructive.""Some tornadoes are destructive" would qualify as
a proposition, and we would all probably agree it is a true
proposition, having heard of at least one tornado that met our
definition of "destructive."
Terms called quantifiers are available for making propositions.
Quantifiers are words such as every, all, some, none, many, and
few, to name a few. These words allow a partial quantification of
items to be specified. Although words like some, many, and Jew
may provide only a vague quantification (we don't know how
many many is), words like all and none are quite specific.
The English words all and every are called (affirmative) universal quantifiers in logic. They indicate the totality (100 percent) of
something. Sometimes the all is implied, as in "Members in good
standing may vote." However, if we want to emphasize the point,
we may say, "All persons are treated equally under the law." The
word any is sometimes regarded as a universal quantifier. "Any
person who can show just cause why this man and woman
should not be joined in holy wedlock. . . ."The article a may also
be used as a universal quantifier, as in "A library is a place to borrow books" meaning "All libraries are places to borrow books."
Universal affirmative propositions such as these were called de
omni, meaning all, by Latin commentators on Aristotle.
It has been shown that the universality of the word all is
clearer than the universality of any and a. In a 1989 study, David
O'Brien and his colleagues assessed the difficulty of different formulations of the universal all by testing second graders, fourth
graders, eighth graders, and adults.3 Without exception, in every
age group the tendency to err was greatest when the indefinite
article a was used, "If a thing. . . ." For older children and adults,

42

LOGIC MADE E A S Y

errors decreased when any was used, "If any thing . . . ," and
errors virtually vanished when the universality was made
explicit, "all things. . . ."With the youngest children, though the
errors did not vanish, they were reduced significantly when the
universality was made clear with the word all.

All5are/»
In addition to a quantifier, each proposition contains a subject and
a predicate. For example, in the universal affirmation "All men
are human beings," the class of men is called the subject of the
universal proposition and the class of human beings is called the
predicate. Consequently, in logic books, the universal affirmation is often introduced to the reader as "All S are P."
Although not truly an "all" statement, one other type of
proposition is classified as a universal affirmation: "Socrates was
a Greek." "I am a teacher." These propositions do not, on the
surface, appear to be universal propositions. They are called singular or individual and are treated as universal claims. Even
though the statements speak of a single individual, they are
interpreted as constituting an entire class that has only a single
entity in it.4 Classical logic construes the propositions as, "All
things that are identical with Socrates were Greek" or "All things
that belong to the class of things that are me are teachers."

Vice Versa
Given the right example, it is clear that the statement "AU S are
P" is not the same as the statement "All P are S." We would
probably agree "All mothers are parents" is a true statement

ALL

43

whereas "All parents are mothers" is not. Yet this conversion is a
common mistake. These two statements, "All S are P " and "All P
are S," are called converse statements. They do not mean the same
thing. It is possible that one is true and the other is not. It is also
possible that both are true or neither is true. You might think of
the converse as the vice versa. All faculty members are employees
of the university, but not vice versa. All dogs love their owners
and vice versa. (Although I'm not sure either is true.)
According to Barbel Inhelder and Jean Piaget, children aged 5
and 6 have trouble with the quantifier all even when information
is graphic and visual. In their experiments, they laid out red
square counters and blue circle counters, adding some blue
squares, all of which the children were allowed to see during
their questioning. Using white and gray counters, their experiment involved a set of objects such as those in Figure 5. Children were then asked questions such as "Are all the squares
white?" (NO) and "Are all the circles gray?" (YES.) More difficult for the younger children were questions such as "Are all the
white ones squares?" (YES.) The youngest subjects converted
the quantification 50 percent of the time, thinking that "All the
squares are white" meant the same as "All the white ones are
squares."5 This may be explained in part by the less developed
language ability of the youngest children, but their mistakes may
also be explained by their inability to focus their attention on
the relevant information.

O n O D D O O D
Figure 5. Which statements are true?
"All squares are white. All white things are squares."
"All circles are gray. All gray things are circles."

44

LOGIC MADE EASY

DOOD

(b)
Figure 6. (a) Are all the white things squares? (b) Are all the
squares white? To correctly answer these questions, we must
focus our attention on the pertinent information.

Inhelder and Piaget noted the difficulty of mastering the idea
of class inclusion in the youngest children (Fig. 6). That is, the
class of white squares is included in the class of squares, but not
vice versa. By ages 8 and 9, children were able to correctly
answer the easier questions 100 percent of the time and produced the incorrect conversion on the more difficult questions
only 10 to 20 percent of the time.
Understanding the idea of class inclusion is important to
understanding "all" propositions. If the statement "All taxicabs
are yellow" is true, then the class of all taxicabs belongs to the
class of all yellow cars. Or, we could say that the set of all taxicabs is a subset of the set of all yellow cars. Sometimes a visual
representation like Figure 7 is helpful, and quite often diagrams
are used as illustrative devices.
The introduction of diagrams to illustrate or solve problems
in logic is usually attributed to the brilliant Swiss mathematician
Leonhard Euler. His diagrams were contained in a series of let-

ALL 45

Figure 7. Graphic representation of "All taxicabs are yellow."

ters written in 1761 to the Princess of Anhalt-Dessau, the niece
of Frederick the Great, King of Prussia. The famous Letters to a
German Princess (Lettres à une Princesse D'Allemagne) were published i
1768, proved to be immensely popular, and were circulated
in book form in seven languages.6 Euler's letters were intended to
give lessons to the princess in mechanics, physical optics,
astronomy, sound, and several topics in philosophy, including
logic. One translator, writing in 1795, remarked on how
unusual it was that a young woman of the time had wished to be
educated in the sciences and philosophy when most young
women of even the late eighteenth century were encouraged to
learn little more than the likes of needlepoint.7
Euler's instruction in logic is not original; rather, it is a summary of classical Aristotelian and limited Stoic logic. It turns out
that his use of diagrams is not original either. The identical diagrams that the mathematical community called Euler's circles had
been demonstrated earlier by the German "universal genius"
Gottfried Leibniz. A master at law, philosophy, religion, history,
and statecraft, Leibniz was two centuries ahead of his time in
logic and mathematics. Most of his work in logic was not published until the late nineteenth century or early twentieth century, but around 1686 (one hundred years before the publication
of Euler's famous Letters), Leibniz wrote a paper called De Formae
Logicae Comprobatione per Linearum Ductus, which contained the

46

LOGIC MADE E A S Y

figures that became known as Euler's circles. The diagrams are
one and the same; there is no way that Euler could not have seen
them previously. Most likely, the idea had been suggested to him
through his mathematics tutor, Johann Bernoulli. The famous
Swiss mathematicians, brothers Jakob and Johann Bernoulli,
had been avid followers of Leibniz and disseminated his work
throughout Europe.
Although his mathematical ability is legendary, Euler was also
noted for his ability to convey mathematical ideas with great
clarity. In other words, he was an excellent teacher. Like any
good teacher, he used any device in his repertoire to instruct his
students. Euler's impact on the mathematical world was so
influential that his style and notation were often imitated. Thus,
the idea of using diagrams in logic was assigned to him.
The Leibniz/Euler circles exhibit the proposition "Every A is
B " in the same way we earlier displayed "All taxicabs are yellow"—with the class of A-things represented as a circle inside
the circle of B-things. Perhaps more familiar to the reader, and
widely considered an improvement on the Leibniz/Euler circles, is the Venn diagram. 8
John Venn, the English logician and lecturer at Cambridge
University, first published his method of diagrams in an 1880
Philosophical Magazine article, "On the Diagrammatic and
Mechanical Representation of Propositions and Reasoning."
Venn would have represented "All taxicabs are yellow" with two
overlapping circles as shown in Figure 8, shading the portion of
the taxicab circle that is outside the yellow-cars circle as an indication that there is nothing there. The shaded portion indicates
that the class of non-yellow taxicabs is empty.
At first glance, Venn's diagram does not seem as illustrative as
the Leibniz/Euler diagram—their diagram actually depicts the
class of taxicabs inside the class of yellow cars. However, as we will

ALL 47
Taxicabs

Figure 8. A Venn diagram of "All taxicabs are yellow."

later see, Venn's diagram has the advantage of being much more
flexible. Many other philosophers and mathematicians have
devised diagrammatic techniques as tools for analyzing propositions in logic. The American scientist and logician Charles Sanders
Peirce (pronounced "purse") invented a system comparable to
Venn's for analyzing more complicated propositions. Lewis Carroll devised a system resembling John Venn's—using overlapping
rectangles instead of circles—and used an O to indicate an empty
cell, as in Figure 9. Both Peirce and Carroll were huge advocates
of teaching logic to schoolchildren through the use of graphs such
as these. Educators must have been paying attention, because
schoolchildren today are taught classification skills from a very
early age by the use ofVenn's overlapping circles.
Euler also found the figures valuable as a teaching tool. He

Yellow Nonyellow
cars
cars
Taxicabs

o

Not-Taxicabs

Figure 9. "All taxicabs are yellow," in the style of Lewis Carroll.

48

LOGIC MADE E A S Y

noted that the propositions in logic may "be represented by figures, so as to exhibit their nature to the eye. This must be a great
assistance, toward comprehending, more distinctly, wherein the
accuracy of a chain of reasoning consists."9 Euler wrote to the
princess,
These circles, or rather these spaces, for it is of no importance what figure they are of, are extremely commodious
for facilitating our reflections on this subject, and for
unfolding all the boasted mysteries of logic, which that art
finds it so difficult to explain; whereas, by means of these
signs, the whole is rendered sensible to the eye. 10
It is interesting that in 1761 Euler mentions the difficulty of
explaining the art of logic. This fact should be of some comfort
to teachers everywhere. Even today, instructors at the university
level see these misunderstandings crop up in math, philosophy,
and computer science classes time after time. While adults
would probably have little difficulty dealing with Inhelder and
Piaget's questions with colored counters, when the information
is presented abstractly, without a visual referent, even adults are
likely to reach the wrong conclusion from a given set of statements. Yet, according to Inhelder and Piaget, by approximately
the twelfth grade, most of us have reached our formal reasoning
period and should have the ability to reason logically.

Familiarity—Help or Hindrance?
Unlike the visual clues provided in Inhelder and Piaget's study of
logical reasoning in children or the pictures provided in
Politzer's study as mentioned earlier, we are usually required to

ALL

49

reason without access to direct evidence. Without evidence at
hand, we must recall information that is often remote and
vague. Sometimes our memory provides us with counterexamples to prevent our faulty reasoning, but just as often our memory leads us astray.
The rules of inference dictating how one statement can follow
from another and lead to logical conclusions are the same regardless of the content of the argument. Logical reasoning is supposed
to take place without regard to either the sense or the truth of the
statement or the material being reasoned about.Yet, often reasoning is more difficult if the material under consideration is obscure
or alien. As one researcher put it, "The difficulty of applying a
principle of reasoning increases as the meaningfulness of the content decreases."11 The more abstract or unfamiliar the material,
the more difficult it is for us to draw correct inferences.
In one of the earliest studies examining the content or material being reasoned about, M. C. Wilkins in 1928 found that
when given the premise, "All freshmen take History I," only 8
percent of her subjects erroneously accepted the conversion,
"All students taking History I are freshmen." However, 20 percent of them accepted the equally erroneous conclusion, "Some
students taking History I are not freshmen ."With strictly symbolic material (All S are P ) , the errors "All P are S" and "Some P
are not S" were made by 25 percent and 14 percent of the subjects, respectively. One might guess that in the first instance students retrieved common knowledge about their world—given
the fact that all freshmen take History I does not mean that only
freshmen take it. In fact, they may have themselves observed
nonfreshmen taking History I. So their conclusion was correct
and they were able to construct a counterexample to prevent
making the erroneous conversion. However, as they continued
thinking along those lines, knowledge about their own world

so

L06K MADE EASY

encouraged them to draw a (possibly true) conclusion that was
not based on correct logical inference. "Some students taking
History I are not freshmen" may or may not be true, but it does
not logically follow from "All freshmen take History I." Interestingly enough, when abstract material was used and subjects
could not tap into their own experience and knowledge about
the material, more of them made the conversion mistake (for
which there are countless concrete examples that one can
retrieve from memory—"All women are human" doesn't mean
"All humans are women") while fewer made the second inference mistake.
"All horn players have good chops." My husband, a singer
extraordinaire, can see right through this trap. He will not
accept the converse statement "All people with good chops play
the horn." He's not a horn player but he does have good chops.
With evidence at hand he avoids the common fallacy because he
recognizes a counterexample or inconsistency in accepting the
faulty conclusion.

Clarity or Brevity?
There seem to be two different systems of language—one is that
of natural language and the other that of logic. Often the information we convey is the least amount necessary to get our points
across.
Dr. Susanna Epp of De Paul University uses the example of a
classroom teacher who announces, "All those who sit quietly
during the test may go outside and play afterward."12 Perhaps
this is exactly what the teacher means to say. And, if so, then she
means that those who will get to go out and play will definitely
include the quiet sitters, but might well include those who make

ALL si
noise. In fact her statement says nothing at all about the noisemakers one way or the other. I doubt that the students interpret
her this way.
Is the teacher intentionally deceiving the students? Is she hoping that students will misconstrue the statement? Chances are
good that most of the students believe she is actually making the
converse statement that all those who make noise will not get to
play outside. Had the teacher made the statement "All those
who do not sit quietly during the test may not go out and play
afterwards," then the warning doesn't address the question of
what will happen to the quiet sitters. She probably means, "All
those who sit quietly during the test may go outside and play
afterwards, and those who don't sit quietly may not go outside
and play afterwards." In the interest of brevity, we must often
take the speaker's meaning from the context of his or her language and our own life experiences.
Since logic defines strict rules of inference without regard to
content, we may be forced to accept nonsensical statements as
true due to their correct form. How is one to evaluate the truth
of "All my Ferraris are red" if I have no Ferraris? In ordinary language, we might say that it is neither true nor false—or that it is
nonsense. Yet, the classical rules of logic require propositions to
be either true or not true (law of the excluded middle). Some
logicians have ignored this kind of proposition. They have made
an existential assumption, that is, an assumption that the subject of
any universal proposition exists. Others make no existential
assumption, claiming that the diagrams of Leibniz/Euler and
Venn serve us well to represent the universal proposition regardless of whether the class of my Ferraris has any members or not.
"All angels are good" and "All devils are evil" can be allowed as
true propositions whether or not angels or devils exist.13
Of course, things could get much more complicated. We have

52

LOGIC MADE E A S Y

only considered universal quantifiers and have only quantified
the subject of the proposition. In ordinary language, we put
quantifiers anywhere we want. And what if we put the word
"not" in front of "all"? Not all drastically changes the proposition, not only changing it from an affirmation to a negation but
also changing its universal nature. Even when the rules of logic
were being developed, Aristotle recognized that negation makes
reasoning a good deal more difficult. So naturally he addressed
rules of negation. Let's examine them next.

3
A NOT TANGLES EVERYTHING UP
"No"is only "yes"to a different question.
BOB

PATTERSON

If every instinct you have is wrong,
then the opposite would have to he right.
JERRY

SEINFELD

We encountered negations very early on while examining the law
of the excluded middle and the law of noncontradiction. While
Aristotle reminded us that it is impossible that the same thing
both is and is not at the same time, he also recognized that we can
construct both an affirmation and a negation that have identical
meanings. Aristotle said that there are two types of propositions
that are called simple—the affirmation, which is an assertion, and
the negation, which is a negative assertion or a denial. All others
are merely conjunctions of simple propositions.
"All humans are imperfect" is an affirmation, while "No
human is perfect" is a denial with the same meaning. "Tuesday
you were absent" is an affirmation, and "Tuesday you were not
present" is a denial conveying the same information. "Four is not
an odd number" is a true negation and "four is an even number"
is a true affirmation expressing the same information from a different perspective. Inasmuch as it is possible to affirm the
absence of something or to deny the presence of something,

53

54

LOGIC MADE EASY

the same set of facts may be stated in either the affirmative or
the negative.
So what does the negation of an "all" statement look like? Consider the negation of a simple sentence such as "All the children
like ice cream." Its negation might well read, "It is not the case
that all the children like ice cream." But even long ago Aristotle
suggested that the negation be posed as the contradictory statement, such as "Not every child likes ice cream" or "Some children
don't like ice cream." We could negate using the passive voice—
"Ice cream isn't liked by every child" or "Ice cream isn't liked by
some of the children ."The underlying structure of any of these
negations is simply not-(all the children like ice cream).

The Trouble with Afe^
The noted logic historians William and Martha Kneale state that
from the time of Parmenides in the fifth century B.C., the Greeks
found something mysterious in negation, perhaps associating it
with falsehood.1 In modern times, some researchers have argued
that negation is not "natural" since it is hardly informative to
know what something is not. However, more often than we may
realize the only way to understand what something is is to have a
clear understanding of what it isn't. How would we define an odd
number other than by saying it is a number that is not divisible by
2? What is peace but the absence of war?
Another argument put forth relative to the difficulty of
reasoning with negation concerns the emotional factor. This
position argues that the prohibitive nature of words such as "no"
and "not" makes us uncomfortable. Some psychologists have
suggested that since negation is fraught with psychological prob-

A NOT T A N È L E S EVERYTHING UP

55

lems, negation necessarily increases the difficulty inherent in
making inferences.2
Cognitive psychologists Peter C. Wason and Philip JohnsonLaird have written several books and dozens of articles on how
we reason. They point out that negation is a fundamental concept in reasoning, a concept so basic to our everyday thinking
that no known language is without its negative terms. 3 Negation
ought to be an easy, perhaps the easiest, form of deduction.
However, making even a simple inference involving a negative is
a two-step process. If I say, "I am not an ornithologist," two statements must be absorbed. First, we must grasp what it means to
be an ornithologist, then what it means not to be one. In our
day-to-day communication, the extra step involved in reasoning
with negation may well go unnoticed.
In one of their studies, Wason and Johnson-Laird performed a
series of experiments focusing on the reasoning difficulties associated with negation. When asked questions that involved affirmation and negation, their subjects were slower in evaluating
the truth of a negation than the falsity of an affirmation and got
it wrong more often-—a clear indication that negation is a more
difficult concept to grasp. 4
Negation may be either implicit or explicit. There is evidence
that in some instances an implicit negative is easier to correctly
process than an explicit negative. Implicit negatives are words that
have negative meaning without using the word "not." Implicit
negatives, such as "absent" rather than "not present," "reject"
rather than "not accept," and "fail" rather than "not pass," may be
easier to deal with than their explicitly negative counterparts. In
other instances, implicit negatives may be too well hidden. For
example, researchers have indicated that it is easier to see that
the explicit negative, "The number is not 4," negates "The num-

56

LOGIC MADE E A S Y

ber is 4 " but more difficult to see that the implicit negative, "The
number is 9," also negates "The number is 4." 5
Researcher Sheila Jones tested the ease with which differently
worded instructions were handled by individuals. Three sets of
directions were tested that all had the same meaning—one set
of instructions was an affirmative, one a negative, and one an
implicit negative.6 The subjects were presented a list of digits, 1
through 8, and given one of the following sets of instructions:
Mark the numbers 1, 3, 4 , 6, 7. (affirmative)
Do not mark the numbers 2 , 5 , 8 , mark all the rest,
(negative)
Mark all the numbers except 2, 5, 8. (implicit negative)
The test was set up in a manner similar to that shown in Figure
10. The subjects' speed and accuracy were measured as indicators of difficulty. The subjects performed the task faster and with
fewer errors of omission following the affirmative instruction
even though the list of numbers was considerably longer. Subjects performing the task using "except" were clearly faster than
those following the "not" instruction, signifying that the implicit
negatives were easier to understand than the instructions containing the word "not."

12 3 4 5 6 7 8

1 2 34 5 6 7 i

12 3 4 5 6 7 8

Mark the numbers
1,3,4,6,7.

Do not mark the
numbers 2, S, 8,
mark all the rest.

Mark all the numbers
except 2, S, 8.

Figure 10. Task measuring the difficulty of the affirmative,
negative, and implicit negative.

A NOT TANGLES EVERYTHING UP

57

Some negatives do not have an implicit negative counterpart,
and those negatives are more difficult to evaluate. The statement
"The dress is not red" is harder to process than a statement like
"Seven is not even," because the negation "not even" can be easily exchanged for the affirmative "odd," but "not red" is not
easily translated. "Not red" is also very difficult to visualize. The
difficulties involved with trying to visualize something that is
not may well interfere with one's ability to reason with negatives. If I say that I did not come by car, what do you see in your
mind's eye?
It may be that, wherever possible, we translate negatives into
affirmatives to more easily process information. To make this
translation an individual must first construct a contrast class, like
the class of not-red dresses or the class of modes of transportation that are not-car. The size of the contrast class and the ease
with which a contrast class can be constructed have been shown
to affect our ability to reason with negatives. 7
Wason and Johnson-Laird suggest that in everyday language a
denial often serves as a device to correct a preconceived notion.
Although it is true that I am not an ornithologist, I am not likely
to make that statement unless someone was under the misconception that I was. The statement "Class wasn't boring today"
would probably not be made if the class were generally not boring. This kind of statement is usually made when the class is frequently or almost always boring. The statement functions to
correct the listener's previously held impression by pointing out
an exception.
An experiment by Susan Carry indicated that negatives used
on an exceptional case were easier than negatives used on unexceptional cases. In her experiment, individuals were exposed to
and then questioned about an array of circles, numbered 1
through 8. All of the circles except one were the same color, and

s8

LOGIC MADE E A S Y

the circle of exceptional color varied in its position number.
Presumably, most of us would remember the array of circles by
remembering the exceptional circle since this requires retaining
the least amount of information. Her experiment confirmed
that it is easier to negate an exceptional case in terms of the
property that makes it exceptional than to negate the majority
cases in terms of the property of the exception.8
In addition, the results of a study by Judith Greene showed
that negatives used to change meaning were processed more
easily than negatives used to preserve meaning. Subjects were
asked to determine whether two abstract sentences had the
same or different meanings. A series of tasks paired sentences
sometimes with the same meaning, one involving a negation and
the other not, and other times paired sentences with different
meanings, one involving a negation and the other not. Greene
labeled a negative that signified a change in meaning natural,
while a negative that preserved meaning was dubbed unnatural.
For example, ux exceeds j " and ux does not exceed j " are easily
processed by the brain as being different in meaning (thus the
negative is performing its natural function), while "x exceeds/"
and uj does not exceed x* are more difficult to assess as having
the same meaning. Her studies support the notion that we more
easily digest negatives that change a preconception rather than
negatives that confirm a previously held notion.9

Scope of the Negative
Aristotle went to great lengths in his treatises to point out that
the negation of "All men are just" is the contradictory "It is not
the case that all men are just," rather than the contrary "No men
are just." In the negation, "It is not the case that all men are just,"

A NOT TANGLES EVERYTHING UP

59

the scope of the negative is the entire assertion, "all men are
just." The scope of the negative in the contrary "No men are
just" is simply "men."The difference between the contradictory
and the contrary is that the contradictory is the negation of an
entire proposition and that is why the proposition and its contradictory are always opposite in truth value. When one is true,
the other is false, and vice versa.
Aristotle recommended that the statement "It is not the case
that all men are just" was more naturally communicated as
"Some men are not just." Several studies have borne out the fact
that this form may indeed be more natural. The smaller the
scope of the negative, the easier the statement is to understand.
Studies have shown that it takes systematically longer to process
the type of denial involving "It is not the case that . . ." and "It is
false that . . ." than ordinary negation. Indications are that statements where the scope of the negative is small, like "Some people do not like all ice cream flavors," are easier to process than
ones such as "It is not the case that all people like all ice creams
flavors." 10

A and [Propositions
Medieval scholars of logic invented schemes and labels that
became common terminology for students studying Aristotle's
classification of propositions. The universal affirmation, "All S
are P," was named a type-A proposition. The universal negation
or denial, "No S are P," was named a type-E proposition. This
pair of A and E statements are the contrary statements. As such,
they cannot both be true, but exactly one could be true or both
could be false. The type-A universal affirmation, "All people are
honest in completing their tax forms," and the type-E universal

6o

LOGIC MADE EASY

00
Figure 11. A Leibniz/Euler diagram of "No S are P."
denial, "No people are honest in completing their tax forms,"
are contraries. In this case, both are probably false.
The Leibniz/Euler logic diagrams represent the universal
negation, "No S are P," as two spaces separate from each
other—an indication that nothing in notion S is in notion P. The
proposition, "No S are P," is seen in Figure 11.
John Venn's diagrams once again employed the use of overlapping circles to denote the subject and the predicate. In fact, all of
Venn's diagrams use the overlapping circles, which is one of its
most attractive features. Using Venn's graphical method, all of
the Aristotelian propositions can be represented by different
shadings of the same diagram—using one piece of graph paper,
so to speak. Again, Venn's shaded region indicates emptiness—
nothing exists there. So in representing "No S are P," the region
where S and P overlap is shaded to indicate that nothing can be
there, as shown in Figure 12.
Earlier, we witnessed the error in logic called conversion that is
commonly made with the universal affirmative (type-A) proposition. It is a mistake to think "All S are P " means the same thing as

00'
Figure 12. A Venn diagram of "No S are P."

A NOT TANGLES EVERYTHING UP

61

"AU P are 5." Quite frequently one is true and the other is not.
Just because all zebras are mammals doesn't mean that all mammals are zebras. Yet, converting a type-E proposition (a universal
negation) is not an error. "No chickens are mammals" and "no
mammals are chickens" are both true. In fact, any time "No S are
P" is true, so is "No P are S ."This fact becomes crystal clear by
looking at either the Leibniz/Euler diagram or the Venn diagram.
In the Leibniz /Euler diagram, nothing in space S is in space P and
nothing in space P is in space S. In John Venn's diagram, nothing S
is in P and nothing P is in S. Imagine what the diagrams for "No P
are S" would look like. Using either diagram, it is clear that the
figure for "No P are S" would look exactly the same as "No S are
P" with perhaps the labels on the circles interchanged.

When No Means Yes —The "Negative Pregnant"
and Double Negative
In his On Language column, William Safire discussed a fascinating
legal term called the negative pregnant derived from fifteenthcentury logicians.11 The Oxford English Dictionary notes that a
negative pregnant means "a negative implying or involving an affirmative." If asked, "Did you steal the car on November 4?" the
defendant replying with the negative pregnant "I did not steal it
on November 4" leaves the possibility (maybe even the implication) wide open that he nonetheless stole the car on some day.
Early on in life, young children seem to master this form of
avoiding the issue. When asked, "Did you eat the last cookie yesterday?" we might well hear, "I did not eat it yesterday" or,
"Yesterday? . . . No."
Double negatives fascinate us from the time we first
encounter them in elementary school. They cropped up earlier

62

LOGIC MADE E A S Y

in the discussion of proofs by contradiction, where we begin by
assuming the opposite of that which we want to prove. If I want
to prove proposition P , I assume not-P. Proceeding by impeccable logic, I arrive at a contradiction, an impossibility, something like 0 = 1 . What went wrong? My initial assumption must
be false. I conclude, "not-P is false" or "it is not the case that
not-P" or "not-not-P."The equivalence of the statements "not
(not-P)n and "P"—that the negation of a negation yields a affirmation—was a principle in logic recognized by the Stoics as
early as the second century B.C.12
All too frequently for the electorate we see double negatives
in referendum questions in the voting booth. This yes-means-no
and no-means-yes wording is often found in propositions to
repeal a ban on something. A vote "yes" on the repeal of term
limits means you do not favor term limits. A vote "no" on the
repeal of the ban on smoking means you favor smoking restrictions. A vote "no" to repeal a ban on gay marriages means you
favor restrictions on gay marriages, but a "yes" vote to repeal the
ban on assault weapons means you do not favor restrictions on
assault weapons. I recently received a ballot to vote for some
proposals in the management of my retirement funds. The ballot
question is in Figure 13. If you are like people who want their
money invested in issues they favor (some folks don't care), voting "for" means you are against gun control and voting "against"
means you favor gun control.
Proposal: To stop investing in companies supporting gun control.

For

Against

Abstain

Figure 13. Example of when voting "for" means against.

A NOT TANGLES EVERYTHING UP

63

Studies have shown that reasoners find it difficult to negate a
negative. 13 If the process of negation involves an extra mental
step, a double negative can be mind boggling. Statements such as
"The probability of a false negative for the pregnancy test is 1
percent" or "No non-New Yorkers are required to complete
form 2 0 3 " or "The statistical test indicates that you cannot reject
the hypothesis of no difference" can cause listeners to scratch
their heads (or give them a headache).
As we mentioned earlier, a statement like "It is not the case
that all men are honest" is more naturally communicated as
"Some men are not honest." But some is not universal. So Aristotle defined propositions dealing with some are and some are not.
Do they really need definition? You may be surprised to learn
that they mean different things to different people. Read on.

4
SOME Is

PART OR ALL OF ALL

If every boy likes some girl and every girl likes some boy,
does every boy like someone who likes him?
JONATHAN

BARON,

Thinking and Deciding

Although statements about "all" of something or "none" of something are powerful and yield universal laws in mathematics,
physics, medicine, and other sciences, most statements are not
universal. More often than not, our observations about the
world involve quantifiers like "most" and "some."There was an
important niche for nonuniversal propositions in Aristotle's system of logic.
Ordinarily, if I were to assert, "Some parts of the lecture were
interesting," I would most likely be implying that some parts
were not interesting. You would certainly not expect me to say
that some parts were interesting if all parts were. However, the
assertion "Some of you will miss a day of work due to illness"
does not seem to forgo the possibility that, at some point or
another, all of you might miss a day. Oftentimes, in everyday
language, "some" means "some but not all," while at other times
it means "some or possibly all." To a logician, "some" always
means at least one and possibly all.

64

SOME Is PART OR ALL OF ALL

65

Some Is Existential
Whereas all and none are universal quantifiers, some is called an
existential quantifier, because when we use some we are prepared
to assert that some particular thing or things exist having that
description. "Some" propositions are said to be particular in
nature, rather than universal.
Much like the universal affirmative and negative propositions involving all and none, Aristotle defined and examined
affirmative and negative propositions involving some. Whereas
the universal affirmative "All people are honest" and the universal negative "No people are honest" cannot both be true, particular affirmations and their negative counterparts are oftentimes
both true. The propositions "Some people are honest" and
"Some people are not honest" are both most likely true.
Medieval scholars named the particular affirmative proposition
of the form "Some S are P " a type-I proposition, and they
named the particular negation of the form "Some S are not P " a
type-O proposition.
With the universal affirmative and negative propositions
named A and E, respectively, students of logic used the mnemonic device—ARISTOTLE—to remember these labels. A
and I propositions were affirmations and come from the Latin
Afflrmo (meaning "I affirm"), and E and O propositions were
negations from nEgO (meaning "I deny").1 The outer two vowels, A and E in ARISTOTLE, name the universal propositions,
while the inner two, I and O, name the existential or particular
propositions. Medieval scholars also devised a diagram known as
the Square of Opposition (Fig. 14) to illustrate the contrary or
contradictory relationship between propositions.2 As seen in the
diagram, I and O are contraries, as are A and E. For example,

66

LOGIC MADE EASY

Contradictories

r

o
Contraries

Figure 14. Square of Opposition.
"Some of you are making noise .""But some of us are not
making noise." "Don't be contrary."
The diagonals in the diagram represent the contradictories, A
with O, and E with I.
Whereas John Venn used overlapping circles for propositions
of any type (with different shadings), Gottfried Leibniz, and
later Leonhard Euler, used overlapping circles only for expressing particular propositions. To illustrate "Some S are P," the
Leibniz /Euler diagram required the label for S be written into
that part of S that is in P , whereas for Venn, an asterisk indicated the existence of something in S that is in P, as shown in
Figure 1 5 . 3
Although the Leibniz/Euler diagram might look a little different if the proposition were "Some P are S" (the P would be in

S

P

Leibniz/Euler diagram
Figure 15. "Some S are P."

Venn diagram

SOME Is PART OR ALL OF ALL 67

the overlapping region instead of the S ) , the logicians themselves were well aware that in logic the two propositions are
equivalent. Just as "No S are P " and "No P are S " are equivalent,
"Some S are P " and "Some P are S " are interchangeable because
their truth values are identical. If "Some women are lawyers" is
true, then it is also true that "Some lawyers are women." Venn's
diagram helps to illustrate this relationship. The asterisk merely
indicates that something exists that is both S and P — a s in "Some
(one or more) people exist who are both women and lawyers."
However, in our everyday language we do not really use these
statements interchangeably. We might hear "Some women are
lawyers" in a conversation about possible career choices for
women. The statement "Some lawyers are women" might arise
more naturally in a conversation about the composition of the
population of lawyers. Nonetheless, logic assures us that whenever one statement is true, the other is true, and whenever one
is false, the other is false.
Every now and then, "some" statements seem rather peculiar,
as in the statements "Some women are mothers" and "Some
mothers are women ."The first statement is true because some
(though not all) women are mothers. The second statement is
true because definitely some (and, in fact, all if we restrict ourselves to the discussion of humans) mothers are women. But we
must remember that, logically speaking, any "some" statement
means "some and possibly all."
For example, we would not normally say "Some poodles are
dogs," since we know that all poodles are dogs. During the normal course of conversation, a speaker likes to be as informative
as humanly possible. If the universal "all poodles" holds, we generally use it. 4 However, we might say "Some teachers are
licensed" if we weren't sure whether all were licensed. Author
Jonathan Baron offers the example that when traveling in a new

68

LOGIC MADE EASY

city we might notice that taxicabs are yellow. It would be truthful to say "Some cabs are yellow," withholding our judgment that
all are until we know for sure. 5

Some Are; Some Are Not
An O proposition of the "Some are not" form can also be illustrated by two overlapping circles as in Figure 16. Venn's diagram
is clearly superior (in fact, the Leibniz/Euler diagram has some
serious problems), since "Some S are not P " and "Some P are
not S " are not interchangeable. Just because the proposition
"Some dogs are not poodles" is true does not mean that "Some
poodles are not dogs" is. In fact, it is false.
Peter C. Wason and Philip Johnson-Laird have performed
studies that seem to indicate that individuals illicitly process
"Some X are notY" to conclude "Therefore, some X are Y," believing they are just two sides of the same coin—in much the same
vein as whether the glass is half full or half empty. But in logic the
existential quantifier some means at least one and possibly all. If it
turns out that all X are notY, then "Some X areY" cannot possibly
be true. Their studies indicated that whether an individual gives
the material this interpretation depends primarily on the material. Even though subjects were instructed to interpret some in its
logical fashion, most were able to do so only with material that

Leibniz/Euler diagram
Figure 16. "Some S are not P."

Venn diagram

SOME Is PART OR ALL OF ALL

69

hinted at possible universality. For example, "Some beasts are
animals" was interpreted to mean "Some, and possibly all, beasts
are animals," whereas "Some books are novels" was not generally
interpreted as "Some, and possibly all, books are novels."
Might this be an indication that we are rational and reasonable
after all? A computer could not distinguish between the contexts
of these "some" statements in the way that the human subjects
did. The subjects in these experiments were reading meaning
into the statements given even though they weren't really supposed to. Humans have the unique ability to sometimes interpret what another human meant to say. On the other hand, this
tendency to interpret can get us into a good bit of trouble when
the interpretation is wrong.
In ordinary language "some" can mean "some particular thing"
or "some thing or other from a class of things" and, depending
on its use, will signify completely different statements. Compare the statement "Some ice cream flavor is liked by every student" to "Every student likes some ice cream flavor." The first
statement indicates a particular flavor exists that is liked by all,
while the second statement suggests that each and every student
has his or her favorite. 6
Take a look at Figure 1 7 . Here we have a question taken from
the ETS Tests at a Glance to introduce prospective teachers to the
general knowledge examination required by many states for elementary teacher certification. This question contains examples
of many of the concepts we have seen so far. For example, the
sentence given is "Some values of x are less than 100" and the
examinee is asked to determine which of the answers is NOT
consistent with the sentence. The given sentence is a "some"
proposition, and the question invokes the notion of consistency
with the interference of negation.
The first choice among the answers "5 is not a value of x" is

jo

LOGIC MADE EASY

S o m e v a l u e s o f x a r e less t h a n 100.
W h i c h o f t h e f o l l o w i n g is NOT c o n s i s t e n t w i t h t h e
sentence above?
A. 5 is not a value of x.
B. 95 is a value of x.
C. Some values of x are greater than 100.
D. All values of x are less than 100.
E. No numbers less than 100 are values of x.
F i g u r e 17. S a m p l e q u e s t i o n from Tests at a Glance (ETS).
(Source: The PRAXIS Series: Professional Assessments for Beginning
Teachers, Mathematics (0730)Tests at a Glance at http://www.ets.org/
praxisItaagslprx0730.html. Reprinted by permission of Educational Testing
Service, the copyright owner.)

not inconsistent with the fact that x might have some other value
that is less than 100. The second choice stipulates "95 is a value
of x." Indeed, 95 could be a value of x since some of the x-values
are less than 100. The third choice "Some values of x are greater
than 100" could be true; it is not inconsistent with the fact that
some x-values are less than 100. Many individuals will probably
be tempted to choose choice D as the inconsistent answer, but
not if they know that "some" means "some and possibly all." That
leaves choice E, which is in direct contradiction to the given
statement. If "Some values of x are less than 100," then it can't
be true that "No numbers less than 100 are values of x."

A, E, I,andO
The four types of propositions, A, E, I, and O, were the foundation for Aristotle's logic and all that he deemed necessary to
develop his rules of logical argument. Aristotle disregarded state-

SOME Is PART OR

ALL OF

ALL

7i

ments with more than one quantifier—statements like: "Every
critic liked some of her films" and "Some critics liked all of her
films." Matters could get even more complex if we introduce negation along with more than one quantifier. Consider the following:
Not all of the family enjoyed all of her recipes.
Some of the family did not enjoy all of her recipes.
Some of the family did not enjoy some of her recipes.
All of the family did not enjoy all of her recipes.
By distributing the quantifiers and the negations appropriately,
the same basic facts can be articulated in a number of different
ways. Although these statements are synonymous, some are easier to grasp than others.7
In 1846, Sir William Hamilton of Edinburgh tried to improve
on Aristotle's four types of propositions by allowing quantification of the predicate.8 In his New Analytic of Logical Forms, he distinguished eight different forms, defining "some" as "some but
only some."
1.
2.
3.
4.
5.
6.
7.
8.

AIM is all B .
All A is some B .
Some A is all B .
Some A is some B .
Any A is not any B .
Any A is not some B .
Some A is not any B .
Some A is not some B . 9

While this system seemed more complete than Aristotle's, there
were many difficulties associated with Hamilton's system. His
work led to a famous controversy with the English mathemati-

72

LOCK MADE EASY

cian Augustus De Morgan. One point of disagreement was over
Hamilton's definition of some. Should "some" mean "some at
most" or "some at least" or "some but not the rest"? De Morgan
insisted that some is vague and should remain so. "Here some is a
quantity entirely vague in one direction: it is not-none; one at
least; or more; all, it may be. Some, in common life, often means
both not-none and not-all; in logic, only not-none?™ The American logician Charles Sanders Peirce agreed with De Morgan,
saying that "some" ought to mean only "more than none."11
Hamilton could not really improve upon Aristotle's system;
its simplicity had enabled it to remain basically unchanged for
two thousand years. With only four types of propositions (A, E,
I, and O), Aristotle described a structure for logical argument
that could be relied upon to yield valid conclusions. His arguments became known as syllogisms. Not only would the syllogistic structure always lead to valid conclusions, but as we'll see in
Chapter 5, the system could be used to detect rhetoric that led
to invalid conclusions.

5
SYLLOGISMS
For a complete logical argument, we need
two prim Misses—
And they produce—A delusion.
But what is the whole argument called?
A Sillygism.
LEWIS

CARROLL,

Sylvie and Bruno

With the Greek Age of Enlightenment and the rise of
democracy, every Greek citizen became a potential politician.
By as early as 440 B.C., the Sophists had become the professional educators for those aspiring to a political career and
provided them with the requisite instruction for public life. The
Sophists were not particularly interested in truth but in
intellectual eloquence—some say they were only interested in
intellectual anarchy.1 Plato and later his most famous student,
Aristotle, were concerned about those who might be confused
by the "arguments" of the Sophists, who used obfuscation and
rhetorical ruses to win over an audience. To expose the errors
of the Sophists, Aristotle laid down a doctrine for logical
argument in his treatise, Trior Analytics, or Concerning Syllogisms.
Indeed, many have said that these laws of inference are Aristotle's greatest and most original achievement.2

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LOGIC MADE EASY

In Prior Analytics, Aristotle investigated the methods by which
several propositions could be linked together to produce an
entirely new proposition. Two propositions (called the premises)
would be taken to be true, and another (called the conclusion)
would follow from the premises, forming a three-line argument, called a syllogism. "A syllogism," according to Aristotle, "is
discourse in which, certain things being stated, something other
than what is stated [a conclusion] follows of necessity from their
being so."3 In other words, a syllogism accepts only those conclusions that are inescapable from the stated premises.
In a syllogism, each proposition is one of Aristotle's four
proposition types later classified as types A, E, I, or O. The
propositions in the first two lines are the premises; the proposition in the third line is the conclusion. If the argument is valid
and you accept the premises as true, then you must accept the
conclusion as true. In his Letters to a German Princess, Leonhard
Euler said of the syllogistic forms, "The advantage of all these
forms, to direct our reasonings, is this, that if the premises are
both true, the conclusion, infallibly, is so."4
Consider the following syllogism:
All poodles are dogs.
All dogs are animals.
Therefore, all poodles are animals.
The three propositions above form a valid argument (albeit a
simplistic and obvious one). Since the conclusion follows of
necessity from the two (true) premises, it is inescapable.
Over time, syllogisms were classified as to their mood. Since
each of the three propositions can be one of four types (an A or
an I or an E or an O), there are 4 X 4 X 4 , or 64, different syllogism moods. The first mood described a syllogism with two

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75

universal affirmative premises and a universal affirmative conclusion—named AAA for its three type-A propositions. The
poodle/dog/animal syllogism is an example of a syllogism in
mood AAA.
A syllogism was further classified as to itsfigure. The figure of
a syllogism involved the arrangement of terms within the
propositions of the argument. For example, "All dogs are poodles" and "All poodles are dogs" are different arrangements of
the terms within a single proposition. In every figure, the terms
of the conclusion are designated as the subject and the predicate.
If a conclusion reads "All
are
," the term following
"All" is called the subject term (S) and the term following "are"
is called the predicate term (P). 5 A conclusion in mood AAA
reads like "All S are P." One of the premises includes S and the
other, P, and both include another term common to the two
premises, called the middle term (M).6 A syllogism is classified
according to its figure depending on the ordering of the terms,
S, P, and M, in the two premises. Aristotle recognized three figures, but the noted second century A.D. physician Galen
recognized a fourth figure as a separate type.7 The figures are
indicated in Table 1.
Although we could interchange the order of the first and second premises without injury, what we see in Table 1 is the tradi-

Table 1. Syllogism Classifications by Figure
FIRST

SECOND

THIRD

FOURTH

FIGURE

FIGURE

FIGURE

FIGURE

Second premise

M-P
S-M

P-M
S-M

M-P
M-S

M-S

Conclusion

S-P

S-P

S-P

S-P

First premise

P-M

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LOGIC MADE EASY

tional ordering that was adopted by logicians and brought down
to us over the centuries. In fact, psychologists have found that
the ordering of the first and second premise can make a difference in how well we perform when reasoning syllogistically.
One could even argue that it seems more natural to put the S in
the first premise.
In Prior Analytics, Aristotle offered the first systematic treatise on formal logic as an analysis of valid arguments according to their form—the figures and moods—of the syllogism.
Historians have noted that in this work Aristotle appears to
have been the first to use variables for terms. The idea may
have been suggested by the use of letters to name lines in
geometry; it is a device that allows a generality that particular
examples do not. William and Martha Kneale maintain that
this epoch-making device, used for the first time without
explanation, appears to be Aristotle's invention.8 It is not the
least bit surprising that the ancient Greeks never developed
the use of letters as numerical variables (as we do in algebra)
given that it was their practice to use Greek letters to represent numbers.
Aristotle considered only syllogisms of the first figure to be
perfect or complete. The first syllogism he discussed was the
AAA mood in the first figure. The AAA mood in the first figure
acquired the name Barbara in medieval times from the Latin for
"foreigners" or "barbarians," with the vowels reminding the
scholar or student of the mood—bArbArA. In fact, the 14 valid
syllogisms identified by Aristotle, along with 5 more added by
medieval logicians, were each given mnemonic Latin names to
simplify the task of remembering them. When Aristotle
explained his first valid syllogism (AAA), he generalized the
syllogism using Greek letters but for our ease, we'll use the
English translation:

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77

All B are A.
All C are B .
Therefore, all C are A.
It is somewhat surprising to the modern mind that Aristotle chose the ordering of the two premises that he did. For
example, the f